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--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2011-cornell.bak
@@ -0,0 +1,12161 @@
+%%  2011 dec 31 home suny  REVSIONS OF PRIOR DEC21 7.1 am
+%%%  MINOR REVSIONS OF PRIOR DEC21
+%% primarily in Theorem 2.1 and Remark 6.13
+%% Also few other miniscule changes on pages 9, 37, 52 etc.
+
+\documentstyle[11pt]{article}
+
+
+
+
+
+\addtolength{\oddsidemargin}{-0.75in}
+
+
+
+
+\setlength{\textheight}{9.4 in}
+\setlength{\textheight}{9.0 in}
+
+
+
+
+\setlength{\textwidth}{5.7 in}
+
+
+
+
+\setlength{\textwidth}{6.7 in}
+\setlength{\textwidth}{6.5 in}
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{6.3 in}
+
+\setlength{\textwidth}{5.8 in}
+%% PRINT
+
+%\setlength{\textwidth}{5.6 in}
+
+
+
+
+
+
+
+
+
+
+\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+
+
+
+
+
+
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+ \title{A Detailed Examination
+of Methods for Unifying, Simplifying and Extending 
+Several
+Results About Self-Justifying Logics}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+Email = dew@cs.albany.edu.}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+
+
+\begin{abstract}
+ \baselineskip = 1.5 \normalbaselineskip \large 
+
+This paper will develop a single framework for unifying, simplifying
+and extending our prior results about axiom systems that retain a
+partial knowledge of their own consistency, via an axiomatic
+declaration of self-consistency.  Its perhaps single most surprising
+new result will be its exploration of a viable alternative to
+conventional reflection principles.
+\end{abstract}
+
+
+\normalsize
+
+\parskip 8pt
+
+\baselineskip = 1.4 \normalbaselineskip 
+
+\setcounter{page}{0}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+{\bf Keywords:}
+G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+Open Question,
+Semantic Tableaux
+
+\bigskip
+\bigskip
+
+{\bf Mathematics Subject Classification:}
+03B52; 03F25; 03F45; 03H13 
+
+
+
+\bigskip
+\bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+\newpage
+
+\setcounter{page}{1}
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+
+\def\f55{  \baselineskip = 1.59 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.59 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\def\nop{ }
+\def\nop{\newpage}
+
+\cvl
+\cvnew
+
+
+\def\nop{\newpage}
+
+
+\cvnew
+\cvl
+
+\parskip 3pt
+
+
+
+\def\hgskip{ \medskip }
+
+\def\nop{ }
+\def\njp{\newpage}
+\def\njp{ }
+\def\nop{\newpage}
+
+\cvl
+\cvnew
+
+\def\nskip{\bigskip}
+
+\cvl
+\cvnew
+
+
+\section{Introduction}
+
+%% CHANGES REMOVE NEW PAGE IN BIB before willard 2001
+
+%% CHANGES = - 3 footnote -ACKN -APPF -TABLE NUMBERS
+
+%% CHANGES + ww11 Reminder-to-reader Footnote 24 state TABLE
+
+\label{secc1}
+\label{B1-lem}
+\label{D1-def}
+
+\parskip 2pt
+
+
+Let
+$~\alpha~$ 
+denote an axiom system, 
+and $~d~$  
+denote
+ a
+deduction method.
+The ordered pair
+ $~(  \alpha  , d  )$
+will 
+be called  {\bf Self Justifying} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+states that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a second
+axiom system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+Part-i of 
+this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus the following added
+sentence,  that we call
+{\bf SelfRef$(\alpha,d)~$}:
+\topsep -3pt
+\begin{quote}
+$\bullet~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union} of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of
+SelfRef$(\alpha,d)$'s
+ self-referential statement.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition. 
+
+\smallskip
+
+This problem arises in
+settings
+more general than 
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic.
+There 
+are 
+many
+settings 
+where the Second Incompleteness Theorem does
+generalize
+\cite{Ad2,AB1,AZ1,BS76,Bu86,BI95,Fe60,Go31,HP91,HB39,Ko6,KT74,Lo55,PD83,PW81,Pu84,Pu85,Pu96,Ro67,Sa11,Sm85,So94,Sv7,Ta0,VV94,Vi92,Vi5,WP87,ww2,wwapal,wwlogos,ww7}.
+Each such result formalizes a 
+paradigm where 
+self-justification is infeasible,
+due to a diagonalization issue.
+Most
+logicians 
+have
+hesitated to
+thus
+ employ 
+a SelfRef$(\alpha,d)$
+ axiom
+because
+$\alpha+$SelfRef$(\alpha,d)  $
+is usually 
+ inconsistent     \footnote{ \baselineskip = 1.3 \normalbaselineskip  \label{troub} 
+     Typical ordered pairs $(\alpha,d)$
+     will have the property that
+     the broader axiom system
+     $~\alpha^d~=~\alpha \,+ \,$SelfRef$(\alpha,d)$ will
+     be inconsistent, even
+     when  $~\alpha~$ is consistent. This is because
+     a  
+     standard
+      G\"{o}del-like self-referencing 
+     %diagonalization 
+     construction
+  will 
+     %usually
+     typically 
+     % be able to 
+     produce a proof of $0=1$ from
+     $~\alpha^d\,$, irregardless of whether or not $~\alpha$ is 
+%     formally
+     consistent.}.
+
+
+
+
+
+
+\smallskip
+
+
+ 
+
+
+Our research
+explored special
+circumstances
+\cite{ww1,ww5,ww6,wwapal}
+where it is feasible to
+construct self-justifying formalisms.
+These paradigms involved weakening
+the properties a system can prove about
+addition and/or 
+multiplication
+(to avoid the preceding
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+two 
+3-way predicates
+ indicating
+$x$, $y$ and $z$ satisfy 
+$x+y=z$ and
+$x*y=z$.
+A 
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+1-3 as axioms.
+
+
+\vspace*{- 0.8 em}
+{\  
+\cvl
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 0.6 em}
+
+\cvl
+
+We will say
+a 
+logic
+system 
+$\alpha$
+is 
+{\bf Type-M} iff it contains
+each of \eq{totdefxs} -- \eq{totdefxm}
+as axioms,  
+{\bf Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+and {\bf Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom. 
+A system is called 
+{\bf Type-NS} iff it {\it does not} contain
+any of these axioms.
+
+
+
+
+
+Our investigations
+ \cite{ww1}--\cite{ww7}
+began by observing
+some 
+Type-A systems can  recognize 
+their 
+consistency under semantic tableaux deduction,
+and 
+several
+Type-NS systems 
+can 
+recognize their
+ Hilbert consistency.
+Many of
+these systems were capable of 
+proving 
+%all 
+Peano Arithmetic's
+ $\Pi_1$ theorems
+in a language
+that represents addition and multiplication 
+as
+the
+3-way predicates
+of
+ Add$(x,y,z)$ and  Mult$(x,y,z)$.
+
+
+
+
+
+
+\parskip 1pt
+
+Our self-justifying 
+ evasions of the 
+Incompleteness
+Theorem are difficult to further extend
+primarily because the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} showed
+natural 
+Type-S systems cannot recognize  their own Hilbert consistency.
+Also,  Willard
+ \cite{ww2,ww7,ww9}  
+%
+% wwooo
+%
+strengthened earlier results of 
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  natural 
+Type-M  system cannot recognize their semantic
+tableaux consistency.
+
+\cvs
+
+
+
+
+
+
+\medskip
+
+
+A related
+ class of evasions of the
+ Second Incompleteness
+Theorem
+was discovered
+in \cite{ww9}.
+Let us say 
+ $~\alpha~$ 
+is
+a
+{\bf  Type-Almost-M} 
+axiom system
+iff  $~\alpha~$  can prove
+statements \eq{totdefsymba} and 
+\eq{totdefsymbm}
+as theorems
+while treating
+{\it  
+none of  sentences
+ \eq{totdefxs} -- 
+   \eq{totdefsymbm}
+as axioms.} 
+(Many axiom systems,
+that 
+use
+ function symbols ``$~+~$'' and
+``$~*~$'' for
+formalizing addition and multiplication, 
+fall technically into the
+% obviously  
+Type-Almost-M
+% (rather than Type-M) 
+category.)
+\vspace*{- .2 em}
+\beq 
+\label{totdefsymba}
+\forall x ~\forall y~ \exists z ~~~~x+y=z
+\enq
+\vspace*{- 1.5 em}
+\beq 
+\label{totdefsymbm}
+\forall x ~\forall y ~\exists z 
+ ~~~~x*y=z
+\enq
+\vspace*{-1.4 em}
+
+\noindent
+The preceding is of interest because
+some surprisingly strong 
+(albeit unusual)
+Type-Almost-M systems
+\cite{ww9}
+ have an ability to 
+verify their Herbrand but not
+also semantic tableaux consistency. 
+
+
+
+
+The 
+proofs
+in our prior        papers
+were 
+challenging
+primarily
+ because 
+they required
+one to separate the
+local combinatorial
+methods
+employed in
+\cite{ww93,ww5,wwapal,ww9}'s 
+ particular applications 
+from
+the 
+common principles
+that underlied behind all
+these 
+works.
+Our 
+Theorems \ref{ppp1}, \ref{ppp2} and \ref{pqq4}
+will  rectify this problem by
+identifying 
+common components
+that unite these
+four paradigms.
+(Theorems 
+ \ref{pqq3}, \ref{pqq5},
+  \ref{ppp6}, 
+E.1, G.2 and G.3 will then carry on
+in further
+directions.)
+
+
+%%%iii 1
+
+
+
+All these theorems will 
+contain severe limits on their generality,  so that the
+Second Incompleteness Theorem does not contradict them.
+It is clearly perplexing to imagine
+how humans
+are able to 
+motivate themselves to cogitate, 
+without
+ their thought processes possessing
+some type of
+{\it  at least tentative}
+presumption of 
+their own consistency.
+%It is for this reason that our
+Our
+research
+has thus consisted of an approximately equal effort
+in exploring 
+both 
+\cite{ww2,wwlogos,wwapal,ww7}'s
+new
+% types  of  
+generalizations of the
+Second Incompleteness Theorem
+and 
+%in examining the unusual perspectives of 
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}'s unusual
+boundary-case exceptions to it.
+
+%\parskip 1pt
+
+It is clear 
+every boundary-case exception
+to the  Second Incompleteness Theorem
+has limited scope because
+the  Incompleteness Theorem
+is a broadly encompassing result.
+This
+ paper
+ will, thus, 
+be addressing
+a challenging 
+near-paradoxical
+ question
+about the maximal 
+nature of 
+self-justification
+that
+ can
+ never be resolved
+in a
+fully satisfying
+manner.
+The Second
+ Incompleteness Theorem is
+clearly sufficiently
+central to  
+logic
+for it to be desirable
+to know  
+what {\it partial roads of success} a 
+self-justifying axiom     system can 
+obtain.
+
+%\cvs
+
+%\vspace*{- 0.6 em}
+
+\section{Literature Survey}
+
+
+\label{survey}
+\label{B2-lem}
+\label{D2-def}
+
+
+%\vspace*{- 0.6 em}
+
+
+
+
+
+
+
+
+Two 5-page surveys of the prior literature about the
+Second Incompleteness
+Theorem were provided in our
+articles \cite{ww5,wwapal}. 
+This section will present a 
+more abbreviated survey,
+focusing
+on 
+only 
+those
+developments that are
+particularly germane to
+ the current article.
+
+
+
+The study of incompleteness
+began with 
+four classic papers  by
+G\"{o}del, L\"{o}b, Rosser and Tarski 
+\cite{Go31,Lo55,Ro36,Ta36} and
+with the
+Hilbert-Bernays
+ exploration of 
+their 
+derivability conditions
+\cite{HP91,HB39,Ka91}.
+Generalizations of these results for weak
+axiom systems, such as Q,  began with 
+the work of Tarski-Mostowski-Robinson \cite{TMR53}
+and 
+Bezboruah-Shepherdson \cite{BS76}.
+
+
+
+
+
+
+
+Some 
+more
+notation is needed to describe more
+recent developments.
+Let $~x'~$ denote 
+the ``successor'' operation that maps 
+$x$ onto  $x+1$.
+A formula
+ $ \varphi(x) $ is called \cite{HP91} a
+{\bf Definable Cut} for an
+axiom system $~ \alpha~$   iff
+$~\alpha~$ can prove:
+\begin{equation}
+\label{initdefx}
+\varphi(0) \mbox{  AND  }
+\forall~x~ \{~\varphi(x)\Rightarrow\varphi(~x'~)~ \}
+ \mbox{  AND  }
+ \forall~x ~\forall~y<x
+ ~~ \{ ~\varphi(x)\Rightarrow\varphi(y) ~ \}
+\end{equation}
+Definable cuts 
+and their cousins 
+have been studied by
+an
+extensive literature
+\cite{Ad2,AZ1,Be97,Bu86,BI95,HP91,Ka91,Ko6,Kr87,Ne86,Pa71,PD83,PW81,Pa72,Pu84,Pu85,Pu96,Sm85,Sv83,Sv7,VV94,Vi92,Vi93,Vi5,VH73,WP87}.
+(They
+are
+ unrelated to Gentzen's 
+notion of
+ sequent calculus
+deductive ``cut rule'',
+which uses the word ``cut'' in a  
+different
+context).
+
+\smallskip
+
+A Definable Cut 
+$~\varphi(x)~$
+is called {\bf Non-Trivial}
+relative to
+an axiom system $~\alpha~$ iff
+$\alpha~$ cannot prove  $~\forall ~x~\varphi(x)~,~$
+ {\it although it can prove 
+\eq{initdefx}.}
+Every axiom system $\,\alpha$,
+strictly weaker than Peano Arithmetic, 
+will contain some non-trivial Definable Cut.
+This cut will have the property that 
+$\,\alpha\,$
+ can   verify $~\varphi(n)$ for each 
+fixed integer $~n$,
+although it cannot prove
+ $~\forall ~x~\varphi(x)~$.
+
+\smallskip
+
+
+Let
+ $ \, \lceil  \,  \Psi  \,  \rceil  \, $
+denote
+$   \Psi $'s
+G\"{o}del number,
+and 
+Prf$\, _\alpha^d \, \,(t,p)$
+denote
+that $   p   $ is a proof
+of the theorem $   t   $ from the axiom system $   \alpha   $ 
+using $\, d \,$'s
+deduction method.
+An axiom system
+$   \alpha   $ will
+then
+ be said to recognize its own 
+{\bf Cut-Localized  d-Consistency} 
+relative to a Definable Cut 
+ $ \,  \varphi \,   $
+iff $   \alpha   $ can prove:
+\begin{equation}
+\label{dcon}
+\forall~p~~~~~~ \{ ~~~\varphi(p)~~\Rightarrow~~
+\neg~\mbox{Prf}\,_\alpha^d \, ~(~\lceil 0=1 \rceil~,~p~)~~~ \}
+\end{equation}
+The recent literature 
+has 
+sought to identify which
+triples $   (   \varphi   ,   d   ,    \alpha   )   $ have this property.
+A crucial negative 
+ result about
+ Cut-Localized  d-Consistency, discovered by
+ Pudl\'{a}k
+\cite{Pu85},
+established a 
+significant 
+generalization of G\"{o}del's Second Incompleteness Theorem.
+It
+ showed 
+that
+an axiom system
+$   \alpha   $ must be unable to prove
+\eq{dcon}'s  statement 
+about 
+any of its definable cuts  
+ $\,\varphi \,$, $\,$when
+$   d   $  represents
+ Hilbert deduction and 
+$   \alpha   $ is  any consistent extension of $   Q   $.
+
+
+
+
+
+Solovay \cite{So94} noted how 
+Pudl\'{a}k's
+result could be combined
+  with the techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87} to establish the following 
+theorem
+ that will often be
+cited in this paper:
+\begin{quote}
+  \baselineskip = 1.2 \normalbaselineskip 
+%notre \small
+\small
+\baselineskip = 1.25 \normalbaselineskip 
+\baselineskip = 1.25 \normalbaselineskip 
+{\normalsize \bf  Theorem 2.1 }
+{\it 
+(Solovay's  1994 Generalization \cite{So94}
+of a 1985 theorem of Pudl\'{a}k \cite{Pu85} 
+using some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods )} :
+Let $ \, \alpha \, $ denote any axiom system which
+contains 
+\ep{totdefxs}'s
+Type-S statement
+and which assures 
+%that 
+the successor operation
+always satisfies 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+will be unable to recognize its
+own  Hilbert
+consistency,
+whenever  it
+ treats addition and multiplication
+as 3-way relations 
+satisfying 
+their usual identity,
+associative, commutative and 
+distributive  properties.
+\end{quote}
+Solovay never
+published any
+precise
+proof of Theorem 2.1's hybridizing of
+the work of Pudl\'{a}k, Nelson and Wilkie-Paris 
+\cite{Pu85,Ne86,WP87},  which he 
+privately
+communicated
+ \cite{So94} to us.
+A reader can find 
+generalizations of the Second Incompleteness Theorem 
+that are closely 
+related to
+Theorem 2.1
+in papers by 
+Pudl\'{a}k, Buss-Ignjatovic,
+\v{S}vejdar and Willard
+\cite{BI95,Pu85,Sv7,wwlogos}, as well
+as in 
+Appendix A of 
+\cite{ww1}.
+
+Other interesting observations
+that 
+preceded our research were that
+Wilkie-Paris \cite{WP87} 
+demonstrated  that
+I$\Sigma_0+Exp$ cannot prove the
+Hilbert consistency of even the axiom system $~Q, \,~$
+and that Adamowicz-Zbierski \cite{Ad2,AZ1} showed that
+I$\Sigma_0+\Omega_1$ 
+satisfied the Herbrandized version
+of the Second Incompleteness Theorem.
+Both 
+these results helped  stimulate 
+\cite{ww2,ww7}'s
+ semantic tableaux generalizations
+of  
+the Second Incompleteness Theorem for
+I$\Sigma_0$.    
+
+
+
+
+A fascinating observation by 
+L. A.   Ko{\l}odziejczyk
+\cite{Ko5,Ko6},
+about the difference in lengths between
+semantic tableaux and Herbrandized proofs, 
+also motivated
+our
+investigation \cite{ww9}
+into some surprising properties  of
+unorthodox encodings for I$\Sigma_0$.
+  Ko{\l}odziejczyk 
+observed
+\cite{Ko5,Ko6}
+that various generalizations of
+the Second Incompleteness Theorem
+for  I$\Sigma_0$ and
+I$\Sigma_0+\Omega_i$ in
+\cite{Ad2,AZ1,Sa11,ww2,ww7,ww9} 
+imply
+that the proof of 
+ the Herbrandized version
+of the Second Incompleteness Theorem
+can be
+more complicated 
+than
+ its semantic tableaux counterpart.
+This is because there can be an exponential difference
+between semantic tableaux and Herbrandized proof lengths
+under extremal circumstances.
+It was due to 
+Ko{\l}odziejczyk's
+insightful 
+ communications
+\footnote{ \baselineskip = 1.3 \normalbaselineskip \b55 
+ \label{putglue} The Herbrandized and semantic tableaux
+definitions of an axiom
+ system $\alpha$'s consistency 
+ are 
+different 
+from each other
+because the former
+requires  skolemizing $\,\alpha \,$'s axioms, $\,$while the
+latter permits
+\cite{Fi90} 
+an existential quantifier
+elimination rule
+to
+replace Skolemization.
+This distinction   can
+create a
+ potential
+exponential difference
+ between
+the lengths of
+Herbrand and Semantic Tableaux proofs.
+This 
+insightful observation,
+%distinction,
+due to private communications from
+L. A.  Ko{\l}odziejczyk \cite{Ko5},
+was used in
+ \cite{ww9}
+to create an axiom system that satisfied the
+semantic tableaux but not
+also Herbrandized version of the
+Second Incompleteness Theorem.}
+that \cite{ww9}
+developed 
+an axiom system that was a boundary-case
+exception to the Herbrandized but not also 
+the
+semantic tableaux version
+of the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+
+The literature on Definable Cuts has
+centered its evasions of 
+the Second Incompleteness Theorem around 
+\ep{dcon}'s localization formalism
+(rather than  
+employing  analogs of
+ Section \ref{secc1}'s
+SelfRef$(\alpha,d)$ axiom,
+as we did in \cite{ww93,ww1,ww5,ww6,wwapal,ww9}).
+Pudl\'{a}k \cite{Pu85}
+proved that essentially
+every axiom system
+$    ~     \alpha   ~      $ 
+ of
+finite cardinality can be associated with
+a definable cut  $  ~      \varphi   ~      $ such that 
+$    ~     \alpha   ~      $ 
+can prove
+sentence \eq{dcon}'s  validity for $    ~     \varphi   ~  $
+when $    ~     d   ~      $ is either
+the semantic tableaux or Herbrand-styled
+deductive method.
+
+
+
+Pudl\`{a}k's 
+theorem is related to 
+Friedman's
+observation \cite{Fr79b}
+that 
+for many finite theories
+$S$ and $T$,
+the theory $  S  $ has an interpretation in $  T  $
+if and only if
+I$\Sigma_0+Exp$ can prove that
+$T$'s Herbrand consistency implies
+$S$'s Herbrand consistency.
+Several generalizations of these results 
+by 
+ Kraj\'{i}cek, 
+Pudl\`{a}k,
+ Smory\'{n}ski,
+\v{S}vejdar 
+  and Visser appear  in
+\cite{Kr87,Pu85,Pu96,Sm85,Sv7,Vi92,Vi93,Vi5}.
+Visser's article \cite{Vi5} 
+contains 
+an excellent review of this literature,
+as well as
+many 
+additional
+new results.
+Also,
+we will see how 
+% some of
+some of the reflection machines of
+Beklemishev,
+Kreisel-Takeuti and
+ Verbrugge-Visser
+\cite{Be95,Be97,Be3,KT74,VV94,Vi5}
+nicely complement
+\thx{ppp6}'s
+ reflection mechanisms
+in alternate types of intended applications.
+
+
+It was established by 
+H\'{a}jek,  
+\v{S}vejdar and
+Vop\v{e}nka \cite{Sv83,VH73}
+that GB Set Theory can
+construct
+a definable cut $\varphi$ where
+it can prove
+the statement  \eq{dcon} is  valid when
+$~d~$ denotes Hilbert deduction and
+$~\alpha~$ 
+is ZF Set Theory.
+This result was
+ surprising because Pudl\'{a}k \cite{Pu85}
+showed GB can never verify its own 
+Hilbert consistency 
+localized
+on a definable cut. (Thus, GB will  view its Hilbert consistency as
+equivalent to ZF's Hilbert consistency
+in a global sense 
+ {\it but not} in a  cut-localized
+respect.).
+
+
+%% 
+%% H\'{a}jek,  
+%% \v{S}vejdar and
+%% Vop\v{e}nka \cite{Sv83,VH73}
+%% constructed
+%% a definable cut $\varphi$ where
+%%  GB Set Theory
+%% can prove
+%% that the statement  \eq{dcon} is  valid when
+%% $~d~$ denotes Hilbert deduction and
+%% $~\alpha~$ 
+%% is ZF Set Theory.
+%% This is
+%%  surprising because Pudl\'{a}k \cite{Pu85}
+%% proved 
+%% GB can never verify its 
+%% Hilbert consistency 
+%% localized
+%% on a definable cut.
+%% Thus, GB will  view its Hilbert consistency as
+%% equivalent to ZF's Hilbert consistency
+%% in a global sense 
+%%  {\it but not} in a  cut-localized
+%% respect.
+%% 
+
+\bigskip
+
+
+In some sense, 
+ Kreisel and Takeuti
+  \cite{KT74,Ta53}
+can be viewed as
+the 
+first authors to develop a
+%%type of   system 
+logic
+recognizing its own
+ consistency
+using a
+variant of \ep{initdefx}'s
+formula.
+Their
+results
+for typed logics
+%  have  
+formalized a second-order generalization of 
+Gentzen's sequent calculus
+that can verify
+% a declaration of
+ its own consistency, 
+%in the special case where 
+when
+no sequent calculus deductive
+cuts are performed.
+A key aspect of their formalism
+can be seen as using
+an analog of
+\eq{dcon}'s sentence in an implicit 
+manner. 
+It thus begins
+by using a set of objects, which we shall call $~I,~$
+that includes all the standard integers 
+plus  some
+allowed
+ {\it non-standard integers}
+(that can 
+permissibly
+represent 
+contradictory  proofs).
+  %, and
+Their second-order logic
+ then uses Dedekind's definition of the natural numbers to 
+construct a 
+subset of $~I~$, $\,$called
+ ``$~N~$'', which
+includes all the standard integers
+and which is disallowed to contain any
+contradiction proof.
+We will not go into the details here, but this
+transition from $\,I\,$ to $\,N\,$ (with an 
+accompanying
+relativization of the provability predicate onto
+$\,N\,$'s more restricted domain) can be viewed
+as Kreisel-Takeuti's
+analog
+ of %    sentence  
+\eq{dcon}'s 
+ local consistency 
+statement
+for
+\cite{KT74,Ta53}'s
+``CFA''
+ second-order logic.
+
+
+
+
+
+
+It is difficult to compare our   research
+(which 
+has
+relied upon an analog of 
+SelfRef$(\alpha,d)$'s
+Kleene-like
+{\it ``I am consistent''} axiom)
+with the
+preceding literature
+that has
+used 
+various forms of
+Localized  d-Consistency 
+statements.
+This is because every effort to 
+evade the Second Incompleteness
+Theorem
+employs
+some built-in weakness
+to evade G\"{o}del's classic paradigm.
+
+
+\smallskip
+
+Our 
+work
+in \cite{ww93,ww1,ww5,ww6,ww9}  represented
+less than a 
+full-scale evasion of the
+Second Incompleteness Theorem
+mostly
+because 
+it was
+ incompatible with treating 
+{\it as  formal
+axioms}
+the statements in
+Equations \eq{totdefxm}
+and \eq{totdefsymbm}
+that
+ multiplication is a total function
+\footnote{ \b55 This 
+caveat applies
+also  to our article \cite{ww9},
+although its
+Herbandized form of
+self-justification
+differs from our other papers
+by retaining a capacity to treat 
+\eq{totdefsymbm}'s
+statement about multiplication's totality as a
+derived theorem {\it that is
+not an} 
+%  a formalized  
+axiom.
+The key point is that  
+theorems are
+ weaker
+than  axioms
+under 
+Herbrand deduction 
+ because only
+axioms are used as intermediate steps 
+during proofs.
+This
+explains intuitively
+how    \cite{ww9}'s formalism
+was able to recognize
+its 
+%own 
+Herbrandized
+consistency, $\,$while 
+treating
+\eq{totdefsymbm}'s statement about
+the totality of
+   multiplication 
+ {\it
+as a theorem}.
+(We will return to this subject in 
+Appendix D.) } .
+%(More will be said about this subject at the end of
+%Appendix D.) } .
+Some 
+reasons
+why
+it was helpful for
+\cite{ww93,ww1,ww5,ww6,ww9}
+to employ 
+ analogs of    
+Section \ref{secc1}'s
+SelfRef 
+axiom are that:
+\bed
+\topsep -3pt
+\itemsep +2pt
+\item[A  ]
+An axiom's self-referential
+{\it ``I am consistent''} declaration
+allows a formalism to recognize its consistency
+in 
+a
+global sense, rather than in the 
+$\varphi -$localized sense
+used by
+sentence
+\eq{dcon} and 
+the analogous 
+  Kreisel-Takeuti relativization of their
+second-order
+proof predicate.  
+\item[B  ]
+If a logic is employing a deductive method
+$d$ that lacks a modus ponens rule, as occurs in
+nearly all
+self-justifying
+systems, then it is
+          preferable for  it           to 
+view its {\it ``I am consistent''} 
+statement as an axiom rather
+than as a theorem.
+(This is
+ because 
+weak deductive methods are
+ capable of
+drawing 
+logical inferences
+only 
+from
+axioms
+when
+modus ponens 
+is absent.) 
+\item[C  ]
+Analogs of
+Section \ref{secc1}'s
+SelfRef$(\alpha,d)$'s
+{\it  ``I am consistent''} axiom
+have been shown
+by 
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}
+to at least partially
+formalize the notion of a 
+logic  
+possessing  {\it  an 
+ almost}
+instinctive  
+form of 
+faith in its
+own internal consistency.
+(This paper will make it apparent that
+such an instinctive faith is less than a full-scale
+proof. Yet, Theorem \ref{ppp6} and
+ Remarks 
+\ref{f88},
+\ref{remhappy} 
+and \ref{recc1}
+will make it apparent that such formalizations
+of instinctive faith are also
+useful.) 
+\ennd
+We emphasize
+that both  virtues and drawbacks of
+SelfRef$(\alpha,d)$'s
+{\it  ``I am consistent''} 
+axiom statements 
+have been 
+cited
+in this paragraph because
+every effort to
+evade the Second Incompleteness
+Theorem can obtain no more than
+limited levels of
+success.
+
+
+% \nop
+
+\cvmew
+
+The scope of the challenge we face becomes apparent when one
+realizes 
+ $\alpha \, + \,$SelfRef$(\alpha,d)$ 
+is 
+ inconsistent 
+for most
+$(\alpha,d).~$
+  This is because
+   $\alpha+$SelfRef$(\alpha,d)$ 
+typically
+satisfies Part-i
+  {\it but not also }
+  Part-ii of
+Section 
+\ref{secc1}'s  definition 
+ of a ``self-justifying'' logic.
+(Thus, 
+a
+% G\"{o}del-like 
+diagonalization
+paradigm will typically 
+imply 
+$\alpha \, + \,$SelfRef$(\alpha,d)$ 
+is inconsistent,
+$\,$as a 
+consequence of 
+it containing $ ~$SelfRef$(\alpha,d)~$ 
+as an axiom.)
+This is the
+reason      Kleene, 
+ Rogers and Jeroslow \cite{Kl38,Ro67,Je71}
+were 
+hesitant
+%skeptical
+about 
+the utility of
+SelfRef$(\alpha,d)$'s
+mirror-like
+axiom sentence.
+Our goal 
+in \cite{ww93}-\cite{ww9} 
+has 
+been
+ to 
+develop generalizations and boundary-case exceptions for
+the Second Incompleteness Theorem, so as 
+determine exactly
+which paradigms
+%%\
+%%\{\it 
+%%\paradigms
+%%\are sufficiently
+%%\weak}
+%%\so that 
+%%\they 
+%%\
+can
+support, for example,
+Theorem \ref{ppp6}'s limited notion
+of self-justification.
+
+
+
+
+
+
+The 
+%intuitive
+ reason 
+one would  anticipate some 
+{\it limited}
+%partial
+exceptions to the Second Incompleteness Theorem 
+to
+exist is 
+it is  hard to 
+imagine how humans can motivate
+themselves to cogitate without 
+using 
+some
+variant 
+of 
+self-justification.
+
+%%
+%%Thus, our %recent 
+%%research 
+%%project
+%%has sought to {\it both} generalize
+%%  the Second Incompleteness
+%%Theorem and explore when it can be evaded.
+
+
+
+
+
+
+
+
+
+
+
+\cvl
+
+\section{Generic Configurations}
+
+\label{3uuuu1}
+
+
+% \vspace*{- 0.2 em}
+
+
+
+%Our main results about self-justification will
+%appear in Sections 
+%\ref{3uuuu2} -- \ref{sect64}.
+The
+ phrase  {\bf Bounded Quantifier}
+will refer
+to
+expressions of the form
+``$\, \exists~v  \leq  T \,$'' or
+``$ \,\forall~v  \leq  T\,$''
+where $T$ is a term.
+A formula  
+is
+called
+{\bf Fully-Bounded$\,$} when all its quantifiers are
+   so bounded.
+\lem{lex22} will soon explain how Definition \ref{xd+1x1}'s 
+formalism can encode conventional arithmetic:
+ 
+
+
+
+
+
+
+
+
+
+\medskip
+
+
+\bxbxdd
+\label{xd+1x1}
+\rm
+Let $~\xi~$ denote some 
+non-integer
+indexing superscript
+(whose  properties will 
+be 
+discussed
+ later by
+Definition
+\ref{def3.3}). 
+Then the symbol
+$~\Delta_0^\xi~$
+will 
+denote some 
+fixed
+special
+set of fully-bounded formulae
+that
+is
+closed 
+under negation, 
+in a 
+language
+that will be later
+ called $~L^\xi~$.
+(Thus, if some
+formula $~ \Psi~$ is a member of
+             $~\Delta_0^\xi~$
+then so is 
+$~ \neg ~ \Psi~$.)
+Items 1-3 
+formalize how
+$~\Pi_n^\xi~$ 
+and 
+$~\Sigma_n^\xi$ formulae are
+built 
+in a straightforward manner
+out of
+these
+ $~\Delta_0^\xi~$
+sub-components:
+\bee
+\topsep -15pt
+\item
+Every $ \, \Delta_0^\xi \, $ formula is
+considered to be also a
+$ \, \Pi_0^\xi \, $ 
+and 
+$ \, \Sigma_0^\xi$ formula.
+\item
+For $~n \, \geq \, 1~~,~$
+a formula 
+will be 
+called
+$ ~~  \Pi_n^\xi~~$
+iff it can be
+written 
+in the canonical form of
+$\forall v_1 ~ \forall v_2 ~...~ \forall v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$  where
+$\Phi$ is  $~\Sigma_{n-1}^\xi.~$
+\item
+Likewise for $~n \, \geq \, 1~~,~$
+ a formula will be
+called
+$~~ \Sigma_n^\xi~~ $
+iff it can be
+written in the 
+form of
+$~\exists v_1 ~ \exists v_2 ~...~ \exists v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$
+  where
+$\Phi$ is  $~\Pi_{n-1}^\xi.~$
+\ene
+\eedd
+
+\medskip
+\noindent
+{\bf Notation Convention:}
+Our rules for defining $\,\xi$, specified later in this section,
+will never have this  superscript 
+designate
+an integer quantity. This is because 
+integer superscripts have a special meaning under 
+a typed-based hierarchy,
+not intended here.
+
+
+
+
+\nop
+
+\begin{example}
+\label{ex3-1}
+\rm
+Let
+$~L~$  denote a 
+conventional arithmetic
+language that uses
+function symbols 
+for denoting 
+addition and multiplication.
+Below are two
+examples  
+of $\Delta_0-$like formulae that
+invoke
+Definition \ref{xd+1x1}'s
+notation:
+\bed
+\itemsep 1pt
+\item[  a ] The 
+symbol ``$~\Delta_0^A~$''
+will denote any 
+fully bounded
+formula that uses the addition, multiplication and maximum
+function symbols in an arbitrary manner.
+(Thus,  $~\Delta_0^A~$ 
+corresponds to what 
+many
+textbooks
+\cite{HP91,Ka91,Kr95}
+simply call a ``$~\Delta_0~$'' formula.)
+\medskip
+\item[  b ] The 
+symbol ``$~\Delta_0^{R}~$'' 
+will denote a class of 
+formulae in $L$'s language
+whose bounded quantifiers are  allowed
+to use
+only 
+  the Maximum function symbol. 
+Their
+bodies,
+however, may contain
+any combination of 
+addition, multiplication
+and maximum function symbols.
+\ennd
+Formulae \eq{ex1}  
+and \eq{ex123} 
+illustrate the
+distinction
+between the  $\Delta_0^A$ 
+and  $\Delta_0^{R}$ classes. Thus, \eq{ex1} satisfies the first
+but not second condition (on account of the presence of the
+multiplication symbol used by its bounded quantifiers).
+ In contrast,
+\eq{ex123}
+is an example of a  
+ $~\Delta_0^{R}~$ formula.
+\beq
+\label{ex1}
+\exists \, y\leq  x*x ~~~~
+\forall  \, z\leq  y*y  ~~~~
+\exists \, w\leq  y*z ~~~~
+~~~~~: ~~~~~ 
+ \{ ~~~~ x  *y=z+w ~~~~ \}
+\enq 
+\vspace*{-0.5 em}
+\beq
+\label{ex123}
+\exists \, y\leq  x ~~~~
+\forall  \, z\leq  y  ~~~~
+\exists \, w\leq  \mbox{Max}(y,z) ~~~~
+~~~~~: ~~~~~ 
+ \{ ~~~~ x  *y=z+w ~~~~ \}
+%% x  *y=z+w
+\enq 
+The distinction between 
+ $ \Delta_0^{A}  $
+arithmetic formulae and the unconventional
+ $ \Delta_0^{R}  $ class may first convey the impression
+that these
+two 
+classes 
+have
+fundamentally 
+different natures. Actually,
+\lem{lex22} will
+show
+that their
+relationship is more subtle.
+This is because
+its formalism
+will map  $ \Delta_0^{A}  $ formulae
+onto     $ \Delta_0^{R}  $
+expressions that are equivalent to it under Definition
+\ref{snn}'s Standard-M model
+$~$---$~$ in a context where
+only the length of these
+formulae
+is allowed to 
+possibly
+grow.
+This 
+% result
+% Standard-M 
+equivalence
+ enabled
+ \cite{ww9} to construct
+a natural axiomatic formalism
+that could recognize its
+own
+Herbrandized consistency  
+{\it but which nevertheless  satisfied} 
+the 
+idealized form   
+\footnote{  \b55 An axiom system $\alpha$ 
+is defined to
+ satisfy
+the 
+{\bf ``idealized form''}
+of the
+semantic tableaux
+version 
+ Second Incompleteness Theorem
+when no  $\beta  \supseteq  \alpha$ can
+prove a
+ semantic tableaux
+proof
+ of 0=1
+ from 
+itself is incapable of existing.
+We will summarize
+\cite{ww9}'s formalism
+and the distinction between
+Herbrandized and semantic tableaux 
+deduction 
+at the 
+of 
+Appendix D.} 
+of
+the 
+semantic tableaux
+version of the 
+Second Incompleteness Theorem.
+\end{example}
+
+
+
+
+
+
+
+
+
+\medskip
+
+\begin{definition}
+\label{snn}
+{\bf ``Standard-M''} will denote
+the
+standard model of integers.
+\end{definition}
+
+\medskip
+
+The reason for our interest in Standard-M is
+that
+many 
+pairs of formulae
+are equivalent
+under the Standard-M
+model, while
+weak axiom systems often
+cannot formally prove they are
+equivalent.
+For instance,  this 
+will occur
+when  Example \ref{ex3-2}
+examines
+Definition \ref{def3.3}'s properties.
+
+
+%\notre-only \medskip
+
+
+\nop
+
+\bxbxdd
+\label{def3.3}
+A  {\bf Generic Configuration},
+often identified by 
+ the superscript symbol of
+ $~\xi~$, is defined to be a 5-tuple
+$~(\, L^\xi \, , \, \Delta_0^\xi \,  , \, B^\xi \, , \, d \, , \, G \, )~$
+where:
+\bee
+\topsep -12pt
+\rm
+\item
+$ L^\xi $  
+is a 
+language that includes 
+logical symbols
+for ``$0$'',  ``$1$'',  ``$2$'',
+``$=$'', ``$\leq$'' 
+ and
+for the operation of ``Maximum(x,y)''.
+$ L^\xi $   also includes a 
+ sufficient number of 
+function  and constant symbols 
+so that every integer $~k~$ can be encoded by
+some term 
+$~T_k~$ 
+specifying
+$~k\,$'s value.
+\item
+$~ \Delta_0^\xi ~$ 
+corresponds to 
+any variation
+of
+Definition \ref{xd+1x1}'s 
+class of 
+``fully-bounded'' formulae
+that is rich enough to assure that there exists
+ two  $~\Delta_0^\xi~$ 
+formulae, henceforth
+called ``Add$(x,y,z)$''
+and ``Mult$(x,y,z)$'', $\,$for formalizing 
+the graphs of
+addition and multiplication.
+(It will  generate 
+$\xi$'s set of $\Pi_n^\xi$
+and $\Sigma_n^\xi$ sentences,
+using Definition \ref{xd+1x1}'s 3-part formalism.)
+\item 
+$ \, B^\xi \,$  
+denotes 
+a$~$ {\normalsize {\bf ``Base Axiom System''}}, 
+$~ \,$whose axiom-sentences are true under
+the 
+Standard-M model
+and which is
+$\Sigma_1^\xi$ complete.
+(Thus,
+$  B^\xi $  can  prove every true $\Sigma_1^\xi$ sentence, and
+it can
+likewise refute all false $\Pi_1^\xi$ sentences.)
+\item
+$~d~$   denotes  $ \, \xi \,  \, $'s  
+ method of deduction.
+It is required to be sufficiently conventional to
+satisfy the usual indirect-implication 
+property \footnote{  \b55   
+$~~$This is that irregardless of whether or not
+$~d~$ contains a built-in
+modus
+ponens rule, it does support some
+form of
+a (possibly
+quite 
+lengthy)
+ proof of a
+theorem $Z$,
+when it is able to prove
+$X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as
+theorems. }
+associated with
+ G\"{o}del's Completeness Theorem. 
+\item
+$~g~$   denotes  a method for
+encoding the  G\"{o}del numbers
+of proofs.
+\ene
+\eedd
+
+
+
+
+
+\medskip
+
+\begin{example}
+\label{ex3-2}
+\rm
+Let us recall
+Example \ref{ex3-1} defined
+``$~\Delta_0^A~$'' 
+as essentially the conventional
+textbook notion
+\cite{HP91,Kr95}
+ of an arithmetic 
+ ``$~\Delta_0~$'' formula.
+This example  will  outline how
+well-known techniques can map
+every $\Delta_0^A$  formula 
+onto a semantically 
+equivalent 
+ $\Delta_0^\xi$ 
+formula under the Standard-M model.
+
+
+
+
+
+
+
+
+
+Our discussion will have
+Seq$(x)$   denote a 
+function that
+maps 
+non-negative integers onto 
+binary strings
+in
+lexicographic order.
+Thus Seq$(x)$  maps
+0 onto the empty
+string, 
+$\,$the integers
+1 and 2 onto the strings of ``0'' and ``1'', $\,$the integers 3--6
+onto$\,$  ``00'',   ``01'',   ``10'',   ``11'', etc.   
+(Formally,
+Seq$(x)$
+is
+an operation
+that maps integer $~x~$ onto the bit-string 
+that occurs to the immediate right of the leftmost ``1'' bit in
+the binary encoding of $~x+1.~~)$
+
+  
+
+Given any $\,k-$tuple  $~(~x_1\, , \, x_2\, , ~ ... ~ x_k~),~$
+let  $~$STRING$(   x_1   ,   x_2   ,\, ... \,   x_k   )$
+denote the
+ concatenation of
+Seq$(x_1), \, ... ~$Seq$(x_k)$.
+For any 
+integers $v$ and $w$ satisfying $v  \leq w^2$, 
+it is clear that 
+there  exists $(x_1,x_2,x_3)$ where
+STRING$(x_1,x_2,x_3)$ 
+represents $v  $'s
+binary encoding and
+each $x_i  \leq   \mbox{Max}(w,4)$.
+
+\smallskip
+
+An example will now illustrate
+the approximate structure of an inductive methodology
+for mapping
+$\Delta_0^A$ formulae 
+onto their equivalent   $\Delta_0^\xi$  counterparts
+in the Standard-M model.
+Let SQUARE$(x_1,x_2,x_3,w)$ be a 
+$\Delta_0^\xi$ formula which 
+specifies 
+that
+STRING$(x_1,x_2,x_3)$ represents  an integer 
+$ \leq \, w^2~.~ \,$ Also,  
+$\,$let $\phi^*(x_1,x_2,x_3)$
+and $\phi(v)$ represent 
+a pair of  $\Delta_0^\xi$ and
+$\Delta_0^A$ formulae 
+that
+are equivalent
+under the Standard-M model
+$\,$when STRING$(x_1,x_2,x_3)$ is an encoding for  $~v~$. 
+Then one
+possible method
+for mapping $\Delta_0^A$ formulae onto their 
+equivalent $\Delta_0^\xi$ 
+counterparts 
+(in the Standard-M model)
+could 
+map the
+formula \eq{exam21} 
+onto 
+\eq{exam22}'s alternate form:
+\beq
+\label{exam21}
+\forall ~ v\, \leq \, w^2~~~~~\phi(v)
+\enq
+\smallskip
+\begin{center}
+$\forall ~ x_1 \leq~ \mbox{Max}(w,2)~~~~~\forall ~ x_2 \leq ~ \mbox{Max}(w,2)
+  ~~~~~\forall ~ x_3 \leq ~ \mbox{Max}(w,2)$
+\end{center}
+
+\vspace*{-0.5 em}
+
+\beq
+\label{exam22}
+  \{~~ \mbox{SQUARE}(x_1,x_2,x_3,w)
+  ~~ \Rightarrow~~
+\phi^*(x_1,x_2,x_3)~~\}
+\enq
+\lem{lex22}
+indicates
+Example \ref{ex3-2}'s
+translational methodology generalizes easily to
+       all combinations of   $\Delta_0^A$  inputs
+and
+generic configurations $\, \xi \,$,
+via
+an 
+approximate
+inductive generalization of the transition from
+sentence \eq{exam21} to \eq{exam22}. (Its 
+procedure essentially
+performs
+iteratively a finite number of such transitions, so as to
+translate   all 
+the
+clauses of
+ an
+initial
+$\Delta_0^A$ 
+% format
+formulae
+into their 
+$\Delta_0^\xi$ 
+counterparts via an inductive methodology.
+The intuition behind these transitions is 
+they will repeatedly replace a single variable,
+such as $~v~$ in 
+sentence \eq{exam21},
+with a multiplicity of variables,
+such as $(x_1,x_2,x_3)$ in 
+\eq{exam22}.)
+\end{example} 
+
+
+
+\smallskip
+
+\begin{lemma}
+\label{lex22}
+{\rm (Paris-Dimitracopoulos
+\cite{PD82} )}
+For every generic configuration $\xi$, each
+$\Delta_0^A$ formula
+can be translated
+into an  equivalent 
+$\Delta_0^\xi$ 
+formula
+in the
+Standard-M
+model
+via a generalization of Example \ref{ex3-2}'s
+process.
+{\rm (This 
+ clearly
+ implies 
+$\Pi_j^A$ 
+formulae
+can
+also be
+translated
+into
+$\Pi_j^\xi$ 
+expressions.)}
+\end{lemma}
+
+
+
+\smallskip
+
+
+
+
+Paris-Dimitracopoulos \cite{PD82}
+sketched   
+an 
+%approximate
+analog of Lemma \ref{lex22}'s
+translation 
+algorithm,
+using
+only
+ slightly different notation,
+ that
+is applicable to 
+any formalism that satisfies Parts (1) and (2)
+of 
+ Definition \ref{def3.3}.
+Their formalism
+thus uses an 
+inductively-iterated
+analog of the prior example's
+replacement of a single variable $~v~$ in formula
+\eq{exam21} with \eq{exam22}'s multiplicity of variables 
+% in the tuple 
+$~(x_1,x_2,x_3)~,~$ 
+so as to perform
+ Lemma \ref{lex22}'s translation 
+task.
+It will  be unnecessary for a reader to 
+consider the
+details behind 
+% either 
+\cite{PD82}'s Theorem 1 or
+Lemma \ref{lex22}'s 
+similar translation 
+mechanism
+%%% 
+%%% 
+because
+ the remainder of this article will never
+% explicitly
+ use
+them again.
+%its mechanism.
+Instead, their sole purpose 
+% of  Lemma \ref{lex22}
+has been
+% merely
+ to provide an {\it implicit backdrop}
+for our results by illustrating how
+the study of the $\Pi_1^\xi$ sentences of
+Definition \ref{def3.3}'s generic configurations
+% also
+provides information
+about   $\Pi_1^A$ sentences 
+(after the needed translating is done).
+
+\smallskip
+
+Four  examples of 
+self-justifying
+systems that employ  
+Definition \ref{def3.3}'s 
+ $\Pi_1^\xi$ sentences 
+ will be
+illustrated in 
+\appD.
+These examples   are 
+too
+complicated
+to be 
+examined
+ before Sections \ref{3uuuu1} -- \ref{sect64}
+are 
+read.
+However, the next example 
+should convey some useful  intuitions:
+
+\medskip
+
+\nop
+
+\cvwew
+
+\begin{example}
+\label{ex3-3}
+\rm
+Let
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+denote sequences defined by:
+\vspace*{- 0.7 em}
+\beq
+\label{zs}
+x_0~~~=~~~2~~~=~~~y_0
+\enq
+\beq
+x_{i}~~~=~~~x_{i-1}~+~x_{i-1}
+\label{as}
+\enq
+\beq
+y_{i}~~~=~~~y_{i-1}~*~y_{i-1}
+\label{bs}
+\enq
+\end{example}
+For $\, i \, > \,0 \,$, 
+$\,$let $ \, \phi_{i} \, $ 
+and $ \, \psi_{i} \, $ 
+denote the  
+sentences in 
+\eq{as} and \eq{bs}
+respectively.
+Also, 
+ let
+  $ \, \phi_{0} \, $ and
+$ \, \psi_{0} \, $ 
+denote \eq{zs}'s
+sentence.
+Then
+ $ \, \phi_0, \, \phi_1, \, ... \, \phi_n \, $
+imply
+ $ \, x_n \, = \, 2^{ n+1} \, , \, $ and 
+ $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
+ imply $ \, y_n \, = \, 2^{2^n} \, $.
+Thus, the  latter sequence
+grows at a 
+faster
+rate than 
+the former.
+Much of our research has used the difference between the
+growth rates of 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+as a motivating example explaining why 
+\ep{totdefxa}'s Type-A axiom systems can
+support a stronger form of boundary-case exception
+to the semantic tableaux version of the Second Incompleteness
+theorem than
+can 
+Type-M systems.
+
+\parskip 2pt
+
+
+\smallskip
+
+
+Let
+Log$(\, y_n \,) \, = \, 2^{n} \, $ 
+and
+Log$(\, x_n \,) \, = \, {n+1} \, $ 
+thus
+designate the lengths of the binary codings for
+$ \, y_n \, $
+ and 
+$ \, x_n \, $.
+Then $ \, y_n\,$'s coding 
+has a length
+$\,  2^{n} \,  $, which is
+ {\it much larger} than
+the $  n+1  $
+steps that  $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
+use to
+define its existence.
+However,
+ $ \, x_n\,$'s 
+length  has a 
+ smaller
+ size of
+$ \, {n+1} \, $.
+These observations are useful because every proof 
+of the 
+Incompleteness Theorem
+involves 
+a 
+G\"{o}del
+ number $ \, z \,$ 
+coding a sentence
+that has a  capacity
+to self-reference its own definition.
+The faster
+growing series $y_0,\,y_1,\,,\,...\,y_n$
+should  
+be intuitively 
+anticipated
+to have
+ this 
+self-referencing
+capacity because 
+ $~y_n\,$'s binary encoding 
+has a 
+$~2^{n+1}~$  length that 
+dwarfs the
+size of the $O(n)$
+steps 
+used
+ to define its
+value. Leaving aside 
+\cite{ww2,ww7}'s
+%
+% wwoooo
+%
+many details,
+this
+fast growth 
+explains
+roughly
+ why many Type-M 
+logics
+satisfy the semantic tableaux version of 
+the Second Incompleteness Theorem.
+
+
+\smallskip
+
+This paradigm also 
+illustrates intuitively
+why some 
+ Type-A  systems, employing
+\cite{ww93,ww1,ww5}'s
+semantic tableaux formalism,
+can 
+represent
+boundary-case exceptions to the
+ Second
+Incompleteness
+Theorem.
+ This is because such formalisms 
+lack access to 
+\ep{totdefxm}'s axiom that multiplication is a total function.
+(They are
+unable,
+thus,
+ to
+easily
+ construct numbers $ \, z \, $ that can
+self-reference their own definitions
+because they have access  only
+to the slower growing
+              addition primitive.)
+In particular
+assuming only that each sentence in
+the axiom-sequence
+  $   \phi_0,   \phi_1,   ...   \phi_n   $
+(from  \ep{as} )
+requires  a mere 
+two bits 
+for its encoding, 
+the length    $     n+1     $ of 
+ $   x_n   $'s
+binary encoding 
+will be 
+smaller
+than the
+length of its
+defining 
+ sequence.
+
+\smallskip
+
+This short length for  $   x_n   $
+had
+motivated 
+\cite{ww93,ww1,ww5,ww6}'s
+evasion of  
+the semantic tableaux 
+version
+of the
+Second Incompleteness Theorem. 
+It
+suggested that the self-referencing
+needed in a
+G\"{o}del-like diagonalization argument would stop being
+feasible
+when \ep{as}'s slow-growing 
+$x_1,\,x_2,\,x_3,\,...$
+sequence represents the fastest growth that is possible.
+
+
+
+
+One of the several goals in this article 
+will be
+ to formalize 
+a generalizations of
+\cite{ww93,ww1,ww5,ww6}'s
+self-justifying methodologies by using
+Definition \ref{def3.3}'s
+generic configurations.
+The  proofs
+of  our main theorems
+will, of course, be
+more subtle than
+the hand-waving intuitions appearing in this example.
+For instance, the combined work of
+Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87}
+(summarized by Theorem 2.1)
+raised the subtle issue that
+  no Type-S   system
+can
+prove a theorem affirming its
+own Hilbert consistency. 
+Another complication 
+is that the
+\ep{bs}'s implication for proofs that use
+the multiplication operative has
+different side effects for
+Herbrandized and  
+semantic tableaux
+deduction
+(on account of
+ Ko{\l}odziejczyk
+\cite{Ko5,Ko6}'s
+previously mentioned
+observations about the potential
+exponential difference between the lengths of these
+proofs
+under extremal
+circumstances). 
+
+
+
+
+\smallskip
+
+Our
+main theorems
+will show
+that
+self-justifying systems, 
+using four
+deduction methods,
+are
+capable of  proving 
+all of Peano Arithmetic's  $\Pi_1^\xi$ 
+theorems.
+Interestingly,
+self-justification
+will
+be compatible with \cite{ww5}'s modification
+of semantic tableaux deduction, 
+that includes a
+ modus ponens  rule 
+ for 
+ $\Pi_1^\xi$  and  $\Sigma_1^\xi$ type sentences.
+However,
+ \cite{wwlogos}
+has shown 
+an analogous
+  modus ponens rule for
+ $\Pi_2^\xi$  and  $\Sigma_2^\xi$  sentences
+is incompatible
+with self justification.
+(Thus, the contrast between our
+main 
+results and the
+Second Incompleteness
+Theorem's generalizations will be quite tight.)
+
+
+
+\cvl
+
+
+\section{Five Helpful Definitions and An Informative Lemma}
+
+\label{3uuuu2}
+
+\label{D.4-theorx} 
+
+
+This section will  introduce 
+five definitions and 
+ prove 
+a 
+Lemma \ref{lemex4} 
+about
+self justification.
+This lemma 
+will be 
+% much 
+weaker
+than
+Sections \ref{3uuuu3}
+and \ref{sect64}'s
+main results.
+Its
+main
+purpose
+ will be to provide
+a
+useful
+starting example. 
+
+
+
+
+
+\smallskip
+
+\bxbxdr
+\label{xd+1x3}
+The symbol
+``E$(n)$'' 
+will denote some
+term 
+in Definition \ref{def3.3}'s
+language  $ \, L^\xi \, $
+that represents the
+value  
+ $ \, 2^n \, . \, $ 
+In using 
+this symbol, we  do
+not presume that
+$ \, L^\xi \, $ possesses a function
+symbol for 
+the exponent operation.
+Thus if  $ \, L^\xi \, $
+has only a function symbol for
+multiplication,
+then  
+ E$(n)$ 
+could 
+designate
+the term of $ \, $``$ \, 2*2* \, ... \, *2 \, $''$ \, $ with
+$ \, n \, $ repetitions of ``2'' $. \, $ (Alternatively,
+ E$(n)$ 
+can be
+ defined via
+applying
+ $\, 2^n \,$ iterations
+of the successor function to zero,
+or by having a
+special constant symbol  designating
+ $ \, 2^n \, $'s value.
+Essentially, any reasonable method can
+be used to define  E$(n)$'s value)
+
+
+\eedd
+
+\cvrew
+
+\nop
+
+\smallskip
+
+\bxbxdr
+\label{xd+1x4}
+Let
+$\,\Uxp \,$ denote
+a prenex normal sentence.
+Then 
+{\bf Scope$_E$($\Uxp,$N)$~$} will  denote
+a sentence identical to $~\Uxp~$ except that
+every unbounded universal quantifier  ``$~\forall~v~$''
+is changed to  ``$ \, \forall \, v \, < \, E(N) \, $'',
+and every
+ unbounded existential quantifier  ``$ \, \exists \, v \, $''
+is changed to ``$ \, \exists \, v \, < \, E(N) \, $''.
+(No change is made among the bounded quantifiers within
+the
+$~\Delta_0^\xi$ 
+part of the sentence
+$\,\Uxp \,$.) 
+For example, if 
+$\,\Uxp \,$ denotes
+the $\Pi_1^\xi$ 
+sentence of
+$~\forall \, v_1~\forall \, v_2~...~\forall \, v_k~~~\phi(v_1,v_2,...v_k)~$
+then
+\eq{scopede} 
+illustrates
+Scope$_E$($\Uxp,$N)'s form.
+Likewise if
+$~\Uxp ~$  is
+the $~\Sigma_1^\xi~$ 
+sentence of
+$~\exists \, v_1~\exists \, v_2~...~\exists \, v_k~~~\phi(v_1,v_2,...v_k)~$
+then
+\eq{scopedx} illustrates
+Scope$_E$($\Uxp,$N)'s form.
+
+\vspace*{- 0.4 em}
+
+{ 
+\small  
+\cvl
+\beq
+\label{scopede}
+\forall ~ v_1~ < ~E(N)~~\forall ~ v_2~ < ~E(N)~~ ...
+\forall ~ v_k~ < ~E(N) ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~
+\enq
+\beq
+\label{scopedx}
+\exists ~ v_1~ < ~E(N)~~\exists ~ v_2~ < ~E(N)~~ ...
+\exists ~ v_k~ < ~E(N) ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~
+\enq}
+\eedd
+
+
+{\bf Special Note about Definition \ref{xd+1x4}'s Meaning.} 
+If
+$\Uxp$ is a 
+$\Delta_0^\xi~$ 
+sentence then
+Scope$_E$($\Uxp,N)$ 
+will 
+be equivalent to  $~\Uxp~$
+for every $N \geq 0$
+by definition.
+(This is  because 
+$\Delta_0^\xi~$  formulae
+contain
+no unbounded  quantifiers
+that undergo change when 
+  $\Uxp$ is mapped onto  
+Scope$_E$($\Uxp,N).~~)$ 
+
+
+\medskip
+
+
+
+{\bf More About
+this Notation:}
+The
+potentially lengthy
+syntactic object of
+``$\,$Scope$_E$($\Uxp,$N)$\,$'' {\it
+ will actually 
+ not}
+be used 
+in our physical encodings
+of proofs.
+Instead, these encodings will
+ use the 
+more 
+desirably
+compressed
+object of
+``$\,\Uxp\,$''
+(which has no
+possibly bulky
+ $E(N)$ term). The {\it sole function} of 
+$\,$Scope$_E$($\Uxp,$N)$\,$ 
+will be for us to speculate about what 
+Boolean value
+$\,\Uxp\,$
+ {\it would theoretically assume}
+(under the Standard-M
+ model)
+if  $\,\Uxp\,$'s quantifiers
+were  modified
+ so that their ranges
+were changed to be
+bounded by $E(N).$
+$~\,$(It  turns out that  
+Scope$_E$($\Uxp,$N)$\,$'s
+finitized quantifier-range
+will 
+help
+%greatly
+simplify our
+analysis.)
+
+%proofs.)
+
+
+\medskip
+
+
+\bxbxdr
+\label{xd+1x5}
+A
+$\Pi_1^\xi$
+or $\Sigma_1^\xi$
+sentence
+ $ \Uxp $ will be called
+{\bf Good(N)} 
+when the entity
+Scope$_E$($\Uxp,N)$ is 
+true
+under the
+Standard-M model
+\footnote{\f55  \label{fgood}  A
+quite
+unusual aspect of Definition \ref{xd+1x5} is that its
+Good$(N)$ condition 
+has opposite properties when it is applied to
+$\Pi_1^\xi$ 
+and $\Sigma_1^\xi$ 
+sentences
+in one particular respect.
+This is because for each  $~N,~$ the
+ Good$(~N~)$ condition  is weaker than
+the
+Good$(~\infty~)$ condition 
+for $\Pi_1^\xi$ 
+sentences, while it is stronger than it for 
+ $\Sigma_1^\xi$ 
+sentences. (For instance,
+$~\forall \, x~\phi(x)~$ is
+stronger than
+$~\forall \, x <E(N)~\phi(x)~$,
+but 
+$~\exists \, x~\phi(x)~$ is weaker than
+$~\exists  \, x<E(N)~\phi(x)~$.)}.
+Also, a set of 
+$\Pi_1^\xi$
+sentences, 
+denoted as $\theta$, is called  Good(N)
+iff 
+all of its
+sentences are Good(N).
+ 
+\eedd
+
+
+
+\bxbxdr
+\label{chg}
+If $\Uxp $ 
+is a $\Pi_1^\xi$ sentence then
+ $\zzz \Uxp \,)$ 
+will denote the largest integer $J$ such
+that $\Uxp $ satisfies the 
+Good$(J)$ condition.
+(It will equal $\infty$ if 
+$\Uxp $ satisfies 
+Good$(J)$ for all $J$.)
+Also,
+if $\theta$ is a set of 
+$\Pi_1^\xi$ sentences, then
+ $\zzz \theta \,)$ 
+will denote the largest $J$ where each sentence in $\theta$
+is
+Good$(J)$.
+\eedd
+
+
+
+
+\medskip
+
+{\bf A Very Helpful Start :}
+Several 
+more
+definitions will 
+be needed
+before Section \ref{sect64}
+  can present our strongest results.
+The remainder of this section will  illustrate
+how the current formalism   is already
+ sufficient for introducing a
+useful
+starting
+lemma.
+ 
+
+\smallskip
+
+\begin{definition}
+\label{deftight} \rm
+Let
+      \gggen  
+again   denote 
+a generic configuration called $~\xi~$,
+and let us
+presume that
+its 
+ base axiom system
+ $~B^\xi ~$ 
+is comprised exclusively of 
+ $ \, \Pi_1^\xi \, $ sentences.
+Also,
+let
+ $ \, \beta \,  \supset  \,  B^\xi  \, $
+      denote a second axiom system,  comprised
+also of 
+ $ \, \Pi_1^\xi \, $ sentences,
+that 
+(unlike $ \, B^\xi \,$)
+can
+possibly  be
+inconsistent.
+(If  $ ~  \beta ~ $
+ is inconsistent then 
+ let $ \,  q_\beta  \, $ denote the shortest proof of
+$ \, 0=1  \,$
+from $~\beta~$.$~$)
+Then the
+generic
+configuration
+ $ \, \xi \, $ 
+will be called
+ {\bf Tight} if 
+iff {\it every 
+inconsistent} set of 
+ $ \, \Pi_1^\xi \, $ sentences
+ $~\beta \, \supset \, B^\xi  ~ $
+satisfies the following
+constraint:
+\beq
+\label{piss1}
+\mbox{ Log(}\,q_\beta \, )  ~ ~ \geq ~~ \zzz \beta ~)
+ ~~+~~2
+\enq
+\end{definition}
+
+
+
+Lemma \ref{lemex4} will 
+prove 
+$B^\xi+$SelfRef$(B^\xi,d)$
+satisfies
+Section \ref{secc1}'s
+ self-justification
+criteria
+ whenever
+$ \, \xi \,$
+is tight.  
+This ``tightness''
+will clearly fail to be satisfied by many
+generic configurations.
+This is
+ because 
+the Second Incompleteness Theorem is 
+a widely encompassing result,
+which imposes severe restrictions on its
+allowed
+ exceptions.
+Lemma \ref{lemex4}'s 
+mini-result 
+will be of interest 
+primarily
+because it will be generalized
+substantially in
+Sections \ref{3uuuu3} 
+and \ref{sect64}.
+
+
+
+
+\medskip
+
+\begin{lemma}
+\label{lemex4}
+If a 
+generic configuration
+  \gggen  
+is tight then
+ $B^\xi+$SelfRef$(B^\xi,d)$
+will be a consistent
+self-justifying axiom system.
+\end{lemma}
+
+\medskip
+
+{\bf Proof Sketch:} Our justification of \lem{lemex4}'s
+mini-result  will be
+simpler than 
+ the
+next section's 
+proof of 
+\thx{ppp1}'s stronger result.
+ The current proof 
+will also be kept
+brief and informal because the 
+same topic will be visited  more 
+rigorously during
+Section  \ref{3uuuu3}'s
+discourse.
+
+Let $~\Psi~$ 
+denote 
+Section \ref{secc1}'s
+SelfRef$(B^\xi,d)$  sentence.
+We will  
+omit formalizing
+ $~\Psi\,$'s exact 
+ $\Pi_1^\xi$ 
+encoding here
+because Appendix A will 
+provide a more general
+fixed-point
+ construction,
+using
+Definition \ref{xd+1x7}'s stronger paradigm.
+%% 
+%% Let $~\Psi~$ 
+%% denote 
+%% Section \ref{secc1}'s
+%% SelfRef$(B^\xi,d)$  sentence.
+%% We will  
+%% omit formalizing
+%%  $~\Psi\,$'s exact 
+%%  $\Pi_1^\xi$ 
+%% encoding 
+%% here because Appendix A will later
+%% provide an encoding for a more general fixed-point
+%% sentence in the context of
+%% Definition \ref{xd+1x7}'s stronger paradigm.
+%% 
+The
+current
+proof
+will 
+simply 
+presume 
+ $~\Psi\,$'s fixed point statement 
+can receive a
+ $\Pi_1^\xi$ 
+encoding
+under
+sentence
+\eq{fixnew},
+where 
+``$~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~)~$''
+is a $~\Delta_0^\xi$  formula, indicating that
+$~p~$ is a proof
+of 0=1 from 
+$~B^\xi~+~$SelfRef$(B^\xi,d)~$
+under  $~d \, $'s
+deduction method.
+\beq
+\label{fixnew}
+\forall ~p~~~\neg ~ 
+~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~)
+\enq
+
+%\cvs
+
+
+The
+Definition \ref{chg}'s symbol 
+ $~\sharp~$ 
+will be
+% become 
+helpful
+at this juncture. The application of
+$\xi$'s tightness to 
+% sentence
+\eq{fixnew}'s
+$\Pi_1^\xi$ styled encoding  
+will 
+%thus 
+imply
+ \footnote{  \b55 \g55 \ep{piss2} is 
+easy to justify when
+one  presumes
+there is an
+available 
+$\Pi_1^\xi$ encoding of
+\eq{fixnew}'s statement,
+which we call  $ \, \Psi \, $.
+(This 
+ $\Pi_1^\xi$ 
+presumption
+is reasonable because an analog of
+$ \, \Psi\,$'s
+exact
+  $\Pi_1^\xi$ encoding 
+will be discussed later by  Definition \ref{xd+1x7}
+and 
+in
+Appendix A.)
+The invalidity  
+of $ \, \Psi \, $
+% in this context,  
+will 
+thus assure the existence of 
+a
+proof of $0=1$ from
+ $ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. Moreover, 
+Definition \ref{chg}'s notation  implies
+$\zzz \Psi  \, )$ will equal Log$(q) \, - \, 1 \, $ when 
+$q$ denotes the  
+ shortest proof of $0=1$ from
+ $ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. 
+The latter shows \eq{piss2} is
+valid.} that
+\eq{piss2} must be true 
+when
+ $~\Psi~$ 
+is 
+false
+under the Standard-M model
+and
+% 
+when
+the shortest proof of $0=1$ from
+ $~B^\xi~+~$SelfRef$(B^\xi,d)$
+is denoted as $~q$.
+\beq
+\label{piss2}
+\mbox{ Log(}\,q~)  ~~ = ~~ \zzz \Psi ~) ~~+~~1
+\enq
+Also
+ Definition \ref{chg}
+trivially implies 
+$~\zzz B^\xi+\Psi ~) 
+~ = ~ 
+\zzz \Psi ~)~$ 
+(because  all of
+ $~B^\xi \,$'s axioms are
+true  under the Standard-M model).  Thus, 
+\eq{piss2}
+yields
+\eq{piss3}.
+\beq
+\label{piss3}
+\mbox{ Log(}\,q~) ~~ = ~~ 
+\zzz B^\xi+\Psi ~) 
+ ~~ +~~1
+\enq
+But the point is that
+the 
+Tightness constraint's  \ep{piss1},
+$~$used in the 
+%%nope particular 
+context where $~\beta~$
+is the axiom system
+of $~ B^\xi+\Psi ~,~$
+implies
+%%nope  that
+$~\mbox{ Log(}\,q \, ) ~ \geq ~ \zzz B^\xi+\Psi ~)~+~2~$.
+This  directly contradicts \ep{piss3}'s equality, 
+{\it whenever} the proof  ``$~q~$'' of $~0=1~$ cited 
+in our discussion 
+{\it does formally exist.}
+This 
+contradiction
+%latter observation
+is precisely what 
+is needed to corroborate 
+Lemma \ref{lemex4}'s claim that 
+the axiom system $~B^\xi~+~$SelfRef$(B^\xi,d)$
+must be
+   consistent. (Thus,  
+ $~B^\xi~+~$SelfRef$(B^\xi,d)$
+must be  consistent 
+because otherwise
+a proof $~q~$ of  $~0=1~$ would exist and have its
+ $~\mbox{ Log(}\,q \, ) ~ \geq ~ \zzz B^\xi+\Psi ~)~+~2~$ inequality
+contradict \ep{piss3}.)  $~~\Box$
+
+
+\medskip
+
+
+%\cvs
+%\cvl
+
+{\bf How  Lemma \ref{lemex4}
+May Be Interpreted :}
+We 
+%firstly 
+remind the reader that many
+(but not all)
+generic configurations will
+fail to satisfy 
+Lemma \ref{lemex4}'s 
+tightness hypothesis. This
+is because any configuration
+satisfying this hypothesis  represents one of those unusual
+boundary-case
+exceptions to the Second Incompleteness Theorem
+that are feasible.
+
+Lemma \ref{lemex4}
+was intended to capture
+the simplest variant of a self-justifying
+phenomena (that employs 
+Definition \ref{chg}'s machinery).
+Its proof was
+ kept
+informal
+because 
+more sophisticated
+ self-justifying formalisms will be explored
+in Sections \ref{3uuuu3} and \ref{sect64}.
+They will apply to
+ four different types
+of generic configurations, each of whose base axiom systems
+$~B^\xi~$  
+can be made capable of proving
+all
+Peano Arithmetic's
+$~\Pi_1^\xi~$ theorems --- 
+in a context where these  systems use a
+broader variant
+of {\it ``I am consistent''} axiom-statement
+than does
+Lemma \ref{lemex4}'s
+``SelfRef$(B^\xi,d)$''
+sentence.
+
+
+\section{The First Two Meta-Theorems about Self-Justification}
+
+
+\label{3uuuu3}
+
+
+The core theorems in this section will employ
+the following notation:
+\bee
+\item
+The symbol $~\theta$ will 
+denote 
+any
+recursively enumerable (r.e.)
+set of
+$\Pi_1^\xi$ sentences, 
+henceforth called a {\bf R-View}.
+(An
+R-View
+does not need to be valid under the
+Standard-M model. It 
+only needs to be r.e.)
+
+\medskip
+
+\item
+ {\bf RE-Class$(\xi)$} 
+shall denote the
+set of all possible ``R-Views''
+$~\theta~$ that 
+can be built out of $~\xi \,$'s language
+$~L^\xi~$. (This 
+ permits
+ both valid and invalid
+R-Views  to appear in
+RE-Class$(\xi)$. 
+We choose this unrestricted definition because
+no recursive 
+decision
+procedure can
+identify
+all  the true  $\Pi_1^\xi$ sentences
+in the Standard-M model.)
+\ene
+
+
+\nop
+
+
+
+\bxbxdr
+\label{astab}
+Let   $~\xi~$
+denote 
+the 5-tuple
+  \gggen 
+representing
+one of
+Definition \ref{def3.3}'s 
+generic configurations,
+and let RE-Class$(\xi)$
+and its R-Views ``$\theta$''
+be defined as in the previous paragraph.
+Then
+$ \,\xi \, $ is called
+{\bf A-Stable\sss } iff 
+each
+$ \, \theta \, \in \, $RE-Class$(\xi)$
+satisfies the 
+following 
+invariant:
+
+
+
+\begin{description}
+\item[  *$~$   ]
+If $ \, \Uxp \, $ is a
+$\Pi_1^\xi$ theorem 
+of
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1 $
+then  $ \, \Uxp \, $ 
+will 
+satisfy  Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\end{description}
+\eedd
+
+
+\begin{remark}
+\label{re3-1}
+\rm
+The invariant
+ $ \, * \,$ 
+states
+short proofs
+(with lengths 
+ $ \leq \,  \zzz \theta)~+1~~) $
+  will 
+produce at least 
+{\it partially useful} deductions,
+in that
+their $\Pi_1^\xi$
+theorems 
+will always
+ satisfy
+  Good$\{ \,  \, \tftt  \zzz \theta) \,  \, \} \, $,
+{\it irregardless of whether or not}
+$\theta$'s axioms are {\it technically} true.
+(This makes the study of  A-stability
+very
+interesting
+unto itself,
+ apart from its
+applications
+        in the
+current article).
+\end{remark}
+
+
+
+
+
+\phx{ppp2} will show
+the presence of
+ A-stability,
+alone,
+is sufficient
+for 
+constructing 
+self-justifying systems. 
+This  will imply 
+every A-stable configuration
+must contain 
+some
+embedded weakness
+(as every evasion
+of the Second Incompleteness Theorem 
+always does).
+On the other hand,
+Appendix F will explain how
+A-stability 
+and its 
+``EA-stable'' cousin 
+(defined later)
+are both
+epistemologically interesting.
+Thus, A-stability has 
+redeeming features.
+
+
+\medskip
+
+\bxbxdr
+\label{estab}
+A Generic Configuration 
+$\,\xi \,$
+ will be called
+ {\bf E-Stable}  iff 
+all of the
+$\theta  \in \,$RE-Class$(\xi)~$
+satisfy
+  $~**~$.  $~$(This 
+construct is the counterpart
+for $\Sigma_1^\xi$ sentences of
+the Item  $ *$ in
+Definition \ref{astab}.)
+
+\begin{description}
+\item[ **   ]
+If $ \, \Uxp \, $ is a
+$\Sigma_1^\xi$ theorem 
+ derived
+from the
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1 $
+$~$then  $ \, \Uxp \, $ 
+will 
+automatically satisfy 
+Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$.
+{ (This invariant
+{\it further
+implies} 
+\footnote{ \g55 
+This point is easy to confirm when one
+remembers that $\Sigma_1^\xi$ sentences $~\Upsilon~$
+have the property that $~A<B~$ implies 
+ Scope$_E$($\Uxp,A$)
+is stronger than  Scope$_E$($\Uxp,B$).
+It is then obvious
+that the
+ Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$
+criteria implies the validity of
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$
+for $\, \Sigma_1^\xi \,$ sentences
+because the invariant $\,** \, $ presumes 
+its  $\, \Sigma_1^\xi \,$ theorems have proofs $\, p \,$
+satisfying 
+`` Log$(p)~\leq \zzz \theta)~+1 $ ''. }  
+that  $ \Uxp  $  
+will also
+satisfy 
+ the Good$\{ \, \tftt  \zzz \theta) \,~\} $
+criteria.)} 
+\end{description}
+\eedd
+
+
+
+
+\smallskip
+
+
+
+\begin{remark}
+\label{re3-2}
+\rm
+The invariants $ \, * \, $ and $ \, ** \,  \,  $ 
+are partially analogous 
+to each other
+because both imply that
+if $ \, p \, $ is a proof short enough to
+satisfy  Log$(p) \, \leq \zzz \theta) \, +1 \,  $ 
+then their resulting theorem will satisfy
+Good$\{ \,  \, \tftt  \zzz \theta) \,  \, \} \, $.
+However, there is 
+a
+% an important 
+distinction between 
+Definitions
+\ref{astab} and \ref{estab},
+as well.
+This is because the
+prior section's
+ Footnote \ref{fgood}
+observed that $ \, \Sigma_1^\xi \, $ sentences are stronger
+when they meet a 
+Good$( \, N \, )$ 
+rather than a
+Good$( \, \infty \, )$ threshold, $ \, ${\it while the reverse is true for 
+  $ \, \Pi_1^\xi \, $ sentences.}
+Thus $ \, ** \,$'s short proofs
+$ \, p \, $  
+(satisfying  Log$(p) \,\leq \zzz \theta)~+1~ $)
+will have the
+special
+ property that
+their theorems  $ \, \Uxp~$ 
+will satisfy a 
+``Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$''
+constraint 
+that is actually stronger than their formal
+$~\Sigma_1^\xi~$ 
+statements.
+\end{remark}
+
+
+
+
+\smallskip
+
+
+
+
+The \appD will provide four examples
+of generic configurations that are either E-stable
+or A-stable (or often 
+% very often 
+both).
+Its most prominent example will be a 
+configuration
+$~\xi^*~$ 
+that uses  semantic tableaux deduction and
+recognizes addition as a total function.
+Theorem D.\ref{D.4-theorx} will 
+imply such systems can be made
+% both 
+self-justifying
+and 
+able to
+ prove 
+% all
+ Peano Arithmetic's
+ $\Pi_1^*$ theorems.
+
+\njp
+%\smallskip
+
+Three more  definitions
+are needed 
+to help introduce 
+our first theorem
+
+
+
+
+\smallskip
+
+\bxbxdr
+\label{eastab}
+A generic configuration $~\xi~$
+will be called
+{\bf EA-stable} iff it 
+is both 
+E-stable
+and A-stable.
+(It will thus
+satisfy both 
+$~*~$ and $~**~$.)
+\eedd
+
+\smallskip
+
+
+
+
+Our next definition is related to the fact
+ that  
+many definitions of consistency
+are logically 
+ equivalent 
+from the perspective
+of strong enough logics,
+but they 
+are {\it  often not provably equivalent}
+from the perspectives of 
+weak logics.
+
+\medskip
+
+\bxbxd
+\label{lev}
+\rm
+Let $   \xi   $
+denote the generic configuration
+  \gggcp, 
+and $   \alpha   $ be an axiom system satisfying
+$   \alpha  \supseteq  B^\xi   $.
+Then
+$   \alpha   $ is called
+{\bf Level$(k^\xi)$ 
+Consistent} when there
+exists  no proofs from $\alpha$ via
+$ d   $'s deduction method  
+of both a $\Pi^\xi_k$ sentence and 
+of  the
+$\Sigma^\xi_k$ sentence that is its  negation
+\eedd
+
+\medskip
+
+Most of this article will focus on 
+self-justifying systems that recognize their
+Level$(k^\xi)$ consistency 
+when 
+ $k$ 
+equals 0 or 1.
+Our next definition will
+be applied mostly to these two cases.
+
+
+
+\smallskip
+
+
+\bxbxdr
+\label{xd+1x7}
+Given any  
+$k \geq 0$, 
+a
+generic configuration
+  \gggen     
+and an
+axiom system
+ $\beta  \supset  B^\xi$,
+the symbol
+SelfCons$^k(\beta,d)$
+will
+denote a 
+self-referencing
+$\Pi_1^\xi$ sentence declaring
+$\,  \beta   +  $SelfCons$^k(\beta,d)      $'s
+formal
+Level(k$^\xi)$ consistency,
+as is 
+illustrated
+below
+by  
+the 
+statement $ ~ +~ $.  (An  encoding for
+SelfCons$^k(\beta,d)$ 
+will be provided
+by Appendix A.)
+\begin{quote}
+$+~~~$ 
+There exists  no two proofs 
+(using deduction method $   d   $)
+of {\it both}
+some $\Pi_k^\xi$ sentence and
+of the 
+$\Sigma_k^\xi$ sentence, 
+$\,$that 
+represents its negation,
+$\,$from the  union of
+the axiom system $~\beta~ $
+with $~this~$ added
+sentence  ``SelfCons$^k(\beta,d) \,$''
+{\it (looking at itself).}
+\end{quote}
+\eedd
+
+
+\begin{remark}
+\label{re3-3}
+\rm
+We will focus on
+Definition \ref{xd+1x7}'s 
+SelfCons$^k(\beta,d) \,$ axiom 
+mostly in
+ the settings
+where $\,k=0$ or 1.
+This is because 
+SelfCons$^k(\beta,d) \,$ 
+will 
+typically 
+be too strong for it to generate boundary-case exceptions
+to the Second Incompleteness Theorem when
+ $~k \geq 2~$.
+It turns out that even when
+ $~k=1~,~$  Definition \ref{xd+1x7}'s
+SelfCons$^k(\beta,d) \,$ statement
+will be significantly stronger than the
+axiomatic
+declaration $\, \bullet \,$
+used by 
+Section
+\ref{secc1}'s
+ SelfRef$(\beta,d)$
+axiom.
+This is 
+because 
+SelfCons$^1(\beta,d) \,$ asserts 
+ non-existence of 
+simultaneous proofs
+for a  $~\Pi_1^\xi~$ sentence and its negation,
+$\,$while  SelfRef$(\beta,d)$
+establishes
+merely
+the
+ non-existence of a proof of $0=1$.
+\end{remark}
+
+
+\smallskip
+
+
+\begin{theorem}
+\label{ppp1}
+Let $~\xi~$
+denote a
+generic configuration
+  \gggen   
+that is 
+EA-stable.  Then 
+the corresponding  axiom system of 
+$B^\xi+$SelfCons$^1(B^\xi,d)$
+must satisfy Section
+\ref{secc1}'s  definition of
+self-justification.
+\end{theorem} 
+
+\parskip 3pt
+
+\smallskip
+
+
+
+{\bf Proof.}
+The justification 
+of \phx{ppp1} will
+be 
+a more elaborate  version
+ of 
+\lem{lemex4}'s
+mini-proof.
+It will
+replace
+Definition \ref{deftight}'s
+Tightness constraint with
+an EA-stability requirement.
+It will also
+ replace
+SelfRef$(\beta,d)$'s 
+``I am consistent''
+axiom
+with a 
+{\it stronger}
+SelfCons$^1(\beta,d)$
+statement.   
+
+
+\cvb
+\parskip 5pt
+
+
+
+Our proof will focus on
+showing 
+  $~B^\xi+$SelfCons$^1(B^\xi,d)~$ 
+is consistent (and thus 
+% it
+satisfies 
+the subtle
+Part-ii component
+of Section \ref{secc1}'s
+definition of
+``Self-Justification'').
+It will be awkward 
+during our discussion
+to 
+write repeatedly  the
+expression ``$~B^\xi+$SelfCons$^1(B^\xi,d)~$''.
+Therefore,
+ ``$S$'' will be the abbreviated
+name for this axiom system.
+We will also employ the following notation:
+\bee
+\itemsep +3pt
+\item
+$~\mbox{Prf}_S(t,p)~$ will denote 
+that $~p~$ 
+is
+ a proof of the theorem $~t~$ from
+the above mentioned formalism ``$~S~$''
+(using 
+$~\xi \,$'s
+deduction method of $d$ ).
+\item 
+$~\mbox{Neg}^1(x,y)~$ will denote
+that $~x~$  is
+the G\"{o}del encoding of a
+ $\Pi_1^\xi$ sentence and
+that $~y~$ is a $\Sigma_1^\xi$ 
+sentence which represents
+$~x \,$'s formal negation.
+\ene 
+Appendix A explains how to combine the theory of LinH
+functions \cite{HP91,Kr95,Wr78} with \cite{ww1}'s
+fixed point methods to provide 
+$\mbox{Prf}_S(t,p)$ 
+and 
+$\mbox{Neg}^1(x,y)$ with
+$\Delta_0^\xi$ encodings.
+(A reader can
+ omit examining Appendix A,
+if he 
+just
+accepts this fact.)
+Thus, \eq{pencode} 
+can
+be 
+viewed \footnote{ \s55 
+Our notation convention has the 
+abbreviated
+formula of
+``$~\mbox{Prf}_S(t,p)~$''
+in \ep{pencode} 
+corresponding 
+to
+ Appendix A's 
+``$~\mbox{SubstPrf} \, _{\beta}^d  \, 
+( \bar{n}    ,    t    ,    p   ) ~$'' formula,
+in a context where $~n~$ specifies the G\"{o}del number of
+the expression
+$~\Gamma^k(g)~$  
+in
+Appendix A's 
+ \ep{encode12} and where
+the superscript $~k~$ 
+within this
+ formula
+``$~\Gamma^k(g)~$''
+is set equal to 1.  
+This 
+implies that sentence
+\eq{pencode} does
+assert the Level$(1^\xi)$ 
+consistency
+of S.  \fend }
+in this context
+as being   SelfCons$^1(B^\xi,d)\,$'s 
+formalized
+$\Pi_1^\xi$ 
+statement,
+declaring 
+the Level$(1^\xi)$ 
+consistency
+of S:
+\begin{equation}
+\label{pencode}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^1(x,y) ~ \wedge~ 
+\mbox{Prf}_S  \, (     x    ,    p   ) ~
+\wedge  ~
+\mbox{Prf}_S \,    (       y    ,    q   )  \, \}
+\end{equation}
+
+
+Let   $ \Phi $   denote 
+\eq{pencode}'s
+ sentence. 
+We will
+use it to
+prove
+\phx{ppp1}'s claim that  $ S $
+is consistent. Our proof 
+will be a proof by contradiction.
+It will 
+begin with
+the 
+assumption that $ S $ is inconsistent.
+This
+implies 
+$\Phi$ is
+false under the Standard-M model.
+Hence, 
+Definition \ref{chg}
+implies
+\begin{equation}
+\label{punk}
+ \zzz \Phi~) 
+   ~~ < ~~ \infty
+\end{equation}
+
+
+
+\noindent
+\ep{punk} 
+thus
+indicates there  exists
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying \eq{pencodepunk}
+(because such a tuple
+will be a counter-example to
+\eq{pencode}'s assertion).  
+The particular
+$(p,q,x,y)$
+ satisfying \eq{pencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ 
+can 
+then
+be easily shown$\,$
+\footnote{ \sm55 \label{footp1} 
+Let $~L~$ denote the 
+minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$  for a tuple
+$(p,q,x,y)$
+ satisfying \ep{pencodepunk}. Then by definition,
+$~\Phi~$ satisfies Good$(L-1)$ but not
+Good$(L)$.
+From Definition \ref{chg}, this 
+establishes
+ the validity of
+\ep{punkless}  (because 
+the minimal
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying sentence \eq{pencodepunk}
+has
+$ \mbox{Log}\{\mbox{Max}[  \bar{p}  ,  \bar{q}  ,  \bar{x}  ,  \bar{y}  ] \}
+ \, = \, L.~$) }
+  $\,$to also
+satisfy
+\ep{punkless}.
+\begin{equation}
+\label{pencodepunk}
+  \,\mbox{Neg}^1(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ 
+\mbox{Prf}_S  \, (     ~ \bar{x}~    ,    ~ \bar{p}~   ) ~
+\wedge  ~
+\mbox{Prf}_S \,    (       ~ \bar{y}~    ,    ~ \bar{q}~   )  
+\end{equation}
+\begin{equation}
+\label{punkless}
+ \mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
+   ~~~ =~~~\zzz \Phi~) ~~+~~1
+\end{equation}
+
+$~~~$ We will now use 
+\eq{pencodepunk} and \eq{punkless} 
+ to
+bring our proof-by-contradiction to its conclusion.
+Let $ \, \Upsilon \, $ denote the $\Pi_1^\xi$ sentence specified
+by $ \, \bar{x} \, $. Then
+$ \, \neg  \,  \Upsilon \, $ 
+will
+correspond to the $\Sigma_1^\xi$ sentence denoted
+by $ \, \bar{y} \, $.
+Also, 
+\eq{pencodepunk} and \eq{punkless} imply that 
+both
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $
+have  proofs 
+such that the logarithms of
+their G\"{o}del numbers are bounded
+by
+$\zzz \Phi \,)  \,+ \,1 \,$.
+These facts
+imply
+ that
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $
+{\it both} 
+satisfy 
+ Good$\{ \,  \, \tftt  \zzz \Phi) \,  \, \} \, $
+under our formalism.
+(This is because
+if we take $ \, \theta \, $ in Definitions
+\ref{astab} and \ref{estab} to be simply $ \, \Phi \,$'s
+1-sentence statement, $\,$then the invariants of
+ $ \, *  \,$ and $ \, **  \,$ from these two definitions
+both impose the
+same
+Good$\{ \,  \, \tftt  \zzz \Phi) \,  \, \} \, $
+ constraint
+on 
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $.)
+ 
+
+It is infeasible, 
+however,
+for a sentence and its
+negation to both satisfy the same goodness
+constraint.
+ This completes \phx{ppp1}'s
+proof-by-contradiction
+because 
+the initial
+ assumption that 
+$S$ was inconsistent has led to an
+infeasible conclusion.
+$~~\Box$
+
+
+
+\bigskip
+
+\cvl
+
+\parskip 3pt
+
+Our next definition will help formalize
+a useful cousin of
+\phx{ppp1}.
+
+\smallskip
+
+
+\bxbxdr
+\label{ostab}
+A Generic Configuration of 
+$\xi $
+will be called
+ {\bf $~$0-Stable$~$} when every
+particular
+$\theta \, \in $ RE-Class$(\xi)$
+satisfies the 
+invariant of  $***$. 
+(This invariant 
+is strictly weaker than 
+its counterparts   $*$  and $**$
+in Definitions 
+   \ref{astab} and \ref{estab}.) 
+\begin{description}
+\item[ ***   ]
+If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1~,~ $
+then  $ \, \Uxp \, $
+is 
+true under the Standard-M model.
+\end{description}
+\eedd
+
+
+\begin{theorem}
+\label{ppp2}
+If 
+the configuration
+$\xi$ is 
+0-stable 
+then
+$B^\xi+$SelfCons$^0(B^\xi,d)$
+is      a self-justifying 
+formalism. {\rm (Appendix C 
+shows this result  applies
+also to
+E-stable and  A-stable  configurations.)}
+\end{theorem} 
+
+\medskip
+
+
+
+
+
+
+\phx{ppp2}'s proof
+is similar
+to  \phx{ppp1}'s
+ proof. 
+The difference between these two
+propositions is that
+\phx{ppp2}
+has reduced
+       SelfCons's
+ superscript  
+ from 1 to 0, so that its
+ hypothesis
+can encompass a 
+theoretically
+ broader 
+set of applications.
+(The Appendix C summarizes how
+ \phx{ppp1}'s proof can be
+easily modified to
+also prove  \phx{ppp2}.)
+
+\medskip
+
+\begin{remark}
+\label{newrem}
+{\rm
+Theorems 
+\ref{ppp1} and \ref{ppp2}
+should clarify 
+the  nature of 
+\cite{ww93}--\cite{ww9}'s
+formalisms.
+This is because proofs-by-contradictions 
+are notorious 
+in the mathematical literature
+for being
+confusing.
+They should be simplified whenever possible.
+This has been done
+mainly through
+\phx{ppp1}'s
+short proof.
+(It 
+ applies to
+three of Appendix D's
+four examples of 
+generic 
+configurations,
+and
+\phx{ppp2} applies to 
+Appendix D's
+fourth example.)
+Furthermore, Section \ref{sect64} 
+will
+show how more elaborate
+self-justification systems can verify all
+Peano Arithmetic's $\Pi_1^\xi$ theorems.}
+\end{remark}
+
+ 
+\section{Four Further Meta-Theorems}
+\label{sect64}
+
+
+We need one preliminary lemma
+before  exploring 
+how strong self-viewing
+                logics may become
+before 
+they   cross
+  the inevitable
+boundary between 
+self-justification
+and 
+ inconsistency,
+$\,$implied by 
+G\"{o}del's
+Theorem.
+
+
+
+
+\begin{lemma}
+\label{llpp3}
+Let  $  \xi  $ 
+denote a generic configuration  
+   \gggcp  
+$\,$, and $  \mheta  $ denote an r.e. set of 
+$\Pi_1^\xi$ sentences,
+each of which holds true in the Standard-M model.
+Let $  \mxi  $ denote a 
+5-tuple that differs from 
+ $  \xi  $  
+in that its base axiom system
+is  $  B^\xi \, \cup \, \mheta  $
+(rather than
+$  B^\xi  $).
+These conditions imply that
+$  \mxi  $ is a
+generic configuration,
+and 
+it
+will satisfy
+the following
+four invariants:
+\bed
+\item[     i   ]
+If $~\xi~$ is  0-stable\sss 
+then 
+$~\mxi~$  will also be
+0-stable\sss.
+\item[      ii    ]
+If $~\xi~$ is  A-stable\sss 
+then 
+$~\mxi~$ will also be
+A-stable\sss.
+\item[     iii   ]
+If $~\xi~$ is  E-stable\sss 
+then 
+$~\mxi~$  will also be
+E-stable\sss.
+\item[     iv   ]
+If $~\xi~$ is  EA-stable\sss 
+then 
+$~\mxi~$  will also be
+EA-stable\sss.
+\ennd
+\end{lemma}
+
+
+Lemma  \ref{llpp3}'s proof
+is fairly straightforward. It
+has been placed in
+the Appendix B. 
+This section will
+use  Lemma  \ref{llpp3}
+to prove
+four
+meta-theorems
+ that are consequences of
+its
+formalism.
+
+
+\smallskip
+
+\begin{definition}
+\label{dap4-1}
+\rm
+Let  $\xi$  again
+denote
+a generic configuration  
+  \gggcp,  
+and
+$\theta$ denote some r.e. set of 
+$\Pi_1^\xi$ sentences
+(which are not
+required to be  true  under the Standard-M model).
+For the cases where $\,k \,$ is either
+0 or 1,
+the symbol 
+$G^\xi_k(\, \theta \,)$
+will
+denote the following axiom system:
+\vspace*{- 0.2 em}
+\begin{equation}
+\label{4gedef}
+G^\xi_k(~ \theta ~)~~= ~~
+\theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}
+\vspace*{- 0.2 em}
+\end{equation}
+Also when
+$ \, k \, = \, 0 \,$
+or 1,   the function 
+$ \, G^\xi_k ~ $
+(which maps $ \, \theta \, $ onto 
+$G^\xi_k(\, \theta \,)~~~)$ is 
+called 
+{\bf Consistency Preserving} iff 
+$ \, G^\xi_k(\, \theta \,) \, $is 
+assured to be consistent whenever 
+all the sentences in $~\theta ~$
+are  
+true  under the Standard-M model.
+\end{definition}
+
+\smallskip
+
+We emphasize 
+ consistency preservation is 
+unusual in 
+logic. This is because
+$ \, G^\xi_k(\, \theta \,) \, $
+comes from adding a self-justifying axiom to an
+initially consistent
+formalism $~B^\xi+\theta~$, and the 
+Second Incompleteness Theorem
+demonstrates that sufficiently powerful
+formalisms are simply incompatible with
+such an axiom.
+ However, 
+there will   be
+four
+specialized 
+paradigms, defined in Appendix D,
+that are exceptions to this rule.
+They will be related to our next result:
+
+
+
+
+
+\smallskip
+
+\begin{theorem}
+\label{pqq3} 
+The function 
+$~G^\xi_1~$ 
+shall satisfy Definition \ref{dap4-1}'s
+consistency preservation
+property
+when $~\xi~$ is
+EA-stable. Likewise 
+the function 
+$~G^\xi_0~$ 
+will be consistency preserving when $~\xi~$ is
+any one of A-stable, E-stable or 0-stable.
+{\rm 
+(Thus 
+in each case,
+$~G^\xi_k(\, \theta \,)~$
+will  
+be consistent
+when
+all the sentences in $~\theta ~$
+are  
+true in the Standard-M model.)}
+\end{theorem}
+
+\medskip
+
+{\bf Proof}.
+It will be
+ convenient
+for our proof
+ to use a
+ dot-style
+notation,
+ analogous to
+Lemma \ref{llpp3}'s
+terminology. 
+ Thus, 
+\bee
+\item
+ $~\mheta~$ denotes any r.e. set of 
+$\Pi_1^\xi$ 
+sentences {\it that are 
+each
+true} under the Standard-M model.
+\item
+ $\mxi$
+is the tuple
+$ \, ( \, L^\xi  \, ,  \, \Delta_0^\xi  \,  ,
+ \,  B^\xi \cup \mheta  \, ,  \, d   \, ,  \,  G  \,  )$.
+(It
+differs from 
+ $\xi$
+by replacing
+ $\xi\,$'s  base axiom system
+of $\,B^\xi \,$ with $\,B^\xi  \cup  \mheta$.)
+\ene
+ Part-iv of Lemma \ref{llpp3}
+indicates
+the EA-stability of 
+$\xi$ 
+ implies
+the EA-stability of 
+$  \mxi $.
+ Moreover  \phx{ppp1} 
+applies to all EA-stable
+configurations, 
+including  
+$\mxi $.
+Thus,
+$G^\xi_1(\, \mheta \,)$ 
+is  consistent
+because $\xi$ is EA-stable
+and all of $~\mheta \,$'s sentences are true in
+the Standard-M model.
+ This
+proves
+\phx{pqq3}'s 
+first 
+claim.
+
+An 
+almost
+identical proof,
+where 
+\phx{ppp2} 
+simply
+ replaces \phx{ppp1}
+as the central self-justifying
+engine,
+will corroborate 
+\phx{pqq3}'s second claim .
+Thus,
+$~G^\xi_0~$ 
+is consistency preserving when $~\xi~$ is
+one of A-stable, E-stable or 0-stable. $~~~\Box$
+
+
+
+\medskip
+
+\begin{remark}
+\label{re4-1}
+\rm
+Most
+ generic configurations
+$~\xi~$ 
+will not satisfy \phx{pqq3}'s hypothesis
+because
+$~G^\xi_k(\theta)~$ will typically be
+inconsistent,
+irregardless 
+\footnote{\sm55 Conventional generic configurations $\xi$
+will satisfy the Hilbert-Bernays derivability conditions
+\cite{HB39,HP91}.
+Their
+$~G^\xi_k(\theta)~$ will thus
+be automatically inconsistent
+because of
+a G\"{o}del-like diagonalization argument.}
+of whether or not
+all of $~\theta\,$'s axioms are valid
+under the Standard-M model.
+The significance of \phx{pqq3} 
+is that it shows that some
+outlying
+ exceptions to this
+general rule do prevail when
+$~\xi~$ satisfies one of 
+\phx{pqq3}'s 
+four stability conditions.
+These
+exceptions
+are related to the
+3-page abbreviated philosophical discussion
+that will appear later in Appendix F.
+They  
+will 
+assure 
+that
+Theorem \ref{pqq3}'s 
+formalisms
+of
+ $G^\xi_1(\theta)$ 
+and $G^\xi_0(\theta)$
+are {\it always
+consistent}
+whenever 
+$~\theta\,$'s axioms are valid
+in the Standard-M model.
+\end{remark}    
+
+\smallskip
+
+
+Our next definition will enable
+self-justifying formalisms
+to prove
+the
+$\Pi_1^\xi$ 
+theorems 
+of  any
+consistent 
+r.e. axiom system
+that uses  $~L^\xi \,$'s language.
+
+
+
+\smallskip
+
+\begin{definition}
+\label{dap4-2}
+\rm
+Let  $\,\xi\,$  denote
+a generic configuration  
+  \gggcp.   
+Let $   \nal{B}  $ 
+denote any recursive axiom system whose language
+is an extension of 
+$\, L^\xi \,$.
+For an arbitrary 
+deduction method $ \,  \nal{D} \,  $
+(which may be 
+possibly
+different from $\,\xi \,$'s deduction method
+$ \,d \,$ ),
+ let
+$\mbox{Prf}_{ \nal{B}}^D \,( \, \lceil \, \Psi \, \rceil \,  , \,  q \, )\,  $
+denote a $\Delta_0^\xi$ formula indicating 
+$  q  $ is a proof of the theorem $  \Psi  $ from 
+axiom system
+$ \,  \nal{B}\,$, 
+using
+deduction method
+ $ \,  \nal{D} \,  $.
+Then the  {\bf Group-2 Schema} for
+$  (   \nal{B}  ,  D  )  $ is
+defined as an infinite set of axioms that includes
+one instance of \eq{group2}'s axiom 
+for each 
+$\Pi_1^\xi$ sentence $\, \Psi$. 
+\beq
+\label{group2}
+\forall ~q~~~ \{~
+\mbox{Prf}_{ \nal{B}}^D \,( \,  \lceil \, \Psi \, \rceil \,  , \,  q \, )
+ ~~~\longrightarrow ~~~ \Psi ~~ \}
+\enq
+\end{definition}
+{\bf  $~~~~$Comment about this Notation:}
+The Definition \ref{dap4-2}
+had called 
+\eq{group2} a 
+ ``Group-2 Schema''  so as to keep our terminology consistent
+with
+\cite{ww93,ww1,ww5,wwapal}'s notation.
+
+\bigskip
+
+%% Several of our articles
+%% \cite{ww93,ww1,ww5,wwapal}
+%% have  used the term
+%%  ``Group-2 Schema'' to refer to
+%% Definition \ref{dap4-2}'s
+%% formalism. The same notation is
+%% thus 
+%% used here.
+
+\smallskip
+
+
+
+\begin{theorem}
+\label{pqq4} 
+Let  $~\xi~$ denote any
+arbitrary generic configuration,
+and 
+$  (   \nal{B}  ,  D  )  $ 
+denote any pair consisting of
+an axiom system and a deduction method
+(which, once  
+again, are allowed to be 
+different from
+ $~\xi \,$'s deduction method
+and axiom system).
+Then if all of $  (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems are 
+true in the Standard-M model,
+the following two invariants will hold:
+\bed
+\topsep -4pt
+\itemsep -2pt
+\item[   i ] 
+If $~\xi~$ is EA-stable then
+ there will exist an
+r.e.$~$self-justifying system that can prove
+all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems 
+and 
+recognize its own Level($~1^\xi~)~$ consistency.
+\item[  ii ] 
+Likewise,
+if
+ $~\xi~$ is
+one of A-stable, E-stable or 0-stable,
+ there
+will  exist an
+r.e.$~$self-justifying system that can confirm
+all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems 
+and 
+which can
+recognize its own Level($~0^\xi~)~$ consistency.
+\ennd
+\end{theorem}
+
+\smallskip
+
+
+
+{\bf Proof.}
+\phx{pqq4} 
+follows from
+\phx{pqq3}.
+Thus let
+ $\theta$
+denote
+ the set of 
+all
+$\Pi_1^\xi$ sentences 
+that are members of
+Definition \ref{dap4-2}'s Group-2 schema.
+Then every one
+of $\theta\,$'s 
+Group-2 axioms 
+must be
+true  under the Standard-M model
+(because 
+the hypothesis of  
+\phx{pqq4} 
+ indicated
+all of $  (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems are
+true in this model).
+Hence,  \phx{pqq3}  implies:
+\bee
+\item
+$G^\xi_1(\, \theta \,)$
+ is consistent when $\xi$ is EA-stable.
+\item
+$G^\xi_0(\, \theta \,)$ is consistent when $\xi$ is one of
+ A-stable,  E-stable or  0-stable.
+\ene
+Since $G^\xi_0(\, \theta \,)$ and $G^\xi_1(\, \theta \,)$
+are self-justifying systems that prove all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems, 
+Items 1 and 2
+will substantiate
+\phx{pqq4}'s
+two claims. $\Box$ 
+
+
+
+\bigskip
+
+An
+ awkward aspect of Definition \ref{dap4-2}'s
+``Group-2'' schema is that 
+it employs an
+infinite number of instances
+of \eq{group2}'s Group-2-like axiom sentences.
+It turns out 
+that this
+Group-2 scheme
+can be
+compressed
+ into a {\it single axiom sentence,}
+if one is willing to 
+settle for a
+slightly 
+diluted 
+variant of
+ $ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ knowledge.
+To 
+formalize this  concept,
+the following  notation shall be used:
+\bee
+\item
+Check$^\xi(t)$ will denote a $\Delta_0^\xi$ formula
+that checks to see whether $t$ represents the G\"{o}del
+number of a $\Pi_1^\xi$ sentence. 
+\item
+Test$^\xi(t,x)$ 
+will denote any $\Delta_0^\xi$ formula
+where
+\eq{testsim}'s invariant 
+is 
+true   under
+the Standard-M model
+for
+every $  \Pi_1^\xi  $ sentence 
+$   \Psi   $
+simultaneously.
+There are infinitely many 
+different  $\Delta_0^\xi$ 
+formulae that can
+serve as 
+Test$^\xi(t,x)$ predicates
+satisfying 
+this condition.
+(Example \ref{new-exam} will illustrate
+one such encoding of a
+Test$^\xi(t,x)$ predicate.)
+\beq
+\label{testsim}
+\Psi ~~~ \longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}^\xi(~\lceil~\Psi~\rceil~,~x~)
+\enq
+\ene
+The expression \eq{globsim}
+will be called 
+a
+{\bf Global Simulation Sentence}
+for representing
+$( \nal{B},D)$
+via $\xi \,$.
+Its $\mbox{Test}^\xi(t,x)$ clause essentially allows
+$\xi$ to simulate the $\Pi_1^\xi$ knowledge of
+$( \nal{B},D)$'s 
+set of
+theorems.
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{ \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}^\xi(t,x)~~~ \}
+\enq
+
+ 
+\bigskip
+
+\begin{example}
+\label{new-exam}
+\rm
+For any generic configuration $\,\xi~=\,$ \gggcp,
+$\,$let NegPrf$^\xi(t,x)$ denote
+a  $\Delta_0^\xi$ formula
+specifying
+that
+$ \, t \, $ is a $\Pi_1^\xi$ sentence and
+that
+$ \, x \, $ is a proof under
+ $\,d\,$'s deduction
+method  of the 
+ $\Sigma_1^\xi$ sentence that represents $ \, t\,$'s negation.
+Also, let   Test$^\xi_0(t,x)$ 
+be defined as follow:
+\beq
+\label{new1-exam}
+\mbox{Test}^\xi_0(t,x)~~~~ =_{\mbox{def}}~~~~ \neg~ \mbox{NegPrf}^\xi(t,x)
+\enq
+For each $\Pi_1^\xi$ sentence $\Psi$,
+%%  vvvv  
+it is easy to verify 
+\footnote{  \b55 Part-3 of Definition \ref{def3.3}
+indicated that $~\xi\,$'s base axiom system is
+``$~\Sigma_1^\xi~$ complete''. $~$(It is
+thus able to prove
+all $~\Sigma_1^\xi~$ sentences that are true in the true in
+Standard-M model,
+and it will likewise refute all 
+$~\Pi_1^\xi~$ sentences that are false.)
+The statement \eq{new1-exam}
+then
+immediately implies that
+\eq{newtestsim}
+must  be true
+ under the Standard-M model
+for
+ every $~\Pi_1^\xi~$ sentence $~\Psi~$.}
+that
+the statement 
+\eq{newtestsim}
+is true
+under
+the 
+Standard-M model.
+\beq
+\label{newtestsim}
+\Psi ~~~ \longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}^\xi_0(~\lceil~\Psi~\rceil~,~x~)
+\enq
+\end{example}
+\vspace*{- 1.3 em}
+$~~~$ 
+%Thus, 
+
+The latter
+% immediately
+ implies 
+that
+\eq{new2-exam} 
+is 
+% an example of
+a global simulation sentence
+for  $  (   \nal{B}  ,  D  )  $.
+\beq
+\label{new2-exam} 
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{ \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}_0^\xi(t,x)~~~ \}
+\enq
+We emphasize that there are countably infinite different examples
+of  $\mbox{Test}_i^\xi(t,x)~$ predicates that generate
+global simulation sentences and
+that statement \eq{new2-exam} 
+illustrates only one such example.
+
+
+%\bigskip
+
+\begin{definition}
+\label{gsim}
+\rm
+Let $( \nal{B},D)$ denote any ordered pair whose set
+of $~\Pi_1^\xi$ theorems are true under the Standard-M model.
+Let
+Test$^\xi_1 \, , \, $ 
+Test$^\xi_2 \, , \, $ Test$^\xi_3  \,~....~ $
+denote the set of 
+$\Delta_0^\xi$ formulae 
+where
+statement
+\eq{testsim}
+is true
+under the Standard-M model
+for every
+$\Pi_1^\xi$ sentences
+$\Psi$.
+Then TestList$^\xi$ will denote
+a
+list of all  these 
+Test$^\xi_i \,$ predicates.
+Also for each
+Test$^\xi_j $ 
+formula
+in  TestList$^\xi\,$,
+$\,$the symbol
+{\bf GlobSim$^D_{ \nal{B}} \,(\xi,j)$} 
+will denote 
+the special version  
+of \eq{globsim}'s global simulation formalism
+that employs
+Test$^\xi_j \,  $'s machinery.
+ \end{definition}
+
+%\smallskip 
+
+
+
+\begin{remark}
+\label{re4-2}
+\rm
+A comparison between 
+Definition \ref{dap4-2}'s
+Group-2 schema with  \ref{gsim}'s
+global simulation sentences
+will
+reveal
+neither 
+is 
+strictly
+ better than the other. 
+Both         have their own separate advantages.
+Thus,
+the attractive aspect about Definition \ref{gsim}'s
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$  sentence is that it is a
+finite-sized object that can simulate
+the
+infinite set of  axioms
+associated with
+Definition \ref{dap4-2}'s Group-2 schema. 
+The accompanying
+drawback
+\footnote{ \f55  The
+chief
+ difficulty arises
+essentially
+        because
+a  $\Pi_1^\xi$
+theorem of
+ $ (   \nal{B}  ,  D  )  $ 
+may contain an arbitrarily long combination
+of bounded universal and bounded existential quantifiers.
+Thus,
+some generic configurations will have 
+base axiom systems $B^\xi$ 
+that are so weak that
+their combination with \eq{globsim}'s
+Global Simulation Sentence 
+is insufficient 
+{\it to prove} the validity of \eq{testsim}'s
+equivalence statement
+{\it  for all}
+ $\Pi_1^\xi$ sentences $\Psi$ simultaneously.
+In particular, such   proofs
+will often be infeasible
+when  $\Psi \,$'s
+sequence of bounded  quantifies
+has a length greatly exceeding the length of
+the G\"{o}del encoding for  \eq{globsim}'s
+global simulation
+statement.}
+$\,$of 
+a global simulation sentence
+ is that the union of it
+with the base-axiom system  $B^\xi$ 
+will typically be inadequate  to prove every $\Pi_1^\xi$
+sentence  that is a theorem of
+ $ (   \nal{B}  ,  D  )  $.
+$\,$Instead, in a context where
+  $\Psi \,$ is a $\Pi_1^\xi$
+theorem of
+ $ (   \nal{B}  ,  D  )  $,
+the sentence 
+ GlobSim$^D_{ \nal{B}} \,(\xi,j)$ 
+will usually
+provide only
+enough {\it  fragmented information} to prove
+the statement \eq{quasisim} (which is equivalent to
+$\Psi$ 
+{\it under the Standard-M model).}
+\beq
+\label{quasisim}
+ \forall ~x~~
+\mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~)
+\vspace*{- 0.4 em}
+\enq
+While \eq{quasisim} may 
+be insufficient
+%not 
+% provide
+%enough information 
+to prove $ \, \Psi \, $ from  
+$ \, B^\xi \, $, it 
+still 
+(according to the 
+statement \eq{testsim} )
+has  the desired property of being
+equivalent to $\Psi$
+under the Standard-M model.
+(This means that
+the knowledge of \eq{quasisim}'s
+truth is helpful, {\it even if it is unknown
+from $ \, B^\xi\,$'s perspective} to be
+equivalent to
+$ \, \Psi \, $ . )
+\end{remark}
+
+%\smallskip
+
+
+Our prior 
+articles
+\cite{ww93,ww1,ww5,wwapal} 
+did not use
+Definition \ref{gsim}'s
+global simulation formalism.
+They
+employed,
+instead,
+Definition \ref{dap4-2}'s
+Group-2 axiom schema.
+\phx{pqq5} 
+will be
+   the 
+ analog of  \phx{pqq4} 
+for
+global simulation.
+It will be
+ useful 
+when
+%in applications where 
+one desires
+to compress all the information held
+by a Group-2 schema into a
+single 
+finite-sized object.
+
+
+
+
+\smallskip
+
+\begin{theorem}
+\label{pqq5}
+Let  $ \,\xi \,$ 
+denote the generic configuration  
+  \gggcp,   
+ $\, \nal{B}\,$ denote a
+ recursively enumerable
+axiom system
+and $ \,  \nal{D} \,$ 
+denote any
+deduction method
+(which can be different than $\xi \,$'s deduction method    $~d~~)~$.
+Suppose that all the 
+$\Pi_1^\xi$
+theorems generated by
+ $ \, ( \,  \nal{B} \, , \, D \, ) \, $ are
+true  under the Standard-M model.
+Then the following  invariants
+do hold:
+\bed
+\item[   i ]
+If $ \, \xi \, $ is EA-stable 
+then 
+for each  $\, j \,$
+there 
+exists a 
+{\bf finitized
+extension}
+$~\beta_j~$ of $~B^\xi$ that 
+recognizes its 
+Level$(1^\xi$)
+self-consistency
+ and which
+contains
+the sentence
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$. 
+\item[  ii ] 
+Likewise
+for each $ j $,
+if $\xi$ is 
+  E-stable, A-stable  or 0-stable
+then there 
+exists a 
+{finitized
+extension}
+$~\beta_j~$ of $~B^\xi$ that 
+recognizes its 
+Level$(0^\xi$)
+self-consistency
+ and which
+contains
+the sentence
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$. 
+\ennd
+\end{theorem}
+
+%\medskip
+
+
+{\bf Proof:}
+Let
+$\theta$ denote the 1-sentence R-View of 
+``GlobSim$^D_{ \nal{B}} \,(\xi,j)$''
+formalized by  Definition
+\ref{gsim},
+$~$and let 
+$~\beta_j^1~$  and 
+$~\beta_j^0~$ denote 
+$ ~ \, B^\xi \, \cup \, \theta  \, +  \, $SelfCons$^1(
+\,B^\xi \, \cup \, \theta  \, )  \,~ $
+and
+$  \, B^\xi \, \cup \, \theta  \, +  \, $SelfCons$^0(
+\,B^\xi \, \cup \, \theta  \, )  \, $,
+respectively.
+These axiom systems
+correspond 
+to
+the objects 
+that Definition 
+\ref{dap4-1} 
+had called $G^\xi_1(\theta)$ 
+and $G^\xi_0(\theta)$. 
+
+\medskip
+
+\phx{pqq5}'s  hypothesis 
+indicates
+%had specified that
+all the 
+$\Pi_1^\xi$
+theorems of
+ $( \nal{B},D)$  are true
+ under the Standard-M model.
+Thus,
+it follows that $\, \theta \,$ is 
+also
+true in the Standard-M model.
+Hence \phx{pqq3}
+implies that 
+$ \, \beta_j^1 \, = \, G^\xi_1(\theta)$ 
+is a consistent  system 
+satisfying \phx{pqq5}'s 
+claim (i).
+Likewise, \phx{pqq3}
+implies 
+$ \, \beta_j^0 \, = \, G^\xi_0(\theta)$ 
+satisfies
+\phx{pqq5}'s 
+ second claim.
+$~~\Box$
+
+
+\bigskip
+
+\begin{remark}
+\label{re4-n}
+\rm
+Theorems  \ref{pqq4} 
+and \ref{pqq5} raise a fascinating 
+%open 
+question:
+{\it Is the trade-off between these 
+% two 
+formalisms needed?}
+That is, 
+%Thus,
+can self-justifying systems use only a 
+ a finite number of added axioms
+beyond those lying in $~\xi\,$'s base 
+%axiom 
+system of $B^\xi$
+and 
+also
+%{\it simultaneously} 
+duplicate all 
+ $( \nal{B},D)$'s $\Pi_1^\xi$ theorems in a pure 
+sense (i.e. without simulation) ?
+We will return to this topic in Appendix G.
+\end{remark}
+
+
+%%nov150ut 
+%%nov150ut \begin{remark}
+%%nov150ut \label{old-re4-n}
+%%nov150ut \rm
+%%nov150ut Theorems  \ref{pqq4} 
+%%nov150ut and \ref{pqq5} 
+%%nov150ut are difficult to compare
+%%nov150ut  because
+%%nov150ut neither result is uniformly better than the other.
+%%nov150ut The advantage to Theorem  \ref{pqq4} 
+%%nov150ut is that it duplicates all of
+%%nov150ut  $( \nal{B},D)$'s $\Pi_1^\xi$ theorems in a pure sense,
+%%nov150ut rather than using Theorem  \ref{pqq5}'s
+%%nov150ut weaker simulated construct.
+%%nov150ut The virtue of Theorem  \ref{pqq5}, 
+%%nov150ut in contrast,
+%%nov150ut  is that its self-justifying systems 
+%%nov150ut include only
+%%nov150ut Our suspicion is that Theorem  
+%%nov150ut \ref{pqq5} is of greater  pragmatic interest,
+%%nov150ut while
+%%nov150ut \thx{pqq4} is the more mathematically surprising result.
+%%nov150ut \end{remark}
+
+
+
+
+
+%% \begin{remark}
+%% \label{recc1}
+%% \rm
+%% This article deliberately
+%% used the phrase ``boundary-case exception''
+%% to characterize our limited  evasions
+%% the Second Incompleteness Theorem. 
+%% This
+%% cautious terminology was employed
+%% because there are two fundamental barriers
+%% limiting the generality 
+%% of all such
+%% results:  
+
+
+
+\bigskip
+\bigskip
+\bigskip
+\bigskip
+
+%{\bf $~~~~$ A BROADER PERSPECTIVE:}
+
+% {\bf $~~~~$ A Yet Further Issue$~$:}
+
+
+
+
+% {\bf $~~~~$ A Yet Further Issue$~$:}
+{\bf $~~~~$ REFLECTION PARADIGMS$~$:}
+$~$Our  
+last
+%final 
+goal
+is to show how
+self-justifying systems 
+%can 
+support
+%% 
+%% will be to formalize
+%% how Theorems \ref{ppp1},
+%%  \ref{pqq3}, \ref{pqq4} 
+%% and \ref{pqq5} support 
+%% 
+{\it unusually}
+strong 
+reflection principles.
+% for their $\Pi_1^\xi$ theorems. 
+Let $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ 
+%thus
+denote  \eq{reflect}'s statement 
+when
+%in a context where
+$~\Psi~$ is a sentence with G\"{o}del number
+ $~\lceil \, \Psi \, \rceil~$, and 
+$~(\alpha,d)~$ denotes an axiom system
+and deduction method.
+\beq
+\label{reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+ ~~~ \Rightarrow ~~~ \Psi~~]
+\enq
+L\"{o}b's Theorem
+\cite{HP91,Lo55,So76b}
+ implies that conventional
+systems $\, (\alpha,d) \,$, 
+possessing at least
+ Peano Arithmetic's strength, are unable to
+prove 
+% the sentence 
+ $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ except for in the degenerate cases
+where they can prove $~\Psi$.
+
+
+% itself.
+
+%\cvt
+
+\medskip
+
+Moreover, it is easy to generalize
+L\"{o}b's Theorem
+(via say  \cite{ww1}'s
+Theorem 7.2) 
+so that
+a wide class of formalisms $~\alpha~$, weaker than
+%PA,
+Peano Arithmetic,
+are also
+ unable to prove
+% the validity of 
+ $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ 
+for
+% essentially 
+all $\Pi_1^\xi$
+sentences $~\Psi~$ simultaneously. 
+
+\medskip
+
+
+
+The intuition
+behind this generalization is quite simple.
+Let
+$~\mho~$
+denote 
+a $\Pi_1^\xi$ encoding for
+the classic
+ G\"{o}del sentence
+declaring: 
+{ \it  `` There is no proof of me 
+%{\it this sentence}
+from the axiom system $~\alpha~$ using $~d\,$'s deduction method''.}
+Then \cite{ww1}'s
+Theorem 7.2  
+uses a very routine 
+diagonalization argument
+to show 
+most formalisms
+$~\alpha~$ will be
+ inconsistent 
+%(and thus useless)
+if they  prove  $\mbox{Reflect}_{\alpha,d}(~\mho~)$'s
+statement. 
+
+\medskip
+
+%% 
+%% The intuition
+%% behind this result is quite simple.
+%% Let
+%% $~\mho~$
+%% denote 
+%% a $\Pi_1^\xi$ encoding for
+%% the classic
+%%  G\"{o}del sentence
+%% declaring: 
+%% { \it  `` There is no proof of me 
+%% %{\it this sentence}
+%% from the axiom system $~\alpha~$ using $~d\,$'s deduction method''.}
+%% Then 
+%% Theorem 7.2  
+%% used an essentially 
+%% conventional
+%% diagonalization argument
+%% to show 
+%% its formalisms
+%% $~\alpha~$ would be 
+%%  inconsistent (and thus useless)
+%% if they proved  $\mbox{Reflect}_{\alpha,d}(~\mho~)$'s
+%% statement. 
+
+
+
+Our next theorem will show, surprisingly, that
+the preceding limitation is much less 
+stringent 
+% severe
+than 
+it may
+initially  appear to be.
+This is
+because 
+%EA-stable 
+Level$(~1^\xi~)$ self-justifying 
+axiom systems
+are capable of proving  
+very close analogs to
+\eq{reflect}'s
+impermissible
+
+\nop
+
+\noindent
+ reflection principle,
+using a ``translational'' methodology.
+
+\smallskip
+
+Thus, let $~T~$  denote an
+algorithm that maps a
+% initial 
+$~\Pi_1^\xi~$ sentence $~\Psi~$ onto a 
+``translated''
+sentence  $~\Psi^T~$ that is equivalent
+to  $~\Psi~$ 
+under the Standard-M model
+{\it and which is also written in a  $~\Pi_1^\xi~$ 
+format.} 
+(See footnote\footnote{\label{imper} Part-3 of Definition
+\ref{def3.3} indicated that generic configurations are
+$\Sigma_1^\xi$ complete. 
+%In this context, 
+Our requirement that
+ $~\Psi^T~$ 
+must have a       $\Pi_1^\xi$ format
+thus
+causes 
+$~T~$ 
+to
+gain  much added
+meaning.
+This is 
+because 
+the
+axiom system $~B^\xi~$ will then  
+automatically  disprove
+$~\Psi^T~$ 
+ whenever
+it is false under the Standard-M model.
+(Thus, $T$'s mapping of 
+$~\Psi~$  onto $~\Psi^T~$ 
+gains much  significance
+when 
+$~\Psi~$ and
+$~\Psi^T~$ do rest on the same  $\Pi_1^\xi$ 
+level of the arithmetic
+hierarchy.)}
+for why 
+it is absolutely imperative that
+{\bf both these} requirements be included in
+$~T\,$'s definition.)
+Also, let
+$\mbox{Reflect}^T_{\alpha,d}(~\Psi~)~$ denote the
+%  corresponding
+translational
+modification of \eq{reflect}'s reflection principle that
+replaces $\Psi$ with $\Psi^T$. 
+%in its
+%modified {\bf ``Translation'' Reflection Principle.}
+\beq
+\label{T-reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+  ~~~ \Rightarrow ~~~ \Psi^T~~]
+\enq
+
+
+
+
+
+
+\begin{theorem}
+\label{ppp6}
+Let  $ \,\xi \,$ 
+denote the EA-stable configuration of  
+  \gggcp,   
+and let
+$~\alpha~=~   B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$
+denote  $ \,\xi \,$'s corresponding Level(1) self-justifying
+axiom system. Then
+there will exist 
+a translation methodology $~T~$ where
+ $~\alpha~$ can prove the validity
+of \eq{T-reflect} for all its $\Pi_1^\xi$ sentences simultaneously.
+%% 
+%% {\rm (In other words, this means that
+%% $~\alpha~$ will  know that
+%% its 
+%% proof of $~\Psi~$ automatically
+%% implies a counterpart
+%% of $~\Psi~$ is true in
+%% the Standard-M model ! )}
+%% 
+\end{theorem}
+
+% {\bf Abbreviated Proof:}
+
+\bigskip
+
+
+{\bf Proof:} Let us use
+Example  \ref{new-exam}'s notation.
+It observed that  $~\Psi\,$'s
+$\Pi_1^\xi$ statement
+was equivalent 
+under the Standard-M model
+to
+ ``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$''.
+Thus, 
+let us view $\,T \,$  as 
+being
+a mapping of
+the first sentence onto the second. 
+
+Our proof of Theorem \ref{ppp6} will next  use
+the following observations:
+\bee
+\item The non-existence of a proof of $~ \neg ~ \Psi~$ from
+$~    B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$
+trivially implies the non-existence of a proof of the same theorem
+from $~B^\xi~$
+ (because the latter axiom system is 
+simply
+a subset of
+the former).
+\item 
+Moreover, Example  \ref{new-exam}'s notation treats
+``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$'' as being equivalent to the declaration that no proof of
+$~\neg~\Psi~$ from $~B^\xi~$ exists. 
+\ene
+Hence, $~\alpha~$ can prove 
+\eq{T-reflect}'s  statement by noting
+that 
+$~p\,$'s proof of 
+a $\Pi_1^\xi$ sentence
+$~\Psi~$ implies 
+(via $\, \alpha \,$'s $~$SelfCons$^1~$  axiom) 
+the non-existence
+of a proof of $~\neg ~\Psi~$, which 
+(via Items 1 and 2)
+implies 
+ ``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$''
+$~~\Box$.
+
+\cvs
+
+\bigskip
+\bigskip
+
+\begin{remark}
+\rm \label{f88}:    
+$~$\phx{ppp6} and the statement
+\eq{T-reflect}'s 
+Translational Reflection Principle
+may possibly be useful devices in unraveling some of the
+mystery that has enshrouded G\"{o}del's Second Incompleteness
+Theorem, since its inception.
+This is
+partly
+ because G\"{o}del was 
+explicitly uncertain
+about the 
+generality of the
+Second Incompleteness Theorem 
+in his initial 
+ 1931 
+seminal
+paper \cite{Go31}
+about this subject.
+% to appreciate the preceding 
+%oint.
+His centennial paper
+about Incompleteness
+ thus included 
+%
+%
+the following 
+quite
+poignant
+caveat:
+\begin{quote}
+%\small
+%\baselineskip = 1.0 \normalbaselineskip 
+$~\bullet~~~$ : 
+``It must be expressly noted that
+Theorem XI
+(i.e the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and {\it there might
+conceivably be finite proofs} which cannot
+be stated in ... ''
+%
+%%%%%%%%%%%%%%%%%%%%%%%% 
+%
+\end{quote}
+Some of the issues
+that troubled 
+G\"{o}del 
+in the 
+statement 
+$\bullet$
+can 
+perhaps 
+be 
+partially
+resolved if one
+compares the reflection principles of sentences
+\eq{reflect} 
+and \eq{T-reflect}.
+This is because 
+\eq{reflect} is probably
+unnecessary to explain how thinking beings can
+appreciate their
+ $~\Pi_1^\xi~$
+ theorems 
+%$~$---$~$ 
+when 
+\phx{ppp6}'s specialized
+logics can,
+instead,
+ use the fact that
+its $\Pi_1^\xi$ sentences  satisfy at least
+\eq{T-reflect}'s
+modified
+ reflection principle. 
+%% 
+%% (with
+%% $~\Psi^T~$ being required to be both 
+%% $~\Pi_1^\xi~$ and equivalent to $~\Psi~$ in the
+%% Standard-M model). 
+%% 
+%%
+%
+%In other words, 
+Thus,
+some of the mystery surrounding the
+Second Incompleteness
+Effect can be
+clarified
+ when one notices that 
+%the validity of
+\eq{T-reflect}'s translational reflection princible 
+is a useful precept, that was
+shown by \phx{ppp6} to be
+technically
+%essentially
+ unrelated to G\"{o}del's observation
+that no
+reasonable formalism  
+can prove
+%proves 
+\eq{reflect}'s 
+{\it purist}
+%a reflection 
+principle
+ for all 
+ $~\Pi_1^\xi~$
+sentences
+simultaneously.
+\end{remark}
+
+
+%%% 
+%%% 
+%%% Some of the issues 
+%%% G\"{o}del 
+%%% had raised in statement $~\bullet~$
+%%% can be
+%%% partially
+%%%  resolved when one
+%%% compares the reflection principles of sentences
+%%% \eq{reflect} 
+%%% and \eq{T-reflect}.
+%%% This is because 
+%%% \eq{reflect}'s reflection principle is probably
+%%% unnecessary to explain how thinking beings can
+%%% appreciate their
+%%%  $~\Pi_1^\xi~$
+%%%  theorems $~$---$~$ 
+%%% when some logics can,
+%%% instead,
+%%%  use the fact that
+%%% every $~\Pi_1^\xi~$ statement  satisfies at least
+%%% \eq{T-reflect}'s
+%%% modified
+%%%  reflection principle. 
+%%% %% 
+%%% %% (with
+%%% %% $~\Psi^T~$ being required to be both 
+%%% %% $~\Pi_1^\xi~$ and equivalent to $~\Psi~$ in the
+%%% %% Standard-M model). 
+%%% %% 
+%%% %%
+%%% In other words, some of the mystery surrounding the
+%%% Second Incompleteness
+%%% Effect can be
+%%% resolved
+%%%  when one notices that the validity of
+%%% \eq{T-reflect}'s translational reflection princible 
+%%% (for some {\it specialized} 
+%%% logics)
+%%% is unrelated to G\"{o}del's observation
+%%% that no
+%%% reasonable formalism can feasibly prove 
+%%% \eq{reflect}'s 
+%%% {\it purist form} of
+%%% a
+%%% reflection principle
+%%%  for all 
+%%%  $~\Pi_1^\xi~$
+%%% sentences
+%%% simultaneously.
+%%% %% \end{remark}
+
+%% 
+%% ) $~$---$~$
+%% {\it while at the same time} \phx{ppp6}
+%% permits a 
+%%  formalism
+%% to,
+%% nevertheless, 
+%% support  \eq{T-reflect}'s modified reflection principle and
+%% simultaneously
+%%  prove all Peano Arithmetic's  $~\Pi_1^\xi~$
+%% theorems.
+
+
+
+\bigskip
+
+
+
+\medskip
+
+\begin{remark}
+\label{remhappy}
+\rm
+Theorem \ref{ppp6} is
+also
+significant
+because it explains how
+its  specialized logics 
+ can grapple with a 
+$\, \Pi_1^\xi \, $ encoded
+G\"{o}del sentence
+$~\mho~$ 
+which asserts {\it ``There is no proof of me''}.
+This issue is challenging because
+routine constructions, such as
+\cite{ww1}'s Theorem 7.2,  
+demonstrate that no natural logic
+can 
+verify
+statement
+  \eq{reflect}'s validity 
+for all $~\Pi_1^\xi~$ 
+sentences $~\Psi~$
+(on account of the
+well-known
+syllogism
+posed by 
+$~\mho$'s  G\"{o}del sentence).
+%% 
+%% 
+%% on account of the challenge posed by 
+%% $~\mho\,$'s available  
+%%  $~\Pi_1^\xi~$ encoding. 
+%% 
+%% 
+Theorem \ref{ppp6} 
+constructs,
+%demonstrates,
+however,
+a 
+%method to 
+%serious
+reply to this challenge. 
+%% 
+%%% 
+%%%  that this 
+%%% challenge
+%%% %difficulty
+%%% is not quite as disturbing
+%%% as  might first 
+%%% appear
+%%% %be anticipated.
+%%% 
+This is
+because 
+%%%Theorem \ref{ppp6}'s 
+its self-justifying systems do
+%%% can
+surprisingly
+%{\it at least,}
+prove,
+without difficulty \footnote{Since 
+$\Psi$ and $\Psi^T~$ are equivalent under
+the Standard-M Model {\it but not also
+equivalent}  
+from the perspective of the 
+%axiom 
+system 
+$~\alpha~=~   B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$,
+the conventional contradictions, produced by
+% sentence 
+\eq{reflect}'s reflection principle, disappear when it
+is replaced by 
+\eq{T-reflect}.
+%'s
+% theoretical
+% alternative principle. 
+(Thus, there is no danger that
+$\alpha$ could
+use \eq{T-reflect}'s reflection principle to 
+prove 
+an analog of $\,\mho \,$'s 
+forbidden G\"{o}del sentence.) \fend
+%the forbidden statement
+%{\it ``There is no proof of me''} 
+%using
+%%
+%% 
+%% The 
+%% ``difficulty'' that conventionally arises is that an
+%% axiom system 
+%% $~\alpha~$
+%% becomes inconsistent when it proves
+%% $\mbox{Reflect}_{\alpha,d}(~\mho~)~$'s sentence because this causes
+%% it to prove $~\mho\,$'s sentence, which asserts {\it ``There is no
+%% proof of me''.}  
+%% This difficulty vanishes,
+%% however, 
+%%  when $~\alpha~$
+%% proves the translational reflection sentence of 
+%% $\mbox{Reflect}^T_{\alpha,d}(~\mho~)$.
+%% This is because the latter replaces
+%% $~\mho~$ in  \ep{reflect}
+%% with \eq{T-reflect}'s sentence $~\mho^T~$.
+%% This second sentence  is equivalent to $~\mho~$
+%% under the Standard-M
+%% model.
+%% However,
+%% $~\alpha~$ is unable to recognize
+%% this equivalence (and  its
+%% thus escapes the syllogism of
+%% a G\"{o}del-like diagonalization argument).
+%% 
+%% 
+}, the validity of
+\eq{T-reflect}'s 
+{\it translated modification}
+of  \eq{reflect}'s
+% formally 
+unobtainable
+$\Pi_1^\xi$ 
+styled
+variant of
+a reflection principle.
+\end{remark}
+
+
+\bigskip
+
+\begin{remark}
+\label{new14}
+\rm
+We encourage the reader to examine the work of
+Beklemishev, Kreisel-Takeuti
+ and Verbrugge-Visser
+\cite{Be95,Be97,Be3,KT74,VV94,Vi5}
+to see
+alternative
+reflection principles
+and 
+their uses.
+(The  constraint on proof-length,
+by Verbrugge-Visser,
+ is certainly
+one 
+alternative to
+\thx{ppp6}'s machinery.
+Likewise, 
+%\cite{KT74}'s 
+Kreisel-Takeuti's
+Second Order
+Logic {\it CFA}
+reflection is another alternative,
+although it will not \footnote{Kreisel-Takeuti indicate on
+page 25 of \cite{KT74} that CFA's reflection principle for
+a first order formula ``A'' infers the validity of
+{\it only the relativized formula} ``$~A^N~$'' from $A$'s proof.
+Also, their proof predicate is similarily relativized. 
+Thus CFA's second-order logic
+reflection principle, while fascinating, does
+not generalize to first-order logic environments. }
+generalize to first-order 
+logics.) 
+One 
+complicating aspect of
+%drawback to
+our  \thx{ppp6}'s
+%translational 
+reflection
+%principle
+method
+ is
+%  quite 
+that
+Appendix E 
+% formally 
+proves it becomes
+%%% unfortunately
+% fully
+ inoperable
+when an axiom system is 
+% simply
+sufficiently conventional
+to satisfy G\"{o}del's Second Incompleteness Theorem.
+In essence, \thx{ppp6}'s translational reflection
+principle is a
+specialized
+methodology,
+% an attractive type of
+%machinery,
+%reflection methodology, 
+% that is
+intended for logics using
+Definition \ref{xd+1x7}'s
+Level(1) style
+of self-justification. 
+% ``SelfCons''
+% self-justifying axiom.
+\end{remark}
+
+
+%%%% cccccccccc
+%%%% Some readers may wish to examine the work of
+%%%% Beklemishev, Verbrugge and Visser
+%%%% \cite{Be95,Be97,Be3,VV94,Vi5}
+%%%% to see
+%%%% some  different 
+%%%% perspectives about 
+%%%% reflection principles.
+%%%% (The Verbrugge-Visser mechanism,
+%%%% with its  constraints  \cite{VV94,Vi5} on proof-length,
+%%%%  is 
+%%%% a
+%%%% % an especially
+%%%% fascinating alternative to
+%%%% \thx{ppp6}'s machinery.) 
+%%%% One 
+%%%% complicating aspect of
+%%%% %drawback to
+%%%% our  \thx{ppp6}'s
+%%%% %translational 
+%%%% reflection
+%%%% principle is
+%%%% %  quite 
+%%%% that
+%%%% Appendix E 
+%%%% % formally 
+%%%% proves it becomes
+%%%% %%% unfortunately
+%%%% % fully
+%%%%  inoperable
+%%%% when an axiom system is 
+%%%% % simply
+%%%% sufficiently conventional
+%%%% to satisfy G\"{o}del's Second Incompleteness Theorem.
+%%%% Thus, self-justifying systems support a type
+%%%% of 
+%%%% %surprising 
+%%%% reflection principle that is 
+%%%% uniquely intended 
+%%%%  for their particular
+%%%% % special types of
+%%%% applications.
+%%%% %inapplicable in conventional settings.
+
+
+\bigskip
+
+%\smallskip
+
+\begin{remark}
+\label{recc1}
+\rm
+The preceding discussion clearly shows 
+%that 
+self-justifying logics are 
+% very 
+% quite 
+tempting.
+At the same time,
+it is necessary 
+to be 
+very
+cautious
+because there are also two 
+fundamental 
+barriers
+limiting 
+such results:
+\bed
+\topsep -4pt
+\itemsep +2pt
+\item[   a  ] 
+The
+first is the
+ Theorem 2.1 arising from the joint work of
+Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87}.
+It
+showed
+no 
+reasonable 
+system recognizing successor
+as a total function can verify its own Hilbert consistency.
+%Moreover,
+Also,
+Willard \cite{ww2,ww7} established 
+%% ww0
+analogous results 
+under semantic tableaux consistency 
+for 
+% most
+systems recognizing multiplication
+as a total function. 
+%These results show that every effort to evade
+Thus, each effort to evade
+the  Second Incompleteness Theorem 
+must
+encounter 
+%inevitable
+ robust
+barriers.
+\item[   b  ]
+A second issue is that Definition \ref{xd+1x7}'s
+``SelfCons''
+{\it ``I am consistent''} 
+axiom
+sentence is less than ideal because 
+it causes axiom systems to produce
+essentially
+a
+1-line proof of their
+own 
+ consistency.
+Such an excessively compressed proof corresponds more
+closely to an axiom system formulating
+ an
+{\it instinctive faith} in its own consistency
+(rather than it  supporting 
+a 
+% more elaborate 
+full-length proof-justification
+of this fact).
+\ennd
+Part of the reason 
+self-justifying systems are of
+interest, despite these limitations, is that they illustrate
+how some formalisms {\it are compatible} with at least an 
+{\it instinctive faith} in their own self-consistency.
+(This compatibility issue is non-trivial
+because Item (a) 
+implies
+there are
+many circumstances where a generalization of the Second Incompleteness
+Theorem will make it infeasible for a formalism to satisfy
+{\it both} 
+Parts (i) and (ii) of Section 1's definition
+of Self-Justification.)
+Moreover, three of Appendix D's four sample self-justifying
+configurations,
+called $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$,
+ will be Type-A systems that
+ recognize
+addition as a total function. These configurations will
+thus
+possess
+the following three 
+significant
+finitized 
+ features:
+\bee
+\topsep -4pt
+\itemsep +2pt
+\item
+They will be able to construct the entire infinite set of integers
+{\it by finite means} because they recognize addition as a total function.
+\item
+For any r.e. logical
+configuration $  (   \nal{B}  ,  D  )  $, 
+it will be possible to develop a 1-sentence
+ {\it finitized} 
+extension for the base axiom systems of any of  
+the configurations of  $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~,~$
+which deploy
+ \eq{globsim}'s
+ Global Simulation Sentence
+to simulate the 
+ $~\Pi^\xi_1~$ knowledge of  $  (   \nal{B}  ,  D  )  $.
+This means 
+that some fully
+finite-sized
+extensions of the base-formalisms of 
+  $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$ will contain a
+non-trivial amount of 
+ $~\Pi^\xi_1~$ styled
+knowledge,
+since  $  (   \nal{B}  ,  D  )  $ can 
+correspond to, say,
+%represent for example
+Peano Arithmetic. 
+\item
+The 
+key point is
+ that a 1-sentence extension of an
+axiom system  containing  features (1) and (2)
+can formalize 
+how a
+logic
+ can possess
+an instinctive faith in its own
+consistency
+via \thx{pqq5}'s explicitly
+{\it finitized} structure.
+(Moreover, \thx{ppp6}'s 
+Translational  Reflection Principle
+is applicable to 
+Appendix D's
+ generic configurations of
+ $~\xi^*~$ and $~\xi^{**}~$.
+It will 
+thus imply that  their single 
+{\it finitized}
+ Level-1 self-justifying
+axioms enable 
+% each of
+them to prove
+an  {\it infinite number} of incarnations
+of \eq{T-reflect}'s
+translational 
+reflection principle, where  each
+$\Pi_1^*$ sentence  
+$\Psi$ 
+is mapped onto one such 
+%well-definded unique 
+unique instance.)
+%%% 
+%%% 
+%%% Appendix G's
+%%%  more 
+%%% sophisticated
+%%%  Theorems G.2 and G.3,
+%%% %%% 
+%%% %%% accompanied by 
+%%% %%% Appendix F's
+%%% %%% epistemological  
+%%% %%% discussion,
+%%% %%% in{re
+%%% will
+%%% further strengthens   
+%%% these results.)
+%%% 
+%%% 
+\ene
+The contrast between Items 1-3's positive remarks
+about ``finitized'' cogitation with
+Items (a) and (b)'s 
+opposing comments 
+is 
+obviously
+formidable. 
+It is clearly preferable
+to
+view these
+positive results 
+cautiously and treat them
+as being no more than
+boundary-case  exceptions  to
+the Second Incompleteness Theorem. 
+The essential reason why 
+these exceptions are of
+interest
+is that G\"{o}del's famous centennial paper has
+implicitly raised
+the following 
+%unresolved 
+puzzling issue:
+\begin{quote}
+ \#  How is it that Human Beings
+ manage 
+to muster 
+the physical 
+drive
+to think (and prove theorems) when the many
+generalizations of 
+G\"{o}del's Second 
+Incompleteness Theorem assert 
+conventional logics
+lack knowledge of
+their own consistency?  
+\end{quote}
+There will, of course, never be any perfect answer to the 
+puzzle
+posed by
+$~\# ~$ 
+because philosophical paradoxes and ironical dilemmas never yield
+perfect answers. 
+However, 
+ part of an
+imperfect
+ answer to 
+$~\# ~$ is that 
+Items 1-3 reply to
+ Challenges (a) and (b) by formalizing how
+a thinking being can muster an
+% at least
+approximate partial
+{\it instinctive faith} in its own self-consistency.
+(Moreover, the
+tight
+ contrast 
+between various 
+generalizations of the Second
+Incompleteness Theorem
+\cite{Ad2,AZ1,BS76,BI95,HP91,HB39,Ko6,Lo55,Pu85,Sa11,So94,Sv7,WP87,ww2,wwlogos,ww7}
+with the
+self-justifying systems appearing
+in Appendixes D and G
+suggests 
+that these
+come close to
+being
+maximal forms
+of
+feasible   
+results.)
+\end{remark}
+
+%\smallskip
+
+Our remaining discussion 
+will consist of four optional sections, called
+ Appendixes D, E,  F and G,
+% consists of three 
+% optional
+% sections,
+which
+can be skimmed,
+omitted or  examined
+in 
+any
+% whichever 
+order 
+ the reader 
+%so 
+prefers.
+A 
+%brief 
+summary of 
+their contents 
+is given 
+%provided
+below:
+\bed
+\itemsep +4pt
+\item[   I  ] 
+The Appendix D provides four examples of
+generic configurations
+ that
+utilize
+Theorems \ref{ppp1}, \ref{ppp2},
+\ref{pqq3}, \ref{pqq4} 
+and \ref{pqq5}.
+Its most prominent
+examples involve 
+\ep{totdefxa}'s
+Type-A axiom systems
+where the deduction method is
+either semantic tableaux
+or a modified version of tableaux that permits
+a modus ponens rule for $\Pi_1^\xi$ and $\Sigma_1^\xi$ sentences.
+\item[ II  ]
+The Appendix E introduces a generalization of
+the  Second Incompleteness
+Theorem which shows that \thx{ppp6}'s
+Translational Reflection Principle
+applies {\it only to} 
+self-justifying logics. (It is thus  fully inoperative
+for conventional logics.
+This may explain why 
+\thx{ppp6}'s
+self-justifying systems are an interesting topic.)
+\item[ III  ]
+The Appendix F differs from the rest of
+this paper
+by having a 
+philosophical
+slant.
+It will 
+offer a 3-page summary about why we suspect 
+\thx{pqq3}'s 
+self-justification
+formalism and Remark \ref{recc1}'s
+notion of ``instinctive faith'' are useful.
+%and related to Theorem \ref{ppp}'s reflection principle.
+\item[  IV  ]
+The Appendix G 
+introduces 
+a ``Braced$^\xi(  \Phi  ,j  )$''
+construct
+ and two new theorems that hybridize
+the methodologies of
+ Theorems \ref{pqq4} 
+and \ref{pqq5}.
+These  results will
+improve upon
+ Theorem \ref{pqq4} because 
+their self-justifying  systems 
+contain only a finite 
+number of axiom-sentences beyond those lying in
+$~\xi \,$'s base formalism
+of  $B^\xi $.
+They 
+will
+improve upon 
+ Theorem \ref{pqq5} because
+they can prove the important  
+ Braced$^\xi(  \Phi  ,j  )$  
+subset of
+ $ (   \nal{B} , D )  $'s $\Pi_1^\xi$ theorems
+in a full sense
+ (rather than in
+Remark  \ref{re4-2}'s 
+weaker 
+simulated respect).
+Appendix G's results are useful
+because {\it for arbitrary $k$}
+and 
+for
+any of Appendix D's four sample configurations,
+every
+$\Pi_1^\xi$ theorem
+of  $  (   \nal{B}  ,  D  )  $
+ containing $~k~$ or
+fewer bounded and unbounded quantifiers will 
+be proven 
+by its Theorem G.3 to be self-justifying
+in an undiluted pure sense. (This is because each
+such $\Pi_1^\xi$ sentence, with fewer than $~k~$
+quantifiers, will 
+lie in
+ some
+ fixed  Braced$^\xi(  \Phi  ,j  )$  
+set $~$---$~$  where solely 
+the value of
+$~k~$ determines the values for
+ $~\Phi~$ and $~j ~$. )
+% \nop
+\ennd 
+
+%\njp
+
+\cvmew
+\cvs
+
+
+
+It is 
+probably
+desirable
+ to 
+ concentrate 
+primarily
+ on
+ Theorems \ref{ppp1}, \ref{ppp2},
+\ref{pqq3}, \ref{pqq4}, \ref{pqq5}
+and  \ref{ppp6}
+during 
+one's first reading of this paper.
+This is because
+Appendixes A-G
+are
+less central than these 
+% five 
+core theorems,
+although their material
+does
+add 
+several
+useful
+further
+perspectives to this subject.
+
+
+%%notre
+  
+\medskip
+
+
+\cvb
+\parskip 3pt
+
+
+%%notre-only  
+%%notre-only    {\bf Reminder to the Referee and Editor:} The cover
+%%notre-only    page of this article indicated any (or all) of 
+%%notre-only    Appendixes A-F can be removed from this article if it
+%%notre-only    is deemed too long. In that case, the manuscript could
+%%notre-only    cite the Cornell Archive paper 
+%%notre-only    \cite{ww11}
+%%notre-only    containing these appendixes.
+
+
+%% 
+%%   {\bf Reminder to the Referee:}
+%% % and Editor:} 
+%% The cover
+%%   page of this article indicated any (or all) of 
+%%   Appendixes A-F can be removed from this article if it
+%%   is deemed too long. 
+%% (This is because 
+%% these results
+%% are stored in \cite{ww11}'s archives.)
+
+
+%% In that case, the manuscript could
+%%   cite the Cornell Archive paper 
+%%   \cite{ww11}
+%%   containing these appendixes.
+
+
+
+
+\section*{7.  Concluding Remarks}
+
+
+The research in 
+this article 
+has been
+ a continuation of our prior
+research \cite{ww93}-\cite{ww9} that simultaneously
+has simplified, unified
+and extended the prior results. 
+It
+has explored self-justification
+with a 3-part approach
+where:
+\bee
+% \itemsep +3pt
+\item
+Sections \ref{3uuuu2} and \ref{3uuuu3}
+introduced
+three different
+  stem components
+that can be used to generate
+ self-justifying
+ systems. (These are
+     the relatively simple Lemma  \ref{lemex4}
+and the mathematically more sophisticated
+Theorems  \ref{ppp1}  and \ref{ppp2}.)
+\item
+Section \ref{sect64} and Appendix G then
+generalized our initial stem-like theorems
+in the
+six 
+% five 
+different directions 
+formalized
+by
+Theorem  \ref{pqq3},  \ref{pqq4},
+  \ref{pqq5}, 
+  \ref{ppp6}, 
+G.2 and G.3
+\item
+Appendix D 
+subsequently
+provided four examples of generic configurations
+that are applications of 
+Section \ref{sect64}'s results.
+\ene
+This 3-part approach is
+very
+different from the methods
+used
+in our  prior articles
+\cite{ww93,ww5,wwapal,ww9}. The latter
+examined
+ particular isolated applications
+in 
+thorough
+detail (rather than
+compartmentalize
+and separate
+ the analysis
+into three stages). The virtue of this 3-stage analysis
+is 
+it leads to many new theorems, in addition to
+unifying our prior results.
+
+%%cc \bigskip
+
+It is desirable to 
+categorize
+the maximal
+generality and 
+strongest form of
+boundary-case exceptions
+for the Second Incompleteness Theorem
+that are feasible because
+G\"{o}del's centennial discovery 
+ beckons the 
+scholarly community
+to sharpen their understanding of his 1931 
+landmark discovery,
+that
+has
+fundamentally
+ reshaped mathematics.
+
+
+
+
+%%cc \bigskip
+
+
+It should be emphasized that
+our over-all 
+research
+in \cite{ww93} -- \cite{ww9}
+ has spent an approximately
+equal 
+effort
+in
+ exploring generalizations of
+the Second Incompleteness Theorem
+\cite{ww2,wwlogos,wwapal,ww7} 
+and 
+in
+examining
+its 
+viable 
+boundary-case exceptions
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}
+(although the current article
+focused on
+the latter topic).
+ This is because
+the
+Second 
+Incompleteness Theorem
+is 
+a starkly 
+robust result that
+ imposes 
+sharp
+limits
+on how strong self-justifying systems may become.
+
+%%cc \bigskip
+
+
+Finally, we encourage the reader to take another brief glance
+at Remarks \ref{f88} -- \ref{recc1}.
+They
+offer a brief
+summary
+of both the
+strengths and limitations
+of our
+chief
+ results.
+They also explain how Theorem \ref{ppp6}'s reflection
+principle for
+$\Pi_1^\xi$ styled theorems is a  
+very  unexpected
+ result.
+
+%quite unexpected
+%and pleasing result.
+
+% implications for  
+% Remark \ref{remhappy} additionally explains
+% what types of
+
+%reflection principles. 
+
+%%%%% that are associated with 
+%%%%boundary-case
+
+
+
+
+  
+%%%%%  \bigskip
+%  
+
+
+%%cccorn-ff  
+%%cccorn-cc  
+%%cccorn-cc  
+
+
+  \bigskip
+  \bigskip
+    
+  
+  
+    
+    {\bf ACKNOWLEDGMENTS: }
+    I thank
+    Bradley Armour-Garb,
+    Seth Chaiken and
+    Kenneth W. Regan
+    for many useful suggestions about how to
+    % significantly
+     improve
+    the presentation 
+    style
+    of some earlier drafts of this article.
+    As was noted in the text of this article, some
+    of my research
+    during the last 17 years was influenced by some private 
+    communications
+    that I had with Robert M. Solovay 
+    in 1994 and
+     Leszek 
+    Ko{\l}odziejczy  in 2005.
+
+
+
+\newpage
+
+\parskip 2pt
+
+\cvt
+
+\section*{Appendix A: The
+$~\Pi_1^\xi~$ encoding for 
+SelfCons$^k(\beta,d)$}
+
+
+
+
+
+
+This appendix will  
+summarize how to 
+formalize
+a $\Pi_1^\xi$ encoding for
+Definition \ref{xd+1x7}'s 
+SelfCons$^k(\beta,d)$ predicate.
+It
+will
+use
+the following notation:
+\bee
+\itemsep +1pt
+\baselineskip = 1.0 \normalbaselineskip
+\topsep -3pt
+\itemsep 4pt
+\item
+Neg$^k(x,y)$,
+will denote 
+a
+$\Delta_0^{\xi}$
+formula 
+indicating that $~x~$ is the G\"{o}del
+number of a 
+$\Pi_k^\xi$ sentence and
+that $~y~$
+represents the  
+$\Sigma_k^\xi$ sentence
+which is its logical negation.
+\item
+$\mbox{Prf} \, _\beta^d~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that
+ $~p~$ is  a proof of theorem $~t~$ from the axiom
+system  $~\beta~$ using the deduction method $~d.~$
+\item
+$\mbox{ExPrf} \, _\beta^d\,( \, h \, , \, t \, , \, p \,)$  
+will denote that
+$ \, p \, $ is a proof
+(using $d$'s deduction method)
+ of
+a theorem $t$ 
+from the union 
+of the axiom system
+$\beta$ with the added
+sentence whose G\"{o}del number equals
+$\, h \,$. 
+\item
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ encodes a sentence
+that replaces all occurrence of free variables in $g \,$ with
+a term of $\bar{g}$ (that specifies $g$'s
+G\"{o}del number). 
+\item
+$\mbox{SubstPrf} \, _\beta^d~( \, g \, , \, t \, , \, p \,)$  
+will denote the 
+hybridization of Items 3 and 4
+that yields a Boolean value of TRUE 
+when there
+exists an integer $~h~$ 
+satisfying 
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf} \, _\beta^d~( \, h \, , \, t \, , \, p \,)$.
+\ene
+
+
+It is easy to apply \cite{ww1}'s methodologies
+to confirm 
+Items 1-5 
+can be encoded as $\Delta_0^\xi$ formulae.
+Thus,  
+Appendixes C and D of  \cite{ww1}
+explained how  the theory of
+LinH functions 
+\cite{HP91,Kr95,Wr78}
+implied  there existed 
+ $\Delta_0$ encodings 
+for  formulae 1-4,
+and
+these 
+ $\Delta_0$ encodings can be
+easily rewritten
+ \footnote{  \b55 This rewriting
+of conventional $\Delta_0$ formulae into
+a  $\Delta_0^\xi$ format
+ is possible because  
+Part 2  of Definition \ref{def3.3}
+indicated that
+two 3-way predicates 
+of
+Add$(x,y,z)$
+and Mult$(x,y,z)$ do 
+encode  addition and multiplication
+in a $\Delta_0^\xi$ styled form.}
+as
+ $ ~ \Delta_0^\xi ~$ expressions.
+\ep{encode} 
+ uses this information to formulate a
+$\Delta_0^{\xi}$ encoding for 
+$\mbox{SubstPrf} \, _{~\beta}^d \,(   g   ,   t   ,   p  )$'s 
+graph. It is
+equivalent to 
+$~$``$~\exists ~h~ \{ ~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf} \, _{\beta}^d(    h    ,    t    ,    p   )\,  \}  \, \,$''$,~$
+  but \ep{encode} is written in
+ a $\Delta_0^{\xi}$ format --- {\it unlike} the quoted
+expression.
+\begin{equation}
+\label{encode}
+\mbox{Prf} \, _{\beta}^d~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~ \{ ~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf} \, _{\beta}^d~( \, h \, , \, t \, , \, p \,)~ \}   
+\end{equation}
+
+Using \eq{encode}'s
+  $\Delta_0^{\xi}$ encoding for 
+$\mbox{SubstPrf} \, _{\beta}^d(   g   ,   t   ,   p  )$, it is
+easy to encode 
+SelfCons$^k(\beta,d)$
+ as  a $\Pi_1^{\xi}$ 
+axiom-sentence.
+Thus,  let
+$ \, \Gamma^k(g) \, $ 
+denote  \eq{encode12}'s formula, and
+let  $ \, \bar{n} \, $ denote $ \, \Gamma(g)$'s
+G\"{o}del number.
+\begin{equation}
+\label{encode12}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^k(x,y) ~ \wedge~ 
+\mbox{SubstPrf} \, _{\beta}^d  \, (  g    ,    x    ,    p   ) ~
+\wedge  ~
+\mbox{SubstPrf} \, _{\beta}^d  \,    (    g    ,    y    ,    q   )  \, \}
+\end{equation}
+Then  $~$``$~\Gamma^k(~ \bar{n}~)~$''$~$ 
+is a $\Pi_1^{\xi}$ 
+encoding 
+for SelfCons$^k(\beta,d)$'s formalization of
+the statement $+$ from Definition
+\ref{xd+1x7}.
+Thus,  $~\Gamma^k(~ \bar{n}~)~$
+is 
+encoded is as follows:
+\begin{equation}
+\label{encode12n}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^k(x,y) ~ \wedge~ 
+\mbox{SubstPrf} \, _{\beta}^d  \, (  \bar{n}     ,    x    ,    p   ) ~
+\wedge  ~
+\mbox{SubstPrf} \, _{\beta}^d  \,    (     \bar{n}   ,    y    ,    q   )  \, \}
+\end{equation}
+
+
+
+{\bf Reminder about \ep{encode12n} :}
+This sentence's
+definition for 
+SelfCons$^k(\beta,d)$ 
+does not assure 
+\eq{encode12n}
+is
+true under the Standard-M model. Indeed for nearly
+all $(\beta,d)$, it will be false
+when $k \geq 2$. This
+is the reason that the study of
+SelfCons$^k(\beta,d)$,
+under Theorems \ref{ppp1} and \ref{ppp2},
+has focused on the cases where $~k~$ equals 0
+or 1.
+Moreover,
+the preceding
+construction did assure that 
+SelfCons$^k(\beta,d)$
+had a
+$~\Pi_1^\xi~$ encoding
+because 
+such
+an
+{\it ``I am consistent''} axiom 
+carries 
+ more meaning
+than  a 
+$\Pi_2^{\xi}$ encoded axiom.
+
+
+
+
+
+
+\parskip 0 pt
+
+
+\section*{Appendix B: The Proof of Lemma \ref{llpp3}}
+
+
+\cvt
+
+Lemma \ref{llpp3} is
+a crucial interim step used to verify
+each of
+Theorems \ref{pqq3}, \ref{pqq4} and \ref{pqq5}.
+Its proof will employ
+the following three 
+straightforward observations:
+\bed
+\item[Fact B.1 ]
+Lemma  \ref{llpp3}'s 
+hypothesis
+implies that
+ $\mxi$ 
+is a
+generic configuration. (This is because 
+it specified that
+ $\xi$ 
+was a generic configuration  
+and that
+all the $\Pi^\xi_1$ sentences of
+$~\mheta~$ were
+true
+in the
+ Standard-M model.
+Thus,
+ $\mxi$ 
+must also be a
+generic configuration.)
+\item[Fact B.2 ]
+The
+associative identity of 
+ $ \, \theta \, \cup \, (\mheta \,  \cup B^\xi) \,  \, = \,  \, 
+( \, \theta \, \cup \, \mheta \, ) \cup B^\xi \, $ 
+obviously holds. It 
+implies
+a sentence
+$ \, \Uxp \, $
+is a theorem of
+ $ \, \theta \, \cup \, (\mheta \,  \cup B^\xi) \, $ 
+if and only if 
+it is a theorem of 
+ $ \, ( \, \theta \, \cup \, \mheta \, ) \cup B^\xi \, $.
+
+
+
+\item[Fact B.3 ]
+Lemma \ref{llpp3}'s hypothesis 
+directly
+\footnote{\s55 The identity of 
+$~ \zzz \mheta ~)~=~\infty~$
+must be true
+because the 
+hypothesis 
+of Lemma  \ref{llpp3}
+ indicated that 
+all the $\Pi^\xi_1$ sentences in
+$~\mheta~$ are 
+true  under the Standard-M model.
+By  Definition \ref{chg}, 
+this
+ implies $~ \zzz \theta ~)~= ~ \zzz \mheta \cup \theta ~)~$.}
+implies $ \zzz \theta )=  \zzz \mheta \cup \theta )$.
+\ennd
+
+\smallskip
+
+The justification of
+claims (i)-(iv)
+are
+ consequences of 
+Facts B.1 through B.3.
+We
+will
+ provide a detailed proof of only Claim (i) here
+ because all
+four claims have 
+similar proofs.
+
+
+
+
+
+
+
+\medskip
+
+{\bf Proof of Claim (i) :}
+The hypothesis of Claim (i) indicated that $~\xi~$ 
+was 0-stable. Therefore, it
+satisfies
+ Definition \ref{ostab}'s
+invariant of $~***~$. 
+In a context where
+$~\phi ~$ is a variable
+designating a r.e. set of $\Pi_1^\xi$ sentences 
+and  $ \, \Uxp \, $ is a variable corresponding to a 
+$\Delta_0^\xi$ sentence, 
+ the invariant  $~***~$
+can be rewritten in a  quasi-rigorous form as :
+\beq
+\label{whatshit1}
+\forall ~ \phi ~~ \forall ~ \Uxp 
+ ~~~~~  \mbox{\it the below
+statement, called $~ \Psi_1(\phi,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $ \, \phi \,  \cup B^\xi \, $ 
+via a proof $ \, p \, , \, $ 
+whose length satisfies
+ Log$(p) \, \leq \zzz \phi \, ) \, +1 \, , \,  $
+ then  $ \, \Uxp \, $ 
+is true under the Standard-M model.
+\end{quote}
+Since \eq{whatshit1}'s
+ universally quantified variable 
+$~\phi~$
+can designate any r.e. set of $\Pi_1^\xi$ sentences, it 
+may
+designate the object  ``$~\theta ~\cup~\mheta~$'', where
+$~\mheta~$ is the r.e. set of  $\Pi_1^\xi$ sentences
+defined by Lemma \ref{llpp3}'s hypothesis
+(and $~\theta~$ is any second r.e. set of sentences).
+Thus,  \eq{whatshit1} 
+directly
+implies :
+\beq
+\label{whatshit2}
+\forall ~ \theta ~~ \forall  \Uxp
+ ~~~~~  \mbox{\it the below
+statement, called $~ \Psi_2(\theta,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$~\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $~(~\theta~\cup~\mheta~)~ \cup B^\xi~$ 
+via a proof $~p~,~$ 
+whose length satisfies
+ Log$(p) \, \leq \zzz  \, \theta\cup\mheta)~   +1, $
+ then  $ \, \Uxp \, $ 
+is true  under  Standard-M.
+\end{quote}
+Facts B.2 and B.3 enable one to 
+simplify 
+\eq{whatshit2}'s
+ terms of  $( \, \theta \, \cup \, \mheta \, ) \,  \cup B^\xi$ 
+and  $\zzz ( \, \theta \, \cup \, \mheta \, ) \, ~    )$  
+and thus to derive \eq{whatshit3} as a consequence.
+\beq
+\label{whatshit3}
+\forall ~ \theta ~~ \forall  \Uxp ~~~~
+   \mbox{\it the below
+statement, called $~ \Psi_3(\theta,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$~\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from 
+axiom system $\theta~\cup~(~\mheta~ \cup B^\xi~)$ 
+via a proof $~p$, 
+whose length satisfies
+ Log$(p) \, \leq \zzz \theta) \, + \,1 $,
+ then  $  \Uxp  $ 
+is true under the Standard-M model.
+\end{quote}
+We will now use Fact B.1's observation that
+$\mxi$ 
+is a generic configuration.
+The sentence
+\eq{whatshit3} 
+indicates
+this configuration
+satisfies  
+Definition \ref{ostab}'s invariant of
+$***$.
+Hence, 
+$\mxi $
+ is
+0-stable.
+$\Box$
+
+%%cccorn-ff
+
+
+
+  
+  \medskip
+  
+  \cvs
+  
+  {\bf Brief Comments about the Justifications of Claims (ii)-(iv):}
+  This appendix has  omitted 
+  proving 
+  (ii)-(iv) 
+  for the sake of brevity.
+  Their proofs are   similar
+   to Claim (i)'s proof.
+  For instance,   Claim (ii)'s proof differs
+  from  Claim (i)'s proof  by having
+  the 0-stability invariant in $***$
+   replaced
+  by the 
+  A-stability invariant of $~*~$.
+  This will  cause the analogs of
+  \eq{whatshit1} -- \eq{whatshit3} 
+  to undergo the 
+  following two 
+  simple
+  changes under Claim (ii)'s proof :
+  \bee
+  \topsep -10pt
+  \itemsep +2pt
+  \item 
+  $ \, \Uxp \, $ 
+  will  represent a 
+  $\Pi_1^\xi$ 
+  (rather than a
+  $\Delta_0^\xi$ ) theorem-statement under the 
+  revised versions of $\Psi_1$, 
+  $\Psi_2$ and $\Psi_3$  
+   used in Claim (ii)'s proof
+  \item 
+  The requirement
+  (in 
+  sentences \eq{whatshit1}-\eq{whatshit3} of Claim i's proof)
+   that
+  $  \Uxp  $  be
+  true in the Standard model
+  is
+   changed 
+  to the stipulation that 
+  $  \Uxp  $ 
+     satisfies 
+   the Good$\{ \, \tftt  \zzz \phi) \, \}$
+  and
+  Good$\{  \tftt  \zzz \theta)  \}$
+  conditions
+   under the revised forms of
+   $\Psi_1$, 
+  $\Psi_2$ and $\Psi_3$  
+  used   to prove Claim (ii).
+  \ene
+  Other
+  minor adjustments in Claim (i)'s proof
+  shall verify Claims iii and iv.
+  
+  
+  
+
+\nop
+
+
+\section*{Appendix C: The Proof of \phx{ppp2} }
+
+
+
+Our  proof of  \phx{ppp2}
+is a straightforward modification of
+\phx{ppp1}'s 
+ proof. 
+It
+will be divided into
+two lemmas.
+
+
+\medskip
+
+{\bf Lemma C.\ref{B1-lem}.}
+{\it 
+Every generic configuration that is either E-stable or
+A-stable will automatically satisfy
+Definition \ref{ostab}'s 0-stability condition.}
+
+\medskip
+
+
+
+{\bf Proof.} $~$
+Lemma C.\ref{B1-lem} is 
+a 
+consequence of
+the 
+``Special Note''  appearing at the end of 
+Definition \ref{xd+1x4}.
+For every  $N\geq 0$,
+it
+ indicated that
+if
+$\Uxp$ is a 
+$\Delta_0^\xi~$ 
+sentence then
+Scope$_E$($\Uxp,N)$
+is  equivalent to  $~\Uxp~$.
+This implies 
+(via Definition \ref{xd+1x5})
+that if 
+$\Uxp$ is a  
+Good(N)
+$\Delta_0^\xi~$ formula
+then Scope$_E$($\Uxp,N)$ is automatically
+true under the Standard-M model.
+
+
+
+
+
+
+The latter observation makes it easy to confirm
+Lemma C.\ref{B1-lem}. This is 
+because  every
+$\Delta_0^\xi$ sentence
+is a
+$\Pi_1^\xi$ and
+$\Sigma_1^\xi$ statement.
+Hence, the application of the invariants
+$~*~$  and $~**~$
+from Definitions 
+\ref{astab} and \ref{estab}, in the degenerate case
+where
+$\Uxp$ is a
+ $\Delta_0^\xi$ theorem,
+corroborates 
+Lemma C.\ref{B1-lem}'s claim
+(by showing that
+ Definition \ref{ostab}'s 
+ invariant of
+ $\,***\,$ does hold).
+$~~\Box$
+
+\medskip
+
+The remainder of this appendix will  focus on
+Definition \ref{ostab}'s 0-stability condition.
+(This is sufficient to justify \thx{ppp2}
+because Lemma C.\ref{B1-lem}
+showed all 
+ E-stable and A-stable
+configurations are
+ 0-stable.)
+
+\smallskip
+
+{\bf Lemma C.\ref{B2-lem}.}
+{\it 
+Let $~\xi~$
+denote a
+generic configuration
+ \gggen 
+that is
+0-Stable.  Then 
+the axiom system of
+$B^\xi+$SelfCons$^0(B^\xi,d)$
+will be consistent (and 
+hence self-justifying).}
+
+\smallskip
+
+
+
+\smallskip
+
+{\bf Proof:}
+It will be awkward 
+to write repeatedly
+the expression   ``$~B^\xi+$SelfCons$^0(B^\xi,d)~$''
+during our proof.
+Therefore,  $\zhz $ will
+be an abbreviated
+name for this 
+system. 
+Our justification
+ of Lemma C.\ref{B2-lem},
+is similar to \phx{ppp1}'s proof,
+$\,$except  it replaces a 
+Level$(1^\xi)$  form of self-justification
+with  
+a 
+Level($0^\xi)$.
+It
+will thus
+  be
+ abbreviated and 
+use the following notation:
+\bee
+\itemsep +4pt
+\item
+$~\mbox{Prf}_\zhz (t,p)~$  is a
+$\Delta_0^\xi$ formula 
+specifying
+$~p~$ 
+is
+ a proof of the theorem $~t~$ from
+the  axiom system $~\zhz ~$
+(using 
+$~\xi \,$'s
+deduction method of $d$ ).
+\item
+$~\mbox{Neg}^0(x,y)~$ 
+ is a
+$\Delta_0^\xi$ formula 
+ indicating
+$~x~$ is
+the G\"{o}del encoding of a
+ $\Delta_0^\xi$ 
+sentence and
+$~y~$ is a   $\Delta_0^\xi$ 
+sentence  representing
+$~x \,$'s negation.
+\ene 
+Expression \eq{qpencode} 
+denotes
+ $~\zhz \,$'s 
+ Level$(0^\xi)$
+self-justification axiom.
+It is encoded using
+Appendix A's
+methodology,
+similar to 
+its counterpart 
+used in \phx{ppp1}'s proof (i.e.  
+\ep{pencode} ).
+\begin{equation}
+\label{qpencode}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^0(x,y) ~ \wedge~ 
+\mbox{Prf}_\zhz   \, (     x    ,    p   ) ~
+\wedge  ~
+\mbox{Prf}_\zhz  \,    (       y    ,    q   )  \, \}
+\end{equation}
+
+
+Our proof of Lemma C.\ref{B2-lem}
+will be a proof by contradiction.
+It will thus
+ begin with
+the 
+contrary
+assumption that $~\zhz ~$ is inconsistent
+and have  $~\Phi~$ denote
+\eq{qpencode}'s sentence.
+The inconsistency of $~\zhz ~$ 
+ implies that 
+$~\Phi~$ is
+false under the Standard-M model.
+Hence via
+Definition \ref{chg}, we get:
+\begin{equation}
+\label{qpunk}
+ \zzz \Phi~) 
+   ~~ < ~~ \infty
+\end{equation}
+\noindent
+\ep{qpunk} 
+implies there  exists
+a tuple $(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying \eq{qpencodepunk}.
+(This is because such a 
+$(\bar{p},\bar{q},\bar{x},\bar{y})$ 
+corroborates \eq{qpunk}'s implication that a 
+counter-example to
+\eq{qpencode}'s sentence 
+does
+exist.)
+\begin{equation}
+\label{qpencodepunk}
+  \,\mbox{Neg}^0(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ 
+\mbox{Prf}_\zhz   \, (     ~ \bar{x}~    ,    ~ \bar{p}~   ) ~
+\wedge  ~
+\mbox{Prf}_\zhz  \,    (       ~ \bar{y}~    ,    ~ \bar{q}~   )  
+\end{equation}
+% Furthermore, the  
+The
+$(p,q,x,y)$
+ satisfying \eq{qpencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{\, \mbox{Max}[p,q,x,y] \, \}$ 
+will
+additionally
+%also
+satisfy
+\eq{qpunkless}.
+(This  observation follows
+from the analog of the
+Footnote \ref{footp1}
+appearing 
+in \phx{ppp1}'s proof. 
+Thus, Section \ref{3uuuu3}'s Equations
+of
+\eq{pencodepunk}
+and
+\eq{punkless}
+are 
+the
+analogs of the current
+ Equations
+\eq{qpencodepunk}
+and
+\eq{qpunkless}.
+The 
+Footnote \ref{footp1} 
+showed
+the particular
+$(p,q,x,y)$
+ satisfying \eq{pencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ 
+satisfied
+\eq{punkless}.
+By the same reasoning,  
+the minimal 
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying
+\eq{qpencodepunk} will     satisfy
+\eq{qpunkless}.)
+\begin{equation}
+\label{qpunkless}
+ \mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
+   ~~~ =~~~\zzz \Phi~) ~~+~~1
+\end{equation}
+
+
+
+
+Equations 
+\eq{qpencodepunk} and \eq{qpunkless} 
+shall
+bring 
+Lemma C.\ref{B2-lem}'s
+ proof-by-contradiction to its 
+sought-after
+ end. Thus,
+let $~\Upsilon~$ denote the
+ $\Delta_0^\xi$ 
+sentence specified
+by $~\bar{x}~$. Then
+$~\neg ~ \Upsilon~$ 
+corresponds to the 
+ $\Delta_0^\xi$ 
+ sentence denoted
+by $~\bar{y}~$. 
+\ep{qpunkless}
+indicates
+that
+both
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+have  proofs 
+such that the logarithms of
+their G\"{o}del numbers are bounded
+by
+$~\zzz \Phi~) ~+~1~$.
+Using
+Definition \ref{ostab}'s
+invariant of $~***~$,
+these facts establish  
+\footnote{\s55 To  
+apply Definition \ref{ostab}'s
+ invariant $~***~$ in the present setting,
+one simply sets $~\theta\,$'s R-View 
+equal to $~\Phi \,$'s
+1-sentence statement. 
+Then $~***~$ implies that both 
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+must
+be 
+true under the Standard-M model
+because both  their proofs had lengths 
+$\, \leq ~\zzz \Phi~) ~+~1~$.
+} that both 
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+are
+true under the Standard-M model.
+
+ 
+But it is impossible for
+  a sentence and its
+negation to be both
+true.
+This finishes
+Lemma C.\ref{B2-lem}'s
+proof
+because the
+temporary
+ assumption that 
+$\zhz $ was inconsistent
+has
+led to a
+contradiction.
+$~~\Box$
+
+\medskip
+
+\phx{ppp2} is a
+ consequence of 
+the  Lemmas C.\ref{B1-lem} and
+ C.\ref{B2-lem} because
+Lemma
+ C.\ref{B2-lem}'s formalism 
+generalizes
+to all
+E-stable and A-stable configurations
+via Lemma C.\ref{B1-lem}'s reduction methodology. 
+
+
+\nop
+
+
+\section*{Appendix D: Applications and Examples}
+
+
+
+This appendix will illustrate
+four examples of generic configurations 
+that satisfy
+Theorems \ref{ppp1} and \ref{ppp2} 
+(and 
+which
+therefore  are self-justifying).
+It will be divided into three parts.
+Section D-1 will define our first example of
+self-justifying configuration, called $~\xi^*~$.
+It will use
+semantic tableaux
+ deduction.
+Section D-2 will 
+prove that          $~\xi^*~$ is  EA-stable. 
+Section D-3 will 
+briefly sketch
+        three additional examples of
+   stable generic      configurations
+
+
+It is likely preferable to examine
+Sections \ref{3uuuu1} -- 
+\ref{sect64} before  this appendix.
+However, Section
+ D-1's 
+ short 2-page discussion 
+can be
+ read
+quite easily
+either before or
+ after 
+Section \ref{sect64}.
+
+
+
+
+
+
+
+\subsection*{D-1. $~$Definition of
+the EA-Stable Configuration  
+ $~\xi^*~$ }
+
+
+Our first example of an 
+EA-Stable configuration,
+ called \xxi,
+will be defined in
+this section.
+Its deduction method will be
+semantic tableaux.
+Its 
+ base axiom 
+system $\, B^* \,$ will be a
+ Type-A  formalism,
+which 
+treats addition but not
+multiplication as a total function
+(i.e. see  \ep{totdefxa} $~).$
+
+
+
+
+The closest analog of  $B^*$
+and  \xxi 
+ in our prior work
+% had
+appeared in Section 5 of \cite{ww5}.
+%(Its formalism was
+(It 
+differed
+from  \xxi 
+partly because
+it
+%\cite{ww5}
+did not 
+use Definition  \ref{def3.3}'s
+unifying
+notation.)
+In 
+\cite{ww5},
+% \cite{ww5}'s  discussion,
+a function 
+ $\, F \, $ 
+was     called
+ {\bf Non-Growth} 
+iff  
+$ F(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$. Six  examples of  
+non-growth functions are:
+\begin{enumerate}
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {\it the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0,~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lceil ~x^{1/y}~ \rceil$ when $~y\geq 1,~$
+and
+it equals $~x~$ when $~y=0.$
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+and  zero otherwise. 
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\end{enumerate}
+These operations were called
+{\bf Grounding Functions}
+in \cite{ww5}.
+The term
+{\bf U-Grounding Function}  
+referred 
+to a
+set of
+functions that included the
+Grounding  operations
+ plus 
+the further 
+primitives  of addition and
+{\it Double$(x)=x+x$.} 
+(The Double operation  
+is helpful because it significantly
+ enhances
+\cite{ww5}'s linguistic efficiency \footnote{\s55 The symbol
+{\it Double$(x)$}
+was technically unnecessary in \cite{ww5}'s formalism
+because
+$x+x$ can encode
+ Double$(x)$.
+However,
+its notation
+adds expressive power to
+\cite{ww5}'s language because, for example,
+Double(Double(Double(Double(x))) 
+requires less memory space to encode than
+than
+$~x~$ added to itself 16 times.
+} .)
+
+ 
+
+\medskip
+
+The symbol $~\Delta_0^*~$ 
+will be the
+analog of Definition \ref{xd+1x1}'s 
+ $\Delta_0^\xi$
+construct under
+the U-Grounding function language.
+(It will be defined to be any formula in a U-Grounding 
+language where all its quantifiers are bounded.)
+It is easy in this context to encode
+a $\Delta_0^{    *}$ 
+formula $Mult(x,y,z)$ 
+for 
+representing
+  multiplication's graph.
+For instance,
+\ep{neweq1} 
+is one such $\, \Delta_0^{    *} \, $ formula
+(which actually does not employ any bounded quantifiers):
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+Expression
+\eq{neweq1} 
+is significant because Part 2
+of Definition \ref{def3.3}
+indicated every generic configuration must have 
+ available some 
+method to represent the graphs of
+addition and multiplication 
+in a
+ $\Delta_0^\xi$ 
+styled
+format, similar to     
+\eq{neweq1}'s paradigm.
+ (Addition can be
+treated trivially
+because the 
+U-grounding language
+           possesses 
+an addition function symbol.)
+The
+footnote
+ \footnote{\s55  Section \ref{3uuuu1} 
+explained
+during
+its discussion
+of  
+Lemma \ref{lex22}
+ that
+every generic configuration $  \xi  $
+must have a means to encode 
+the graphs of
+addition and multiplication
+as $  \Delta_0^{\xi}  $ formulae
+(visavis Parts 1 and 2 
+of 
+ Definition \ref{def3.3}).
+This enabled Lemma \ref{lex22}'s
+procedure to translate
+all of conventional arithmetic's 
+$\Sigma_j$ and $\Pi_j$
+formulae 
+into
+equivalent 
+$\Sigma_j^{\xi}$ and $\Pi_j^\xi$
+expressions. \fend }
+% in $ \, \xi \,$'s language. 
+serves as a reminder about why
+these $\Delta_0^*$ encodings are 
+needed. 
+Our next goal is to define 
+the generic configuration 
+ \xxi that
+ Section D-2 will prove  
+is EA-stable.
+
+%\hgskip
+
+\bigskip
+
+
+{\bf Definition
+D.\ref{D1-def}.}
+The 
+language
+ $~L^{*}~$ of the generic configuration 
+ \xxi 
+will be built in a natural manner
+out of 
+the eight 
+U-grounding  function operations,
+the usual
+atomic predicate
+symbols of ``$~=~$'' and 
+ ``$~\leq~$'',
+and the
+three constant symbols 
+ $~K_0,~ K_1\,$ and $\,K_2\,$ (that define the
+integers of 0, 1 and 2).
+The
+other components
+of  \xxi 's configuration
+  are defined
+below:
+\bed
+\cvh
+\itemsep 4pt
+\item[    i  ]
+As previously noted,
+  $~\Delta_0^{*}~$
+is defined to
+represent the set of all
+formulae in $ \, L^* \,$'s 
+language, 
+whose  quantifiers are bounded 
+in an
+arbitrary
+ manner by terms employing 
+the U-Grounding function
+symbols.
+(It will thus generate via Definition \ref{xd+1x1}
+the  $\Pi_n^*$ and $\Sigma_n^*$ sentences of
+$ \, L^* \,$.)
+\item[   ii  ]
+The base axiom system 
+ $~B^*~$ 
+for \xxi 
+will be allowed to be any
+consistent
+set of   $\Pi_1^{\xi^*}~$ sentences
+that is capable of proving every
+ $\Delta_0^*$ sentence that is valid
+in Standard-M.
+It will 
+also
+include   sentence
+ \eq{d1a}'s 
+{\it very precise} \footnote{\g55 \label{fodii}
+The two appearances of the term ``$~x+y~$'' in sentence
+\eq{d1a} 
+may at first appear to be redundant. (This
+statement is 
+equivalent to sentence  \eq{totdefsymba}'s
+declaration that addition is a total function,
+which had avoided such redundancy.) The virtue of
+\eq{d1a}'s format is that it is a    
+ $\Pi_1^*$ styled 
+statement, unlike \eq{totdefsymba}'s
+ $\Pi_2^*$ styled format. 
+This sharpened
+ $\Pi_1^*$ perspective
+will help simplify some of 
+our proofs. }
+$\Pi_1^*$  styled
+declaration that addition is a total function.
+\beq 
+\label{d1a}
+\small
+\forall x ~~~~~~\forall y~~~~~~ \exists~ z \leq  x+y ~~:~~~ ~~~
+  \{~~~z=x+y~~~\}
+\enq
+\item[  iii  ]
+\xxi's deduction method will be the semantic tableaux
+method.
+\item[   iv  ]
+\xxi' s 
+G\"{o}delized
+method $~g~$
+for encoding
+a semantic tableaux proof
+ can be essentially
+any natural method 
+ that satisfies 
+the 
+minor stipulation
+that
+ at least $~5 \, J~$ bits
+are required
+ to encode a semantic tableaux
+proof that has $~J~$ function symbols. 
+This stipulation is called the 
+{\bf ``Conventional Tableaux Encoding Requirement''}.
+It is trivial 
+\footnote{\g55 The Conventional Tableaux Encoding Criteria
+requires that the 
+ G\"{o}del 
+number of a semantic tableaux proof,
+with  $~J~$ function symbols,
+ must be 
+least as large as $~32^J~$.
+It is clear that
+all
+the usual methods for 
+generating the G\"{o}del codes
+satisfy this criteria.
+This is because 
+any proof that has  $~J~$ function symbols will contain at least
+ $~2  \, J~$ logical symbols and thus employ 
+at least
+ $~5  \, J~$ bits. \fend
+} to corroborate   
+ that all the
+usual methods for encoding semantic tableaux proofs
+satisfy this 
+criteria.
+(The  Appendix A of \cite{ww5} provides
+one 
+example of 
+a possible tableaux encoding method.
+Any other natural mechanism for encoding 
+tableaux proofs is equally suitable.)
+\ennd
+
+\smallskip
+
+
+Section D-2
+will,
+interestingly,
+prove  that   $\xi^*$ is EA-stable.
+This 
+will imply (via \phx{ppp1})
+that
+$~B^*~\cup~\mbox{SelfCons}^1\{~B^*~,d~\}$ 
+is 
+self-justifying.
+Theorems \ref{pqq4},  \ref{pqq5}, G.2 and G.3 will,
+formalize, in this context,
+four different methods in which 
+$~B^*~$ can be extended to 
+construct self-justifying formalisms that are able to prove
+Peano Arithmetic's 
+$~\Pi_1^*$ theorems.
+
+\nskip 
+
+\medskip
+
+ Thus while  self-justifying axiom systems
+contain unavoidable weaknesses, they  also possess
+the nice feature that they are able to prove many
+of the useful theorems of mathematics.
+
+
+\nskip
+
+\subsection*{D-2. $~$Proof of the  EA-Stability of
+ $~\xi^*~$ }
+
+\nskip
+
+This section
+ will prove \xxi 
+is EA-stable and thus
+satisfies the paradigms of 
+Theorems \ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5}
+and  \ref{ppp6}.
+Our proof will be based on modifying some of the methodologies
+from \cite{ww5}, so that they
+become applicable to \xxi .
+Many readers may prefer to omit examining
+both this part of Appendix D and Section 
+ D-3 because they 
+are unnecessary for understanding
+the 
+% further
+ material in
+Section \ref{sect64} and
+Appendixes E and F. (Our recommendation is that the latter
+material be read first.)
+
+
+
+
+\nskip
+\smallskip
+
+
+Our notation for defining a
+  semantic tableaux
+proof
+        in the next paragraph
+will be similar to the 
+conventional
+      definitions
+appearing in 
+Fitting's and Smullyan's textbooks  \cite{Fi90,Sm68}.
+It will employ
+\cite{ww5}'s
+notation
+so that we can employ
+two of its
+lemmas during our analysis of semantic tableaux proofs.
+
+\nskip
+
+%\cvmew
+
+{
+%notre \cvh
+In our discussion,
+$~\Phi~$
+will be called
+a {\bf  Prenex-Level($\,$m$^* \,$)}
+  sentence
+ iff it is a 
+ $\Pi_m^*$ or $\Sigma_m^*$ 
+expression
+that satisfies the usual prenex requirement (that
+all its unbounded 
+quantifiers   lie  
+in its leftmost part).
+If $\Phi$ is  Prenex-Level$(m^* )$
+then {\bf Reverse($\Phi$)} 
+shall 
+denote a second
+ Prenex-Level$( \, m^* \,)$
+sentence that is equivalent to
+$~\neg~\Phi~$  rewritten 
+\footnote{\s55  For example,
+if $\Phi$ denotes ``$~\forall \, x ~ \exists \,y ~~\psi(x,y)~$''
+then Reverse$(\Phi)$ 
+would be written  as
+ ``$~ \exists \, x ~ \forall \,y ~~\neg~\psi(x,y)~$''.}
+%% \footnote{\s55 Thus  
+%% % For example,
+%% if $\Phi$ denotes ``$~\forall \, x ~ \exists \,y ~~\psi(x,y)~$''
+%% then Reverse$(\Phi)$ is 
+%% %would be written  as
+%%  ``$~ \exists \, x ~ \forall \,y ~~\neg~\psi(x,y)~$''.}
+in a
+ Prenex Level$( \, m^* \, )$ form.
+For a fixed axiom system $ \, \alpha ,  \, $ its
+{\bf $\Phi$-Based Candidate Tree}
+will  be defined
+to be a  tree structure 
+whose root is
+the sentence 
+Reverse$(\Phi)$ 
+ and whose all other nodes are
+either axioms of $ \, \alpha \, $ or deductions from higher
+nodes of the tree,
+via the rules 1--8 
+given
+below. (The symbol
+``{\bf $ \,  $A$  ~  \Longrightarrow  ~  $B$  \, $}''
+in  rules 1-8
+will mean
+that {\bf $ \,   $B$  \, $} is a valid deduction
+from its ancestor {\bf $ \,  $A$  \, $}
+in the germane 
+ deduction tree.)
+\begin{enumerate}
+\itemsep 5pt
+%notre \small
+%%corbl \baselineskip = 1.05 \normalbaselineskip
+\item $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Gamma ~$ .
+$~~~$
+\item $ \,  \neg  \,\neg \, \Upsilon  \,  \Longrightarrow  \,  \Upsilon. \, $  
+Other rules for
+the ``$ \, \neg \,$'' symbol  are:
+$ \, \neg ( \Upsilon \vee \Gamma )  \,  \Longrightarrow  \,  \neg \Upsilon
+\wedge \neg \Gamma$,
+%\newline
+$ \, \neg ( \Upsilon \rightarrow \Gamma )  \,  \Longrightarrow  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \Longrightarrow  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $
+ and $ ~~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed
+when their ancestor is
+$~\Upsilon \, \vee \, \Gamma.~$
+\item A pair of sibling nodes $ \,  \neg \Upsilon  \, $ and $ \,  \Gamma  \, $ is
+allowed
+when their ancestor is
+$ \, \Upsilon \, \rightarrow \, \Gamma$.
+\item $~ \exists v \, \Upsilon  (v) ~ \Longrightarrow ~ \, \Upsilon(u) ~$
+where $\,u \,$ is a newly introduced ``Parameter Symbol''. 
+\item 
+ $~ \exists v \leq s ~ \, \Upsilon  (v) ~~ \Longrightarrow ~ ~
+u \leq s ~ \wedge~ \Upsilon(u) ~$
+is the
+ variation of Rule 5 for  bounded existential quantifiers
+of the form $~$``$~ \exists v \leq s ~$''.
+\item $\forall v \, \Upsilon  (v)  \,  \Longrightarrow    \, \Upsilon(t)  \, $
+where $t$ denotes a U-Grounded 
+term. These terms may be any one of
+a constant symbol,
+a  parameter symbol (defined  by a 
+prior application of  Rules 5 or 6 to some
+ some ancestor of the current node),
+or a U-Grounding function-symbol with  
+recursively defined
+inputs.
+\item 
+ $\forall v \leq s  \, \Upsilon  (v) ~~ \Longrightarrow ~~
+t \leq s \, \rightarrow \, \Upsilon(t) $ is 
+the variation of Rule 7 
+for a
+ bounded  quantifier such as
+ ``$~ \forall v \leq s ~ $''
+\end{enumerate} }
+
+%%corbl  \baselineskip = 1.1 \normalbaselineskip 
+\noindent
+Let us say a leaf-to-root branch (in a candidate
+tree) is {\bf Closed} iff it contains both some sentence
+$ \,  \Upsilon  \, $ and its negation ``$ \,  \neg \, \Upsilon  \, $''.  
+Then a
+ {\bf  Semantic
+Tableaux Proof} of $  \Phi  $,
+from the axiom system  $  \alpha $,
+is defined to be a 
+$\Phi$-Based Candidate Tree
+whose every
+ leaf-to-root branch is closed.
+
+\cvt
+
+\hgskip
+
+%\cvwew
+
+\nskip
+
+
+%bbbbb {
+
+It is next helpful to define
+the notion of a 
+{\bf Z-Based Deduction Tree},
+in a context where $~Z~$ represents 
+an
+ axiom system,
+typically different from 
+the prior paragraph's
+$~\alpha~$.
+This object 
+will be
+defined to be
+identical to
+a semantic tableaux proof, except for the following
+changes:
+\bed
+%%corbl  \baselineskip = 1.1 \normalbaselineskip 
+\itemsep 5pt
+\item[   i   ]
+{\it Every node} in a $Z-$Based 
+deduction tree
+must be either an axiom of  $~Z~$
+or a deduction from a higher node of the tree via
+the 
+rules 1-8.
+(This
+applies
+also to the root of
+a $Z-$Based  deduction tree.
+It will  store an 
+axiom of  $~Z~$
+ in its root, unlike a semantic tableaux proof
+ which had
+stored
+ Reverse$(\Phi)$ 
+in its
+root.)
+\item[  ii   ]
+There will be no requirement that each 
+ leaf-to-root branch
+be
+ closed
+in a 
+$Z-$Based  deduction tree.
+(Indeed, 
+some branch will 
+automatically
+not be closed
+if  $~Z~$ is  consistent.) 
+\ennd
+Items (i) and (ii) make it 
+apparent
+ that
+$Z-$Based  deduction trees
+are different 
+% properties
+from 
+semantic tableaux proofs.
+It will
+turn out,
+nevertheless, 
+ that the study of
+$Z-$Based  deduction trees
+will clarify the nature of 
+ semantic tableaux proofs.
+
+
+\nskip
+\nskip
+
+\medskip
+
+
+
+{\bf Definition 
+D.\ref{D2-def}.}
+Let $ ~   a ~   $ and $ ~   b ~   $ denote two
+integers that are powers of $\,  2 \,  $ 
+satisfying
+$  a \, > b \,
+\geq \, 2 \, $
+Then
+ an axiom system $ \, Z \, $
+(employing $ \, L^* \,$'s language)
+will be called
+a {\bf Normed(a,b)} formalism  iff:
+\begin{enumerate}
+\item All  $Z$'s 
+axioms  are 
+either $\Pi_1^{*}$
+or  $\Sigma_1^{*}$ sentences.
+\item 
+Each  $\Pi_1^{*}$ axiom of $~Z~$
+will satisfy Definition \ref{xd+1x5}'s
+Good($~$Log$_2a~)$ 
+criteria, and
+ each
+ $\Sigma_1^{*}$ axiom of $~Z~$
+will
+likewise
+  satisfy Good($~$Log$_2b~)$. 
+\ene
+
+\nskip
+\nskip
+
+
+{\bf  Clarification about Definition D.\ref{D2-def}
+: } The
+``Normed(a,b)''  
+concept (above)
+is obviously 
+equivalent to the same-named notion
+appearing in
+Definition 4 of \cite{ww5}.
+It uses,
+however,
+ a 
+different notation
+to make it compatible 
+with
+Section \ref{3uuuu2}'s formalism.
+Thus, Item 2's assertion
+that the
+  $\Pi_1^{*}$ axiom 
+$ \, \forall  \,  v_1 \,  \, \forall  \,  v_2 \,  \,  ...
+\forall  v_k   \,  \,  \,  \, \phi(v_1,v_2,...v_k) \,  \, $
+satisfies
+Good($~$Log$_2a~)$  is equivalent to 
+\eq{normscopede}'s statement.
+The
+Good($\,$Log$_2b\,)$ property of 
+$\exists \,v_1\,\exists \,v_2\, ...\,\exists \,v_k\, ~~\phi(v_1,v_2,...v_k)~$
+is,
+likewise,
+ equivalent to \eq{normscopedx}.
+
+\nskip
+
+\beq
+\label{normscopede}
+\forall ~ v_1~ < ~a~~\forall ~ v_2~ < ~a~~ ...
+\forall ~ v_k~ < ~a
+   ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~~.
+\enq
+
+\nskip
+
+\beq
+\label{normscopedx}
+\exists ~ v_1~ < ~b~~\exists ~ v_2~ < ~b~~ ...
+\exists ~ v_k~ < ~b
+ ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~~.
+\enq
+
+\parskip 5pt
+
+\nskip
+
+Our interests in
+this notation
+will center around Fact D.3's invariant:
+
+
+
+{
+%%corbl  \baselineskip = 1.17 \normalbaselineskip 
+
+\medskip
+
+\nop
+
+{\bf Fact D.3$~.$}
+Let
+$ \, \xi^* \, $ 
+denote 
+ Definition D.\ref{D1-def}'s
+generic configuration, and 
+$  Z $ be an extension of $ \, \xi^* \,$'s
+base axiom
+system  $ B^*  $ 
+which
+satisfies Definition D.\ref{D2-def}'s
+Normed$(a,b)$ constraint.
+Then any
+$Z-$Based  deduction tree
+$\, T \, $that
+has a G\"{o}del
+number 
+smaller than
+ $ \, (a/b)^4 \,$
+must 
+contain
+at least one 
+root-to-leaf branch, called $ \, \peta \,  \, , \, $
+that is not
+``closed''. {\rm
+(In other words, this path   $ \, \peta \, $
+will be contradiction-free, insofar as it
+does not contain  both  some
+sentence $~\Psi~$ and its formal
+negation).}
+}
+
+%%notre
+  
+%%notre-only  {\bf Proof:} $~$ The  justification of
+%%notre-only  Fact D.3 is essentially 
+%%notre-only  an
+%%notre-only  %a direct and
+%%notre-only  immediate
+%%notre-only   consequence
+%%notre-only  of the
+%%notre-only  %% no  
+%%notre-only  %% no \newpage
+%%notre-only  %% no \noindent
+%%notre-only  %% no 
+%%notre-only  Lemmas 1 and 2 from \cite{ww5}
+%%notre-only  after one maps these two lemmas's notation into the
+%%notre-only  context of Fact D.3's
+%%notre-only  hypothesis. (We provide a detailed justification
+%%notre-only  of this 
+%%notre-only  technical
+%%notre-only  fact in the Footnote 23 of the slightly longer
+%%notre-only  unabridged version of this article 
+%%notre-only  that
+%%notre-only  resides in the Cornell Archives \cite{ww11}.) 
+  
+
+  
+  \smallskip
+  
+\nskip
+
+%%cccorn-ff
+
+%%cccorn-cc  
+%%cccorn-cc  
+   {\bf Proof:}
+   $~$ The 
+   justification of
+   Fact D.3 is
+% essentially 
+%   an
+   a direct 
+% and    immediate
+    consequence
+   of 
+   %% no  
+   %% no \newpage
+   %% no \noindent
+   %% no 
+the   Lemmas 1 and 2
+appearing in article \cite{ww5} 
+   (see footnote 
+   \footnote{\label{newtry}
+   \baselineskip = 1.3 \normalbaselineskip  
+   A 
+   %formal 
+   proof of Fact D.3 from first principles would
+   be 
+   quite
+   complicated because there are eight elimination rules employed
+   by semantic tableaux deduction, $~$each
+   of which needs to
+   be 
+   % simultaneously 
+   examined by such a proof's umbrella
+   formalism. Fortunately, we do not need provide such a
+   complicated analysis here 
+   because  
+   a 4-page proof of
+   the Lemmas 1 and 2 
+   in Section 5.2
+   of \cite{ww5}
+   had 
+    already
+   visited these issues. 
+   Thus,
+    Fact D.3 
+   turns out to be an easy consequence 
+   of these two lemmas
+   %the Lemmas 1 and 2 of \cite{ww5}
+   after the following two straightforward issues are addressed: 
+   \bee
+   \topsep -4pt
+   \itemsep -2pt
+   \item Section 5.2 of \cite{ww5} 
+   had
+   defined the $\,$``U-Height''$\,$ of a deduction tree to
+   be the 
+   largest
+   number of 
+   U-Grounding function symbols
+   that appear in any of its root-to-leaf
+   branches. Its Lemma 1
+   proved that
+   every deduction tree with a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b \, $
+   will contain at least one branch
+   satisfying a condition,
+   which \cite{ww5}
+    called ``Positive(a,b)''.
+   The Lemma 2 in \cite{ww5} then showed that this
+   Positive(a,b) property  implies that the germane 
+   deduction tree must contain some branch that is 
+   contradiction-free.
+   The combination of these two lemmas
+   thus amounts to the establishing of
+   the following 
+   %slightly 
+   rephrased 
+   hybridized statement:
+   %invariant:
+   \begin{quote}
+   $\bullet~~~$
+   If a Z-based deduction
+   tree  has 
+                   a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b \, $,
+   then some branch of it 
+   is contradiction-free
+   (i.e.  this branch cannot
+   contain  both  some
+   % $\Delta_0^*$
+   sentence $~\Psi~$ and its
+   negation).
+   \end{quote}
+   \item
+   %% 
+   %% Item $\, \bullet \,$
+   %% applies to Fact D.3 's  
+   %% deduction trees
+   %% because 
+   %% 
+   Fact D.3 's   hypothesis indicated the
+   G\"{o}del number $g$ for its deduction tree
+   satisfied the following 
+   conditions:
+   \bed
+   \topsep -7pt
+   \itemsep -1pt
+   \item[   I.   ]
+   $~~g~\leq ~ (a/b)^4~$
+   \item[  II.   ]
+   The   U-Height of $~g \,$'s deduction tree
+   is less than $~\frac{1}{5}~$Log$_2~g~$. (This
+   is
+   simply
+    because
+    Fact D.3 presumes that
+   the ``Conventional Tableaux Encoding'' methodology
+   from Part-iv of 
+   Definition D.\ref{D1-def}
+   was used to encode $~g$'s
+   G\"{o}del number.)
+   \ennd
+   Items
+   I and II imply 
+   $g \,$'s  tree has  
+    a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b  $.
+   The invariant  $\, \bullet \,$ 
+   %then, in turn, implies
+   %that this deduction tree
+   then implies
+   this deduction tree
+   has at least one branch
+   that is contradiction-free (as  Fact D.3
+   claimed). $\Box$
+   \ene
+   We emphasize that the 
+   %The
+   above 
+   % 2-part 
+   justification
+    of Fact D.3 is 
+   {\it much simpler}
+   than a proof from first principles.
+   %
+   %This is  because the 
+   %
+   The
+   latter would
+   require examining 
+    eight different 
+   tableaux 
+   elimination rules,
+   as the detailed
+   proofs of \cite{ww5}'s
+    Lemmas 1 and 2 
+   actually
+   did do.}
+   %
+   %\noindent
+   %-----------------------------------------------------------------}
+for more details).  $~~\Box$
+
+
+%%%% 
+%%%% provides   some
+%%%%    %some
+%%%% %   precise 
+%%%% % details 
+%%%%    details about how 
+%%%%    the   Lemmas 1 and 2
+%%%% of \cite{ww5} do
+%%%%  imply Fact D.3).
+%%%% 
+
+
+\nop
+
+
+
+We will now apply Fact D.3 to prove 
+Theorem D.\ref{D.4-theorx}.
+Its invariant will, 
+interestingly,
+collapse entirely
+\footnote{\h55 $~$The
+difficulty posed by 
+%the 
+multiplication 
+%operation
+can be easily understood when one compares
+two 
+integers sequences  
+   $ x_0, x_1, x_2,  ... $ 
+and  $ y_0, y_1, y_2,  ... $,
+defined 
+as follows:
+\begin{center}
+$x_{i}~~=~~x_{i-1}+x_{i-1}~~~~~$
+$~~$ AND  $~~$ 
+$~~~~~y_{i}~~=~~y_{i-1}*y_{i-1}$
+\end{center}
+It turns out that the faster growth rate of  multiplication
+under
+the series  $~ y_0, y_1, y_2,  ...~ $
+enables
+one to to construct  tiny 
+$Z-$Based  deduction trees
+$\, T$ that
+violate
+the analog of Fact D.3 's paradigm.
+(This is 
+because 
+such trees
+ can have
+G\"{o}del
+numbers 
+smaller than
+ $ \, (a/b)^4 \,$,
+while 
+all their 
+root-to-leaf branches 
+can be simultaneously   
+``closed'' via contradictions.)
+This 
+property of
+            multiplication is 
+analogous
+to Example \ref{ex3-3}'s
+observations about
+          how        the
+differing growth rates of  
+$ x_0, x_1, x_2,  ... $ 
+and  $ y_0, y_1, y_2,  ... $
+are related to the
+threshold where the semantic tableaux version of 
+ Second Incompleteness Theorem can be evaded. \fend }
+%% 
+%% It
+%% explains
+%% intuitively 
+%%  why   Fact D.3 and 
+%% Theorem D.\ref{D.4-theorx} 
+%% use
+%% paradigms
+%% that recognize addition but not multiplication
+%% as total functions
+%% during their evasions of
+%% the semantic tableaux version of
+%% the Second Incompleteness Theorem.  \fend }
+%% 
+ if one were to
+merely  add
+a multiplication function symbol to 
+the
+    U-Grounding language.
+This is why
+our 
+boundary-case exceptions
+to the semantic tableaux version of the Second Incompleteness
+allow 
+a Type-A axiom system 
+to
+recognize addition
+as a total function (but
+suppress a similar 
+ treatment of
+ multiplication).  
+
+
+
+
+\hgskip
+
+
+{\bf Theorem
+D.\ref{D.4-theorx}.}
+{\it The generic configuration $\xi^*$ 
+is both A-stable and
+E-stable.} {\rm (This 
+implies  many different 
+self-justifying formalisms
+exist
+via Theorems 
+\ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5},
+ \ref{ppp6},
+ G.2 and G.3.)}
+
+\cvs
+
+\hgskip
+
+Our proof of
+Theorem D.\ref{D.4-theorx} will 
+separately
+show 
+ $\xi^*$  is 
+A-stable and E-stable. 
+
+\hgskip 
+
+\parskip 3pt
+
+{\bf Proof of 
+ $ \, \xi^* \,$'s A-stability :}
+Suppose for the sake of establishing a
+proof by
+ contradiction
+that  $ \, \xi^* \, $ was not A-stable.
+Then the constraint $ \, * \, $ 
+of Definition \ref{astab}
+would  be violated
+by at least
+%%%%%  bbbbb
+some
+$\theta \, \in \,$RE-Class$(\xi)$.
+This violation
+ will cause the statement $ \, + \, $ to be true
+for 
+such a
+ $\theta~$:
+\begin{description}
+\item[  +   ]
+There exists a
+semantic tableaux
+ proof $~p~$
+of a
+ $~\Pi_1^*~ $ theorem,
+called say 
+$ \, \Uxp ~,~ $ 
+from
+ the
+axiom system of $~\theta \cup B^\xi~$
+such that
+ Log$(p)~\leq \zzz \theta)~+1~ $
+and where 
+ $ \, \Uxp \, $ also  
+fails to
+satisfy 
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\end{description}
+Let us recall that
+if  $\Uxp$ is $\Pi_1^*\,$ then
+Reverse$(\, \Uxp \,)~$ 
+is a 
+$\Sigma_1^*$ sentence equivalent to $~\neg \Uxp~$.
+Thus, 
+Reverse$(\, \Uxp \,)~$ will satisfy
+Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$
+criteria (simply because
+it
+% Reverse$(\, \Uxp \,)~$
+  has the opposite 
+goodness
+property as
+$\,\Uxp \,$ ).
+Also, if
+$~Z~$ 
+denotes the axiom system of
+ $~\theta \cup B^\xi \, + \, $Reverse$(\, \Uxp \,)$,
+it is easy to verify
+\footnote{\g55    \label{jnorm}
+The axiom system
+ $Z$  must satisfy 
+\nor1
+because:
+\bee
+\item
+The
+quantity
+$~2^{ \zzz \theta  \, )}$
+is a valid
+first component
+for $Z$'s   norming constraint 
+because
+all the axioms of $~B^\xi~$ are 
+true in the Standard-M model
+and
+because
+Definition \ref{chg}
+implies
+all of
+$ \theta \,$'s axioms satisfy 
+Good $\{\,\zzz \theta) \,\}$.
+\item
+The
+quantity      $ ~\sqrt{~2^{ \zzz \theta \,   )}} ~$
+is a valid
+second component
+for $Z$'s   norming constraint 
+because
+ Reverse$(\, \Uxp \,)~$ 
+is the only $\Sigma_1^*$ sentence belonging to
+Z, and  because  Reverse$(\, \Uxp \,)~$ 
+satisfies 
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\ene}
+that $~Z\,$'s axioms 
+will
+satisfy the 
+\nor1
+criteria.
+
+
+
+\smallskip
+
+It is next helpful to  observe that what is
+a proof from one perspective corresponds to
+being a
+deduction tree from 
+a different
+perspective.
+Thus, Item $~\, + \,$'s proof
+ $~p~$
+of the 
+theorem
+$ \, \Uxp \,$ 
+from
+ the
+axiom system of $~\theta \cup B^\xi~$
+corresponds to being  a Z-based deduction tree,
+with $~Z~$ representing the
+axiom system of
+$~\theta \cup B^\xi \, + \, $Reverse$(\, \Uxp \,)$.
+In this context,
+Item  $\, + \,$'s 
+inequality of 
+ Log$(p)~\leq \zzz \theta  \, )~+1~ $
+implies 
+\footnote{ \g55 \label{fd7} 
+Without loss of generality, we 
+may assume that 
+every non-trivial proof $~p~$ satisfies
+  Log$(p)~\geq ~64~$ (since a 
+string with fewer than 64 bits is too short to be a
+proof).
+Then the footnoted paragraph's 
+ Log$(p)~\leq \zzz \theta)~+1~ $ inequality 
+trivially implies 
+$~p~<~3^{ \zzz \theta  \, )}~$.
+In a context where 
+ $~Z~$ 
+is a
+\nor1
+axiom system,
+the
+ latter inequality 
+certainly
+implies
+$~p~$, viewed as a deduction tree for $~Z~,~$
+has a small enough G\"{o}del number to
+ satisfy the
+hypothesis for Fact D.3. 
+(This is because if one sets
+$~a~=~2^{ \zzz \theta  \, )} ~$ 
+and 
+$~b~=~\sqrt{2^{\zzz \theta  \, )}} ~$ 
+then obviously 
+$ ~~p~<~3^{ \zzz \theta  \, )}~ <~4^{ \zzz \theta  \, )}~ =~
+(a/b)^4~~~).$  \fend } 
+$\,$that 
+$~p~$, viewed as a deduction tree for $~Z~,~$
+ satisfies the
+hypothesis of Fact D.3 . 
+Hence, Fact D.3 establishes
+that $~p~$ must contain
+at least
+ one contradiction-free
+root-to-leaf branch.
+
+\smallskip
+
+
+This last observation 
+is all that is needed
+to confirm
+ $~\xi^* \,$'s A-stability,
+via a proof-by-contradiction.
+This is because the 
+definition of a semantic tableaux proof implies
+every 
+one of its
+root-to-leaf branches 
+must
+end with a pair of
+contradicting nodes.
+However, the last paragraph showed 
+$~p~$
+will not satisfy this
+required
+ property,
+if $~\xi^*~$ is not A-stable.
+Hence 
+our construction has proven the A-stability of
+ $~\xi^*~$ by showing that otherwise an infeasible
+circumstance will arise.
+ $~~\Box$ 
+
+
+
+
+
+\hgskip
+
+{\bf  Proof of 
+ $  \xi^*  $'s E-stability :}
+A
+proof-by-contradiction
+will verify
+ $  \xi^*  $ is
+E-stable, analogous
+ to  the
+ proof of
+its
+    A-stability.
+Thus if
+ $  \xi^*  $ was not
+E-stable, then
+statement $++$ would be true
+for some  $\theta$. (This is because 
+at least one
+$\theta \, \in \,$RE-Class$(\xi)$
+would then
+ violate
+Definition \ref{estab}'s
+requirement of  $~**~~$.)
+\begin{description}
+\item[ ++   ]
+There exists a
+semantic tableaux
+ proof $~p~$
+of a $\Sigma_1^\xi$ theorem 
+$ \, \Uxp \, $ from the
+axiom 
+system $\theta \cup B^\xi$
+such that
+ Log$(p) \,\leq \zzz \theta) +1 $
+and  
+ $  \Uxp  $  also
+fails to
+satisfy 
+Good$\{  \tftt  \zzz \theta) \,\}.$
+\end{description}
+Item $++  $ 
+ implies 
+Reverse$( \Uxp )$ 
+satisfies
+ Good$\{  \tftt  \zzz \theta) \}$
+(because Reverse$( \Uxp )$ again
+   has
+ the opposite 
+goodness property
+as 
+$ \Uxp $ ).
+Let
+$Z$ now
+denote the formal axiom system of  
+ $\, \theta \cup B^\xi  \, +  \, $Reverse$( \Uxp )$.
+The footnote \footnote{ \sm55   
+The axiom system
+ $~Z~$ must satisfy 
+\xor2
+because:
+\bee
+\item The first component of its norming constraint 
+can be set equal to
+ $ ~\sqrt{~2^{ \zzz \theta   \, )}} ~$
+because Reverse$(\, \Uxp \,)~$ 
+is a 
+ Good$\{~ \, \tftt  \zzz \theta  \, ) \, ~\}~$
+$\Pi_1^*$ sentence, and all $~Z\,$'s other 
+$\Pi_1^*$ sentences satisfy more relaxed constraints.
+\item The second component of $Z$'s norming constraint 
+is satisfied by the constant of
+2 because Definition D.\ref{D2-def} implies  this 
+quantity is always permissible
+when $~Z~$ 
+contains no $\Sigma_1^*$ axiom sentences.
+\ene   \fend }
+then
+   uses
+reasoning 
+similar
+to footnote    \ref{jnorm} 
+to
+show 
+$Z$ 
+satisfies 
+\xor2
+
+\parskip 3pt
+
+\smallskip
+
+As before 
+via a simple change in notation, 
+ $~p\,$'s
+semantic tableaux proof of
+$ \, \Uxp \,$ 
+can be viewed 
+as 
+ a deduction tree
+using $~Z\,$'s 
+axioms.
+Also as before, we may 
+use the combination of the facts that 
+ $~Z~$ is a
+\xor2 system
+and that  Item ++ 
+indicated 
+ Log$(p) \,\leq \zzz \theta) +1 $ to 
+deduce\footnote{\sm55 The proof that 
+ $~p~$ is
+ small enough to satisfy 
+Fact D.3 's
+hypothesis in the current E-stable case 
+is almost
+identical to Footnote  \ref{fd7}'s analysis of
+the A-stable case. 
+Thus as in the earlier case,
+Item ++'s inequality of
+ Log$(p)~\leq \zzz \theta)~+1~ $ 
+trivially implies 
+$~p~<~3^{ \zzz \theta  \, )}~$.
+Also, we may again assume that
+  Log$(p)~\geq ~64~$ (since a sequence with fewer than 64 
+bits cannot amount to a proof of any interesting fact under
+all normal coding conventions). 
+An analog of Footnote  \ref{fd7}'s chain of inequalities
+will then allow us to conclude that 
+$~p~$
+is
+ small enough 
+proof from a 
+\xor2
+system
+to
+ satisfy the
+hypothesis for 
+Fact D.3.  \fend }  that
+ $~p~$ is small enough to satisfy 
+Fact D.3 's
+hypothesis.
+ Hence once again,
+Fact D.3  implies that
+$~Z~$ must contain
+at least
+ one contradiction-free
+root-to-leaf branch.
+As before, the existence 
+ of this
+contradiction-free
+path 
+violates the definition of a semantic tableaux proof
+and 
+enables
+ our proof-by-contradiction to
+reach its
+desired end. $~\Box$ 
+
+\hgskip
+
+
+
+{\bf Remark D.5} {\it  (about 
+Theorem  D.\ref{D.4-theorx}'s
+significance) :}
+Part-ii of
+ Definition
+D.\ref{D1-def}
+indicated  $~\xi^* \,$'s base axiom of
+$~B^*~$ was a Type-A formalism that recognized addition
+as a total function. This is significant because
+ \cite{ww0,ww2,ww7,ww9} 
+showed
+nearly all Type-M formalisms, including 
+all the common axiomatizations for
+I$\Sigma_0$,
+are unable to recognize their 
+semantic tableaux consistency.
+Thus, the declaration that multiplication is a total
+function
+is {\it the trigger-point} causing
+\footnote{\sm55 
+We formally proved in \cite{ww0,ww2,ww7,ww9} 
+that
+multiplication's totality property
+causes
+ the semantic tableaux version of the Second
+Incompleteness Theorem to become active.
+The  
+ Example \ref{ex3-3} 
+summarizes the main
+%%%  
+%%%  offers a 
+%%%  % nice brief 
+%%%  summary of the
+%%%  % underlying
+%%%  
+ intuition behind these
+results.}
+the semantic tableaux version of the
+Second Incompleteness Theorem
+to become active. 
+This threshold effect is 
+significant
+% quite tight 
+because
+Theorem D.4, combined with
+Theorems 
+ \ref{pqq4}, \ref{pqq5}, G.2 and G.3,
+formalize       {\it four different respects} in which 
+Type-A
+self-justifying 
+formalisms can
+prove all 
+Peano Arithmetic's
+ $\Pi_1^*$ theorems
+{\it ( after} multiplication's totality axiom is suppressed).
+
+\cvt
+
+
+
+\subsection*{D-3. $~$Three Further Examples of
+Stable Generic Configurations}
+
+
+
+
+Our second 
+example
+of an
+ EA-stable
+        configuration
+is
+called $~\xi^{**}~$. 
+It will be identical to 
+ $~\xi^*~$ except that it will replace
+semantic tableaux with a stronger
+deduction method,
+which 
+\cite{ww5} 
+    called  Tab$-U_1^*$.
+The latter 
+is          a
+revised version of 
+semantic tableaux
+that permits
+ a modus ponens
+rule to perform
+deductive cut 
+ operations on
+$\Pi_1^*$ 
+and $\Sigma_1^*$ sentences.
+(The formal
+definition of 
+ Tab$-U_1^*$ 
+deduction 
+had appeared in \cite{ww5}.
+It will  be
+unnecessary to
+ repeat here.)
+
+
+
+
+The Section 5.3 of 
+\cite{ww5} 
+noted
+Tab$-U_1^*$ 
+has
+ similar self-justification properties
+as conventional semantic tableaux.
+%It is thus unnecessary to discuss
+%Tab$-U_1^*$ in detail here.
+All the results that
+Section D-2 
+proved about
+ $~\xi^{*}~$
+% ,
+% however,
+apply also  to  $~\xi^{**}~$,
+via their natural generalization
+under
+\cite{ww5}'s
+Tab$-U_1^*$
+deduction
+method.
+Thus,
+ $~\xi^{**}~$
+is also EA-stable.
+
+
+%%%%% 
+%%%%% 
+%%%%% Our second 
+%%%%% example
+%%%%% of an
+%%%%%  EA-stable
+%%%%%         configuration
+%%%%% is
+%%%%% called $~\xi^{**}~$. 
+%%%%% It will be identical to 
+%%%%%  $~\xi^*~$ except that it will replace
+%%%%% semantic tableaux with a stronger
+%%%%% deduction method,
+%%%%% which 
+%%%%% \cite{ww5} 
+%%%%%     called  Tab$-U_1^*$.
+%%%%% The latter 
+%%%%% is          a
+%%%%% revised version of 
+%%%%% semantic tableaux
+%%%%% that permits
+%%%%%  a modus ponens
+%%%%% rule to perform
+%%%%% deductive cut 
+%%%%%  operations on
+%%%%% $\Pi_1^*$ 
+%%%%% and $\Sigma_1^*$ sentences.
+%%%%% (The 
+%%%%% definition of 
+%%%%%  Tab$-U_1^*$ 
+%%%%% deduction 
+%%%%% appeared in \cite{ww5}.
+%%%%% It
+%%%%% is unimportant to
+%%%%%  repeat here.)
+%%%%% 
+%%%%% 
+%%%%% 
+%%%%% 
+%%%%% The Section 5.3 of 
+%%%%% \cite{ww5} 
+%%%%% noted
+%%%%% Tab$-U_1^*$ 
+%%%%% has
+%%%%%  similar self-justification properties
+%%%%% as conventional semantic tableaux.
+%%%%% It is thus unnecessary to discuss
+%%%%% Tab$-U_1^*$ in detail here.
+%%%%% %%  
+%%%%% %% 
+%%%%% %% We will not
+%%%%% %%  go  into the full details
+%%%%% %% again,
+%%%%% %% for the sake of brevity.
+%%%%% %% 
+%%%%% %% 
+%%%%% All the results that
+%%%%% Section D-2 had
+%%%%% proved about
+%%%%%  $~\xi^{*}~$
+%%%%% % ,
+%%%%% % however,
+%%%%% apply also  to  $~\xi^{**}~$,
+%%%%% via their natural generalization
+%%%%% under
+%%%%% \cite{ww5}'s
+%%%%% Tab$-U_1^*$
+%%%%% deduction
+%%%%% method.
+%%%%% Thus,
+%%%%%  $~\xi^{**}~$
+%%%%% is also EA-stable.
+
+
+ 
+A key point is that 
+ there is a
+non-trivial
+distinction between  
+$  \xi^{*}  $ 
+and  $  \xi^{**}  $, despite
+the fact that they
+have 
+similar technical qualities.
+This is because 
+ $  \xi^{**}  $ contains  a
+Level-1
+modus ponens rule
+(unlike $  \xi^{*}  $ ).
+ If 
+it were infeasible to expand  $  \xi^{*}  $ 
+into 
+a broader 
+ $  \xi^{**}  $, $  $then both formalisms could,
+perhaps,
+ be easily
+dismissed as having 
+negligible pragmatic 
+significance (since
+modus ponens is central
+to
+cogitation). 
+However in a context where  $  \xi^{**}  $ does permit
+a  Level-1
+modus ponens rule, 
+it is a tempting formalism
+(despite its limited modus ponens rule).
+
+
+
+
+Unlike
+ $~\xi^{*}~$ 
+and  $~\xi^{**}~$,
+our third 
+example of an EA-stable
+configuration,
+called $~\xi^{-}~$,
+will support            an unlimited modus ponens
+rule.
+This will be possible because 
+ $~\xi^-\,$'s
+language of $~L^-~$ 
+will 
+be weaker than the languages of
+ $~\xi^{*}~$ 
+and  $~\xi^{**}~$.
+Thus $~L^-$  will
+include 
+ the six Grounding functions, but
+not the Growth functions of addition and doubling.
+It will thus treat addition and multiplication as
+3-way atomic predicates,
+ Add$(x,y,z)$ 
+and Mult$(x,y,z)$, rather than as
+formal functions.
+
+
+\cvt
+\parskip 2pt
+
+
+
+This
+perspective
+enabled  $~\xi^-~$   to
+support an evasion of 
+the Second Incompleteness Theorem with
+an unlimited  modus ponens rule
+present,
+in a context where
+the other four parts of 
+its
+generic configuration are defined below:
+\bee
+\item
+The 
+  $~\Delta_0^{-}~$
+class for $~\xi^-~$
+will 
+be built in an
+essentially natural
+ manner from
+the Grounding function
+set.
+It will 
+thus 
+include
+all formulae in $~L^- \,$'s language, 
+whose 
+quantifiers
+are bounded
+ in any arbitrary manner 
+using
+the Grounding function
+primitives.
+\item
+The
+base axiom system
+$~B^-~$ of $~\xi^-~$
+ will
+employ  an infinite number of constant symbols,
+denoted as 
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+where 
+ $K_1=1$ 
+and where 
+ $K_{i+1}$ is
+a power of 2
+ defined by the axiom of:
+\beq
+\label{addc}
+\mbox{Add}(~K_{i}~,~K_{i}~,~K_{i+1}~)
+\enq
+Thus, the combination of 
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+with the
+Grounding function  
+of subtraction allows
+the language  $L^-$
+to encode the value of any
+arbitrary natural
+number (as 
+Part 1 
+of Definition \ref{def3.3} 
+had required).
+Essentially,
+$~\xi^- \,$'s 
+base axiom system
+of $~B^-~$ 
+can
+be any 
+consistent
+r.e. set of
+$~\Pi_1^-~$ sentences 
+that includes \eq{addc}'s axiom schema
+and is
+able to
+prove every 
+ $\Delta_0^{-}~$ sentence which is valid
+in the Standard-M model.
+\item
+$~\xi^- \, $'s
+deduction method can be any version of a classic
+Hilbert-style proof methodology. (Thus, it  will  include
+a modus ponens rule with no restrictions.)
+\item
+$~\xi^- \, $'s
+G\"{o}delization method can be essentially
+any natural technique.
+\ene
+An interesting aspect of  $\xi^-$
+is it can be 
+proven to be
+EA-stable
+via an analog of
+Section D-2's treatment of  $\xi^*$.
+Thus,
+Theorem \ref{pqq4}
+implies
+every axiom system
+$\alpha$,
+whose
+ $\Pi_1^-$ theorems 
+hold true
+in the Standard-M model,
+can be mapped
+onto an extension of  $\xi^-\,$'s base
+axiom system
+that can 
+ recognize
+its own 
+Hilbert
+ consistency 
+and
+prove 
+$\alpha $'s  $\Pi_1^-$ theorems.
+Except for
+minor changes in notation,
+this  
+result
+represents a new way of proving
+\cite{wwapal}'s Theorem$\, 3.~$
+
+
+
+
+
+
+\medskip
+
+
+
+The self-justifying features
+of
+ $ \xi^{*} $,  $ \xi^{**} $  and  $ \xi^{-} $
+are of interest
+primarily 
+ because
+the 
+Second 
+Incompleteness Theorem implies 
+that they
+cannot be 
+improved
+much 
+further.
+This tight fit is
+summarized by  Items 1-4.
+\begin{enumerate}
+\topsep -15pt
+\item 
+The Theorem 2.1 
+(due to  the combined work of 
+Nelson,
+Pudl\'{a}k,
+Solovay and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} )
+implies no natural axiom system can
+prove Successor is a total function and recognize its own
+Hilbert consistency. This theorem 
+thus explains why
+the presence of
+growth functions 
+must be omitted from
+ $\xi^{-}\,$'s
+base axiom system of $B^-$.
+\item
+Moreover, \cite{wwapal}
+  proved 
+$~\xi^{-} \,$'s method 
+for evading  
+the Second Incompleteness
+Theorem
+will 
+collapse
+if one replaces  
+\ep{addc}'s  ``addition-based named sequence'' of
+constant symbols
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+with a faster 
+growing
+``multiplicative 
+convention'',
+where 
+the
+constant symbols 
+$    C_1   ,   C_2   ,   C_3   , \,   ...   $
+are formally defined via 
+\eq{multc}'s
+ schema.
+\beq
+\label{multc}
+\mbox{Mult}(~C_{i}~,~C_{i}~,~C_{i+1}~)
+\enq
+Thus,  \cite{wwapal} showed that there
+exists a 
+ $\Pi_1^{-}$ sentence $ \,W \,$ 
+(provable from Peano Arithmetic)
+such that no consistent  system can
+simultaneously prove  $W$,
+contain \eq{multc}'s axiom schema and prove the non-existence
+of proof of $0=1$ from itself.
+There is no space to prove it here, but 
+a generalization of the Second Incompleteness
+Theorem implies  the modification
+of $\xi^-$ that replaces
+\eq{addc}'s  axiom schema with 
+\eq{multc}'s schema 
+{\it is not even 0-stable.}
+\item
+Similarly,
+\cite{ww2,ww7} proved that if 
+ $ \, \xi^{*} \, $'s and  $ \, \xi^{**} \, $'s 
+ base 
+ axiom system of 
+$ \, B^* \, $ 
+was strengthened
+to include the assumption that
+multiplication was a total function then \cite{ww5}'s
+two
+semantic tableaux  evasions of
+the Second Incompleteness Theorem would 
+both collapse.
+\item
+Also, \cite{wwlogos} 
+proved
+that  an analog of $\xi^{**} \,$'s
+evasion of the Second Incompleteness Theorem
+will collapse if its 
+modus ponens rule was expanded to apply to
+either $\Pi_2^*$ or $\Sigma_2^*$ sentences.
+\ene
+The Item 3
+is especially interesting because
+\cite{ww6} 
+proved \cite{ww5}'s evasion of the Second Incompleteness
+Theorem 
+was compatible with its formalism recognizing
+an infinitized generalization of a
+computer's floating point multiplication
+as a total function.
+Thus
+ while the semantic tableaux formalisms
+of 
+ $\xi^{*}$ or  $\xi^{**}$ 
+are provably unable \cite{ww2,ww7} to
+ recognize integer
+multiplication as a total function,
+their relationship to floating point multiplication
+is 
+more subtle.
+
+
+
+
+
+
+\smallskip
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Our fourth example of an
+application of Section \ref{sect64}'s theorems
+was stimulated by
+some
+insightful 
+email  we
+received from
+L. A. Ko{\l}odziejczyk
+\cite{Ko5} in 2005.
+It noted
+there existed a
+potential
+ exponential gap between the lengths
+of semantic tableaux and Herbrand-style proofs 
+under some 
+circumstances.
+Our earlier research \cite{ww2} 
+addressed a 
+1981 Paris-Wilkie open question \cite{PW81} 
+by
+generalizing some 
+Adamowicz-Zbierski techniques 
+\cite{Ad2,AZ1} 
+to show
+a natural  axiomatization of I$\Sigma_0$ 
+ satisfied the semantic tableaux version of the Second
+Incompleteness Theorem.
+In this context,
+Ko{\l}odziejczyk
+asked whether this would
+apply to       all
+plausible axiomatizations for I$\Sigma_0$ ?
+
+
+
+
+
+
+
+We replied in
+\cite{ww9}
+to
+Ko{\l}odziejczyk's
+stimulating question
+ by 
+distinguishing between Example \ref{ex3-1}'s 
+$\Delta_0^A$ 
+and $\Delta_0^R$  formulae
+and by using
+the Paris-Dimitracopoulos \cite{PD82}
+translation algorithm for 
+$\Delta_0$ formulae. (The latter procedure was summarized
+earlier by
+ Lemma \ref{lex22}. It
+demonstrated
+ how to
+ map
+classic arithmetic's
+$\Delta_0^A$ formulae
+onto  equivalent   
+$\Delta_0^R$  formulae
+in  the Standard-M
+model.) Our reply to 
+Ko{\l}odziejczyk's
+question, 
+thus,
+employed this
+translation methodology to show that there existed an axiom system,
+called Ax-3, which proved the identical set of theorems as
+the more common Ax-1 and Ax-2 encodings of 
+ I$\Sigma_0$ and which possessed the following 
+pair of 
+quite
+fascinating 
+  contrasting properties:
+\bed
+\topsep -3pt
+\item[   A  ]
+No
+consistent
+ superset $~\beta~$ of Ax-3's set of axioms is capable of
+proving its
+own
+ semantic tableaux consistency \cite{ww9}.
+\item[   B  ]
+In contrast,  
+if  ``Herb'' denotes
+the next paragraph's
+ Herbrand-styled deduction
+and if ``SelfRef'' denotes 
+the sentence $~\bullet~$ from
+Section \ref{secc1},
+then 
+Ax3$\, + \,$SelfRef(Ax-3,Herb) will
+be a self-justifying axiom system.
+%%% 
+%%% 
+%%% In contrast,  
+%%% if  ``Herb'' denotes
+%%% the next paragraph's
+%%%  Herbrand-styled deduction
+%%% method
+%%% then 
+%%% Ax3$\, + \,$SelfRef(Ax-3,Herb) will
+%%% satisfy both Parts (i) and (ii) of 
+%%% Section \ref{secc1}'s 
+%%%  definition of
+%%% self-justification.
+%%% 
+%%% 
+\ennd
+
+The intuition behind \cite{ww9}'s proof of Items A and B can be 
+easily summarized if 
+we define a
+ ``Herbrandized-style''  proof of a 
+theorem $ \,\Phi \,$ from an axiom system $ \, \alpha \,$
+as being an essentially 
+2-part
+structure where:
+\bee
+\topsep -3pt
+\item Each  of $\alpha$'s axioms and
+also the sentence $~ \neg \Phi~$ are
+first written as  Skolemized
+expressions.
+\item
+A propositional calculus proof 
+is then used to show
+ that some formal conjunction
+of instances of Item 1's Skolemization schema has no satisfying
+truth assignment.
+\ene
+Such a
+ formalism is 
+different from the definition
+of a semantic tableaux proof (appearing in for example Fitting's
+textbook \cite{Fi90} ).
+This is because the latter replaces the use
+of Skolemization in Items 1 and 2 with an existential quantifier
+elimination rule. It turns out that this distinction enables
+some semantic tableaux proofs to be exponentially more compressed
+than their Herbrandized counterparts,
+as Ko{\l}odziejczyk observed 
+\cite{Ko5,Ko6}. This fact
+ enabled \cite{ww9}
+to prove that Herbrandized and semantic tableaux proofs
+have the divergent properties summarized by Items A and B.
+
+
+One reason
+Ax-3's evasion of the Second Incompleteness Theorem
+is of interest is that I$\Sigma_0$ supports many more generalizations
+of the Second Incompleteness Theorem
+than evasions 
+of it.
+ Thus,
+Willard  \cite{ww2,ww7,ww9}
+proved that the semantic tableaux version of
+the Second Incompleteness Theorem 
+was valid for three different encodings of
+I$\Sigma_0$, and
+Adamowicz, Salehi and  Zbierski
+have discussed in great detail
+\cite{Ad2,AZ1,Sa11}
+ various Herbrandized generalizations
+of the Second Incompleteness Theorem for
+particular encodings of  
+I$\Sigma_0$ 
+and I$\Sigma_0+\Omega_i$. 
+Moreover, an added 
+facet of \cite{ww9}'s 
+Ax-3 encoding for I$\Sigma_0$ is that most automated
+theorem provers use a 
+particular
+variant of the Resolution method
+that causes \cite{ww9}'s 
+unusual 
+methodology
+to apply
+also to them \footnote{
+\b55 
+The main theorems in \cite{ww9} generalize for resolution because
+Resolution-based theorem provers
+employ skolemization analogously to Herbrand deduction. \fend }.
+
+
+The reason for our 
+%particular 
+interest in
+\cite{ww9}'s results 
+% in the current article
+is that it represents a fourth example 
+%% of
+where the 
+ meta-theorems from Sections 
+\ref{3uuuu3} and  \ref{sect64} 
+can be useful.
+Thus,
+the footnote
+\footnote{\b55  The 
+discussion in \cite{ww9}
+ did not
+technically  
+use
+ Definition  \ref{estab}'s machinery
+to establish 
+%that 
+there existed an extension of
+its ``Ax-3'' encoding for 
+I$\Sigma_0$ that could recognize its own Herbrand consistency.
+Its formalism,
+however,
+ could be
+easily couched in terms of 
+ Definition  \ref{estab}'s machinery, if one uses a 
+generic 
+configuration $~\xi^R~$ where
+\bee
+\item  $~\xi^R\,$'s 
+base language is the same as the usual language of
+arithmetic, 
+\item  $~\xi^R\,$'s 
+$~\Delta_0^R~$ sub-class is defined by
+Item (b) in Example \ref{ex3-1}, 
+\item  $~\xi^R\,$'s 
+base axiom system 
+is \cite{ww9}'s ``Ax-3'' system, 
+\item  $~\xi^R\,$'s 
+deduction method
+is either a Herbrandized styled-method
+or a Resolution system that relies upon
+Skolemizatin in a similar manner.
+\item  $~\xi^R\,$'s G\"{o}del encoding scheme
+may be any such 
+natural method.
+\ene
+This approach 
+supports a 
+% somewhat
+  stronger form of
+self-justification result
+than
+had 
+appeared in \cite{ww9}.
+This is
+because $~\xi^R~$ can be proven to be E-stable (by a 
+% straightforward 
+generalization of \cite{ww9}'s analysis
+techniques). 
+Thus,
+ \phx{ppp2} 
+implies
+that
+Ax-3 has a well-defined self-justifying extension
+that
+can  recognizes its own
+formalized
+ Level$(0^R)$ consistency.
+(This 
+self-justification result
+is  stronger than
+\cite{ww9}'s main theorem.
+The latter
+merely established
+that some extension of Ax-3
+recognized the non-existence of a
+Herbrandized deduction of $0=1$ from itself.) \fend }
+summarizes how  a fourth 
+type of
+generic configuration,
+called $~\xi^R~$,
+can be defined
+ that both duplicates
+\cite{ww9}'s
+main
+ self-justification results 
+under the above definition of
+Herb-deduction,
+as well as  strengthens
+them. (In particular, 
+  $~\xi^R~$ meets
+Theorem \ref{ppp2}'s
+requirements,
+and  self-justifying 
+extensions of its Ax-3 system thus
+recognize their 
+%own
+ Level$(0^R)$ consistency.)
+
+
+
+
+\parskip 2pt
+
+
+
+
+
+The properties of our
+four generic configurations of $~\xi^R$,
+  $~\xi^*~$,
+ $~\xi^{**}~$ and $~\xi^-~$ are  summarized by Table I.
+These configurations are
+listed in ascending order according to the strength
+of their deduction methods $~d~$. As their deduction methods
+increase in strength, 
+these configurations 
+% the associated configurations 
+have
+their ability
+reduced
+%weakened
+ to recognize 
+the totality of 
+the addition and multiplication operations.
+%% 
+%% 
+%% (This is because the corresponding generic configurations will
+%% violate the requirements of the Second Incompleteness Theorem,
+%% if they simultaneously use too strong a deduction method
+%% and possess too strong an understanding of their
+%% own consistency.)
+
+
+
+
+
+$~\xi^R\,$ is thus a Type
+Almost-M 
+system
+that
+can prove multiplication is a total function
+(but 
+which
+does not contain 
+\ep{totdefsymbm}'s
+ totality statement {\it as an axiom)}.
+On the other hand, $~\xi^-~$ 
+uses a stronger Hilbert-styled
+ deduction methodology,
+which is  incompatible with treating
+the totality of addition or multiplication as either
+axioms {\it or as derived theorems.} 
+%% (This incompatibility is unavoidable because the
+%% Theorem 2.1, due to the joint work of
+%% Nelson,
+%% Pudl\'{a}k,
+%% Solovay and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87}, 
+%% implies that self-justifying systems cannot simultaneously
+%% recognize successor as a total function and prove their
+%% own Hilbert consistency.)
+
+
+
+
+
+
+Each of Table I's
+rows
+ $ \, \xi^R$,
+ $ \, \xi^{**} \, $ and $ \, \xi^- \, $  
+are  maximal (in that an alternate row
+improves upon one   column's measurement 
+only when it is 
+weaker from the perspective of
+another 
+column).
+Only  $ \, \xi^* \, $ 
+is
+an exception to this rule: It is
+strictly
+weaker
+\footnote{\b55 \label{f28} $  \xi^{**}  $ employs a stronger
+deduction method 
+than   $  \xi^*  $
+because it allows a modus ponens rule for $\Pi_1^*$ and
+$\Sigma_1^*$ sentences to be added to semantic tableaux 
+deduction
+(see \cite{ww5} for the precise definition of this  
+``Tab$-U_1^*~$''
+ modification of the  semantic tableaux 
+deductive method).  \fend }
+     than
+     $ \,  \xi^{**}  $. 
+    This appendix has discussed 
+      $ \, \xi^* \, $ 
+%% despite its sub-optimality, 
+%%    %primarily 
+%%    partly 
+because it makes
+    Theorem D.\ref{D.4-theorx}'s proof simpler
+    (and also because 
+     semantic tableaux 
+    % deduction 
+    is a 
+    % very
+    frequent topic in the logic literature).
+
+    %%%footnnnnnnnnn 
+
+
+
+%\smallskip
+
+\begin{center}
+{\bf Table I}
+\end{center}
+
+
+\bigskip
+
+\small
+\baselineskip = 1.1 \normalbaselineskip 
+\noindent
+\begin{tabular}{|c|c|c|c|c|c|c|} \hline
+Name & Deduction Method  & Type & Almost   & Type  & {\bf Axiom} 
+& {Self-Just }  \\
+ &     & {\bf A} 
+& {\bf M} & {\bf M} &  { Format} &  { Level}
+\\ \hline\hline 
+ &  Resolution and/or  & &  &  &  & \\
+ $\xi^R$ & Herbrandized analogs & Yes$^{35}$
+  & Yes  & No &  E-stable &Level $(0^R)$ \\ \hline
+ &  & &  &  &  & \\
+ $\xi^*$ & Semantic Tableaux & Yes  & No  & No & EA-stable & Level $(1^*)$  \\ \hline
+ &  & &  &  &  & \\
+ $\xi^{**}$ & Tab$-U_1^*$ Deduction$\, ^{34}$ & Yes  & No  & No &  EA-stable &Level $(1^*)$   \\ \hline
+ &  & &  &  &  & \\
+ $\xi^{-}$ & Hilbert Deduction & No  & No  & No &  EA-stable &
+ Level $(\, \infty^{-} \,)$   \\ \hline
+\end{tabular}
+
+%\njp
+
+\bigskip
+\normalsize
+\cvl
+
+
+%\cvt
+
+\parskip 0pt
+
+%\nskip
+%\nskip
+
+\bigskip
+The footnote
+\footnote{\b55    For the sake of simplicity,
+    the Ax-3 system of \cite{ww9}  did not
+    use either 
+    Equations
+    \eq{totdefxa} or  
+    \eq{totdefsymba}'s
+  as axiom statements
+(since  they were
+provable as  theorems).
+ All
+    \cite{ww9}'s 
+     results 
+    do,
+however,  generalize
+%    readily
+    when 
+\eq{totdefxa}'s
+statement  about
+addition's totality 
+is included as 
+%added 
+an axiom.
+Thus, it is appropriate to attach the designation
+of ``Yes'' with a caveat to  the ``Type-A'' entry
+in Table I's first row. (This row is called 
+``Resolution and/or Herbrandized analogs'' 
+because it applies
+to 
+essentially 
+any deduction scheme that relies upon Skolemization
+as an alternative to  \cite{Fi90}'s
+semantic tableaux
+existential quantifier elimination rule.)} ,
+ attached to Table I's first row,
+explains why  a caveat is attached to its first
+``Yes'' entry. 
+The theme of Table I is 
+%thus 
+that self-justifying
+axiom systems have some nice redeeming features,
+although
+the
+ Second
+Incompleteness Theorem
+clearly also
+imposes
+severe
+ limits on their abilities.
+This point  will be reinforced 
+when
+Appendix E introduces a generalization of the
+Second Incompleteness Theorem,
+that shows 
+\thx{ppp6}'s translational reflection principle is close
+to being a maximal feasible result, and when Appendix F 
+discusses the epistemological  significance
+of self justification.
+
+
+%% 
+%% the next two sections discuss
+%% hybridizations of
+%% Theorems  \ref{pqq4}
+%% and \ref{pqq5}
+%% and the epistemological  implications
+%% of self justifying systems.
+
+
+
+%% 
+%% an alternate system
+%% and when 
+%% Appendix F discusses the
+%% epistemological  implications
+%% of self justification.
+
+
+%% 
+%% the next two sections discuss
+%% hybridizations of
+%% Theorems  \ref{pqq4}
+%% and \ref{pqq5}
+%% and the epistemological  implications
+%% of self justifying systems.
+
+\nskip
+\nskip
+
+
+%% 
+%% an alternate system
+%% and when 
+%% Appendix F discusses the
+%% epistemological  implications
+%% of self justification.
+
+
+\nop
+
+\section*{Appendix E: A Clarification of
+Theorem \ref{ppp6}'s Significance}
+
+% An Anti-Reflection Principle for
+%\LARGE
+% \baselineskip = 1.8 \normalbaselineskip 
+
+
+It has been known since the time of G\"{o}del that
+most conventional 
+arithmetic 
+axiom systems will satisfy the following two 
+invariants:
+\bee
+\item
+They are {\it physically unable} to prove
+their own consistency
+\item
+They are  $\Sigma_1$ 
+complete. This means they can
+formally prove any  $\Sigma_1$
+arithmetic sentence that holds
+true in the Standard-M model, and
+they can likewise refute any
+$\Pi_1^\xi$ sentence 
+that is false.
+\ene
+Let $~\xi~$ denote any generic configuration of the
+form \gggcp . This appendix will use the
+term {\bf $\xi -$Conventional} to describe any axiom
+system that satisfies analogs of the
+preceding 
+%two 
+conditions for generic configurations.
+Thus
+$~\alpha~$ is
+$\xi -$Conventional 
+iff it satisfies the following two
+criteria:
+\bed
+\item[ a.  ]
+The axiom system $~\alpha~$
+will be {\it  unable} to verify
+its own consistency under $~\xi\,$'s
+deduction
+    method  of $~d~$.
+\item[ b.  ]
+The axiom system $~\alpha~$
+will be an extension of $~\xi \,$'s base axiom of $~B^\xi~$.
+Part-3 of Definition \ref{def3.3} will thus imply it
+is 
+$\Sigma^\xi_1$ 
+complete. (Hence, $~\alpha~$ can
+formally prove any  $\Sigma^\xi_1$ sentence that holds
+true in the Standard-M model, and
+it can likewise refute any
+$\Pi^\xi_1$ sentence 
+that is false.)
+\ennd
+
+This section will prove no analog of \ep{T-reflect}'s
+translational reflection principle is feasible for 
+$\xi -$Conventional axiom systems. Thus,
+Theorem \ref{ppp6} must be close to being a maximal result,
+since it cannot 
+plausibly
+be further extended to hold under conventional axiom systems.
+
+\medskip
+
+{\bf Theorem E.1}  (A New Type of Version of the Second Incompleteness
+Theorem): 
+{\it
+There exists no 
+$\xi -$Conventional 
+axiom
+system $~\alpha~$ 
+that can prove the validity of 
+\eq{G-reflect}'s
+Translational Reflection Principle
+for any translation-mapping T.}
+(In other words,
+there exists no
+ algorithm  $~T~$ that 
+maps 
+$~\Pi_1^\xi$ sentences $~\Psi~$  
+onto alternate  $~\Pi_1^\xi$ 
+sentences
+$~\Psi^T~,~$ 
+which 
+ are equivalent 
+to $~\Psi~$
+in the
+Standard-M model
+and where $~\alpha~$ can
+verify
+\eq{G-reflect}'s reflection principle for every
+$~\Pi_1^\xi$ 
+ sentence $~\Psi .~)$
+%
+%simultaneously
+%for 
+% all  $~\Pi_1^\xi$ sentences
+% $~\Psi~$.) 
+\beq
+\label{G-reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+  ~~~ \Rightarrow ~~~ \Psi^T~~]
+\enq
+
+
+{\bf Proof:} $~$It is  easy to prove Theorem E.1 via a
+proof-by-contradiction. Thus consider the possibility
+that Theorem E.1's translational mapping $~T~$ did exist.
+One can then easily select a  $~\Pi_1^\xi$ sentences $~\Psi~$  
+that is false in the 
+Standard-M model. 
+Then 
+$~\Psi^T~$ is also false under the 
+Standard-M model 
+(since 
+ $~\Psi~$  
+and
+$~\Psi^T~$ 
+are equivalent in this model).
+
+
+%% Moreover, $T$'s definition requires 
+%% $\Psi^T$ 
+%% to have a $\Pi_1^\xi$ format.
+%% Thus,
+
+Hence
+Part-b of the definition of 
+$\xi -$Conventionality implies
+$~\alpha~$ must prove 
+$~\neg~\Psi^T$
+ (on account of $~\Psi^T\,$'s 
+$\Pi_1^\xi$ format).
+
+
+It is at this juncture that our proof-by-contradiction will
+reach its end. This is because if $~\alpha~$ can prove
+\eq{G-reflect}'s statement
+and
+{\it also prove}
+ the sentence $~\neg~\Psi^T~$,
+ then it
+certainly can combine these two facts to prove the
+non-existence of a proof of $~\Psi~$.
+The latter 
+contradicts 
+Part-a of the definition of 
+$\xi -$Conventionality (because it  shows $~\alpha~$
+can verify its own consistency). $~~\Box$ 
+
+
+%%  The latter would
+%% contradict 
+%% Part-a of the definition of 
+%% $\xi -$Conventionality (because it would show $~\alpha~$
+%% can verify its own consistency). $~~\Box$ 
+
+\bigskip
+
+{\bf Remark E.2.} 
+We remind the reader that 
+Footnote \ref{imper} 
+pointed out that
+$T$'s translational mapping 
+would lose its main functionality, if 
+it did not require $\Psi^T$ 
+to have a  $~\Pi_1^\xi~$ format,
+similar to $\Psi$.
+In essence,
+Theorem E.1 is
+of interest
+because it shows that 
+Theorem \ref{ppp6}'s evasion of the Second Incompleteness Theorem 
+is close to
+being a maximal result. 
+%(because 
+(It thus shows that \eq{G-reflect}'s
+translational reflection principle does not generalize 
+to conventional 
+axiom systems.)
+%settings.)
+This 
+dichotomy
+may explain why self-justifying axiom systems,
+along with    Theorem \ref{ppp6}'s
+particular
+ invariant,
+are 
+potentially useful results.
+
+%surprising topics.
+
+
+ % {\bf Remark E.2.} 
+% The 
+% preceding proof 
+% of Theorem E.1 makes it
+% clear
+% that its generalization of the
+% Second Incompleteness Theorem
+% is a 
+% straightforward
+% result. It is 
+
+
+ 
+\cvl
+
+
+
+
+
+\section*{Appendix F: Epistemological 
+Perspective and Speculations}
+
+
+
+\cvs
+
+\parskip 6pt
+
+It is desirable to include a short
+purely
+epistemological
+discussion 
+within this 
+mostly mathematical article so that 
+the more subtle nature of our
+ results 
+cannot be
+misconstrued.
+
+Part of the reason 
+Self Justification
+can lend itself 
+to
+easy
+misinterpretations is that 
+the
+First Incompleteness Theorem 
+demonstrates 
+the
+impossibility of 
+constructing an ideally
+optimal axiomatization of number theory. 
+For
+any initial
+r.e$. \,$axiom system $~\alpha~$
+and deduction
+method$~d$, 
+G\"{o}del 
+thus
+noted
+it is
+easy
+\footnote{  \b55 
+Let $\mho(a,d)$ the
+classic
+ G\"{o}del
+sentence 
+that  asserts:
+{\it ``There is no proof of {\it this sentence} from 
+$\alpha$'s
+axiom system 
+under $\,d$'s deduction method.''}
+G\"{o}del
+\cite{Go31}
+ noted 
+$~\alpha+\mho(\alpha,d)~$ 
+always proves  more theorems than  $\alpha$.}
+ to
+develop
+ an extension of  $~\alpha~$
+that can prove 
+strictly more
+theorems than $~\alpha~$ under 
+ $\,d$'s
+deduction method. 
+Moreover, a large number of generalizations of the Second Incompleteness
+Theorem, starting with its 1939  
+Hilbert-Bernays version
+\cite{HB39}, are known to be 
+%very 
+robust results.
+
+Such considerations 
+naturally lead to  questions
+about whether
+any
+ r.e$. \,$axiom system can encompass the workings
+of the human mind. It may surprise some readers to learn
+that this author 
+shares 
+such
+ skepticism.
+That is, 
+we
+doubt
+{\it any single ISOLATED} self-justifying
+r.e.$\,$logic 
+can {\it  fully} approximate
+the complex workings of the human mind.
+
+
+In this short appendix,
+let us instead
+view  cogitation
+as 
+{\it roughly} a
+process wondering
+though some universe $ \,   \nal{U}$, 
+comprised of
+{\it both} consistent and
+inconsistent axiom systems,
+with a
+trial-and-error evolutionary method 
+focusing 
+ its attention
+over time  
+increasingly  
+onto
+ the members of this universe 
+ $~  \nal{U}~$ that are found to be
+ consistent.
+It is 
+%
+%%notre
+%
+%
+%%notre-only straightforward
+%
+%easy
+%
+%%cccorn-ff  
+%%cccorn-cc  
+  straightforward\footnote{  \b55 It 
+   is trivial from a theoretical perspective
+   to design a 
+   learning heuristic that
+   will utilize all 
+   consistent axiom systems
+   from
+   its available  universe $  \nal{U}$ 
+    eventually, and
+   it will
+   spend only an infinitesimal fraction of its effort on
+   inconsistent systems as time runs to infinity.
+   (This because
+   there exists only
+   a countable number of distinct r.e. sets
+   belonging to the universe
+   $  \nal{U}.~)~$
+   Also, 
+   this
+   learning
+    process can 
+   presumably be made to
+    employ 
+   some type of
+   smart souped-up
+   AI heuristics to enhance its efficiency,
+   whose details will not concern us 
+   % here
+   in this abbreviated 3-page appendix.
+   What is 
+   central to the current discussion
+    is that some type of formally
+   {\it non-recursive} 
+   and presumably trial-and-error
+   method must 
+   obviously
+   be used 
+   by this learning process
+    to find
+   the consistent elements of
+   $~  \nal{U}~,~$ 
+   on account of G\"{o}del's
+   undecidability results.}
+   %%
+ %%
+ to define
+many 
+universes  $~  \nal{U}~$  and
+ evolutionary processes that 
+fall into this gendre.
+Our 
+goal 
+in this section
+will be to
+examine
+ Section \ref{3uuuu3}'s
+``R-View'' $\theta$ and its RE-Class$(\xi)$.
+
+ 
+
+Thus,  $\theta$ will denote an 
+R-View
+that consists of
+an arbitrary
+r.e$. \,$set of
+$\Pi_1^\xi$ sentences.
+Also, 
+RE-Class$(\xi)$ will
+again
+denote the
+set of all
+$~\theta~$ which 
+can be built under $~\xi \,$'s language
+of
+$\,L^\xi$.
+(Section \ref{3uuuu3}
+had
+allowed
+ both valid and invalid
+R-Views  $~\theta$ 
+ to appear in
+RE-Class$(\xi)$ because 
+no recursive 
+decision
+procedure can 
+identify
+all  
+the Standard-M model's true  $\Pi_1^\xi$ sentences.)
+
+%% \nop
+
+The epistemological purpose of this notation was revealed
+in Section \ref{sect64}.
+For the cases where $k  =  0$
+or 1,
+Section \ref{sect64}
+defined 
+$G^\xi_k(\, \theta \,)$
+to be  the 
+axiom system:
+\begin{equation}
+\label{f4gedef}
+G^\xi_k(~ \theta ~)~~= ~~
+\theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}
+\end{equation}
+Also, Definition \ref{dap4-1} indicated that
+  the function 
+$ \, G^\xi_k ~ $
+(which maps $ \, \theta \, $ onto 
+$G^\xi_k(\, \theta \,)~~~)$ would be
+called 
+{\bf Consistency Preserving} iff 
+$ \, G^\xi_k(\, \theta \,) \, $is 
+assured to be consistent whenever 
+all the sentences in $~\theta ~$
+are  
+true  under the Standard-M model.
+\thx{pqq3} 
+indicated,
+in this context,
+ that
+$~G^\xi_1~$ 
+satisfies 
+this 
+property
+whenever $~\xi~$ is
+EA-stable.
+Likewise,
+$~G^\xi_0~$ 
+is      consistency preserving whenever $~\xi~$ is
+one of A-stable, E-stable or 0-stable.
+
+\nskip
+
+
+These results indicate 
+a trial-and-error experimental process
+can, indeed, walk 
+{\it in an 
+unusually
+orderly manner} through an universe
+of self-reflecting 
+candidate
+formalisms, when
+RE-Class$(\xi)$  denotes 
+$~  \nal{U}\,$'s 
+universe  
+and
+ $~\xi~$ satisfies any of
+the  EA-stable, E-stable, 
+A-stable or 0-stable conditions.
+This is because if 
+$~\theta~$ designates
+a 
+set of
+$\Pi_1^\xi$ sentences
+holding true 
+in
+ the Standard-M model, 
+then  
+$~G^\xi_k(\theta)~$ 
+will
+{\bf automatically}
+satisfy both Parts (i) and (ii)
+of Section 1's definition of Self Justification,
+according to \thx{pqq3}.
+
+Such 
+consistency preservation is
+surprising because 
+it is simply inapplicable to
+%does not apply to
+the $\,G^\xi_k \,$ 
+% functions of
+ functions for
+most 
+pairs $~(\xi,k).$
+\thx{pqq3}'s  first
+ contribution 
+is,
+ thus,  
+that it formalizes
+how  $G^\xi_k\,$'s  mapping function 
+can
+represent
+ a type of approximation
+for
+instinctive faith,
+under certain well-defined circumstances.
+
+
+This notion of instinctive faith is, of course, 
+less
+robust than a conventional proof.
+One
+obvious
+  difficulty
+is that a 1-sentence proof,
+using an {\it ``I am consistent''} axiom,
+is 
+less convincing
+than a full-length proof from first principles. Also, if
+the initial formalism $~\theta~$ contains a false
+$~\Pi_1^\xi~$ sentence then  $~B^\xi+\theta~$
+and $~G^\xi_k(\theta)~$ 
+will be  both inconsistent.
+
+
+
+\nop
+
+\cvl
+
+Nevertheless
+for $\,k\,$ equals 0 or 1,
+ if   $~\theta~$ is comprised of the true sentences
+in the Standard-M model, then 
+\thx{pqq3} 
+will
+assure that
+$~G^\xi_k(\theta)~$ is 
+a consistent system that has an ability
+to
+use its {\it ``I am consistent''}
+axiom sentence
+to 
+formalize
+its
+own consistency.
+Moreover, the axiom system
+$~G^\xi_k(\theta)~$ is helpful because
+G\"{o}del's famous centennial paper
+% has certainly 
+implicitly
+raised
+the following
+bedeviling 
+issue:
+%dilemma:
+\begin{quote}
+$\#~~$   How is it that Human Beings
+ manage 
+to muster 
+the physical 
+drive
+to think (and prove theorems) when the many
+generalizations of 
+G\"{o}del's Second 
+Incompleteness Theorem 
+demonstrate
+conventional logics
+lack knowledge of
+their own consistency?  
+\end{quote}
+While  philosophical paradoxes and ironical 
+dilemmas,
+similar to 
+$~\# ~,~$ 
+ never yield
+perfect answers, the preceding discussion is helpful
+because it 
+explores
+a
+certain
+syllogism
+% paradigms
+whereby a logic
+can 
+formalize
+ at least some
+fragmented
+operational
+ appreciation of its own consistency.
+
+Moreover, 
+Part-3 of Appendix D
+ indicated that its 
+four self-justifying configurations were
+ close to being maximal results
+that cannot  be 
+much
+improved,
+on account of various
+barriers imposed by
+ the Second Incompleteness Theorem. 
+Thus, these
+particular
+ positive results,
+combined with Theorems \ref{ppp1}
+\ref{pqq3},   \ref{pqq4},  
+\ref{pqq5},
+ \ref{ppp6},
+D.4, E.1,
+ G.2, G.3 and Remarks
+\ref{re4-1} and
+ \ref{recc1},  
+come 
+close to formalizing the
+maximal variants of instinctive faith that a 
+first-order logic can 
+bolster.
+
+
+
+The theme of the last two paragraphs
+is
+thus
+ that our approximation of 
+ {\it ``instinctive faith''} may be imperfect, but it
+is still a useful partial reply to 
+$~\# \,$'s puzzling
+dilemma
+{\it in a context where}
+unambiguous 
+full 
+resolutions to $\, \# \,$
+{\it are not permitted by}
+the Second Incompleteness
+Theorem.
+Furthermore,
+\ep{T-reflect}'s
+translational reflection principle,
+together with Theorem \ref{ppp6}
+and the Remarks \ref{f88} and \ref{remhappy},
+illustrate how the notion of 
+ an instinctive faith 
+about the usefulness of $\Pi_1^\xi$ theorems
+ can
+be
+almost physically
+{\it hard-wired} into self-justifying formalisms.
+
+
+%%% instinctive faith 
+%%% about the validity of $\Pi_1^\xi$ theorems
+%%%  can
+%%% be
+%%% almost physically
+%%% {\it hard-wired} into self-justifying formalisms.
+
+
+
+
+
+
+%%cccorn-ff 
+
+  \bigskip        
+     {\bf A Yet Further
+      Facet
+       of this Unusual Epistemological Interpretation: } 
+      Let
+      % us use 
+      the term 
+      {\it Epistemological Bundle Theory} 
+      refer to
+       the underlying
+      theory, advanced in this appendix, which 
+      speculates about a 
+       Thinking Agent
+      % as 
+      walking
+      through   
+      RE-Class$(\xi)$'s
+      bundled universe of valid and invalid
+      collections of $\Pi_1^\xi$ sentences
+      and 
+      then applying  some heuristic  to
+      attempt to
+      identify
+      %locate
+      % locate (via  heuristics)
+       those
+      % particular elements
+       $\theta \, \in \,$RE-Class$(\xi)$
+      whose sentences are true under the Standard-M model.
+      
+      Such a 
+      theory has a second virtue, aside from 
+      % its
+      addressing  $ \# \,$'s
+       paradoxical question
+      about the nature  
+      of {\it ``instinctive faith''. }
+      It also clarifies
+      the meaning of 
+      our main theorems
+      %%  
+      %%   
+      %%   and simplifies
+      %%  the 
+      %%  % mathematical
+      %%   structure of 
+      %%  Sections
+      %%  \ref{3uuuu1}-\ref{sect64}'s theorems  
+      %%  % formal
+      %%  
+      %%  
+      and the related
+           E-stability, A-stability,
+       EA-stability and 
+      RE-Class$(\xi)$ constructs.
+      
+      
+      %%
+      %%help 
+      %%analyze 
+      %%%prove their theorems about 
+      %%self-justification.
+      %%It turns out that
+      %%%A nice aspect about the  
+      %% epistemological bundling
+      %%can explain the motivation behind these
+      %%theorems.
+      %%
+      %%
+      %%*is that it can explain much of the intuition behind
+      %%* these theorems.
+      
+      
+      This is because the
+      Items $\, * \,$ and  $\, ** \,$
+      from the
+      definitions of 
+      A-stability and E-stability
+      in Section \ref{3uuuu3} 
+      formalize
+      how a thinking agent $~T~$ can view short
+      proofs from a {\it technically inconsistent} axiom system
+      of $~B^\xi \cup \theta~$ as containing
+      % some 
+      pragmatically
+      useful information
+      {\it under the assumption} that the
+      lengths 
+      %   FIXED ALREADY length's length's length's
+      of $~T\,$'s proofs {\it are shorter}
+      than  
+      the errors in
+      $~\theta\,$'s $\Pi_1^\xi$ styled-statements.
+      The pleasing aspect 
+       about this
+      observation, illustrated by 
+      % for example
+      Remark \ref{re3-1},
+      %
+      % epistemological bundling
+      is that those same invariants,
+      $\, * \,$ and  $\, ** \,$, 
+      which 
+%%%%%      can 
+      tempt a
+       thinking agent $~T~$ 
+      to engage in a trial-and-error walk through
+      RE-Class$(\xi)$'s bundled universe, 
+      %% are 
+      also 
+      %% the invariants that 
+      make
+      viable
+      %% the operating prerequisites for making
+       \thx{ppp1}'s self-justifying formalisms. 
+      
+      
+      %active.
+      
+      Thus aside from 
+      addressing
+       $\, \# \,$'s dilemma about the nature of 
+      instinctive faith, 
+      the 
+      % mathematical
+      meta-formalism in this appendix
+      is
+      %was 
+      % also
+      useful 
+      %in 
+      %venting
+      % for
+      in  explaining
+      the 
+      % underlying
+       motivation behind the 
+      %very
+      %quite
+      % fairly
+      elaborate
+      network 
+      % labyrinth
+      of
+      theorems, proofs and definitions 
+      %raised
+      that were introduced 
+      % had 
+      %appeared 
+      in this 
+      paper.
+In summary,
+EA-stable logics are thus
+ interesting   both
+in their own right
+(as a vehicle 
+ enabling a Thinking Being to partially tolerate
+its own errors), and 
+because
+they are  useful in explaining
+how a Thinking Being 
+can possess a type of instinctive
+faith in its own consistency
+(via the 
+reflection
+principles of
+ Theorem \ref{ppp6}
+and 
+of  Remarks \ref{f88} and \ref{remhappy}).
+
+
+
+
+
+
+\cvl
+
+\section*{Appendix G:  Improvements upon Theorems  \ref{pqq4} 
+and \ref{pqq5}  }
+
+Let us recall that 
+Remark \ref{re4-n} indicated that there was a
+subtle
+trade-off between 
+Theorems  \ref{pqq4} 
+and \ref{pqq5},
+where neither result was 
+strictly
+ better than the other.
+This section
+will introduce
+two hybrid 
+methodologies, using 
+Definition G.1's
+formalism, that 
+improve upon \thx{pqq5}
+while  retaining a large part of \thx{pqq4}'s
+nice features.
+
+\hgskip
+
+% \cvt
+\parskip 2pt
+
+
+{\bf Definition G.1}
+Let $~\xi~$ denote 
+the  generic configuration, 
+whose base axiom system is again denoted as 
+$ \, B^\xi \,$,
+$\,~ \Phi~$ denote any          $\Pi_1^\xi$ 
+sentence 
+that   is true in    the Standard-M model
+and 
+$~j~$ denote an index that represents some
+predicate 
+ Test$^\xi_j \,$ 
+lying 
+in Definition \ref{gsim}'s
+ TestList$^\xi$ sequence. Then
+a
+$\,\Pi_1^\xi \,$ sentences $\Psi $ 
+ will be said to be a 
+{\bf Braced}$^\xi( \, \Phi \, , \, j \, )$ expression when
+$~ B^\xi \, + \, \Phi~$ can prove:
+\beq
+\label{punch}
+\{~~~ \forall ~x~~~
+\mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~) ~~~\}
+ ~~~~\longrightarrow ~~~~ \Psi 
+\enq
+
+
+\medskip
+
+
+
+
+
+{\bf Theorem G.2}
+{\it $~$Let  $ \,\xi \,$ again 
+denote an arbitrary  generic configuration  
+  \gggcp, 
+and let
+ $(  \nal{B},D)$ again denote any second axiom system and deduction
+method whose $\Pi_1^\xi$ theorems are true under the
+Standard-M model.
+Then for any 
+integer $~j~$ and for any 
+  $\Pi_1^\xi$ sentence
+$~\Phi~$ that is  true  in the
+Standard-M model,
+ the following  invariants
+do hold:}
+\bed
+\item[   i ] 
+{\it If $ \, \xi \, $ is EA-stable\sss  
+then there 
+will exist a self-justifying
+$~\beta_j~\supset~B^\xi$ that 
+can recognize its 
+Level$(1^\xi$)
+ consistency,
+contains
+only {\bf a finite number} of additional axioms 
+beyond those appearing in 
+$~B^\xi$,
+ and which
+can 
+prove
+all  of $(  \nal{B},D)$'s $\Pi_1^\xi$ theorems that 
+are Braced$^\xi(  \Phi  ,j)$ expressions.}
+\item[  ii ] 
+{\it Likewise, 
+if $\xi$ is 
+  E-stable, A-stable  or 0-stable 
+then 
+ a
+self-justifying
+$\beta_j \supset B^\xi$ 
+will exist
+with the same properties except 
+that it
+recognizes its own
+Level$(0^\xi$)
+ consistency.}
+\ennd
+
+\medskip
+
+
+
+
+{\bf Proof.}
+To justify
+Theorem G.2, we must first define
+the axiom system  $~\beta_j~,~$ 
+whose existence is claimed by
+Items (i) and (ii).
+It will be defined to
+consist of the union of
+the initial base axiom system
+ $B^\xi$ 
+with the following three added axiom-sentences.
+\bed
+\item[   1  ]
+The $\Pi_1^\xi$ sentence $~\Phi~$
+used
+by  Definition G.1's
+ Braced$^\xi(  \Phi  ,j  )$ formula.
+
+\smallskip
+
+\item[   2  ]
+A GlobSim$^D_{  \nal{B}} \,(\xi,j)$ sentence whose indexing
+integer $ \, j \,$ 
+is defined by  Definition G.1.
+This global simulation sentence is
+thus the statement:
+\beq
+\label{glob2}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{  \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}_j^\xi(t,x)  ~~~\}
+\enq
+\item[   3  ]
+A  $\Pi_1^\xi$ sentence
+ of the form
+$\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}~$
+where:
+\bed
+\item[   a   ] 
+$\theta~$ is an R-view
+consisting of the two
+ $\Pi_1^\xi$ sentences defined by
+Items 1 and 2.
+\item[   b   ] 
+$B^\xi\,$ is $~\xi \,$'s base axiom system,
+and
+\item[   c   ] 
+$k~$ equals respectively 
+ 1  and 0 under formalisms  (i) and (ii).
+\ennd
+\ennd
+Thus,
+the  system $\beta_j$
+uses identical definitions
+ under
+formalisms (i) and (ii),      
+except 
+that
+its third sentence will use a different value for
+$~k~$.
+Our proof of 
+Theorem G.2
+will require        first confirming
+the following fact:
+\begin{description}
+\item[  Claim *  ]
+The axiom system
+$\,\beta_j \,$ (which consists of the union of
+$B^\xi$  with the
+  sentences 1-3)
+will have a capacity
+to prove every 
+Braced$^\xi(  \Phi  ,j)$ 
+sentence $~\Psi~$
+that is a
+ $\Pi_1^\xi$
+  theorem of
+$(  \nal{B},D)$.
+\end{description}
+The proof of Claim * is quite simple.
+It will rest on
+the following
+three
+observations:
+\bed
+\item[   a  ]
+For each  $\Pi_1^\xi$ sentence $\Psi$,
+the  system $~\beta_j~$
+must certainly
+have a capacity to prove 
+\eq{glob21}'s sentence (which 
+states 
+that $~\Psi\,$'s
+G\"{o}del number 
+formally
+encodes
+a $\Pi_1^\xi \,$  statement). This  
+is because 
+\eq{glob21}
+ is true
+ in the Standard-M
+model
+ and
+because Part 3 of Definition \ref{def3.3}
+indicated
+that
+ the 
+$~B^\xi~$ 
+sub-component of $~\beta_j~$
+has a capacity to prove 
+ every
+$\Delta_0^\xi$ sentence
+ that is true.
+\beq
+\label{glob21}
+\mbox{Check}^\xi( ~ \lceil \, \Psi \, \rceil ~)
+\enq
+\item[   b  ]
+Since
+Claim $\,* \,$ specifies
+ $~\Psi~$ is a theorem of 
+ $(  \nal{B},D)$, 
+there must certainly exist some integer
+$~N~$ that is the G\"{o}del number of its proof from
+ $(  \nal{B},D)$. This implies that 
+\eq{glob22} must be a true
+$\Delta_0^\xi$ sentence
+ under the Standard-M model.
+As was the case with \ep{glob21}, this implies
+that
+it must be provable from
+$~B^\xi~$ (because it is a valid 
+$\Delta_0^\xi$ sentence).
+\beq
+\label{glob22}
+\mbox{Prf}_{  \nal{B}}^D \,(   ~ \lceil \, \Psi \, \rceil ~  , ~  N ~ )
+\enq
+\item[   c  ]
+It is apparent that 
+Equations \eq{glob2}, \eq{glob21} and \eq{glob22}
+imply the validity of \eq{glob23}.
+Moreover, Part 4 of Definition \ref{def3.3} indicated that
+the generic configuration $~\xi\,$'s deduction method
+does satisfy  G\"{o}del's Completeness Theorem.
+This fact assures that 
+$~\beta_j~$ must be able to prove 
+\eq{glob23} because it contains
+\eq{glob2}
+as an axiom and 
+ \eq{glob21} and \eq{glob22} 
+as derived theorems \footnote{\label{fcomp} \sm55 
+Every deduction method $\,d$,
+$\,$satisfying G\"{o}del's Completeness Theorem,
+will be automatically able to prove
+a theorem $~Z~$
+when it contains
+$X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as
+theorems, irregardless of whether or not it contains
+an explicit built-in
+modus
+ponens rule.
+Thus $~d~$ can prove
+\eq{glob23} because of its knowledge about 
+\eq{glob2}--\eq{glob22}'s validity.}.
+\ennd
+\vspace*{- 0.1 em}
+\beq
+\label{glob23}
+\forall ~x~~~
+\mbox{Test}_j^\xi(   ~ \lceil \, \Psi \, \rceil ~  , ~  x ~ )
+\vspace*{- 0.1 em}
+\enq
+Claim $*$ is a
+consequence of 
+Observations a-c. This is  because
+$ \, \Phi \, $ 
+is one of $ \, \beta_j \,$'s defined axioms,
+and
+Definition G.1 
+indicated 
+ $ \, B^\xi \, + \, \Phi \, $ was capable of proving
+\eq{punch}'s statement
+ for every  Braced$^\xi(  \Phi  ,j  )$ sentence $ \, \Psi \, $.
+These facts corroborate Claim $*$ 
+because they imply
+that $ \, \beta_j \, $
+must be able to verify
+Claim $\,* \,$'s
+ sentence
+ $ \, \Psi \, $ (because
+ $ \, \beta_j \, $
+can verify statements
+\eq{punch} and \eq{glob23}).
+
+
+
+
+
+
+
+\medskip
+
+The remainder of 
+Theorem G.2's proof is 
+analogous
+to
+\phx{pqq5}'s proof.
+This is because the prior paragraph 
+established
+that $~\beta_j~$  can prove 
+every 
+ Braced$^\xi(  \Phi  ,j  )$ 
+theorem of
+ $(  \nal{B},D)$
+(as was required by
+ Claims i and ii ). 
+The only 
+remaining task is to
+show that
+ $~\beta_j~$  is a self-justifying formalism that can
+recognize its 
+Level($1^\xi$) and Level($0^\xi$)
+consistencies,
+as specified by
+ Claims i and ii.  
+This part of 
+Theorem G.2's
+ verification
+is identical
+to
+the methods used
+to prove
+Theorems \ref{pqq3} and \ref{pqq5}.
+It
+will
+thus
+ not be repeated
+here.
+  $~~\Box$
+
+
+
+\medskip
+
+
+
+The last part of this appendix will
+require the
+following additional
+ notation to formalize
+the main intended application
+of
+Theorem G.2's 
+formalism.
+\bee
+\item
+\topsep -7pt
+ Count$(  \Psi )$ will denote
+the number of 
+quantifiers appearing in the  sentence $\Psi$
+(including both its bounded and unbounded quantifiers).
+\item 
+Size$^\xi(c)$ 
+will
+denote the  set of $\Pi_1^\xi$ sentences
+$\Psi$ where  Count$(  \Psi ) \, \leq \, c \,$.
+\ene
+Our next theorem will be   a specialized
+variant of
+Theorem G.2, using the 
+Size$^\xi(c)$  construct.
+It  will explain the 
+ intended application
+of
+this
+formalism:
+
+
+
+
+\medskip
+
+
+{\bf Theorem G.3.}
+{\it 
+$~$Let $~\xi~$ denote any one of Appendix D's four sample
+generic configurations of $~\xi^*~$, 
+$~\xi^{**}~$,  $~\xi^-~$  or   $~\xi^R~$.
+Then 
+ for any  $~c>0~$, 
+Theorem G.2's axiom systems
+of $~\beta_j~$
+can be arranged so that 
+they can prove all of 
+  $  (    \nal{B} ,  D  ) $'s
+Size$^\xi(c)$ 
+ $\Pi_1^\xi$ 
+theorems while
+simultaneously also
+ recognizing their:
+\bee
+\item Level(1)
+consistency  for the cases
+when  $ \, \xi \, $ is one of   $ \, \xi^* \, $,
+$ \, \xi^{**}\,   $ or   $ \, \xi^-$.
+\item
+ Level(0) consistency when $~\xi~$ is
+$~\xi^R~$.
+\ene}
+
+\cvmew
+
+{\bf Proof Sketch:}
+%% 
+%%  There is 
+%% insufficient
+%%  space to prove
+%% Theorem G.3 here, but its intuition 
+%% is easy to summarize.
+%% 
+%% 
+ The  intuition behind
+Theorem G.3's proof is 
+quite easy to summarize.
+For 
+arbitrary $     c>0      $
+and any
+of Appendix D's configurations of
+ $   ~  \xi^*  ~   $, 
+$   ~  \xi^{**}  ~   $,  $  ~   \xi^-  ~   $  and   $  ~   \xi^R  ~   $, 
+it is 
+routine to
+ construct an ordered pair $  ~   (\Phi,j) ~    $ where every
+$\Pi_1^\xi$ sentence of
+Size$^\xi(c)$ is a 
+ Braced$^\xi(  \Phi  ,j  )$ expression.
+Theorem G.3's 
+first claim is,
+thus, a 
+consequence of 
+Part (i)
+of Theorem G.2
+ and 
+the fact that each of
+ $   ~  \xi^*  ~   $, 
+$   ~  \xi^{**}  ~   $ and  $  ~   \xi^-  ~   $  
+are EA-stable.
+Likewise, Theorem G.3's 
+second claim follows from
+Part (ii)
+of Theorem G.2
+ and the fact that 
+$ ~  \xi^R  ~ $ is E-stable,
+$~~\Box$  
+
+\bigskip
+\medskip
+%\parskip 0pt
+
+{\bf Remark G.4.}
+The  Theorems G.2 and G.3 are
+of interest  because the set of 
+$\Pi_1^\xi$ sentences of
+Size$^\xi(c)$ is a natural class to examine.
+It is,
+thus, tempting to consider
+a system that 
+recognizes
+its own
+formal 
+ consistency, uses only a finite
+number of axiom sentences beyond those in $~B^\xi~,~$ and 
+which
+can
+ prove all of
+  $  (    \nal{B} ,  D  ) $'s
+ $\Pi_1^\xi$ theorems 
+of Size$^\xi(c)$.
+Such a system 
+replies to 
+Remark 
+\ref{re4-n}'s challenge by
+% nicely
+ hybridizing the properties of
+Theorems  \ref{pqq4} 
+and \ref{pqq5},
+%%  .
+in a seemingly pragmatic manner.
+
+
+
+
+
+
+\newpage
+
+\begin{thebibliography}{99}
+
+
+\baselineskip =  1.15 \normalbaselineskip
+
+\parskip 6 pt
+
+
+
+
+\bibitem{Ad2}
+Z. Adamowicz,
+``Herbrand Consistency and Bounded
+Arithmetic'', 
+{\it Fundamenta Mathematicae}
+% {\it Fund. Math.}
+171 (2002) pp. 279-292.
+
+
+ 
+ \bibitem{AB1}
+ Z. Adamowicz and T. Bigorajska,
+ ``Existentially Closed Structures and G\"{o}del's Second Incompleteness
+ Theorem'', {\it Journal of Symbolic Logic}
+ 66 (2001), pp. 349-356.
+
+
+
+
+
+
+
+
+\bibitem{AZ1}
+Z. Adamowicz and P. Zbierski,
+``On Herbrand consistency in weak theories'', 
+{\it Archives for Mathematical Logic}
+40(2001)  399-413
+
+
+
+\bibitem{Ar90}
+T. Arai, ``Derivability Conditions on Rosser's Proof Predicates'',
+{\it Notre Dame Journal of Formal Logic} 30(1990) 487-490
+
+
+
+\bibitem{Be95}
+L Berkemishev,
+``Iterated Local Reflection Versus Iterated Consistency'',
+{\it Annals of Pure and Applied Logic}
+75 (1995) pp. 25-48.
+
+
+
+
+
+
+\bibitem{Be97}
+L Berkemishev,
+``Induction Rules, Reflection Principles and Provably
+Recursive Functions'',
+{\it Annals of Pure and Applied Logic}
+85 (1997) pp. 193-242.
+
+
+
+\bibitem{Be3}
+L Berkemishev,
+``Proof Theoretic Analysis
+by Iterated Reflection'',
+{\it Archives on Mathematical Logic}
+42 (2003) pp. 515-552.
+
+
+
+
+\bibitem{BS76}
+A. Bezboruah and J. C. 
+ Shepherdson,
+``G\"{o}del's Second Incompleteness Theorem for Q'',
+{\it Journal of  Symbolic Logic} 41 (1976) pp.  503-512
+
+
+\bibitem{Bu86}
+S. R. Buss, {\it Bounded Arithmetic,} (Ph D Thesis)
+Proof Theory 
+Notes \#3,  Bibliopolis 1986.
+
+
+
+\bibitem{BI95}
+S. R. Buss and A. Ignjatovic,
+``Unprovability of Consistency Statements in Fragments of
+Bounded Arithmetic'', 
+{\it Annals of Pure and Applied Logic} 74 (1995) pp. 221-244.
+
+
+
+\bibitem{Fe60}
+S. Feferman, ``Arithmetization of Mathematics in a General Setting'',
+{\it Fundamenta Mathematicae}
+% {\it Fund. Math.} 
+49  (1960) pp. 35-92.
+
+
+
+
+\bibitem{Fi90}
+M. Fitting, First Order Logic and Automated Theorem Proving,
+Springer-Verlag, 1996.
+ 
+
+
+
+
+
+\bibitem{Fr79b}
+H. M. 
+Friedman, ``Translatability and Relative Consistency'',
+ Ohio State  Tech Report, 1979.
+
+
+
+
+
+
+\bibitem{Go31}
+K. G\"{o}del,
+`` \"{U}ber 
+formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandte Systeme I'',
+{\it Monatshefte f\"{u}r Mathematik und Physik} 37 (1931) pp. 349-360.
+
+
+
+
+
+
+
+
+\bibitem{HP91}
+P. H\'{a}jek and P. Pudl\'{a}k,
+{\it Metamathematics of First Order Arithmetic,}
+Springer Verlag 1991. 
+
+
+
+\bibitem{HB39}
+D. Hilbert and P. Bernays,
+{\it Grundlagen der Mathematic}, Springer 1939.
+
+
+
+
+
+\bibitem{Je71}
+R. 
+ Jeroslow, ``Consistency Statements in 
+Mathematics '', 
+{\it Fundamenta Math}
+ 72 (1971)  pp.17-40.
+ 
+
+
+
+
+\bibitem{Ka91}
+R. Kaye,
+{\it Models of Peano Arithmetic}, Oxford University
+ Press, 1991. 
+
+
+
+
+
+
+
+
+
+
+\bibitem{Kl38}
+S. C.
+Kleene,
+``On the Notation of Ordinal Numbers'',
+{\it Journal of   Symbolic Logic}
+3 (1938), 
+150-156.
+
+
+
+
+
+
+
+\bibitem{Ko5}
+L. A. Ko{\l}odziejczyk, Private Email Communications during November
+of 2005.
+
+
+
+\bibitem{Ko6}
+ L. A. Ko{\l}odziejczyk,
+``On the Herbrand Notion of Consistency for Finitely
+Axiomatizable Fragments 
+of Bound Arithmetic'',
+{\it Journal of Symbolic Logic}
+71 (2006) pp. 624-638.
+
+
+
+
+\bibitem{Kr87} J. Kraj\'{i}cek,
+``A Note on Proofs of Falsehood'', 
+{\it Archives on Math Logic} 26 (1987) pp. 169-176.
+
+
+
+
+ 
+ \bibitem{Kr95} J. Kraj\'{i}cek,
+ {\it Bounded Propositional Logic and Complexity Theory,} Cambridge 
+University
+ Press 1995.
+
+
+\bibitem{KT74}
+G. Kreisel and G. Takeuti,
+``Formally Self-Referential Propositions  for Cut-Free Classical
+Analysis'',
+{\it Dissertationes Mathematicae}
+118
+(1974) pp. 1--55
+
+
+
+
+\bibitem{Lo55} M. H. L\"{o}b, A Solution to a Problem by Leon Henkin,
+{\it Journal of Symbolic Logic}
+20 (1955)  115-118.
+
+
+
+\bibitem{Ne86}
+E. Nelson, {\it  Predicative Arithmetic,}
+Mathematics Notes
+ Princeton U Press,
+  1986.
+
+
+    \bibitem{Pa71}
+  R. Parikh, ``Existence and Feasibility in Arithmetic'',
+  {\it Journal of  Symbolic Logic}
+   36 (1971), pp. 494-508
+
+
+\bibitem{PD82}
+J. B. Paris and C. Dimitracopoulos,
+``Truth Definition for $\Delta_0$
+Formula'',
+{\it Monagraphie de L'Ensignment Mathematique}
+30 (1982) pp. 317-329. (See especially the first two pages
+of this article to
+find the counterpart of   
+Lemma \ref{lex22} in it.)
+
+
+
+\bibitem{PD83}
+J. B.  Paris and C. Dimitracopoulos,
+``A Note on  
+Undefinability of Cuts'', 
+{\it Journal Symbolic  Logic}
+48 (1983)
+  564-569
+
+
+
+
+\bibitem{PW81}
+J. B.  Paris and A. J.  Wilkie,
+``$\Delta_0$ Sets and Induction'', 
+{\it 1981 Jadswin  Conference Proceedings,} 
+pp. 237-248.
+
+
+
+\bibitem{Pa72}
+C. Parsons, 
+On $n-$Quantifier Elimination,
+{\it Journal of Symbolic Logic}
+37 (1972), pp. 466-482.
+
+
+
+
+ \bibitem{Pu84}
+ P. Pudl\'{a}k,
+ ``On Lengths of Proofs of Finistic Consistency Statements in
+ First order Theories'',
+ {\it Logic Colloquium 84}, North Holland (1986)
+ pp. 165-196.
+
+
+
+\bibitem{Pu85}
+P. Pudl\'{a}k,
+``Cuts, Consistency Statements and Interpretations'',
+{\it Journal of Symbolic Logic}
+50 (1985)  423-442
+
+
+
+
+\bibitem{Pu96}
+P. Pudl\'{a}k,
+``On the Lengths of Proofs of Consistency'',
+in {\it Collegium Logicum: 1996 Annals of the Kurt G\"{o}del
+Society} ( Volume 2),  Springer-Wien-NewYork,
+pp 65-86.
+
+
+
+
+
+
+
+
+
+
+
+\bibitem{Ro67}
+H. Rogers, {\it 
+Recursive Functions and Effective 
+Compatibility,} McGrawHill 1967.
+
+
+
+\bibitem{Ro36}
+J. B. Rosser,
+``Extensions of Some
+Theorems by
+G\"{o}del
+and Church'',
+{\it Journal of  Symbolic  Logic}
+1 (1936) pp.  87-91.
+
+
+\bibitem{Sa11}
+S. Salehi,  Herbrandized Consistency of Some 
+Finite Fragments of Bounded
+Arithmetic Theories (CoRR 2011)
+{\rm http://arxiv.org/abs/1110.1848}. 
+
+
+
+ 
+ \bibitem{Sm83}
+ C. A.  Smory\'{n}ski, 
+ The Incompleteness Theorem, 
+ {\it Handbook on Mathematical Logic,}
+ pp. 821--866, 1977.
+ 
+
+
+
+
+
+
+
+\bibitem{Sm85}
+C. A. Smory\'{n}ski, ``Non-standard Models and Related Developments
+in the Work of Harvey Friedman'', in {\it Harvey Friedman's
+Research in the  Foundations of Mathematics},
+NH 1985, pp 179-229.
+
+
+
+
+
+\bibitem{Sm68} 
+R. Smullyan, {\it First Order Logic,} Springer-Verlag, 1968.
+
+
+
+
+\bibitem{So76}
+R. M. 
+Solovay,  ``Interpretability in set theories''.
+Unpublished
+letter to Petr 
+H\'{a}jek
+dated
+ August 17, 1976 and
+available on Petr    H\'{a}jek's home page.
+
+
+
+\bibitem{So76b}
+R. M. 
+Solovay,  ``Provability Interpretations in Modal Logics'',
+{\it Israel Journal of Mathematics} 25 (1976) pp. 278-304.
+
+
+
+\bibitem{So88}
+R. M. 
+Solovay,  ``Injecting Inconsistencies into Models of PA'',
+{\it Annals Pure and Applied Logic''} 44(1988) 102-132
+
+{\baselineskip =  1.1 \normalbaselineskip
+
+
+\bibitem{So94}
+R. M.  Solovay,  Private 
+%telephone  
+communications
+in  1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using the additional formalisms of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+Solovay 
+never published 
+this result (called Theorem 2.1 in Section 2)
+or
+his other
+observations 
+that
+several logicians \cite{HP91,Ne86,PD83,PW81,Pu85,Pu96,So76,Sv7,WP87}
+have 
+attributed to his
+private communications.
+The Appendix A of 
+\cite{ww1} offers our 
+4-page interpretation of
+Solovay's basic suggestion.
+See also
+reference \cite{So76} for a pdf file consisting of
+Solovay's 
+earlier unpublished
+1976
+ letter to Petr 
+H\'{a}jek.}
+
+
+
+
+\bibitem{Sv83}
+V. \v{S}vejdar, ''Modal Analysis of 
+Rosser Sentences'',
+{\it Journal of Symbolic Logic} 48 (1983)  986-999.
+
+
+\bibitem{Sv7}
+V. \v{S}vejdar, ''An Interpretation of Robinson Arithmetic in its
+Grzegorczjk's Weaker Variant''
+{\it Fundamenta Informaticae} 81 (2007)
+pp. 347-354.
+
+
+\bibitem{Ta53} 
+G. Takeuti,
+``On a Generalized Logical Calculi'',
+{\it Japan Journal of Mathematics}
+32 (1953) pp. 39-96. 
+
+
+
+ 
+ \bibitem{Ta87}
+ G. Takeuti, {\it Proof Theory,} 
+ Studies in Logic Volume 81, 
+ North Holland, 1987.
+
+
+
+\bibitem{Ta0} 
+G. Takeuti,
+``G\"{o}del Sentences of Bounded Arithmetic'',
+{\it Journal of Symbolic Logic} 65 (2000)  1338-1346.
+
+
+
+
+
+\bibitem{Ta36}
+A. Tarski,
+``Der Wahrheitsbegriff in den 
+formalisisierten 
+Sprachen'',
+{\it Studia Philosphie.} 1 (1936) 261-405
+
+
+
+
+\bibitem{TMR53}
+A. Tarski, A. Mostowski and R. M. Robinson,
+{\it Undecidable Theories}, North Holland, 1953.
+
+
+\bibitem{VV94}
+R, Verbrugge and A. Visser,
+``A Small Reflection Principle for Bounded Arithmetic'',
+{\it Journal of Symbolic Logic} 59 (1994) pp. 785-812.
+
+
+
+\bibitem{Vi89}
+A. Visser,
+``Peano's Smart Children: A Provability Logical Study of
+Systems with Built-in Consistency'',
+{\it Notre Dame Journal of Formal Logic},
+30 (1987) pp. 161-196.
+
+
+
+
+\bibitem{Vi92}
+A. Visser,
+``An Inside View of Exp'',
+{\it Journal of Symbolic Logic} 57 (1992)
+131--165.
+
+
+
+ 
+ \bibitem{Vi93}
+ A. Visser,
+ ``The Unprovability of Small Inconsistency'',
+ {\it Archive for Math Logic} 32 (1993) pp. 275-298.
+ 
+ 
+
+\bibitem{Vi5}
+A. Visser,
+ ``Faith and Falsity'',
+ {\it Annals of Pure and Applied Logic}
+131 (2005) pp. 103-131.
+
+
+\bibitem{VH73}
+P. Vop\v{e}nka and P. H\'{a}jek,
+``Existence of  a Generalized Semantic Model of
+G\"{o}del-Bernays Set Theory'',
+{\it Bulletin de l'Academie Polonaise des
+Sciences,
+Math, 
+Astromiques et Physiques}
+12 (1973) 
+1079-1086.
+
+
+
+\bibitem{WP87}  
+A. J. Wilkie and J. B. Paris,  
+``On the Scheme of Induction for Bounded
+ Arithmetic'', {\it
+Annals of  Pure and  Applied Logic} (35) 1987, 261-302.
+
+
+
+
+
+ 
+\bibitem{ww93}
+D. Willard,
+``Self-Verifying Axiom Systems'', {\it
+Computational Logic and Proof Theory:$~$
+The   Third Kurt G\"{o}del
+Colloquium}
+(1993),
+Springer-Verlag LNCS\#713,  325-336.
+
+
+  
+  
+  \bibitem{ww0}
+  D. Willard, 
+  ``The Semantic Tableaux Version of the Second
+  Incompleteness Theorem Extends Almost to 
+  Robinson's 
+  Arithmetic 
+  Q'',
+  {\it Automated Reasoning with Analytic Tableaux
+  and Related
+  Methods}, 
+  Springer Verlag (2000) LNCS\#1847, 
+   pp. 415-430. 
+
+
+\bibitem{ww1}
+D. Willard, ``Self-Verifying  Systems, the Incompleteness
+Theorem and Related Reflection
+Principles'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+
+
+\bibitem{ww2}
+D. Willard, 
+``How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem 
+Almost 
+to 
+Robinson's Arithmetic'',
+{\it Journal of Symbolic Logic}
+$~\,67~ (2002)~$pp. 465--496.
+
+
+
+\bibitem{wwlogos}
+D. Willard, 
+``A Version of the
+Second Incompleteness Theorem For Axiom
+Systems that Recognize Addition 
+But Not Multiplication as a Total Function'',
+{\it First Order Logic Revisited,}
+Logos Verlag (Berlin) 2004, pp. 337--368.
+
+\bibitem{ww5}
+D. Willard, 
+``An Exploration of the Partial Respects in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency'', 
+{\it Journal of Symbolic Logic} 70  (2005) pp. 1171-1209. 
+
+\bibitem{ww6}
+D. Willard, 
+``On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency'', 
+{\it Journal of Symbolic Logic} 71 (2006)
+pp. 1189-1199.
+
+\bibitem{wwapal}
+D. Willard,  
+``A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+
+
+
+\bibitem{ww7}
+D. Willard,  
+``Passive induction and a solution to a Paris-Wilkie 
+question'',
+{\it Annals of Pure and Applied Logic}
+146(2007) 
+pp. 124-149.
+
+\bibitem{ww9}
+D. Willard,  
+``Some Specially Formulated Axiomizations for
+I$\Sigma_0$ 
+Manage to
+Evade
+the Herbrandized Version of the Second Incompleteness Theorem'',
+{\it Information and Computation}
+207(2009)  1078-1093
+
+\bibitem{Wr78}
+C. Wrathall, ``Rudimentary Predicates and Relative Computation'',
+{\it Siam Journal on Computing} 7 (1978), pp 194-209
+\end{thebibliography}
+
+
+
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/2011-2019/2011-cornell.tex b/nachlass/collected_dew_materials/2011-2019/2011-cornell.tex
new file mode 100644
index 0000000..89ae53a
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2011-cornell.tex
@@ -0,0 +1,12161 @@
+%%  2011 dec 31 home suny  REVSIONS OF PRIOR DEC21 7.1 am
+%%%  MINOR REVSIONS OF PRIOR DEC21
+%% primarily in Theorem 2.1 and Remark 6.13
+%% Also few other miniscule changes on pages 9, 37, 52 etc.
+
+\documentstyle[11pt]{article}
+
+
+
+
+
+\addtolength{\oddsidemargin}{-0.75in}
+
+
+
+
+\setlength{\textheight}{9.4 in}
+\setlength{\textheight}{9.0 in}
+
+
+
+
+\setlength{\textwidth}{5.7 in}
+
+
+
+
+\setlength{\textwidth}{6.7 in}
+\setlength{\textwidth}{6.5 in}
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{6.3 in}
+
+\setlength{\textwidth}{5.8 in}
+%% PRINT
+
+%\setlength{\textwidth}{5.6 in}
+
+
+
+
+
+
+
+
+
+
+\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+
+
+
+
+
+
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+ \title{A Detailed Examination
+of Methods for Unifying, Simplifying and Extending 
+Several
+Results About Self-Justifying Logics}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+Email = dew@cs.albany.edu.}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+
+
+\begin{abstract}
+ \baselineskip = 1.5 \normalbaselineskip \large 
+
+This paper will develop a single framework for unifying, simplifying
+and extending our prior results about axiom systems that retain a
+partial knowledge of their own consistency, via an axiomatic
+declaration of self-consistency.  Its perhaps single most surprising
+new result will be its exploration of a viable alternative to
+conventional reflection principles.
+\end{abstract}
+
+
+\normalsize
+
+\parskip 8pt
+
+\baselineskip = 1.4 \normalbaselineskip 
+
+\setcounter{page}{0}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+{\bf Keywords:}
+G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+Open Question,
+Semantic Tableaux
+
+\bigskip
+\bigskip
+
+{\bf Mathematics Subject Classification:}
+03B52; 03F25; 03F45; 03H13 
+
+
+
+\bigskip
+\bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+\newpage
+
+\setcounter{page}{1}
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+
+\def\f55{  \baselineskip = 1.59 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.59 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\def\nop{ }
+\def\nop{\newpage}
+
+\cvl
+\cvnew
+
+
+\def\nop{\newpage}
+
+
+\cvnew
+\cvl
+
+\parskip 3pt
+
+
+
+\def\hgskip{ \medskip }
+
+\def\nop{ }
+\def\njp{\newpage}
+\def\njp{ }
+\def\nop{\newpage}
+
+\cvl
+\cvnew
+
+\def\nskip{\bigskip}
+
+\cvl
+\cvnew
+
+
+\section{Introduction}
+
+%% CHANGES REMOVE NEW PAGE IN BIB before willard 2001
+
+%% CHANGES = - 3 footnote -ACKN -APPF -TABLE NUMBERS
+
+%% CHANGES + ww11 Reminder-to-reader Footnote 24 state TABLE
+
+\label{secc1}
+\label{B1-lem}
+\label{D1-def}
+
+\parskip 2pt
+
+
+Let
+$~\alpha~$ 
+denote an axiom system, 
+and $~d~$  
+denote
+ a
+deduction method.
+The ordered pair
+ $~(  \alpha  , d  )$
+will 
+be called  {\bf Self Justifying} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+states that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a second
+axiom system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+Part-i of 
+this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus the following added
+sentence,  that we call
+{\bf SelfRef$(\alpha,d)~$}:
+\topsep -3pt
+\begin{quote}
+$\bullet~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union} of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of
+SelfRef$(\alpha,d)$'s
+ self-referential statement.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition. 
+
+\smallskip
+
+This problem arises in
+settings
+more general than 
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic.
+There 
+are 
+many
+settings 
+where the Second Incompleteness Theorem does
+generalize
+\cite{Ad2,AB1,AZ1,BS76,Bu86,BI95,Fe60,Go31,HP91,HB39,Ko6,KT74,Lo55,PD83,PW81,Pu84,Pu85,Pu96,Ro67,Sa11,Sm85,So94,Sv7,Ta0,VV94,Vi92,Vi5,WP87,ww2,wwapal,wwlogos,ww7}.
+Each such result formalizes a 
+paradigm where 
+self-justification is infeasible,
+due to a diagonalization issue.
+Most
+logicians 
+have
+hesitated to
+thus
+ employ 
+a SelfRef$(\alpha,d)$
+ axiom
+because
+$\alpha+$SelfRef$(\alpha,d)  $
+is usually 
+ inconsistent     \footnote{ \baselineskip = 1.3 \normalbaselineskip  \label{troub} 
+     Typical ordered pairs $(\alpha,d)$
+     will have the property that
+     the broader axiom system
+     $~\alpha^d~=~\alpha \,+ \,$SelfRef$(\alpha,d)$ will
+     be inconsistent, even
+     when  $~\alpha~$ is consistent. This is because
+     a  
+     standard
+      G\"{o}del-like self-referencing 
+     %diagonalization 
+     construction
+  will 
+     %usually
+     typically 
+     % be able to 
+     produce a proof of $0=1$ from
+     $~\alpha^d\,$, irregardless of whether or not $~\alpha$ is 
+%     formally
+     consistent.}.
+
+
+
+
+
+
+\smallskip
+
+
+ 
+
+
+Our research
+explored special
+circumstances
+\cite{ww1,ww5,ww6,wwapal}
+where it is feasible to
+construct self-justifying formalisms.
+These paradigms involved weakening
+the properties a system can prove about
+addition and/or 
+multiplication
+(to avoid the preceding
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+two 
+3-way predicates
+ indicating
+$x$, $y$ and $z$ satisfy 
+$x+y=z$ and
+$x*y=z$.
+A 
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+1-3 as axioms.
+
+
+\vspace*{- 0.8 em}
+{\  
+\cvl
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 0.6 em}
+
+\cvl
+
+We will say
+a 
+logic
+system 
+$\alpha$
+is 
+{\bf Type-M} iff it contains
+each of \eq{totdefxs} -- \eq{totdefxm}
+as axioms,  
+{\bf Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+and {\bf Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom. 
+A system is called 
+{\bf Type-NS} iff it {\it does not} contain
+any of these axioms.
+
+
+
+
+
+Our investigations
+ \cite{ww1}--\cite{ww7}
+began by observing
+some 
+Type-A systems can  recognize 
+their 
+consistency under semantic tableaux deduction,
+and 
+several
+Type-NS systems 
+can 
+recognize their
+ Hilbert consistency.
+Many of
+these systems were capable of 
+proving 
+%all 
+Peano Arithmetic's
+ $\Pi_1$ theorems
+in a language
+that represents addition and multiplication 
+as
+the
+3-way predicates
+of
+ Add$(x,y,z)$ and  Mult$(x,y,z)$.
+
+
+
+
+
+
+\parskip 1pt
+
+Our self-justifying 
+ evasions of the 
+Incompleteness
+Theorem are difficult to further extend
+primarily because the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} showed
+natural 
+Type-S systems cannot recognize  their own Hilbert consistency.
+Also,  Willard
+ \cite{ww2,ww7,ww9}  
+%
+% wwooo
+%
+strengthened earlier results of 
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  natural 
+Type-M  system cannot recognize their semantic
+tableaux consistency.
+
+\cvs
+
+
+
+
+
+
+\medskip
+
+
+A related
+ class of evasions of the
+ Second Incompleteness
+Theorem
+was discovered
+in \cite{ww9}.
+Let us say 
+ $~\alpha~$ 
+is
+a
+{\bf  Type-Almost-M} 
+axiom system
+iff  $~\alpha~$  can prove
+statements \eq{totdefsymba} and 
+\eq{totdefsymbm}
+as theorems
+while treating
+{\it  
+none of  sentences
+ \eq{totdefxs} -- 
+   \eq{totdefsymbm}
+as axioms.} 
+(Many axiom systems,
+that 
+use
+ function symbols ``$~+~$'' and
+``$~*~$'' for
+formalizing addition and multiplication, 
+fall technically into the
+% obviously  
+Type-Almost-M
+% (rather than Type-M) 
+category.)
+\vspace*{- .2 em}
+\beq 
+\label{totdefsymba}
+\forall x ~\forall y~ \exists z ~~~~x+y=z
+\enq
+\vspace*{- 1.5 em}
+\beq 
+\label{totdefsymbm}
+\forall x ~\forall y ~\exists z 
+ ~~~~x*y=z
+\enq
+\vspace*{-1.4 em}
+
+\noindent
+The preceding is of interest because
+some surprisingly strong 
+(albeit unusual)
+Type-Almost-M systems
+\cite{ww9}
+ have an ability to 
+verify their Herbrand but not
+also semantic tableaux consistency. 
+
+
+
+
+The 
+proofs
+in our prior        papers
+were 
+challenging
+primarily
+ because 
+they required
+one to separate the
+local combinatorial
+methods
+employed in
+\cite{ww93,ww5,wwapal,ww9}'s 
+ particular applications 
+from
+the 
+common principles
+that underlied behind all
+these 
+works.
+Our 
+Theorems \ref{ppp1}, \ref{ppp2} and \ref{pqq4}
+will  rectify this problem by
+identifying 
+common components
+that unite these
+four paradigms.
+(Theorems 
+ \ref{pqq3}, \ref{pqq5},
+  \ref{ppp6}, 
+E.1, G.2 and G.3 will then carry on
+in further
+directions.)
+
+
+%%%iii 1
+
+
+
+All these theorems will 
+contain severe limits on their generality,  so that the
+Second Incompleteness Theorem does not contradict them.
+It is clearly perplexing to imagine
+how humans
+are able to 
+motivate themselves to cogitate, 
+without
+ their thought processes possessing
+some type of
+{\it  at least tentative}
+presumption of 
+their own consistency.
+%It is for this reason that our
+Our
+research
+has thus consisted of an approximately equal effort
+in exploring 
+both 
+\cite{ww2,wwlogos,wwapal,ww7}'s
+new
+% types  of  
+generalizations of the
+Second Incompleteness Theorem
+and 
+%in examining the unusual perspectives of 
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}'s unusual
+boundary-case exceptions to it.
+
+%\parskip 1pt
+
+It is clear 
+every boundary-case exception
+to the  Second Incompleteness Theorem
+has limited scope because
+the  Incompleteness Theorem
+is a broadly encompassing result.
+This
+ paper
+ will, thus, 
+be addressing
+a challenging 
+near-paradoxical
+ question
+about the maximal 
+nature of 
+self-justification
+that
+ can
+ never be resolved
+in a
+fully satisfying
+manner.
+The Second
+ Incompleteness Theorem is
+clearly sufficiently
+central to  
+logic
+for it to be desirable
+to know  
+what {\it partial roads of success} a 
+self-justifying axiom     system can 
+obtain.
+
+%\cvs
+
+%\vspace*{- 0.6 em}
+
+\section{Literature Survey}
+
+
+\label{survey}
+\label{B2-lem}
+\label{D2-def}
+
+
+%\vspace*{- 0.6 em}
+
+
+
+
+
+
+
+
+Two 5-page surveys of the prior literature about the
+Second Incompleteness
+Theorem were provided in our
+articles \cite{ww5,wwapal}. 
+This section will present a 
+more abbreviated survey,
+focusing
+on 
+only 
+those
+developments that are
+particularly germane to
+ the current article.
+
+
+
+The study of incompleteness
+began with 
+four classic papers  by
+G\"{o}del, L\"{o}b, Rosser and Tarski 
+\cite{Go31,Lo55,Ro36,Ta36} and
+with the
+Hilbert-Bernays
+ exploration of 
+their 
+derivability conditions
+\cite{HP91,HB39,Ka91}.
+Generalizations of these results for weak
+axiom systems, such as Q,  began with 
+the work of Tarski-Mostowski-Robinson \cite{TMR53}
+and 
+Bezboruah-Shepherdson \cite{BS76}.
+
+
+
+
+
+
+
+Some 
+more
+notation is needed to describe more
+recent developments.
+Let $~x'~$ denote 
+the ``successor'' operation that maps 
+$x$ onto  $x+1$.
+A formula
+ $ \varphi(x) $ is called \cite{HP91} a
+{\bf Definable Cut} for an
+axiom system $~ \alpha~$   iff
+$~\alpha~$ can prove:
+\begin{equation}
+\label{initdefx}
+\varphi(0) \mbox{  AND  }
+\forall~x~ \{~\varphi(x)\Rightarrow\varphi(~x'~)~ \}
+ \mbox{  AND  }
+ \forall~x ~\forall~y<x
+ ~~ \{ ~\varphi(x)\Rightarrow\varphi(y) ~ \}
+\end{equation}
+Definable cuts 
+and their cousins 
+have been studied by
+an
+extensive literature
+\cite{Ad2,AZ1,Be97,Bu86,BI95,HP91,Ka91,Ko6,Kr87,Ne86,Pa71,PD83,PW81,Pa72,Pu84,Pu85,Pu96,Sm85,Sv83,Sv7,VV94,Vi92,Vi93,Vi5,VH73,WP87}.
+(They
+are
+ unrelated to Gentzen's 
+notion of
+ sequent calculus
+deductive ``cut rule'',
+which uses the word ``cut'' in a  
+different
+context).
+
+\smallskip
+
+A Definable Cut 
+$~\varphi(x)~$
+is called {\bf Non-Trivial}
+relative to
+an axiom system $~\alpha~$ iff
+$\alpha~$ cannot prove  $~\forall ~x~\varphi(x)~,~$
+ {\it although it can prove 
+\eq{initdefx}.}
+Every axiom system $\,\alpha$,
+strictly weaker than Peano Arithmetic, 
+will contain some non-trivial Definable Cut.
+This cut will have the property that 
+$\,\alpha\,$
+ can   verify $~\varphi(n)$ for each 
+fixed integer $~n$,
+although it cannot prove
+ $~\forall ~x~\varphi(x)~$.
+
+\smallskip
+
+
+Let
+ $ \, \lceil  \,  \Psi  \,  \rceil  \, $
+denote
+$   \Psi $'s
+G\"{o}del number,
+and 
+Prf$\, _\alpha^d \, \,(t,p)$
+denote
+that $   p   $ is a proof
+of the theorem $   t   $ from the axiom system $   \alpha   $ 
+using $\, d \,$'s
+deduction method.
+An axiom system
+$   \alpha   $ will
+then
+ be said to recognize its own 
+{\bf Cut-Localized  d-Consistency} 
+relative to a Definable Cut 
+ $ \,  \varphi \,   $
+iff $   \alpha   $ can prove:
+\begin{equation}
+\label{dcon}
+\forall~p~~~~~~ \{ ~~~\varphi(p)~~\Rightarrow~~
+\neg~\mbox{Prf}\,_\alpha^d \, ~(~\lceil 0=1 \rceil~,~p~)~~~ \}
+\end{equation}
+The recent literature 
+has 
+sought to identify which
+triples $   (   \varphi   ,   d   ,    \alpha   )   $ have this property.
+A crucial negative 
+ result about
+ Cut-Localized  d-Consistency, discovered by
+ Pudl\'{a}k
+\cite{Pu85},
+established a 
+significant 
+generalization of G\"{o}del's Second Incompleteness Theorem.
+It
+ showed 
+that
+an axiom system
+$   \alpha   $ must be unable to prove
+\eq{dcon}'s  statement 
+about 
+any of its definable cuts  
+ $\,\varphi \,$, $\,$when
+$   d   $  represents
+ Hilbert deduction and 
+$   \alpha   $ is  any consistent extension of $   Q   $.
+
+
+
+
+
+Solovay \cite{So94} noted how 
+Pudl\'{a}k's
+result could be combined
+  with the techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87} to establish the following 
+theorem
+ that will often be
+cited in this paper:
+\begin{quote}
+  \baselineskip = 1.2 \normalbaselineskip 
+%notre \small
+\small
+\baselineskip = 1.25 \normalbaselineskip 
+\baselineskip = 1.25 \normalbaselineskip 
+{\normalsize \bf  Theorem 2.1 }
+{\it 
+(Solovay's  1994 Generalization \cite{So94}
+of a 1985 theorem of Pudl\'{a}k \cite{Pu85} 
+using some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods )} :
+Let $ \, \alpha \, $ denote any axiom system which
+contains 
+\ep{totdefxs}'s
+Type-S statement
+and which assures 
+%that 
+the successor operation
+always satisfies 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+will be unable to recognize its
+own  Hilbert
+consistency,
+whenever  it
+ treats addition and multiplication
+as 3-way relations 
+satisfying 
+their usual identity,
+associative, commutative and 
+distributive  properties.
+\end{quote}
+Solovay never
+published any
+precise
+proof of Theorem 2.1's hybridizing of
+the work of Pudl\'{a}k, Nelson and Wilkie-Paris 
+\cite{Pu85,Ne86,WP87},  which he 
+privately
+communicated
+ \cite{So94} to us.
+A reader can find 
+generalizations of the Second Incompleteness Theorem 
+that are closely 
+related to
+Theorem 2.1
+in papers by 
+Pudl\'{a}k, Buss-Ignjatovic,
+\v{S}vejdar and Willard
+\cite{BI95,Pu85,Sv7,wwlogos}, as well
+as in 
+Appendix A of 
+\cite{ww1}.
+
+Other interesting observations
+that 
+preceded our research were that
+Wilkie-Paris \cite{WP87} 
+demonstrated  that
+I$\Sigma_0+Exp$ cannot prove the
+Hilbert consistency of even the axiom system $~Q, \,~$
+and that Adamowicz-Zbierski \cite{Ad2,AZ1} showed that
+I$\Sigma_0+\Omega_1$ 
+satisfied the Herbrandized version
+of the Second Incompleteness Theorem.
+Both 
+these results helped  stimulate 
+\cite{ww2,ww7}'s
+ semantic tableaux generalizations
+of  
+the Second Incompleteness Theorem for
+I$\Sigma_0$.    
+
+
+
+
+A fascinating observation by 
+L. A.   Ko{\l}odziejczyk
+\cite{Ko5,Ko6},
+about the difference in lengths between
+semantic tableaux and Herbrandized proofs, 
+also motivated
+our
+investigation \cite{ww9}
+into some surprising properties  of
+unorthodox encodings for I$\Sigma_0$.
+  Ko{\l}odziejczyk 
+observed
+\cite{Ko5,Ko6}
+that various generalizations of
+the Second Incompleteness Theorem
+for  I$\Sigma_0$ and
+I$\Sigma_0+\Omega_i$ in
+\cite{Ad2,AZ1,Sa11,ww2,ww7,ww9} 
+imply
+that the proof of 
+ the Herbrandized version
+of the Second Incompleteness Theorem
+can be
+more complicated 
+than
+ its semantic tableaux counterpart.
+This is because there can be an exponential difference
+between semantic tableaux and Herbrandized proof lengths
+under extremal circumstances.
+It was due to 
+Ko{\l}odziejczyk's
+insightful 
+ communications
+\footnote{ \baselineskip = 1.3 \normalbaselineskip \b55 
+ \label{putglue} The Herbrandized and semantic tableaux
+definitions of an axiom
+ system $\alpha$'s consistency 
+ are 
+different 
+from each other
+because the former
+requires  skolemizing $\,\alpha \,$'s axioms, $\,$while the
+latter permits
+\cite{Fi90} 
+an existential quantifier
+elimination rule
+to
+replace Skolemization.
+This distinction   can
+create a
+ potential
+exponential difference
+ between
+the lengths of
+Herbrand and Semantic Tableaux proofs.
+This 
+insightful observation,
+%distinction,
+due to private communications from
+L. A.  Ko{\l}odziejczyk \cite{Ko5},
+was used in
+ \cite{ww9}
+to create an axiom system that satisfied the
+semantic tableaux but not
+also Herbrandized version of the
+Second Incompleteness Theorem.}
+that \cite{ww9}
+developed 
+an axiom system that was a boundary-case
+exception to the Herbrandized but not also 
+the
+semantic tableaux version
+of the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+
+The literature on Definable Cuts has
+centered its evasions of 
+the Second Incompleteness Theorem around 
+\ep{dcon}'s localization formalism
+(rather than  
+employing  analogs of
+ Section \ref{secc1}'s
+SelfRef$(\alpha,d)$ axiom,
+as we did in \cite{ww93,ww1,ww5,ww6,wwapal,ww9}).
+Pudl\'{a}k \cite{Pu85}
+proved that essentially
+every axiom system
+$    ~     \alpha   ~      $ 
+ of
+finite cardinality can be associated with
+a definable cut  $  ~      \varphi   ~      $ such that 
+$    ~     \alpha   ~      $ 
+can prove
+sentence \eq{dcon}'s  validity for $    ~     \varphi   ~  $
+when $    ~     d   ~      $ is either
+the semantic tableaux or Herbrand-styled
+deductive method.
+
+
+
+Pudl\`{a}k's 
+theorem is related to 
+Friedman's
+observation \cite{Fr79b}
+that 
+for many finite theories
+$S$ and $T$,
+the theory $  S  $ has an interpretation in $  T  $
+if and only if
+I$\Sigma_0+Exp$ can prove that
+$T$'s Herbrand consistency implies
+$S$'s Herbrand consistency.
+Several generalizations of these results 
+by 
+ Kraj\'{i}cek, 
+Pudl\`{a}k,
+ Smory\'{n}ski,
+\v{S}vejdar 
+  and Visser appear  in
+\cite{Kr87,Pu85,Pu96,Sm85,Sv7,Vi92,Vi93,Vi5}.
+Visser's article \cite{Vi5} 
+contains 
+an excellent review of this literature,
+as well as
+many 
+additional
+new results.
+Also,
+we will see how 
+% some of
+some of the reflection machines of
+Beklemishev,
+Kreisel-Takeuti and
+ Verbrugge-Visser
+\cite{Be95,Be97,Be3,KT74,VV94,Vi5}
+nicely complement
+\thx{ppp6}'s
+ reflection mechanisms
+in alternate types of intended applications.
+
+
+It was established by 
+H\'{a}jek,  
+\v{S}vejdar and
+Vop\v{e}nka \cite{Sv83,VH73}
+that GB Set Theory can
+construct
+a definable cut $\varphi$ where
+it can prove
+the statement  \eq{dcon} is  valid when
+$~d~$ denotes Hilbert deduction and
+$~\alpha~$ 
+is ZF Set Theory.
+This result was
+ surprising because Pudl\'{a}k \cite{Pu85}
+showed GB can never verify its own 
+Hilbert consistency 
+localized
+on a definable cut. (Thus, GB will  view its Hilbert consistency as
+equivalent to ZF's Hilbert consistency
+in a global sense 
+ {\it but not} in a  cut-localized
+respect.).
+
+
+%% 
+%% H\'{a}jek,  
+%% \v{S}vejdar and
+%% Vop\v{e}nka \cite{Sv83,VH73}
+%% constructed
+%% a definable cut $\varphi$ where
+%%  GB Set Theory
+%% can prove
+%% that the statement  \eq{dcon} is  valid when
+%% $~d~$ denotes Hilbert deduction and
+%% $~\alpha~$ 
+%% is ZF Set Theory.
+%% This is
+%%  surprising because Pudl\'{a}k \cite{Pu85}
+%% proved 
+%% GB can never verify its 
+%% Hilbert consistency 
+%% localized
+%% on a definable cut.
+%% Thus, GB will  view its Hilbert consistency as
+%% equivalent to ZF's Hilbert consistency
+%% in a global sense 
+%%  {\it but not} in a  cut-localized
+%% respect.
+%% 
+
+\bigskip
+
+
+In some sense, 
+ Kreisel and Takeuti
+  \cite{KT74,Ta53}
+can be viewed as
+the 
+first authors to develop a
+%%type of   system 
+logic
+recognizing its own
+ consistency
+using a
+variant of \ep{initdefx}'s
+formula.
+Their
+results
+for typed logics
+%  have  
+formalized a second-order generalization of 
+Gentzen's sequent calculus
+that can verify
+% a declaration of
+ its own consistency, 
+%in the special case where 
+when
+no sequent calculus deductive
+cuts are performed.
+A key aspect of their formalism
+can be seen as using
+an analog of
+\eq{dcon}'s sentence in an implicit 
+manner. 
+It thus begins
+by using a set of objects, which we shall call $~I,~$
+that includes all the standard integers 
+plus  some
+allowed
+ {\it non-standard integers}
+(that can 
+permissibly
+represent 
+contradictory  proofs).
+  %, and
+Their second-order logic
+ then uses Dedekind's definition of the natural numbers to 
+construct a 
+subset of $~I~$, $\,$called
+ ``$~N~$'', which
+includes all the standard integers
+and which is disallowed to contain any
+contradiction proof.
+We will not go into the details here, but this
+transition from $\,I\,$ to $\,N\,$ (with an 
+accompanying
+relativization of the provability predicate onto
+$\,N\,$'s more restricted domain) can be viewed
+as Kreisel-Takeuti's
+analog
+ of %    sentence  
+\eq{dcon}'s 
+ local consistency 
+statement
+for
+\cite{KT74,Ta53}'s
+``CFA''
+ second-order logic.
+
+
+
+
+
+
+It is difficult to compare our   research
+(which 
+has
+relied upon an analog of 
+SelfRef$(\alpha,d)$'s
+Kleene-like
+{\it ``I am consistent''} axiom)
+with the
+preceding literature
+that has
+used 
+various forms of
+Localized  d-Consistency 
+statements.
+This is because every effort to 
+evade the Second Incompleteness
+Theorem
+employs
+some built-in weakness
+to evade G\"{o}del's classic paradigm.
+
+
+\smallskip
+
+Our 
+work
+in \cite{ww93,ww1,ww5,ww6,ww9}  represented
+less than a 
+full-scale evasion of the
+Second Incompleteness Theorem
+mostly
+because 
+it was
+ incompatible with treating 
+{\it as  formal
+axioms}
+the statements in
+Equations \eq{totdefxm}
+and \eq{totdefsymbm}
+that
+ multiplication is a total function
+\footnote{ \b55 This 
+caveat applies
+also  to our article \cite{ww9},
+although its
+Herbandized form of
+self-justification
+differs from our other papers
+by retaining a capacity to treat 
+\eq{totdefsymbm}'s
+statement about multiplication's totality as a
+derived theorem {\it that is
+not an} 
+%  a formalized  
+axiom.
+The key point is that  
+theorems are
+ weaker
+than  axioms
+under 
+Herbrand deduction 
+ because only
+axioms are used as intermediate steps 
+during proofs.
+This
+explains intuitively
+how    \cite{ww9}'s formalism
+was able to recognize
+its 
+%own 
+Herbrandized
+consistency, $\,$while 
+treating
+\eq{totdefsymbm}'s statement about
+the totality of
+   multiplication 
+ {\it
+as a theorem}.
+(We will return to this subject in 
+Appendix D.) } .
+%(More will be said about this subject at the end of
+%Appendix D.) } .
+Some 
+reasons
+why
+it was helpful for
+\cite{ww93,ww1,ww5,ww6,ww9}
+to employ 
+ analogs of    
+Section \ref{secc1}'s
+SelfRef 
+axiom are that:
+\bed
+\topsep -3pt
+\itemsep +2pt
+\item[A  ]
+An axiom's self-referential
+{\it ``I am consistent''} declaration
+allows a formalism to recognize its consistency
+in 
+a
+global sense, rather than in the 
+$\varphi -$localized sense
+used by
+sentence
+\eq{dcon} and 
+the analogous 
+  Kreisel-Takeuti relativization of their
+second-order
+proof predicate.  
+\item[B  ]
+If a logic is employing a deductive method
+$d$ that lacks a modus ponens rule, as occurs in
+nearly all
+self-justifying
+systems, then it is
+          preferable for  it           to 
+view its {\it ``I am consistent''} 
+statement as an axiom rather
+than as a theorem.
+(This is
+ because 
+weak deductive methods are
+ capable of
+drawing 
+logical inferences
+only 
+from
+axioms
+when
+modus ponens 
+is absent.) 
+\item[C  ]
+Analogs of
+Section \ref{secc1}'s
+SelfRef$(\alpha,d)$'s
+{\it  ``I am consistent''} axiom
+have been shown
+by 
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}
+to at least partially
+formalize the notion of a 
+logic  
+possessing  {\it  an 
+ almost}
+instinctive  
+form of 
+faith in its
+own internal consistency.
+(This paper will make it apparent that
+such an instinctive faith is less than a full-scale
+proof. Yet, Theorem \ref{ppp6} and
+ Remarks 
+\ref{f88},
+\ref{remhappy} 
+and \ref{recc1}
+will make it apparent that such formalizations
+of instinctive faith are also
+useful.) 
+\ennd
+We emphasize
+that both  virtues and drawbacks of
+SelfRef$(\alpha,d)$'s
+{\it  ``I am consistent''} 
+axiom statements 
+have been 
+cited
+in this paragraph because
+every effort to
+evade the Second Incompleteness
+Theorem can obtain no more than
+limited levels of
+success.
+
+
+% \nop
+
+\cvmew
+
+The scope of the challenge we face becomes apparent when one
+realizes 
+ $\alpha \, + \,$SelfRef$(\alpha,d)$ 
+is 
+ inconsistent 
+for most
+$(\alpha,d).~$
+  This is because
+   $\alpha+$SelfRef$(\alpha,d)$ 
+typically
+satisfies Part-i
+  {\it but not also }
+  Part-ii of
+Section 
+\ref{secc1}'s  definition 
+ of a ``self-justifying'' logic.
+(Thus, 
+a
+% G\"{o}del-like 
+diagonalization
+paradigm will typically 
+imply 
+$\alpha \, + \,$SelfRef$(\alpha,d)$ 
+is inconsistent,
+$\,$as a 
+consequence of 
+it containing $ ~$SelfRef$(\alpha,d)~$ 
+as an axiom.)
+This is the
+reason      Kleene, 
+ Rogers and Jeroslow \cite{Kl38,Ro67,Je71}
+were 
+hesitant
+%skeptical
+about 
+the utility of
+SelfRef$(\alpha,d)$'s
+mirror-like
+axiom sentence.
+Our goal 
+in \cite{ww93}-\cite{ww9} 
+has 
+been
+ to 
+develop generalizations and boundary-case exceptions for
+the Second Incompleteness Theorem, so as 
+determine exactly
+which paradigms
+%%\
+%%\{\it 
+%%\paradigms
+%%\are sufficiently
+%%\weak}
+%%\so that 
+%%\they 
+%%\
+can
+support, for example,
+Theorem \ref{ppp6}'s limited notion
+of self-justification.
+
+
+
+
+
+
+The 
+%intuitive
+ reason 
+one would  anticipate some 
+{\it limited}
+%partial
+exceptions to the Second Incompleteness Theorem 
+to
+exist is 
+it is  hard to 
+imagine how humans can motivate
+themselves to cogitate without 
+using 
+some
+variant 
+of 
+self-justification.
+
+%%
+%%Thus, our %recent 
+%%research 
+%%project
+%%has sought to {\it both} generalize
+%%  the Second Incompleteness
+%%Theorem and explore when it can be evaded.
+
+
+
+
+
+
+
+
+
+
+
+\cvl
+
+\section{Generic Configurations}
+
+\label{3uuuu1}
+
+
+% \vspace*{- 0.2 em}
+
+
+
+%Our main results about self-justification will
+%appear in Sections 
+%\ref{3uuuu2} -- \ref{sect64}.
+The
+ phrase  {\bf Bounded Quantifier}
+will refer
+to
+expressions of the form
+``$\, \exists~v  \leq  T \,$'' or
+``$ \,\forall~v  \leq  T\,$''
+where $T$ is a term.
+A formula  
+is
+called
+{\bf Fully-Bounded$\,$} when all its quantifiers are
+   so bounded.
+\lem{lex22} will soon explain how Definition \ref{xd+1x1}'s 
+formalism can encode conventional arithmetic:
+ 
+
+
+
+
+
+
+
+
+
+\medskip
+
+
+\bxbxdd
+\label{xd+1x1}
+\rm
+Let $~\xi~$ denote some 
+non-integer
+indexing superscript
+(whose  properties will 
+be 
+discussed
+ later by
+Definition
+\ref{def3.3}). 
+Then the symbol
+$~\Delta_0^\xi~$
+will 
+denote some 
+fixed
+special
+set of fully-bounded formulae
+that
+is
+closed 
+under negation, 
+in a 
+language
+that will be later
+ called $~L^\xi~$.
+(Thus, if some
+formula $~ \Psi~$ is a member of
+             $~\Delta_0^\xi~$
+then so is 
+$~ \neg ~ \Psi~$.)
+Items 1-3 
+formalize how
+$~\Pi_n^\xi~$ 
+and 
+$~\Sigma_n^\xi$ formulae are
+built 
+in a straightforward manner
+out of
+these
+ $~\Delta_0^\xi~$
+sub-components:
+\bee
+\topsep -15pt
+\item
+Every $ \, \Delta_0^\xi \, $ formula is
+considered to be also a
+$ \, \Pi_0^\xi \, $ 
+and 
+$ \, \Sigma_0^\xi$ formula.
+\item
+For $~n \, \geq \, 1~~,~$
+a formula 
+will be 
+called
+$ ~~  \Pi_n^\xi~~$
+iff it can be
+written 
+in the canonical form of
+$\forall v_1 ~ \forall v_2 ~...~ \forall v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$  where
+$\Phi$ is  $~\Sigma_{n-1}^\xi.~$
+\item
+Likewise for $~n \, \geq \, 1~~,~$
+ a formula will be
+called
+$~~ \Sigma_n^\xi~~ $
+iff it can be
+written in the 
+form of
+$~\exists v_1 ~ \exists v_2 ~...~ \exists v_k ~~~ \Phi(v_1,v_2,..v_k \,),~~~$
+  where
+$\Phi$ is  $~\Pi_{n-1}^\xi.~$
+\ene
+\eedd
+
+\medskip
+\noindent
+{\bf Notation Convention:}
+Our rules for defining $\,\xi$, specified later in this section,
+will never have this  superscript 
+designate
+an integer quantity. This is because 
+integer superscripts have a special meaning under 
+a typed-based hierarchy,
+not intended here.
+
+
+
+
+\nop
+
+\begin{example}
+\label{ex3-1}
+\rm
+Let
+$~L~$  denote a 
+conventional arithmetic
+language that uses
+function symbols 
+for denoting 
+addition and multiplication.
+Below are two
+examples  
+of $\Delta_0-$like formulae that
+invoke
+Definition \ref{xd+1x1}'s
+notation:
+\bed
+\itemsep 1pt
+\item[  a ] The 
+symbol ``$~\Delta_0^A~$''
+will denote any 
+fully bounded
+formula that uses the addition, multiplication and maximum
+function symbols in an arbitrary manner.
+(Thus,  $~\Delta_0^A~$ 
+corresponds to what 
+many
+textbooks
+\cite{HP91,Ka91,Kr95}
+simply call a ``$~\Delta_0~$'' formula.)
+\medskip
+\item[  b ] The 
+symbol ``$~\Delta_0^{R}~$'' 
+will denote a class of 
+formulae in $L$'s language
+whose bounded quantifiers are  allowed
+to use
+only 
+  the Maximum function symbol. 
+Their
+bodies,
+however, may contain
+any combination of 
+addition, multiplication
+and maximum function symbols.
+\ennd
+Formulae \eq{ex1}  
+and \eq{ex123} 
+illustrate the
+distinction
+between the  $\Delta_0^A$ 
+and  $\Delta_0^{R}$ classes. Thus, \eq{ex1} satisfies the first
+but not second condition (on account of the presence of the
+multiplication symbol used by its bounded quantifiers).
+ In contrast,
+\eq{ex123}
+is an example of a  
+ $~\Delta_0^{R}~$ formula.
+\beq
+\label{ex1}
+\exists \, y\leq  x*x ~~~~
+\forall  \, z\leq  y*y  ~~~~
+\exists \, w\leq  y*z ~~~~
+~~~~~: ~~~~~ 
+ \{ ~~~~ x  *y=z+w ~~~~ \}
+\enq 
+\vspace*{-0.5 em}
+\beq
+\label{ex123}
+\exists \, y\leq  x ~~~~
+\forall  \, z\leq  y  ~~~~
+\exists \, w\leq  \mbox{Max}(y,z) ~~~~
+~~~~~: ~~~~~ 
+ \{ ~~~~ x  *y=z+w ~~~~ \}
+%% x  *y=z+w
+\enq 
+The distinction between 
+ $ \Delta_0^{A}  $
+arithmetic formulae and the unconventional
+ $ \Delta_0^{R}  $ class may first convey the impression
+that these
+two 
+classes 
+have
+fundamentally 
+different natures. Actually,
+\lem{lex22} will
+show
+that their
+relationship is more subtle.
+This is because
+its formalism
+will map  $ \Delta_0^{A}  $ formulae
+onto     $ \Delta_0^{R}  $
+expressions that are equivalent to it under Definition
+\ref{snn}'s Standard-M model
+$~$---$~$ in a context where
+only the length of these
+formulae
+is allowed to 
+possibly
+grow.
+This 
+% result
+% Standard-M 
+equivalence
+ enabled
+ \cite{ww9} to construct
+a natural axiomatic formalism
+that could recognize its
+own
+Herbrandized consistency  
+{\it but which nevertheless  satisfied} 
+the 
+idealized form   
+\footnote{  \b55 An axiom system $\alpha$ 
+is defined to
+ satisfy
+the 
+{\bf ``idealized form''}
+of the
+semantic tableaux
+version 
+ Second Incompleteness Theorem
+when no  $\beta  \supseteq  \alpha$ can
+prove a
+ semantic tableaux
+proof
+ of 0=1
+ from 
+itself is incapable of existing.
+We will summarize
+\cite{ww9}'s formalism
+and the distinction between
+Herbrandized and semantic tableaux 
+deduction 
+at the 
+of 
+Appendix D.} 
+of
+the 
+semantic tableaux
+version of the 
+Second Incompleteness Theorem.
+\end{example}
+
+
+
+
+
+
+
+
+
+\medskip
+
+\begin{definition}
+\label{snn}
+{\bf ``Standard-M''} will denote
+the
+standard model of integers.
+\end{definition}
+
+\medskip
+
+The reason for our interest in Standard-M is
+that
+many 
+pairs of formulae
+are equivalent
+under the Standard-M
+model, while
+weak axiom systems often
+cannot formally prove they are
+equivalent.
+For instance,  this 
+will occur
+when  Example \ref{ex3-2}
+examines
+Definition \ref{def3.3}'s properties.
+
+
+%\notre-only \medskip
+
+
+\nop
+
+\bxbxdd
+\label{def3.3}
+A  {\bf Generic Configuration},
+often identified by 
+ the superscript symbol of
+ $~\xi~$, is defined to be a 5-tuple
+$~(\, L^\xi \, , \, \Delta_0^\xi \,  , \, B^\xi \, , \, d \, , \, G \, )~$
+where:
+\bee
+\topsep -12pt
+\rm
+\item
+$ L^\xi $  
+is a 
+language that includes 
+logical symbols
+for ``$0$'',  ``$1$'',  ``$2$'',
+``$=$'', ``$\leq$'' 
+ and
+for the operation of ``Maximum(x,y)''.
+$ L^\xi $   also includes a 
+ sufficient number of 
+function  and constant symbols 
+so that every integer $~k~$ can be encoded by
+some term 
+$~T_k~$ 
+specifying
+$~k\,$'s value.
+\item
+$~ \Delta_0^\xi ~$ 
+corresponds to 
+any variation
+of
+Definition \ref{xd+1x1}'s 
+class of 
+``fully-bounded'' formulae
+that is rich enough to assure that there exists
+ two  $~\Delta_0^\xi~$ 
+formulae, henceforth
+called ``Add$(x,y,z)$''
+and ``Mult$(x,y,z)$'', $\,$for formalizing 
+the graphs of
+addition and multiplication.
+(It will  generate 
+$\xi$'s set of $\Pi_n^\xi$
+and $\Sigma_n^\xi$ sentences,
+using Definition \ref{xd+1x1}'s 3-part formalism.)
+\item 
+$ \, B^\xi \,$  
+denotes 
+a$~$ {\normalsize {\bf ``Base Axiom System''}}, 
+$~ \,$whose axiom-sentences are true under
+the 
+Standard-M model
+and which is
+$\Sigma_1^\xi$ complete.
+(Thus,
+$  B^\xi $  can  prove every true $\Sigma_1^\xi$ sentence, and
+it can
+likewise refute all false $\Pi_1^\xi$ sentences.)
+\item
+$~d~$   denotes  $ \, \xi \,  \, $'s  
+ method of deduction.
+It is required to be sufficiently conventional to
+satisfy the usual indirect-implication 
+property \footnote{  \b55   
+$~~$This is that irregardless of whether or not
+$~d~$ contains a built-in
+modus
+ponens rule, it does support some
+form of
+a (possibly
+quite 
+lengthy)
+ proof of a
+theorem $Z$,
+when it is able to prove
+$X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as
+theorems. }
+associated with
+ G\"{o}del's Completeness Theorem. 
+\item
+$~g~$   denotes  a method for
+encoding the  G\"{o}del numbers
+of proofs.
+\ene
+\eedd
+
+
+
+
+
+\medskip
+
+\begin{example}
+\label{ex3-2}
+\rm
+Let us recall
+Example \ref{ex3-1} defined
+``$~\Delta_0^A~$'' 
+as essentially the conventional
+textbook notion
+\cite{HP91,Kr95}
+ of an arithmetic 
+ ``$~\Delta_0~$'' formula.
+This example  will  outline how
+well-known techniques can map
+every $\Delta_0^A$  formula 
+onto a semantically 
+equivalent 
+ $\Delta_0^\xi$ 
+formula under the Standard-M model.
+
+
+
+
+
+
+
+
+
+Our discussion will have
+Seq$(x)$   denote a 
+function that
+maps 
+non-negative integers onto 
+binary strings
+in
+lexicographic order.
+Thus Seq$(x)$  maps
+0 onto the empty
+string, 
+$\,$the integers
+1 and 2 onto the strings of ``0'' and ``1'', $\,$the integers 3--6
+onto$\,$  ``00'',   ``01'',   ``10'',   ``11'', etc.   
+(Formally,
+Seq$(x)$
+is
+an operation
+that maps integer $~x~$ onto the bit-string 
+that occurs to the immediate right of the leftmost ``1'' bit in
+the binary encoding of $~x+1.~~)$
+
+  
+
+Given any $\,k-$tuple  $~(~x_1\, , \, x_2\, , ~ ... ~ x_k~),~$
+let  $~$STRING$(   x_1   ,   x_2   ,\, ... \,   x_k   )$
+denote the
+ concatenation of
+Seq$(x_1), \, ... ~$Seq$(x_k)$.
+For any 
+integers $v$ and $w$ satisfying $v  \leq w^2$, 
+it is clear that 
+there  exists $(x_1,x_2,x_3)$ where
+STRING$(x_1,x_2,x_3)$ 
+represents $v  $'s
+binary encoding and
+each $x_i  \leq   \mbox{Max}(w,4)$.
+
+\smallskip
+
+An example will now illustrate
+the approximate structure of an inductive methodology
+for mapping
+$\Delta_0^A$ formulae 
+onto their equivalent   $\Delta_0^\xi$  counterparts
+in the Standard-M model.
+Let SQUARE$(x_1,x_2,x_3,w)$ be a 
+$\Delta_0^\xi$ formula which 
+specifies 
+that
+STRING$(x_1,x_2,x_3)$ represents  an integer 
+$ \leq \, w^2~.~ \,$ Also,  
+$\,$let $\phi^*(x_1,x_2,x_3)$
+and $\phi(v)$ represent 
+a pair of  $\Delta_0^\xi$ and
+$\Delta_0^A$ formulae 
+that
+are equivalent
+under the Standard-M model
+$\,$when STRING$(x_1,x_2,x_3)$ is an encoding for  $~v~$. 
+Then one
+possible method
+for mapping $\Delta_0^A$ formulae onto their 
+equivalent $\Delta_0^\xi$ 
+counterparts 
+(in the Standard-M model)
+could 
+map the
+formula \eq{exam21} 
+onto 
+\eq{exam22}'s alternate form:
+\beq
+\label{exam21}
+\forall ~ v\, \leq \, w^2~~~~~\phi(v)
+\enq
+\smallskip
+\begin{center}
+$\forall ~ x_1 \leq~ \mbox{Max}(w,2)~~~~~\forall ~ x_2 \leq ~ \mbox{Max}(w,2)
+  ~~~~~\forall ~ x_3 \leq ~ \mbox{Max}(w,2)$
+\end{center}
+
+\vspace*{-0.5 em}
+
+\beq
+\label{exam22}
+  \{~~ \mbox{SQUARE}(x_1,x_2,x_3,w)
+  ~~ \Rightarrow~~
+\phi^*(x_1,x_2,x_3)~~\}
+\enq
+\lem{lex22}
+indicates
+Example \ref{ex3-2}'s
+translational methodology generalizes easily to
+       all combinations of   $\Delta_0^A$  inputs
+and
+generic configurations $\, \xi \,$,
+via
+an 
+approximate
+inductive generalization of the transition from
+sentence \eq{exam21} to \eq{exam22}. (Its 
+procedure essentially
+performs
+iteratively a finite number of such transitions, so as to
+translate   all 
+the
+clauses of
+ an
+initial
+$\Delta_0^A$ 
+% format
+formulae
+into their 
+$\Delta_0^\xi$ 
+counterparts via an inductive methodology.
+The intuition behind these transitions is 
+they will repeatedly replace a single variable,
+such as $~v~$ in 
+sentence \eq{exam21},
+with a multiplicity of variables,
+such as $(x_1,x_2,x_3)$ in 
+\eq{exam22}.)
+\end{example} 
+
+
+
+\smallskip
+
+\begin{lemma}
+\label{lex22}
+{\rm (Paris-Dimitracopoulos
+\cite{PD82} )}
+For every generic configuration $\xi$, each
+$\Delta_0^A$ formula
+can be translated
+into an  equivalent 
+$\Delta_0^\xi$ 
+formula
+in the
+Standard-M
+model
+via a generalization of Example \ref{ex3-2}'s
+process.
+{\rm (This 
+ clearly
+ implies 
+$\Pi_j^A$ 
+formulae
+can
+also be
+translated
+into
+$\Pi_j^\xi$ 
+expressions.)}
+\end{lemma}
+
+
+
+\smallskip
+
+
+
+
+Paris-Dimitracopoulos \cite{PD82}
+sketched   
+an 
+%approximate
+analog of Lemma \ref{lex22}'s
+translation 
+algorithm,
+using
+only
+ slightly different notation,
+ that
+is applicable to 
+any formalism that satisfies Parts (1) and (2)
+of 
+ Definition \ref{def3.3}.
+Their formalism
+thus uses an 
+inductively-iterated
+analog of the prior example's
+replacement of a single variable $~v~$ in formula
+\eq{exam21} with \eq{exam22}'s multiplicity of variables 
+% in the tuple 
+$~(x_1,x_2,x_3)~,~$ 
+so as to perform
+ Lemma \ref{lex22}'s translation 
+task.
+It will  be unnecessary for a reader to 
+consider the
+details behind 
+% either 
+\cite{PD82}'s Theorem 1 or
+Lemma \ref{lex22}'s 
+similar translation 
+mechanism
+%%% 
+%%% 
+because
+ the remainder of this article will never
+% explicitly
+ use
+them again.
+%its mechanism.
+Instead, their sole purpose 
+% of  Lemma \ref{lex22}
+has been
+% merely
+ to provide an {\it implicit backdrop}
+for our results by illustrating how
+the study of the $\Pi_1^\xi$ sentences of
+Definition \ref{def3.3}'s generic configurations
+% also
+provides information
+about   $\Pi_1^A$ sentences 
+(after the needed translating is done).
+
+\smallskip
+
+Four  examples of 
+self-justifying
+systems that employ  
+Definition \ref{def3.3}'s 
+ $\Pi_1^\xi$ sentences 
+ will be
+illustrated in 
+\appD.
+These examples   are 
+too
+complicated
+to be 
+examined
+ before Sections \ref{3uuuu1} -- \ref{sect64}
+are 
+read.
+However, the next example 
+should convey some useful  intuitions:
+
+\medskip
+
+\nop
+
+\cvwew
+
+\begin{example}
+\label{ex3-3}
+\rm
+Let
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+denote sequences defined by:
+\vspace*{- 0.7 em}
+\beq
+\label{zs}
+x_0~~~=~~~2~~~=~~~y_0
+\enq
+\beq
+x_{i}~~~=~~~x_{i-1}~+~x_{i-1}
+\label{as}
+\enq
+\beq
+y_{i}~~~=~~~y_{i-1}~*~y_{i-1}
+\label{bs}
+\enq
+\end{example}
+For $\, i \, > \,0 \,$, 
+$\,$let $ \, \phi_{i} \, $ 
+and $ \, \psi_{i} \, $ 
+denote the  
+sentences in 
+\eq{as} and \eq{bs}
+respectively.
+Also, 
+ let
+  $ \, \phi_{0} \, $ and
+$ \, \psi_{0} \, $ 
+denote \eq{zs}'s
+sentence.
+Then
+ $ \, \phi_0, \, \phi_1, \, ... \, \phi_n \, $
+imply
+ $ \, x_n \, = \, 2^{ n+1} \, , \, $ and 
+ $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
+ imply $ \, y_n \, = \, 2^{2^n} \, $.
+Thus, the  latter sequence
+grows at a 
+faster
+rate than 
+the former.
+Much of our research has used the difference between the
+growth rates of 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+as a motivating example explaining why 
+\ep{totdefxa}'s Type-A axiom systems can
+support a stronger form of boundary-case exception
+to the semantic tableaux version of the Second Incompleteness
+theorem than
+can 
+Type-M systems.
+
+\parskip 2pt
+
+
+\smallskip
+
+
+Let
+Log$(\, y_n \,) \, = \, 2^{n} \, $ 
+and
+Log$(\, x_n \,) \, = \, {n+1} \, $ 
+thus
+designate the lengths of the binary codings for
+$ \, y_n \, $
+ and 
+$ \, x_n \, $.
+Then $ \, y_n\,$'s coding 
+has a length
+$\,  2^{n} \,  $, which is
+ {\it much larger} than
+the $  n+1  $
+steps that  $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
+use to
+define its existence.
+However,
+ $ \, x_n\,$'s 
+length  has a 
+ smaller
+ size of
+$ \, {n+1} \, $.
+These observations are useful because every proof 
+of the 
+Incompleteness Theorem
+involves 
+a 
+G\"{o}del
+ number $ \, z \,$ 
+coding a sentence
+that has a  capacity
+to self-reference its own definition.
+The faster
+growing series $y_0,\,y_1,\,,\,...\,y_n$
+should  
+be intuitively 
+anticipated
+to have
+ this 
+self-referencing
+capacity because 
+ $~y_n\,$'s binary encoding 
+has a 
+$~2^{n+1}~$  length that 
+dwarfs the
+size of the $O(n)$
+steps 
+used
+ to define its
+value. Leaving aside 
+\cite{ww2,ww7}'s
+%
+% wwoooo
+%
+many details,
+this
+fast growth 
+explains
+roughly
+ why many Type-M 
+logics
+satisfy the semantic tableaux version of 
+the Second Incompleteness Theorem.
+
+
+\smallskip
+
+This paradigm also 
+illustrates intuitively
+why some 
+ Type-A  systems, employing
+\cite{ww93,ww1,ww5}'s
+semantic tableaux formalism,
+can 
+represent
+boundary-case exceptions to the
+ Second
+Incompleteness
+Theorem.
+ This is because such formalisms 
+lack access to 
+\ep{totdefxm}'s axiom that multiplication is a total function.
+(They are
+unable,
+thus,
+ to
+easily
+ construct numbers $ \, z \, $ that can
+self-reference their own definitions
+because they have access  only
+to the slower growing
+              addition primitive.)
+In particular
+assuming only that each sentence in
+the axiom-sequence
+  $   \phi_0,   \phi_1,   ...   \phi_n   $
+(from  \ep{as} )
+requires  a mere 
+two bits 
+for its encoding, 
+the length    $     n+1     $ of 
+ $   x_n   $'s
+binary encoding 
+will be 
+smaller
+than the
+length of its
+defining 
+ sequence.
+
+\smallskip
+
+This short length for  $   x_n   $
+had
+motivated 
+\cite{ww93,ww1,ww5,ww6}'s
+evasion of  
+the semantic tableaux 
+version
+of the
+Second Incompleteness Theorem. 
+It
+suggested that the self-referencing
+needed in a
+G\"{o}del-like diagonalization argument would stop being
+feasible
+when \ep{as}'s slow-growing 
+$x_1,\,x_2,\,x_3,\,...$
+sequence represents the fastest growth that is possible.
+
+
+
+
+One of the several goals in this article 
+will be
+ to formalize 
+a generalizations of
+\cite{ww93,ww1,ww5,ww6}'s
+self-justifying methodologies by using
+Definition \ref{def3.3}'s
+generic configurations.
+The  proofs
+of  our main theorems
+will, of course, be
+more subtle than
+the hand-waving intuitions appearing in this example.
+For instance, the combined work of
+Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87}
+(summarized by Theorem 2.1)
+raised the subtle issue that
+  no Type-S   system
+can
+prove a theorem affirming its
+own Hilbert consistency. 
+Another complication 
+is that the
+\ep{bs}'s implication for proofs that use
+the multiplication operative has
+different side effects for
+Herbrandized and  
+semantic tableaux
+deduction
+(on account of
+ Ko{\l}odziejczyk
+\cite{Ko5,Ko6}'s
+previously mentioned
+observations about the potential
+exponential difference between the lengths of these
+proofs
+under extremal
+circumstances). 
+
+
+
+
+\smallskip
+
+Our
+main theorems
+will show
+that
+self-justifying systems, 
+using four
+deduction methods,
+are
+capable of  proving 
+all of Peano Arithmetic's  $\Pi_1^\xi$ 
+theorems.
+Interestingly,
+self-justification
+will
+be compatible with \cite{ww5}'s modification
+of semantic tableaux deduction, 
+that includes a
+ modus ponens  rule 
+ for 
+ $\Pi_1^\xi$  and  $\Sigma_1^\xi$ type sentences.
+However,
+ \cite{wwlogos}
+has shown 
+an analogous
+  modus ponens rule for
+ $\Pi_2^\xi$  and  $\Sigma_2^\xi$  sentences
+is incompatible
+with self justification.
+(Thus, the contrast between our
+main 
+results and the
+Second Incompleteness
+Theorem's generalizations will be quite tight.)
+
+
+
+\cvl
+
+
+\section{Five Helpful Definitions and An Informative Lemma}
+
+\label{3uuuu2}
+
+\label{D.4-theorx} 
+
+
+This section will  introduce 
+five definitions and 
+ prove 
+a 
+Lemma \ref{lemex4} 
+about
+self justification.
+This lemma 
+will be 
+% much 
+weaker
+than
+Sections \ref{3uuuu3}
+and \ref{sect64}'s
+main results.
+Its
+main
+purpose
+ will be to provide
+a
+useful
+starting example. 
+
+
+
+
+
+\smallskip
+
+\bxbxdr
+\label{xd+1x3}
+The symbol
+``E$(n)$'' 
+will denote some
+term 
+in Definition \ref{def3.3}'s
+language  $ \, L^\xi \, $
+that represents the
+value  
+ $ \, 2^n \, . \, $ 
+In using 
+this symbol, we  do
+not presume that
+$ \, L^\xi \, $ possesses a function
+symbol for 
+the exponent operation.
+Thus if  $ \, L^\xi \, $
+has only a function symbol for
+multiplication,
+then  
+ E$(n)$ 
+could 
+designate
+the term of $ \, $``$ \, 2*2* \, ... \, *2 \, $''$ \, $ with
+$ \, n \, $ repetitions of ``2'' $. \, $ (Alternatively,
+ E$(n)$ 
+can be
+ defined via
+applying
+ $\, 2^n \,$ iterations
+of the successor function to zero,
+or by having a
+special constant symbol  designating
+ $ \, 2^n \, $'s value.
+Essentially, any reasonable method can
+be used to define  E$(n)$'s value)
+
+
+\eedd
+
+\cvrew
+
+\nop
+
+\smallskip
+
+\bxbxdr
+\label{xd+1x4}
+Let
+$\,\Uxp \,$ denote
+a prenex normal sentence.
+Then 
+{\bf Scope$_E$($\Uxp,$N)$~$} will  denote
+a sentence identical to $~\Uxp~$ except that
+every unbounded universal quantifier  ``$~\forall~v~$''
+is changed to  ``$ \, \forall \, v \, < \, E(N) \, $'',
+and every
+ unbounded existential quantifier  ``$ \, \exists \, v \, $''
+is changed to ``$ \, \exists \, v \, < \, E(N) \, $''.
+(No change is made among the bounded quantifiers within
+the
+$~\Delta_0^\xi$ 
+part of the sentence
+$\,\Uxp \,$.) 
+For example, if 
+$\,\Uxp \,$ denotes
+the $\Pi_1^\xi$ 
+sentence of
+$~\forall \, v_1~\forall \, v_2~...~\forall \, v_k~~~\phi(v_1,v_2,...v_k)~$
+then
+\eq{scopede} 
+illustrates
+Scope$_E$($\Uxp,$N)'s form.
+Likewise if
+$~\Uxp ~$  is
+the $~\Sigma_1^\xi~$ 
+sentence of
+$~\exists \, v_1~\exists \, v_2~...~\exists \, v_k~~~\phi(v_1,v_2,...v_k)~$
+then
+\eq{scopedx} illustrates
+Scope$_E$($\Uxp,$N)'s form.
+
+\vspace*{- 0.4 em}
+
+{ 
+\small  
+\cvl
+\beq
+\label{scopede}
+\forall ~ v_1~ < ~E(N)~~\forall ~ v_2~ < ~E(N)~~ ...
+\forall ~ v_k~ < ~E(N) ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~
+\enq
+\beq
+\label{scopedx}
+\exists ~ v_1~ < ~E(N)~~\exists ~ v_2~ < ~E(N)~~ ...
+\exists ~ v_k~ < ~E(N) ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~
+\enq}
+\eedd
+
+
+{\bf Special Note about Definition \ref{xd+1x4}'s Meaning.} 
+If
+$\Uxp$ is a 
+$\Delta_0^\xi~$ 
+sentence then
+Scope$_E$($\Uxp,N)$ 
+will 
+be equivalent to  $~\Uxp~$
+for every $N \geq 0$
+by definition.
+(This is  because 
+$\Delta_0^\xi~$  formulae
+contain
+no unbounded  quantifiers
+that undergo change when 
+  $\Uxp$ is mapped onto  
+Scope$_E$($\Uxp,N).~~)$ 
+
+
+\medskip
+
+
+
+{\bf More About
+this Notation:}
+The
+potentially lengthy
+syntactic object of
+``$\,$Scope$_E$($\Uxp,$N)$\,$'' {\it
+ will actually 
+ not}
+be used 
+in our physical encodings
+of proofs.
+Instead, these encodings will
+ use the 
+more 
+desirably
+compressed
+object of
+``$\,\Uxp\,$''
+(which has no
+possibly bulky
+ $E(N)$ term). The {\it sole function} of 
+$\,$Scope$_E$($\Uxp,$N)$\,$ 
+will be for us to speculate about what 
+Boolean value
+$\,\Uxp\,$
+ {\it would theoretically assume}
+(under the Standard-M
+ model)
+if  $\,\Uxp\,$'s quantifiers
+were  modified
+ so that their ranges
+were changed to be
+bounded by $E(N).$
+$~\,$(It  turns out that  
+Scope$_E$($\Uxp,$N)$\,$'s
+finitized quantifier-range
+will 
+help
+%greatly
+simplify our
+analysis.)
+
+%proofs.)
+
+
+\medskip
+
+
+\bxbxdr
+\label{xd+1x5}
+A
+$\Pi_1^\xi$
+or $\Sigma_1^\xi$
+sentence
+ $ \Uxp $ will be called
+{\bf Good(N)} 
+when the entity
+Scope$_E$($\Uxp,N)$ is 
+true
+under the
+Standard-M model
+\footnote{\f55  \label{fgood}  A
+quite
+unusual aspect of Definition \ref{xd+1x5} is that its
+Good$(N)$ condition 
+has opposite properties when it is applied to
+$\Pi_1^\xi$ 
+and $\Sigma_1^\xi$ 
+sentences
+in one particular respect.
+This is because for each  $~N,~$ the
+ Good$(~N~)$ condition  is weaker than
+the
+Good$(~\infty~)$ condition 
+for $\Pi_1^\xi$ 
+sentences, while it is stronger than it for 
+ $\Sigma_1^\xi$ 
+sentences. (For instance,
+$~\forall \, x~\phi(x)~$ is
+stronger than
+$~\forall \, x <E(N)~\phi(x)~$,
+but 
+$~\exists \, x~\phi(x)~$ is weaker than
+$~\exists  \, x<E(N)~\phi(x)~$.)}.
+Also, a set of 
+$\Pi_1^\xi$
+sentences, 
+denoted as $\theta$, is called  Good(N)
+iff 
+all of its
+sentences are Good(N).
+ 
+\eedd
+
+
+
+\bxbxdr
+\label{chg}
+If $\Uxp $ 
+is a $\Pi_1^\xi$ sentence then
+ $\zzz \Uxp \,)$ 
+will denote the largest integer $J$ such
+that $\Uxp $ satisfies the 
+Good$(J)$ condition.
+(It will equal $\infty$ if 
+$\Uxp $ satisfies 
+Good$(J)$ for all $J$.)
+Also,
+if $\theta$ is a set of 
+$\Pi_1^\xi$ sentences, then
+ $\zzz \theta \,)$ 
+will denote the largest $J$ where each sentence in $\theta$
+is
+Good$(J)$.
+\eedd
+
+
+
+
+\medskip
+
+{\bf A Very Helpful Start :}
+Several 
+more
+definitions will 
+be needed
+before Section \ref{sect64}
+  can present our strongest results.
+The remainder of this section will  illustrate
+how the current formalism   is already
+ sufficient for introducing a
+useful
+starting
+lemma.
+ 
+
+\smallskip
+
+\begin{definition}
+\label{deftight} \rm
+Let
+      \gggen  
+again   denote 
+a generic configuration called $~\xi~$,
+and let us
+presume that
+its 
+ base axiom system
+ $~B^\xi ~$ 
+is comprised exclusively of 
+ $ \, \Pi_1^\xi \, $ sentences.
+Also,
+let
+ $ \, \beta \,  \supset  \,  B^\xi  \, $
+      denote a second axiom system,  comprised
+also of 
+ $ \, \Pi_1^\xi \, $ sentences,
+that 
+(unlike $ \, B^\xi \,$)
+can
+possibly  be
+inconsistent.
+(If  $ ~  \beta ~ $
+ is inconsistent then 
+ let $ \,  q_\beta  \, $ denote the shortest proof of
+$ \, 0=1  \,$
+from $~\beta~$.$~$)
+Then the
+generic
+configuration
+ $ \, \xi \, $ 
+will be called
+ {\bf Tight} if 
+iff {\it every 
+inconsistent} set of 
+ $ \, \Pi_1^\xi \, $ sentences
+ $~\beta \, \supset \, B^\xi  ~ $
+satisfies the following
+constraint:
+\beq
+\label{piss1}
+\mbox{ Log(}\,q_\beta \, )  ~ ~ \geq ~~ \zzz \beta ~)
+ ~~+~~2
+\enq
+\end{definition}
+
+
+
+Lemma \ref{lemex4} will 
+prove 
+$B^\xi+$SelfRef$(B^\xi,d)$
+satisfies
+Section \ref{secc1}'s
+ self-justification
+criteria
+ whenever
+$ \, \xi \,$
+is tight.  
+This ``tightness''
+will clearly fail to be satisfied by many
+generic configurations.
+This is
+ because 
+the Second Incompleteness Theorem is 
+a widely encompassing result,
+which imposes severe restrictions on its
+allowed
+ exceptions.
+Lemma \ref{lemex4}'s 
+mini-result 
+will be of interest 
+primarily
+because it will be generalized
+substantially in
+Sections \ref{3uuuu3} 
+and \ref{sect64}.
+
+
+
+
+\medskip
+
+\begin{lemma}
+\label{lemex4}
+If a 
+generic configuration
+  \gggen  
+is tight then
+ $B^\xi+$SelfRef$(B^\xi,d)$
+will be a consistent
+self-justifying axiom system.
+\end{lemma}
+
+\medskip
+
+{\bf Proof Sketch:} Our justification of \lem{lemex4}'s
+mini-result  will be
+simpler than 
+ the
+next section's 
+proof of 
+\thx{ppp1}'s stronger result.
+ The current proof 
+will also be kept
+brief and informal because the 
+same topic will be visited  more 
+rigorously during
+Section  \ref{3uuuu3}'s
+discourse.
+
+Let $~\Psi~$ 
+denote 
+Section \ref{secc1}'s
+SelfRef$(B^\xi,d)$  sentence.
+We will  
+omit formalizing
+ $~\Psi\,$'s exact 
+ $\Pi_1^\xi$ 
+encoding here
+because Appendix A will 
+provide a more general
+fixed-point
+ construction,
+using
+Definition \ref{xd+1x7}'s stronger paradigm.
+%% 
+%% Let $~\Psi~$ 
+%% denote 
+%% Section \ref{secc1}'s
+%% SelfRef$(B^\xi,d)$  sentence.
+%% We will  
+%% omit formalizing
+%%  $~\Psi\,$'s exact 
+%%  $\Pi_1^\xi$ 
+%% encoding 
+%% here because Appendix A will later
+%% provide an encoding for a more general fixed-point
+%% sentence in the context of
+%% Definition \ref{xd+1x7}'s stronger paradigm.
+%% 
+The
+current
+proof
+will 
+simply 
+presume 
+ $~\Psi\,$'s fixed point statement 
+can receive a
+ $\Pi_1^\xi$ 
+encoding
+under
+sentence
+\eq{fixnew},
+where 
+``$~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~)~$''
+is a $~\Delta_0^\xi$  formula, indicating that
+$~p~$ is a proof
+of 0=1 from 
+$~B^\xi~+~$SelfRef$(B^\xi,d)~$
+under  $~d \, $'s
+deduction method.
+\beq
+\label{fixnew}
+\forall ~p~~~\neg ~ 
+~\mbox{Prf}^d_{~B^\xi+{\rm SelfRef}(B^\xi,d)}(~\lceil \, 0=1 \, \rceil~,~p~)
+\enq
+
+%\cvs
+
+
+The
+Definition \ref{chg}'s symbol 
+ $~\sharp~$ 
+will be
+% become 
+helpful
+at this juncture. The application of
+$\xi$'s tightness to 
+% sentence
+\eq{fixnew}'s
+$\Pi_1^\xi$ styled encoding  
+will 
+%thus 
+imply
+ \footnote{  \b55 \g55 \ep{piss2} is 
+easy to justify when
+one  presumes
+there is an
+available 
+$\Pi_1^\xi$ encoding of
+\eq{fixnew}'s statement,
+which we call  $ \, \Psi \, $.
+(This 
+ $\Pi_1^\xi$ 
+presumption
+is reasonable because an analog of
+$ \, \Psi\,$'s
+exact
+  $\Pi_1^\xi$ encoding 
+will be discussed later by  Definition \ref{xd+1x7}
+and 
+in
+Appendix A.)
+The invalidity  
+of $ \, \Psi \, $
+% in this context,  
+will 
+thus assure the existence of 
+a
+proof of $0=1$ from
+ $ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. Moreover, 
+Definition \ref{chg}'s notation  implies
+$\zzz \Psi  \, )$ will equal Log$(q) \, - \, 1 \, $ when 
+$q$ denotes the  
+ shortest proof of $0=1$ from
+ $ \, B^\xi \, + \, $SelfRef$(B^\xi,d)$. 
+The latter shows \eq{piss2} is
+valid.} that
+\eq{piss2} must be true 
+when
+ $~\Psi~$ 
+is 
+false
+under the Standard-M model
+and
+% 
+when
+the shortest proof of $0=1$ from
+ $~B^\xi~+~$SelfRef$(B^\xi,d)$
+is denoted as $~q$.
+\beq
+\label{piss2}
+\mbox{ Log(}\,q~)  ~~ = ~~ \zzz \Psi ~) ~~+~~1
+\enq
+Also
+ Definition \ref{chg}
+trivially implies 
+$~\zzz B^\xi+\Psi ~) 
+~ = ~ 
+\zzz \Psi ~)~$ 
+(because  all of
+ $~B^\xi \,$'s axioms are
+true  under the Standard-M model).  Thus, 
+\eq{piss2}
+yields
+\eq{piss3}.
+\beq
+\label{piss3}
+\mbox{ Log(}\,q~) ~~ = ~~ 
+\zzz B^\xi+\Psi ~) 
+ ~~ +~~1
+\enq
+But the point is that
+the 
+Tightness constraint's  \ep{piss1},
+$~$used in the 
+%%nope particular 
+context where $~\beta~$
+is the axiom system
+of $~ B^\xi+\Psi ~,~$
+implies
+%%nope  that
+$~\mbox{ Log(}\,q \, ) ~ \geq ~ \zzz B^\xi+\Psi ~)~+~2~$.
+This  directly contradicts \ep{piss3}'s equality, 
+{\it whenever} the proof  ``$~q~$'' of $~0=1~$ cited 
+in our discussion 
+{\it does formally exist.}
+This 
+contradiction
+%latter observation
+is precisely what 
+is needed to corroborate 
+Lemma \ref{lemex4}'s claim that 
+the axiom system $~B^\xi~+~$SelfRef$(B^\xi,d)$
+must be
+   consistent. (Thus,  
+ $~B^\xi~+~$SelfRef$(B^\xi,d)$
+must be  consistent 
+because otherwise
+a proof $~q~$ of  $~0=1~$ would exist and have its
+ $~\mbox{ Log(}\,q \, ) ~ \geq ~ \zzz B^\xi+\Psi ~)~+~2~$ inequality
+contradict \ep{piss3}.)  $~~\Box$
+
+
+\medskip
+
+
+%\cvs
+%\cvl
+
+{\bf How  Lemma \ref{lemex4}
+May Be Interpreted :}
+We 
+%firstly 
+remind the reader that many
+(but not all)
+generic configurations will
+fail to satisfy 
+Lemma \ref{lemex4}'s 
+tightness hypothesis. This
+is because any configuration
+satisfying this hypothesis  represents one of those unusual
+boundary-case
+exceptions to the Second Incompleteness Theorem
+that are feasible.
+
+Lemma \ref{lemex4}
+was intended to capture
+the simplest variant of a self-justifying
+phenomena (that employs 
+Definition \ref{chg}'s machinery).
+Its proof was
+ kept
+informal
+because 
+more sophisticated
+ self-justifying formalisms will be explored
+in Sections \ref{3uuuu3} and \ref{sect64}.
+They will apply to
+ four different types
+of generic configurations, each of whose base axiom systems
+$~B^\xi~$  
+can be made capable of proving
+all
+Peano Arithmetic's
+$~\Pi_1^\xi~$ theorems --- 
+in a context where these  systems use a
+broader variant
+of {\it ``I am consistent''} axiom-statement
+than does
+Lemma \ref{lemex4}'s
+``SelfRef$(B^\xi,d)$''
+sentence.
+
+
+\section{The First Two Meta-Theorems about Self-Justification}
+
+
+\label{3uuuu3}
+
+
+The core theorems in this section will employ
+the following notation:
+\bee
+\item
+The symbol $~\theta$ will 
+denote 
+any
+recursively enumerable (r.e.)
+set of
+$\Pi_1^\xi$ sentences, 
+henceforth called a {\bf R-View}.
+(An
+R-View
+does not need to be valid under the
+Standard-M model. It 
+only needs to be r.e.)
+
+\medskip
+
+\item
+ {\bf RE-Class$(\xi)$} 
+shall denote the
+set of all possible ``R-Views''
+$~\theta~$ that 
+can be built out of $~\xi \,$'s language
+$~L^\xi~$. (This 
+ permits
+ both valid and invalid
+R-Views  to appear in
+RE-Class$(\xi)$. 
+We choose this unrestricted definition because
+no recursive 
+decision
+procedure can
+identify
+all  the true  $\Pi_1^\xi$ sentences
+in the Standard-M model.)
+\ene
+
+
+\nop
+
+
+
+\bxbxdr
+\label{astab}
+Let   $~\xi~$
+denote 
+the 5-tuple
+  \gggen 
+representing
+one of
+Definition \ref{def3.3}'s 
+generic configurations,
+and let RE-Class$(\xi)$
+and its R-Views ``$\theta$''
+be defined as in the previous paragraph.
+Then
+$ \,\xi \, $ is called
+{\bf A-Stable\sss } iff 
+each
+$ \, \theta \, \in \, $RE-Class$(\xi)$
+satisfies the 
+following 
+invariant:
+
+
+
+\begin{description}
+\item[  *$~$   ]
+If $ \, \Uxp \, $ is a
+$\Pi_1^\xi$ theorem 
+of
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1 $
+then  $ \, \Uxp \, $ 
+will 
+satisfy  Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\end{description}
+\eedd
+
+
+\begin{remark}
+\label{re3-1}
+\rm
+The invariant
+ $ \, * \,$ 
+states
+short proofs
+(with lengths 
+ $ \leq \,  \zzz \theta)~+1~~) $
+  will 
+produce at least 
+{\it partially useful} deductions,
+in that
+their $\Pi_1^\xi$
+theorems 
+will always
+ satisfy
+  Good$\{ \,  \, \tftt  \zzz \theta) \,  \, \} \, $,
+{\it irregardless of whether or not}
+$\theta$'s axioms are {\it technically} true.
+(This makes the study of  A-stability
+very
+interesting
+unto itself,
+ apart from its
+applications
+        in the
+current article).
+\end{remark}
+
+
+
+
+
+\phx{ppp2} will show
+the presence of
+ A-stability,
+alone,
+is sufficient
+for 
+constructing 
+self-justifying systems. 
+This  will imply 
+every A-stable configuration
+must contain 
+some
+embedded weakness
+(as every evasion
+of the Second Incompleteness Theorem 
+always does).
+On the other hand,
+Appendix F will explain how
+A-stability 
+and its 
+``EA-stable'' cousin 
+(defined later)
+are both
+epistemologically interesting.
+Thus, A-stability has 
+redeeming features.
+
+
+\medskip
+
+\bxbxdr
+\label{estab}
+A Generic Configuration 
+$\,\xi \,$
+ will be called
+ {\bf E-Stable}  iff 
+all of the
+$\theta  \in \,$RE-Class$(\xi)~$
+satisfy
+  $~**~$.  $~$(This 
+construct is the counterpart
+for $\Sigma_1^\xi$ sentences of
+the Item  $ *$ in
+Definition \ref{astab}.)
+
+\begin{description}
+\item[ **   ]
+If $ \, \Uxp \, $ is a
+$\Sigma_1^\xi$ theorem 
+ derived
+from the
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1 $
+$~$then  $ \, \Uxp \, $ 
+will 
+automatically satisfy 
+Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$.
+{ (This invariant
+{\it further
+implies} 
+\footnote{ \g55 
+This point is easy to confirm when one
+remembers that $\Sigma_1^\xi$ sentences $~\Upsilon~$
+have the property that $~A<B~$ implies 
+ Scope$_E$($\Uxp,A$)
+is stronger than  Scope$_E$($\Uxp,B$).
+It is then obvious
+that the
+ Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$
+criteria implies the validity of
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$
+for $\, \Sigma_1^\xi \,$ sentences
+because the invariant $\,** \, $ presumes 
+its  $\, \Sigma_1^\xi \,$ theorems have proofs $\, p \,$
+satisfying 
+`` Log$(p)~\leq \zzz \theta)~+1 $ ''. }  
+that  $ \Uxp  $  
+will also
+satisfy 
+ the Good$\{ \, \tftt  \zzz \theta) \,~\} $
+criteria.)} 
+\end{description}
+\eedd
+
+
+
+
+\smallskip
+
+
+
+\begin{remark}
+\label{re3-2}
+\rm
+The invariants $ \, * \, $ and $ \, ** \,  \,  $ 
+are partially analogous 
+to each other
+because both imply that
+if $ \, p \, $ is a proof short enough to
+satisfy  Log$(p) \, \leq \zzz \theta) \, +1 \,  $ 
+then their resulting theorem will satisfy
+Good$\{ \,  \, \tftt  \zzz \theta) \,  \, \} \, $.
+However, there is 
+a
+% an important 
+distinction between 
+Definitions
+\ref{astab} and \ref{estab},
+as well.
+This is because the
+prior section's
+ Footnote \ref{fgood}
+observed that $ \, \Sigma_1^\xi \, $ sentences are stronger
+when they meet a 
+Good$( \, N \, )$ 
+rather than a
+Good$( \, \infty \, )$ threshold, $ \, ${\it while the reverse is true for 
+  $ \, \Pi_1^\xi \, $ sentences.}
+Thus $ \, ** \,$'s short proofs
+$ \, p \, $  
+(satisfying  Log$(p) \,\leq \zzz \theta)~+1~ $)
+will have the
+special
+ property that
+their theorems  $ \, \Uxp~$ 
+will satisfy a 
+``Good$\{ ~ \frac{1}{2}~\lfloor ~$Log$(p)~\rfloor~-~1     ~\}~$''
+constraint 
+that is actually stronger than their formal
+$~\Sigma_1^\xi~$ 
+statements.
+\end{remark}
+
+
+
+
+\smallskip
+
+
+
+
+The \appD will provide four examples
+of generic configurations that are either E-stable
+or A-stable (or often 
+% very often 
+both).
+Its most prominent example will be a 
+configuration
+$~\xi^*~$ 
+that uses  semantic tableaux deduction and
+recognizes addition as a total function.
+Theorem D.\ref{D.4-theorx} will 
+imply such systems can be made
+% both 
+self-justifying
+and 
+able to
+ prove 
+% all
+ Peano Arithmetic's
+ $\Pi_1^*$ theorems.
+
+\njp
+%\smallskip
+
+Three more  definitions
+are needed 
+to help introduce 
+our first theorem
+
+
+
+
+\smallskip
+
+\bxbxdr
+\label{eastab}
+A generic configuration $~\xi~$
+will be called
+{\bf EA-stable} iff it 
+is both 
+E-stable
+and A-stable.
+(It will thus
+satisfy both 
+$~*~$ and $~**~$.)
+\eedd
+
+\smallskip
+
+
+
+
+Our next definition is related to the fact
+ that  
+many definitions of consistency
+are logically 
+ equivalent 
+from the perspective
+of strong enough logics,
+but they 
+are {\it  often not provably equivalent}
+from the perspectives of 
+weak logics.
+
+\medskip
+
+\bxbxd
+\label{lev}
+\rm
+Let $   \xi   $
+denote the generic configuration
+  \gggcp, 
+and $   \alpha   $ be an axiom system satisfying
+$   \alpha  \supseteq  B^\xi   $.
+Then
+$   \alpha   $ is called
+{\bf Level$(k^\xi)$ 
+Consistent} when there
+exists  no proofs from $\alpha$ via
+$ d   $'s deduction method  
+of both a $\Pi^\xi_k$ sentence and 
+of  the
+$\Sigma^\xi_k$ sentence that is its  negation
+\eedd
+
+\medskip
+
+Most of this article will focus on 
+self-justifying systems that recognize their
+Level$(k^\xi)$ consistency 
+when 
+ $k$ 
+equals 0 or 1.
+Our next definition will
+be applied mostly to these two cases.
+
+
+
+\smallskip
+
+
+\bxbxdr
+\label{xd+1x7}
+Given any  
+$k \geq 0$, 
+a
+generic configuration
+  \gggen     
+and an
+axiom system
+ $\beta  \supset  B^\xi$,
+the symbol
+SelfCons$^k(\beta,d)$
+will
+denote a 
+self-referencing
+$\Pi_1^\xi$ sentence declaring
+$\,  \beta   +  $SelfCons$^k(\beta,d)      $'s
+formal
+Level(k$^\xi)$ consistency,
+as is 
+illustrated
+below
+by  
+the 
+statement $ ~ +~ $.  (An  encoding for
+SelfCons$^k(\beta,d)$ 
+will be provided
+by Appendix A.)
+\begin{quote}
+$+~~~$ 
+There exists  no two proofs 
+(using deduction method $   d   $)
+of {\it both}
+some $\Pi_k^\xi$ sentence and
+of the 
+$\Sigma_k^\xi$ sentence, 
+$\,$that 
+represents its negation,
+$\,$from the  union of
+the axiom system $~\beta~ $
+with $~this~$ added
+sentence  ``SelfCons$^k(\beta,d) \,$''
+{\it (looking at itself).}
+\end{quote}
+\eedd
+
+
+\begin{remark}
+\label{re3-3}
+\rm
+We will focus on
+Definition \ref{xd+1x7}'s 
+SelfCons$^k(\beta,d) \,$ axiom 
+mostly in
+ the settings
+where $\,k=0$ or 1.
+This is because 
+SelfCons$^k(\beta,d) \,$ 
+will 
+typically 
+be too strong for it to generate boundary-case exceptions
+to the Second Incompleteness Theorem when
+ $~k \geq 2~$.
+It turns out that even when
+ $~k=1~,~$  Definition \ref{xd+1x7}'s
+SelfCons$^k(\beta,d) \,$ statement
+will be significantly stronger than the
+axiomatic
+declaration $\, \bullet \,$
+used by 
+Section
+\ref{secc1}'s
+ SelfRef$(\beta,d)$
+axiom.
+This is 
+because 
+SelfCons$^1(\beta,d) \,$ asserts 
+ non-existence of 
+simultaneous proofs
+for a  $~\Pi_1^\xi~$ sentence and its negation,
+$\,$while  SelfRef$(\beta,d)$
+establishes
+merely
+the
+ non-existence of a proof of $0=1$.
+\end{remark}
+
+
+\smallskip
+
+
+\begin{theorem}
+\label{ppp1}
+Let $~\xi~$
+denote a
+generic configuration
+  \gggen   
+that is 
+EA-stable.  Then 
+the corresponding  axiom system of 
+$B^\xi+$SelfCons$^1(B^\xi,d)$
+must satisfy Section
+\ref{secc1}'s  definition of
+self-justification.
+\end{theorem} 
+
+\parskip 3pt
+
+\smallskip
+
+
+
+{\bf Proof.}
+The justification 
+of \phx{ppp1} will
+be 
+a more elaborate  version
+ of 
+\lem{lemex4}'s
+mini-proof.
+It will
+replace
+Definition \ref{deftight}'s
+Tightness constraint with
+an EA-stability requirement.
+It will also
+ replace
+SelfRef$(\beta,d)$'s 
+``I am consistent''
+axiom
+with a 
+{\it stronger}
+SelfCons$^1(\beta,d)$
+statement.   
+
+
+\cvb
+\parskip 5pt
+
+
+
+Our proof will focus on
+showing 
+  $~B^\xi+$SelfCons$^1(B^\xi,d)~$ 
+is consistent (and thus 
+% it
+satisfies 
+the subtle
+Part-ii component
+of Section \ref{secc1}'s
+definition of
+``Self-Justification'').
+It will be awkward 
+during our discussion
+to 
+write repeatedly  the
+expression ``$~B^\xi+$SelfCons$^1(B^\xi,d)~$''.
+Therefore,
+ ``$S$'' will be the abbreviated
+name for this axiom system.
+We will also employ the following notation:
+\bee
+\itemsep +3pt
+\item
+$~\mbox{Prf}_S(t,p)~$ will denote 
+that $~p~$ 
+is
+ a proof of the theorem $~t~$ from
+the above mentioned formalism ``$~S~$''
+(using 
+$~\xi \,$'s
+deduction method of $d$ ).
+\item 
+$~\mbox{Neg}^1(x,y)~$ will denote
+that $~x~$  is
+the G\"{o}del encoding of a
+ $\Pi_1^\xi$ sentence and
+that $~y~$ is a $\Sigma_1^\xi$ 
+sentence which represents
+$~x \,$'s formal negation.
+\ene 
+Appendix A explains how to combine the theory of LinH
+functions \cite{HP91,Kr95,Wr78} with \cite{ww1}'s
+fixed point methods to provide 
+$\mbox{Prf}_S(t,p)$ 
+and 
+$\mbox{Neg}^1(x,y)$ with
+$\Delta_0^\xi$ encodings.
+(A reader can
+ omit examining Appendix A,
+if he 
+just
+accepts this fact.)
+Thus, \eq{pencode} 
+can
+be 
+viewed \footnote{ \s55 
+Our notation convention has the 
+abbreviated
+formula of
+``$~\mbox{Prf}_S(t,p)~$''
+in \ep{pencode} 
+corresponding 
+to
+ Appendix A's 
+``$~\mbox{SubstPrf} \, _{\beta}^d  \, 
+( \bar{n}    ,    t    ,    p   ) ~$'' formula,
+in a context where $~n~$ specifies the G\"{o}del number of
+the expression
+$~\Gamma^k(g)~$  
+in
+Appendix A's 
+ \ep{encode12} and where
+the superscript $~k~$ 
+within this
+ formula
+``$~\Gamma^k(g)~$''
+is set equal to 1.  
+This 
+implies that sentence
+\eq{pencode} does
+assert the Level$(1^\xi)$ 
+consistency
+of S.  \fend }
+in this context
+as being   SelfCons$^1(B^\xi,d)\,$'s 
+formalized
+$\Pi_1^\xi$ 
+statement,
+declaring 
+the Level$(1^\xi)$ 
+consistency
+of S:
+\begin{equation}
+\label{pencode}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^1(x,y) ~ \wedge~ 
+\mbox{Prf}_S  \, (     x    ,    p   ) ~
+\wedge  ~
+\mbox{Prf}_S \,    (       y    ,    q   )  \, \}
+\end{equation}
+
+
+Let   $ \Phi $   denote 
+\eq{pencode}'s
+ sentence. 
+We will
+use it to
+prove
+\phx{ppp1}'s claim that  $ S $
+is consistent. Our proof 
+will be a proof by contradiction.
+It will 
+begin with
+the 
+assumption that $ S $ is inconsistent.
+This
+implies 
+$\Phi$ is
+false under the Standard-M model.
+Hence, 
+Definition \ref{chg}
+implies
+\begin{equation}
+\label{punk}
+ \zzz \Phi~) 
+   ~~ < ~~ \infty
+\end{equation}
+
+
+
+\noindent
+\ep{punk} 
+thus
+indicates there  exists
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying \eq{pencodepunk}
+(because such a tuple
+will be a counter-example to
+\eq{pencode}'s assertion).  
+The particular
+$(p,q,x,y)$
+ satisfying \eq{pencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ 
+can 
+then
+be easily shown$\,$
+\footnote{ \sm55 \label{footp1} 
+Let $~L~$ denote the 
+minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$  for a tuple
+$(p,q,x,y)$
+ satisfying \ep{pencodepunk}. Then by definition,
+$~\Phi~$ satisfies Good$(L-1)$ but not
+Good$(L)$.
+From Definition \ref{chg}, this 
+establishes
+ the validity of
+\ep{punkless}  (because 
+the minimal
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying sentence \eq{pencodepunk}
+has
+$ \mbox{Log}\{\mbox{Max}[  \bar{p}  ,  \bar{q}  ,  \bar{x}  ,  \bar{y}  ] \}
+ \, = \, L.~$) }
+  $\,$to also
+satisfy
+\ep{punkless}.
+\begin{equation}
+\label{pencodepunk}
+  \,\mbox{Neg}^1(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ 
+\mbox{Prf}_S  \, (     ~ \bar{x}~    ,    ~ \bar{p}~   ) ~
+\wedge  ~
+\mbox{Prf}_S \,    (       ~ \bar{y}~    ,    ~ \bar{q}~   )  
+\end{equation}
+\begin{equation}
+\label{punkless}
+ \mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
+   ~~~ =~~~\zzz \Phi~) ~~+~~1
+\end{equation}
+
+$~~~$ We will now use 
+\eq{pencodepunk} and \eq{punkless} 
+ to
+bring our proof-by-contradiction to its conclusion.
+Let $ \, \Upsilon \, $ denote the $\Pi_1^\xi$ sentence specified
+by $ \, \bar{x} \, $. Then
+$ \, \neg  \,  \Upsilon \, $ 
+will
+correspond to the $\Sigma_1^\xi$ sentence denoted
+by $ \, \bar{y} \, $.
+Also, 
+\eq{pencodepunk} and \eq{punkless} imply that 
+both
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $
+have  proofs 
+such that the logarithms of
+their G\"{o}del numbers are bounded
+by
+$\zzz \Phi \,)  \,+ \,1 \,$.
+These facts
+imply
+ that
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $
+{\it both} 
+satisfy 
+ Good$\{ \,  \, \tftt  \zzz \Phi) \,  \, \} \, $
+under our formalism.
+(This is because
+if we take $ \, \theta \, $ in Definitions
+\ref{astab} and \ref{estab} to be simply $ \, \Phi \,$'s
+1-sentence statement, $\,$then the invariants of
+ $ \, *  \,$ and $ \, **  \,$ from these two definitions
+both impose the
+same
+Good$\{ \,  \, \tftt  \zzz \Phi) \,  \, \} \, $
+ constraint
+on 
+ $ \, \Upsilon \, $ and
+$ \, \neg  \,  \Upsilon \, $.)
+ 
+
+It is infeasible, 
+however,
+for a sentence and its
+negation to both satisfy the same goodness
+constraint.
+ This completes \phx{ppp1}'s
+proof-by-contradiction
+because 
+the initial
+ assumption that 
+$S$ was inconsistent has led to an
+infeasible conclusion.
+$~~\Box$
+
+
+
+\bigskip
+
+\cvl
+
+\parskip 3pt
+
+Our next definition will help formalize
+a useful cousin of
+\phx{ppp1}.
+
+\smallskip
+
+
+\bxbxdr
+\label{ostab}
+A Generic Configuration of 
+$\xi $
+will be called
+ {\bf $~$0-Stable$~$} when every
+particular
+$\theta \, \in $ RE-Class$(\xi)$
+satisfies the 
+invariant of  $***$. 
+(This invariant 
+is strictly weaker than 
+its counterparts   $*$  and $**$
+in Definitions 
+   \ref{astab} and \ref{estab}.) 
+\begin{description}
+\item[ ***   ]
+If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $~\theta \cup B^\xi~$
+via a proof $~p~$ 
+whose length satisfies
+ Log$(p)~\leq \zzz \theta)~+1~,~ $
+then  $ \, \Uxp \, $
+is 
+true under the Standard-M model.
+\end{description}
+\eedd
+
+
+\begin{theorem}
+\label{ppp2}
+If 
+the configuration
+$\xi$ is 
+0-stable 
+then
+$B^\xi+$SelfCons$^0(B^\xi,d)$
+is      a self-justifying 
+formalism. {\rm (Appendix C 
+shows this result  applies
+also to
+E-stable and  A-stable  configurations.)}
+\end{theorem} 
+
+\medskip
+
+
+
+
+
+
+\phx{ppp2}'s proof
+is similar
+to  \phx{ppp1}'s
+ proof. 
+The difference between these two
+propositions is that
+\phx{ppp2}
+has reduced
+       SelfCons's
+ superscript  
+ from 1 to 0, so that its
+ hypothesis
+can encompass a 
+theoretically
+ broader 
+set of applications.
+(The Appendix C summarizes how
+ \phx{ppp1}'s proof can be
+easily modified to
+also prove  \phx{ppp2}.)
+
+\medskip
+
+\begin{remark}
+\label{newrem}
+{\rm
+Theorems 
+\ref{ppp1} and \ref{ppp2}
+should clarify 
+the  nature of 
+\cite{ww93}--\cite{ww9}'s
+formalisms.
+This is because proofs-by-contradictions 
+are notorious 
+in the mathematical literature
+for being
+confusing.
+They should be simplified whenever possible.
+This has been done
+mainly through
+\phx{ppp1}'s
+short proof.
+(It 
+ applies to
+three of Appendix D's
+four examples of 
+generic 
+configurations,
+and
+\phx{ppp2} applies to 
+Appendix D's
+fourth example.)
+Furthermore, Section \ref{sect64} 
+will
+show how more elaborate
+self-justification systems can verify all
+Peano Arithmetic's $\Pi_1^\xi$ theorems.}
+\end{remark}
+
+ 
+\section{Four Further Meta-Theorems}
+\label{sect64}
+
+
+We need one preliminary lemma
+before  exploring 
+how strong self-viewing
+                logics may become
+before 
+they   cross
+  the inevitable
+boundary between 
+self-justification
+and 
+ inconsistency,
+$\,$implied by 
+G\"{o}del's
+Theorem.
+
+
+
+
+\begin{lemma}
+\label{llpp3}
+Let  $  \xi  $ 
+denote a generic configuration  
+   \gggcp  
+$\,$, and $  \mheta  $ denote an r.e. set of 
+$\Pi_1^\xi$ sentences,
+each of which holds true in the Standard-M model.
+Let $  \mxi  $ denote a 
+5-tuple that differs from 
+ $  \xi  $  
+in that its base axiom system
+is  $  B^\xi \, \cup \, \mheta  $
+(rather than
+$  B^\xi  $).
+These conditions imply that
+$  \mxi  $ is a
+generic configuration,
+and 
+it
+will satisfy
+the following
+four invariants:
+\bed
+\item[     i   ]
+If $~\xi~$ is  0-stable\sss 
+then 
+$~\mxi~$  will also be
+0-stable\sss.
+\item[      ii    ]
+If $~\xi~$ is  A-stable\sss 
+then 
+$~\mxi~$ will also be
+A-stable\sss.
+\item[     iii   ]
+If $~\xi~$ is  E-stable\sss 
+then 
+$~\mxi~$  will also be
+E-stable\sss.
+\item[     iv   ]
+If $~\xi~$ is  EA-stable\sss 
+then 
+$~\mxi~$  will also be
+EA-stable\sss.
+\ennd
+\end{lemma}
+
+
+Lemma  \ref{llpp3}'s proof
+is fairly straightforward. It
+has been placed in
+the Appendix B. 
+This section will
+use  Lemma  \ref{llpp3}
+to prove
+four
+meta-theorems
+ that are consequences of
+its
+formalism.
+
+
+\smallskip
+
+\begin{definition}
+\label{dap4-1}
+\rm
+Let  $\xi$  again
+denote
+a generic configuration  
+  \gggcp,  
+and
+$\theta$ denote some r.e. set of 
+$\Pi_1^\xi$ sentences
+(which are not
+required to be  true  under the Standard-M model).
+For the cases where $\,k \,$ is either
+0 or 1,
+the symbol 
+$G^\xi_k(\, \theta \,)$
+will
+denote the following axiom system:
+\vspace*{- 0.2 em}
+\begin{equation}
+\label{4gedef}
+G^\xi_k(~ \theta ~)~~= ~~
+\theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}
+\vspace*{- 0.2 em}
+\end{equation}
+Also when
+$ \, k \, = \, 0 \,$
+or 1,   the function 
+$ \, G^\xi_k ~ $
+(which maps $ \, \theta \, $ onto 
+$G^\xi_k(\, \theta \,)~~~)$ is 
+called 
+{\bf Consistency Preserving} iff 
+$ \, G^\xi_k(\, \theta \,) \, $is 
+assured to be consistent whenever 
+all the sentences in $~\theta ~$
+are  
+true  under the Standard-M model.
+\end{definition}
+
+\smallskip
+
+We emphasize 
+ consistency preservation is 
+unusual in 
+logic. This is because
+$ \, G^\xi_k(\, \theta \,) \, $
+comes from adding a self-justifying axiom to an
+initially consistent
+formalism $~B^\xi+\theta~$, and the 
+Second Incompleteness Theorem
+demonstrates that sufficiently powerful
+formalisms are simply incompatible with
+such an axiom.
+ However, 
+there will   be
+four
+specialized 
+paradigms, defined in Appendix D,
+that are exceptions to this rule.
+They will be related to our next result:
+
+
+
+
+
+\smallskip
+
+\begin{theorem}
+\label{pqq3} 
+The function 
+$~G^\xi_1~$ 
+shall satisfy Definition \ref{dap4-1}'s
+consistency preservation
+property
+when $~\xi~$ is
+EA-stable. Likewise 
+the function 
+$~G^\xi_0~$ 
+will be consistency preserving when $~\xi~$ is
+any one of A-stable, E-stable or 0-stable.
+{\rm 
+(Thus 
+in each case,
+$~G^\xi_k(\, \theta \,)~$
+will  
+be consistent
+when
+all the sentences in $~\theta ~$
+are  
+true in the Standard-M model.)}
+\end{theorem}
+
+\medskip
+
+{\bf Proof}.
+It will be
+ convenient
+for our proof
+ to use a
+ dot-style
+notation,
+ analogous to
+Lemma \ref{llpp3}'s
+terminology. 
+ Thus, 
+\bee
+\item
+ $~\mheta~$ denotes any r.e. set of 
+$\Pi_1^\xi$ 
+sentences {\it that are 
+each
+true} under the Standard-M model.
+\item
+ $\mxi$
+is the tuple
+$ \, ( \, L^\xi  \, ,  \, \Delta_0^\xi  \,  ,
+ \,  B^\xi \cup \mheta  \, ,  \, d   \, ,  \,  G  \,  )$.
+(It
+differs from 
+ $\xi$
+by replacing
+ $\xi\,$'s  base axiom system
+of $\,B^\xi \,$ with $\,B^\xi  \cup  \mheta$.)
+\ene
+ Part-iv of Lemma \ref{llpp3}
+indicates
+the EA-stability of 
+$\xi$ 
+ implies
+the EA-stability of 
+$  \mxi $.
+ Moreover  \phx{ppp1} 
+applies to all EA-stable
+configurations, 
+including  
+$\mxi $.
+Thus,
+$G^\xi_1(\, \mheta \,)$ 
+is  consistent
+because $\xi$ is EA-stable
+and all of $~\mheta \,$'s sentences are true in
+the Standard-M model.
+ This
+proves
+\phx{pqq3}'s 
+first 
+claim.
+
+An 
+almost
+identical proof,
+where 
+\phx{ppp2} 
+simply
+ replaces \phx{ppp1}
+as the central self-justifying
+engine,
+will corroborate 
+\phx{pqq3}'s second claim .
+Thus,
+$~G^\xi_0~$ 
+is consistency preserving when $~\xi~$ is
+one of A-stable, E-stable or 0-stable. $~~~\Box$
+
+
+
+\medskip
+
+\begin{remark}
+\label{re4-1}
+\rm
+Most
+ generic configurations
+$~\xi~$ 
+will not satisfy \phx{pqq3}'s hypothesis
+because
+$~G^\xi_k(\theta)~$ will typically be
+inconsistent,
+irregardless 
+\footnote{\sm55 Conventional generic configurations $\xi$
+will satisfy the Hilbert-Bernays derivability conditions
+\cite{HB39,HP91}.
+Their
+$~G^\xi_k(\theta)~$ will thus
+be automatically inconsistent
+because of
+a G\"{o}del-like diagonalization argument.}
+of whether or not
+all of $~\theta\,$'s axioms are valid
+under the Standard-M model.
+The significance of \phx{pqq3} 
+is that it shows that some
+outlying
+ exceptions to this
+general rule do prevail when
+$~\xi~$ satisfies one of 
+\phx{pqq3}'s 
+four stability conditions.
+These
+exceptions
+are related to the
+3-page abbreviated philosophical discussion
+that will appear later in Appendix F.
+They  
+will 
+assure 
+that
+Theorem \ref{pqq3}'s 
+formalisms
+of
+ $G^\xi_1(\theta)$ 
+and $G^\xi_0(\theta)$
+are {\it always
+consistent}
+whenever 
+$~\theta\,$'s axioms are valid
+in the Standard-M model.
+\end{remark}    
+
+\smallskip
+
+
+Our next definition will enable
+self-justifying formalisms
+to prove
+the
+$\Pi_1^\xi$ 
+theorems 
+of  any
+consistent 
+r.e. axiom system
+that uses  $~L^\xi \,$'s language.
+
+
+
+\smallskip
+
+\begin{definition}
+\label{dap4-2}
+\rm
+Let  $\,\xi\,$  denote
+a generic configuration  
+  \gggcp.   
+Let $   \nal{B}  $ 
+denote any recursive axiom system whose language
+is an extension of 
+$\, L^\xi \,$.
+For an arbitrary 
+deduction method $ \,  \nal{D} \,  $
+(which may be 
+possibly
+different from $\,\xi \,$'s deduction method
+$ \,d \,$ ),
+ let
+$\mbox{Prf}_{ \nal{B}}^D \,( \, \lceil \, \Psi \, \rceil \,  , \,  q \, )\,  $
+denote a $\Delta_0^\xi$ formula indicating 
+$  q  $ is a proof of the theorem $  \Psi  $ from 
+axiom system
+$ \,  \nal{B}\,$, 
+using
+deduction method
+ $ \,  \nal{D} \,  $.
+Then the  {\bf Group-2 Schema} for
+$  (   \nal{B}  ,  D  )  $ is
+defined as an infinite set of axioms that includes
+one instance of \eq{group2}'s axiom 
+for each 
+$\Pi_1^\xi$ sentence $\, \Psi$. 
+\beq
+\label{group2}
+\forall ~q~~~ \{~
+\mbox{Prf}_{ \nal{B}}^D \,( \,  \lceil \, \Psi \, \rceil \,  , \,  q \, )
+ ~~~\longrightarrow ~~~ \Psi ~~ \}
+\enq
+\end{definition}
+{\bf  $~~~~$Comment about this Notation:}
+The Definition \ref{dap4-2}
+had called 
+\eq{group2} a 
+ ``Group-2 Schema''  so as to keep our terminology consistent
+with
+\cite{ww93,ww1,ww5,wwapal}'s notation.
+
+\bigskip
+
+%% Several of our articles
+%% \cite{ww93,ww1,ww5,wwapal}
+%% have  used the term
+%%  ``Group-2 Schema'' to refer to
+%% Definition \ref{dap4-2}'s
+%% formalism. The same notation is
+%% thus 
+%% used here.
+
+\smallskip
+
+
+
+\begin{theorem}
+\label{pqq4} 
+Let  $~\xi~$ denote any
+arbitrary generic configuration,
+and 
+$  (   \nal{B}  ,  D  )  $ 
+denote any pair consisting of
+an axiom system and a deduction method
+(which, once  
+again, are allowed to be 
+different from
+ $~\xi \,$'s deduction method
+and axiom system).
+Then if all of $  (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems are 
+true in the Standard-M model,
+the following two invariants will hold:
+\bed
+\topsep -4pt
+\itemsep -2pt
+\item[   i ] 
+If $~\xi~$ is EA-stable then
+ there will exist an
+r.e.$~$self-justifying system that can prove
+all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems 
+and 
+recognize its own Level($~1^\xi~)~$ consistency.
+\item[  ii ] 
+Likewise,
+if
+ $~\xi~$ is
+one of A-stable, E-stable or 0-stable,
+ there
+will  exist an
+r.e.$~$self-justifying system that can confirm
+all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems 
+and 
+which can
+recognize its own Level($~0^\xi~)~$ consistency.
+\ennd
+\end{theorem}
+
+\smallskip
+
+
+
+{\bf Proof.}
+\phx{pqq4} 
+follows from
+\phx{pqq3}.
+Thus let
+ $\theta$
+denote
+ the set of 
+all
+$\Pi_1^\xi$ sentences 
+that are members of
+Definition \ref{dap4-2}'s Group-2 schema.
+Then every one
+of $\theta\,$'s 
+Group-2 axioms 
+must be
+true  under the Standard-M model
+(because 
+the hypothesis of  
+\phx{pqq4} 
+ indicated
+all of $  (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems are
+true in this model).
+Hence,  \phx{pqq3}  implies:
+\bee
+\item
+$G^\xi_1(\, \theta \,)$
+ is consistent when $\xi$ is EA-stable.
+\item
+$G^\xi_0(\, \theta \,)$ is consistent when $\xi$ is one of
+ A-stable,  E-stable or  0-stable.
+\ene
+Since $G^\xi_0(\, \theta \,)$ and $G^\xi_1(\, \theta \,)$
+are self-justifying systems that prove all of 
+$ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ theorems, 
+Items 1 and 2
+will substantiate
+\phx{pqq4}'s
+two claims. $\Box$ 
+
+
+
+\bigskip
+
+An
+ awkward aspect of Definition \ref{dap4-2}'s
+``Group-2'' schema is that 
+it employs an
+infinite number of instances
+of \eq{group2}'s Group-2-like axiom sentences.
+It turns out 
+that this
+Group-2 scheme
+can be
+compressed
+ into a {\it single axiom sentence,}
+if one is willing to 
+settle for a
+slightly 
+diluted 
+variant of
+ $ (   \nal{B}  ,  D  )  $'s
+$\Pi_1^\xi$ knowledge.
+To 
+formalize this  concept,
+the following  notation shall be used:
+\bee
+\item
+Check$^\xi(t)$ will denote a $\Delta_0^\xi$ formula
+that checks to see whether $t$ represents the G\"{o}del
+number of a $\Pi_1^\xi$ sentence. 
+\item
+Test$^\xi(t,x)$ 
+will denote any $\Delta_0^\xi$ formula
+where
+\eq{testsim}'s invariant 
+is 
+true   under
+the Standard-M model
+for
+every $  \Pi_1^\xi  $ sentence 
+$   \Psi   $
+simultaneously.
+There are infinitely many 
+different  $\Delta_0^\xi$ 
+formulae that can
+serve as 
+Test$^\xi(t,x)$ predicates
+satisfying 
+this condition.
+(Example \ref{new-exam} will illustrate
+one such encoding of a
+Test$^\xi(t,x)$ predicate.)
+\beq
+\label{testsim}
+\Psi ~~~ \longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}^\xi(~\lceil~\Psi~\rceil~,~x~)
+\enq
+\ene
+The expression \eq{globsim}
+will be called 
+a
+{\bf Global Simulation Sentence}
+for representing
+$( \nal{B},D)$
+via $\xi \,$.
+Its $\mbox{Test}^\xi(t,x)$ clause essentially allows
+$\xi$ to simulate the $\Pi_1^\xi$ knowledge of
+$( \nal{B},D)$'s 
+set of
+theorems.
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{ \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}^\xi(t,x)~~~ \}
+\enq
+
+ 
+\bigskip
+
+\begin{example}
+\label{new-exam}
+\rm
+For any generic configuration $\,\xi~=\,$ \gggcp,
+$\,$let NegPrf$^\xi(t,x)$ denote
+a  $\Delta_0^\xi$ formula
+specifying
+that
+$ \, t \, $ is a $\Pi_1^\xi$ sentence and
+that
+$ \, x \, $ is a proof under
+ $\,d\,$'s deduction
+method  of the 
+ $\Sigma_1^\xi$ sentence that represents $ \, t\,$'s negation.
+Also, let   Test$^\xi_0(t,x)$ 
+be defined as follow:
+\beq
+\label{new1-exam}
+\mbox{Test}^\xi_0(t,x)~~~~ =_{\mbox{def}}~~~~ \neg~ \mbox{NegPrf}^\xi(t,x)
+\enq
+For each $\Pi_1^\xi$ sentence $\Psi$,
+%%  vvvv  
+it is easy to verify 
+\footnote{  \b55 Part-3 of Definition \ref{def3.3}
+indicated that $~\xi\,$'s base axiom system is
+``$~\Sigma_1^\xi~$ complete''. $~$(It is
+thus able to prove
+all $~\Sigma_1^\xi~$ sentences that are true in the true in
+Standard-M model,
+and it will likewise refute all 
+$~\Pi_1^\xi~$ sentences that are false.)
+The statement \eq{new1-exam}
+then
+immediately implies that
+\eq{newtestsim}
+must  be true
+ under the Standard-M model
+for
+ every $~\Pi_1^\xi~$ sentence $~\Psi~$.}
+that
+the statement 
+\eq{newtestsim}
+is true
+under
+the 
+Standard-M model.
+\beq
+\label{newtestsim}
+\Psi ~~~ \longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}^\xi_0(~\lceil~\Psi~\rceil~,~x~)
+\enq
+\end{example}
+\vspace*{- 1.3 em}
+$~~~$ 
+%Thus, 
+
+The latter
+% immediately
+ implies 
+that
+\eq{new2-exam} 
+is 
+% an example of
+a global simulation sentence
+for  $  (   \nal{B}  ,  D  )  $.
+\beq
+\label{new2-exam} 
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{ \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}_0^\xi(t,x)~~~ \}
+\enq
+We emphasize that there are countably infinite different examples
+of  $\mbox{Test}_i^\xi(t,x)~$ predicates that generate
+global simulation sentences and
+that statement \eq{new2-exam} 
+illustrates only one such example.
+
+
+%\bigskip
+
+\begin{definition}
+\label{gsim}
+\rm
+Let $( \nal{B},D)$ denote any ordered pair whose set
+of $~\Pi_1^\xi$ theorems are true under the Standard-M model.
+Let
+Test$^\xi_1 \, , \, $ 
+Test$^\xi_2 \, , \, $ Test$^\xi_3  \,~....~ $
+denote the set of 
+$\Delta_0^\xi$ formulae 
+where
+statement
+\eq{testsim}
+is true
+under the Standard-M model
+for every
+$\Pi_1^\xi$ sentences
+$\Psi$.
+Then TestList$^\xi$ will denote
+a
+list of all  these 
+Test$^\xi_i \,$ predicates.
+Also for each
+Test$^\xi_j $ 
+formula
+in  TestList$^\xi\,$,
+$\,$the symbol
+{\bf GlobSim$^D_{ \nal{B}} \,(\xi,j)$} 
+will denote 
+the special version  
+of \eq{globsim}'s global simulation formalism
+that employs
+Test$^\xi_j \,  $'s machinery.
+ \end{definition}
+
+%\smallskip 
+
+
+
+\begin{remark}
+\label{re4-2}
+\rm
+A comparison between 
+Definition \ref{dap4-2}'s
+Group-2 schema with  \ref{gsim}'s
+global simulation sentences
+will
+reveal
+neither 
+is 
+strictly
+ better than the other. 
+Both         have their own separate advantages.
+Thus,
+the attractive aspect about Definition \ref{gsim}'s
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$  sentence is that it is a
+finite-sized object that can simulate
+the
+infinite set of  axioms
+associated with
+Definition \ref{dap4-2}'s Group-2 schema. 
+The accompanying
+drawback
+\footnote{ \f55  The
+chief
+ difficulty arises
+essentially
+        because
+a  $\Pi_1^\xi$
+theorem of
+ $ (   \nal{B}  ,  D  )  $ 
+may contain an arbitrarily long combination
+of bounded universal and bounded existential quantifiers.
+Thus,
+some generic configurations will have 
+base axiom systems $B^\xi$ 
+that are so weak that
+their combination with \eq{globsim}'s
+Global Simulation Sentence 
+is insufficient 
+{\it to prove} the validity of \eq{testsim}'s
+equivalence statement
+{\it  for all}
+ $\Pi_1^\xi$ sentences $\Psi$ simultaneously.
+In particular, such   proofs
+will often be infeasible
+when  $\Psi \,$'s
+sequence of bounded  quantifies
+has a length greatly exceeding the length of
+the G\"{o}del encoding for  \eq{globsim}'s
+global simulation
+statement.}
+$\,$of 
+a global simulation sentence
+ is that the union of it
+with the base-axiom system  $B^\xi$ 
+will typically be inadequate  to prove every $\Pi_1^\xi$
+sentence  that is a theorem of
+ $ (   \nal{B}  ,  D  )  $.
+$\,$Instead, in a context where
+  $\Psi \,$ is a $\Pi_1^\xi$
+theorem of
+ $ (   \nal{B}  ,  D  )  $,
+the sentence 
+ GlobSim$^D_{ \nal{B}} \,(\xi,j)$ 
+will usually
+provide only
+enough {\it  fragmented information} to prove
+the statement \eq{quasisim} (which is equivalent to
+$\Psi$ 
+{\it under the Standard-M model).}
+\beq
+\label{quasisim}
+ \forall ~x~~
+\mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~)
+\vspace*{- 0.4 em}
+\enq
+While \eq{quasisim} may 
+be insufficient
+%not 
+% provide
+%enough information 
+to prove $ \, \Psi \, $ from  
+$ \, B^\xi \, $, it 
+still 
+(according to the 
+statement \eq{testsim} )
+has  the desired property of being
+equivalent to $\Psi$
+under the Standard-M model.
+(This means that
+the knowledge of \eq{quasisim}'s
+truth is helpful, {\it even if it is unknown
+from $ \, B^\xi\,$'s perspective} to be
+equivalent to
+$ \, \Psi \, $ . )
+\end{remark}
+
+%\smallskip
+
+
+Our prior 
+articles
+\cite{ww93,ww1,ww5,wwapal} 
+did not use
+Definition \ref{gsim}'s
+global simulation formalism.
+They
+employed,
+instead,
+Definition \ref{dap4-2}'s
+Group-2 axiom schema.
+\phx{pqq5} 
+will be
+   the 
+ analog of  \phx{pqq4} 
+for
+global simulation.
+It will be
+ useful 
+when
+%in applications where 
+one desires
+to compress all the information held
+by a Group-2 schema into a
+single 
+finite-sized object.
+
+
+
+
+\smallskip
+
+\begin{theorem}
+\label{pqq5}
+Let  $ \,\xi \,$ 
+denote the generic configuration  
+  \gggcp,   
+ $\, \nal{B}\,$ denote a
+ recursively enumerable
+axiom system
+and $ \,  \nal{D} \,$ 
+denote any
+deduction method
+(which can be different than $\xi \,$'s deduction method    $~d~~)~$.
+Suppose that all the 
+$\Pi_1^\xi$
+theorems generated by
+ $ \, ( \,  \nal{B} \, , \, D \, ) \, $ are
+true  under the Standard-M model.
+Then the following  invariants
+do hold:
+\bed
+\item[   i ]
+If $ \, \xi \, $ is EA-stable 
+then 
+for each  $\, j \,$
+there 
+exists a 
+{\bf finitized
+extension}
+$~\beta_j~$ of $~B^\xi$ that 
+recognizes its 
+Level$(1^\xi$)
+self-consistency
+ and which
+contains
+the sentence
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$. 
+\item[  ii ] 
+Likewise
+for each $ j $,
+if $\xi$ is 
+  E-stable, A-stable  or 0-stable
+then there 
+exists a 
+{finitized
+extension}
+$~\beta_j~$ of $~B^\xi$ that 
+recognizes its 
+Level$(0^\xi$)
+self-consistency
+ and which
+contains
+the sentence
+GlobSim$^D_{ \nal{B}} \,(\xi,j)$. 
+\ennd
+\end{theorem}
+
+%\medskip
+
+
+{\bf Proof:}
+Let
+$\theta$ denote the 1-sentence R-View of 
+``GlobSim$^D_{ \nal{B}} \,(\xi,j)$''
+formalized by  Definition
+\ref{gsim},
+$~$and let 
+$~\beta_j^1~$  and 
+$~\beta_j^0~$ denote 
+$ ~ \, B^\xi \, \cup \, \theta  \, +  \, $SelfCons$^1(
+\,B^\xi \, \cup \, \theta  \, )  \,~ $
+and
+$  \, B^\xi \, \cup \, \theta  \, +  \, $SelfCons$^0(
+\,B^\xi \, \cup \, \theta  \, )  \, $,
+respectively.
+These axiom systems
+correspond 
+to
+the objects 
+that Definition 
+\ref{dap4-1} 
+had called $G^\xi_1(\theta)$ 
+and $G^\xi_0(\theta)$. 
+
+\medskip
+
+\phx{pqq5}'s  hypothesis 
+indicates
+%had specified that
+all the 
+$\Pi_1^\xi$
+theorems of
+ $( \nal{B},D)$  are true
+ under the Standard-M model.
+Thus,
+it follows that $\, \theta \,$ is 
+also
+true in the Standard-M model.
+Hence \phx{pqq3}
+implies that 
+$ \, \beta_j^1 \, = \, G^\xi_1(\theta)$ 
+is a consistent  system 
+satisfying \phx{pqq5}'s 
+claim (i).
+Likewise, \phx{pqq3}
+implies 
+$ \, \beta_j^0 \, = \, G^\xi_0(\theta)$ 
+satisfies
+\phx{pqq5}'s 
+ second claim.
+$~~\Box$
+
+
+\bigskip
+
+\begin{remark}
+\label{re4-n}
+\rm
+Theorems  \ref{pqq4} 
+and \ref{pqq5} raise a fascinating 
+%open 
+question:
+{\it Is the trade-off between these 
+% two 
+formalisms needed?}
+That is, 
+%Thus,
+can self-justifying systems use only a 
+ a finite number of added axioms
+beyond those lying in $~\xi\,$'s base 
+%axiom 
+system of $B^\xi$
+and 
+also
+%{\it simultaneously} 
+duplicate all 
+ $( \nal{B},D)$'s $\Pi_1^\xi$ theorems in a pure 
+sense (i.e. without simulation) ?
+We will return to this topic in Appendix G.
+\end{remark}
+
+
+%%nov150ut 
+%%nov150ut \begin{remark}
+%%nov150ut \label{old-re4-n}
+%%nov150ut \rm
+%%nov150ut Theorems  \ref{pqq4} 
+%%nov150ut and \ref{pqq5} 
+%%nov150ut are difficult to compare
+%%nov150ut  because
+%%nov150ut neither result is uniformly better than the other.
+%%nov150ut The advantage to Theorem  \ref{pqq4} 
+%%nov150ut is that it duplicates all of
+%%nov150ut  $( \nal{B},D)$'s $\Pi_1^\xi$ theorems in a pure sense,
+%%nov150ut rather than using Theorem  \ref{pqq5}'s
+%%nov150ut weaker simulated construct.
+%%nov150ut The virtue of Theorem  \ref{pqq5}, 
+%%nov150ut in contrast,
+%%nov150ut  is that its self-justifying systems 
+%%nov150ut include only
+%%nov150ut Our suspicion is that Theorem  
+%%nov150ut \ref{pqq5} is of greater  pragmatic interest,
+%%nov150ut while
+%%nov150ut \thx{pqq4} is the more mathematically surprising result.
+%%nov150ut \end{remark}
+
+
+
+
+
+%% \begin{remark}
+%% \label{recc1}
+%% \rm
+%% This article deliberately
+%% used the phrase ``boundary-case exception''
+%% to characterize our limited  evasions
+%% the Second Incompleteness Theorem. 
+%% This
+%% cautious terminology was employed
+%% because there are two fundamental barriers
+%% limiting the generality 
+%% of all such
+%% results:  
+
+
+
+\bigskip
+\bigskip
+\bigskip
+\bigskip
+
+%{\bf $~~~~$ A BROADER PERSPECTIVE:}
+
+% {\bf $~~~~$ A Yet Further Issue$~$:}
+
+
+
+
+% {\bf $~~~~$ A Yet Further Issue$~$:}
+{\bf $~~~~$ REFLECTION PARADIGMS$~$:}
+$~$Our  
+last
+%final 
+goal
+is to show how
+self-justifying systems 
+%can 
+support
+%% 
+%% will be to formalize
+%% how Theorems \ref{ppp1},
+%%  \ref{pqq3}, \ref{pqq4} 
+%% and \ref{pqq5} support 
+%% 
+{\it unusually}
+strong 
+reflection principles.
+% for their $\Pi_1^\xi$ theorems. 
+Let $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ 
+%thus
+denote  \eq{reflect}'s statement 
+when
+%in a context where
+$~\Psi~$ is a sentence with G\"{o}del number
+ $~\lceil \, \Psi \, \rceil~$, and 
+$~(\alpha,d)~$ denotes an axiom system
+and deduction method.
+\beq
+\label{reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+ ~~~ \Rightarrow ~~~ \Psi~~]
+\enq
+L\"{o}b's Theorem
+\cite{HP91,Lo55,So76b}
+ implies that conventional
+systems $\, (\alpha,d) \,$, 
+possessing at least
+ Peano Arithmetic's strength, are unable to
+prove 
+% the sentence 
+ $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ except for in the degenerate cases
+where they can prove $~\Psi$.
+
+
+% itself.
+
+%\cvt
+
+\medskip
+
+Moreover, it is easy to generalize
+L\"{o}b's Theorem
+(via say  \cite{ww1}'s
+Theorem 7.2) 
+so that
+a wide class of formalisms $~\alpha~$, weaker than
+%PA,
+Peano Arithmetic,
+are also
+ unable to prove
+% the validity of 
+ $\mbox{Reflect}_{\alpha,d}(~\Psi~)~$ 
+for
+% essentially 
+all $\Pi_1^\xi$
+sentences $~\Psi~$ simultaneously. 
+
+\medskip
+
+
+
+The intuition
+behind this generalization is quite simple.
+Let
+$~\mho~$
+denote 
+a $\Pi_1^\xi$ encoding for
+the classic
+ G\"{o}del sentence
+declaring: 
+{ \it  `` There is no proof of me 
+%{\it this sentence}
+from the axiom system $~\alpha~$ using $~d\,$'s deduction method''.}
+Then \cite{ww1}'s
+Theorem 7.2  
+uses a very routine 
+diagonalization argument
+to show 
+most formalisms
+$~\alpha~$ will be
+ inconsistent 
+%(and thus useless)
+if they  prove  $\mbox{Reflect}_{\alpha,d}(~\mho~)$'s
+statement. 
+
+\medskip
+
+%% 
+%% The intuition
+%% behind this result is quite simple.
+%% Let
+%% $~\mho~$
+%% denote 
+%% a $\Pi_1^\xi$ encoding for
+%% the classic
+%%  G\"{o}del sentence
+%% declaring: 
+%% { \it  `` There is no proof of me 
+%% %{\it this sentence}
+%% from the axiom system $~\alpha~$ using $~d\,$'s deduction method''.}
+%% Then 
+%% Theorem 7.2  
+%% used an essentially 
+%% conventional
+%% diagonalization argument
+%% to show 
+%% its formalisms
+%% $~\alpha~$ would be 
+%%  inconsistent (and thus useless)
+%% if they proved  $\mbox{Reflect}_{\alpha,d}(~\mho~)$'s
+%% statement. 
+
+
+
+Our next theorem will show, surprisingly, that
+the preceding limitation is much less 
+stringent 
+% severe
+than 
+it may
+initially  appear to be.
+This is
+because 
+%EA-stable 
+Level$(~1^\xi~)$ self-justifying 
+axiom systems
+are capable of proving  
+very close analogs to
+\eq{reflect}'s
+impermissible
+
+\nop
+
+\noindent
+ reflection principle,
+using a ``translational'' methodology.
+
+\smallskip
+
+Thus, let $~T~$  denote an
+algorithm that maps a
+% initial 
+$~\Pi_1^\xi~$ sentence $~\Psi~$ onto a 
+``translated''
+sentence  $~\Psi^T~$ that is equivalent
+to  $~\Psi~$ 
+under the Standard-M model
+{\it and which is also written in a  $~\Pi_1^\xi~$ 
+format.} 
+(See footnote\footnote{\label{imper} Part-3 of Definition
+\ref{def3.3} indicated that generic configurations are
+$\Sigma_1^\xi$ complete. 
+%In this context, 
+Our requirement that
+ $~\Psi^T~$ 
+must have a       $\Pi_1^\xi$ format
+thus
+causes 
+$~T~$ 
+to
+gain  much added
+meaning.
+This is 
+because 
+the
+axiom system $~B^\xi~$ will then  
+automatically  disprove
+$~\Psi^T~$ 
+ whenever
+it is false under the Standard-M model.
+(Thus, $T$'s mapping of 
+$~\Psi~$  onto $~\Psi^T~$ 
+gains much  significance
+when 
+$~\Psi~$ and
+$~\Psi^T~$ do rest on the same  $\Pi_1^\xi$ 
+level of the arithmetic
+hierarchy.)}
+for why 
+it is absolutely imperative that
+{\bf both these} requirements be included in
+$~T\,$'s definition.)
+Also, let
+$\mbox{Reflect}^T_{\alpha,d}(~\Psi~)~$ denote the
+%  corresponding
+translational
+modification of \eq{reflect}'s reflection principle that
+replaces $\Psi$ with $\Psi^T$. 
+%in its
+%modified {\bf ``Translation'' Reflection Principle.}
+\beq
+\label{T-reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+  ~~~ \Rightarrow ~~~ \Psi^T~~]
+\enq
+
+
+
+
+
+
+\begin{theorem}
+\label{ppp6}
+Let  $ \,\xi \,$ 
+denote the EA-stable configuration of  
+  \gggcp,   
+and let
+$~\alpha~=~   B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$
+denote  $ \,\xi \,$'s corresponding Level(1) self-justifying
+axiom system. Then
+there will exist 
+a translation methodology $~T~$ where
+ $~\alpha~$ can prove the validity
+of \eq{T-reflect} for all its $\Pi_1^\xi$ sentences simultaneously.
+%% 
+%% {\rm (In other words, this means that
+%% $~\alpha~$ will  know that
+%% its 
+%% proof of $~\Psi~$ automatically
+%% implies a counterpart
+%% of $~\Psi~$ is true in
+%% the Standard-M model ! )}
+%% 
+\end{theorem}
+
+% {\bf Abbreviated Proof:}
+
+\bigskip
+
+
+{\bf Proof:} Let us use
+Example  \ref{new-exam}'s notation.
+It observed that  $~\Psi\,$'s
+$\Pi_1^\xi$ statement
+was equivalent 
+under the Standard-M model
+to
+ ``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$''.
+Thus, 
+let us view $\,T \,$  as 
+being
+a mapping of
+the first sentence onto the second. 
+
+Our proof of Theorem \ref{ppp6} will next  use
+the following observations:
+\bee
+\item The non-existence of a proof of $~ \neg ~ \Psi~$ from
+$~    B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$
+trivially implies the non-existence of a proof of the same theorem
+from $~B^\xi~$
+ (because the latter axiom system is 
+simply
+a subset of
+the former).
+\item 
+Moreover, Example  \ref{new-exam}'s notation treats
+``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$'' as being equivalent to the declaration that no proof of
+$~\neg~\Psi~$ from $~B^\xi~$ exists. 
+\ene
+Hence, $~\alpha~$ can prove 
+\eq{T-reflect}'s  statement by noting
+that 
+$~p\,$'s proof of 
+a $\Pi_1^\xi$ sentence
+$~\Psi~$ implies 
+(via $\, \alpha \,$'s $~$SelfCons$^1~$  axiom) 
+the non-existence
+of a proof of $~\neg ~\Psi~$, which 
+(via Items 1 and 2)
+implies 
+ ``$ \,  \forall  \, x~ \mbox{Test}^\xi_0( \, \lceil \, \Psi \, \rceil \, , \, x \, )$''
+$~~\Box$.
+
+\cvs
+
+\bigskip
+\bigskip
+
+\begin{remark}
+\rm \label{f88}:    
+$~$\phx{ppp6} and the statement
+\eq{T-reflect}'s 
+Translational Reflection Principle
+may possibly be useful devices in unraveling some of the
+mystery that has enshrouded G\"{o}del's Second Incompleteness
+Theorem, since its inception.
+This is
+partly
+ because G\"{o}del was 
+explicitly uncertain
+about the 
+generality of the
+Second Incompleteness Theorem 
+in his initial 
+ 1931 
+seminal
+paper \cite{Go31}
+about this subject.
+% to appreciate the preceding 
+%oint.
+His centennial paper
+about Incompleteness
+ thus included 
+%
+%
+the following 
+quite
+poignant
+caveat:
+\begin{quote}
+%\small
+%\baselineskip = 1.0 \normalbaselineskip 
+$~\bullet~~~$ : 
+``It must be expressly noted that
+Theorem XI
+(i.e the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and {\it there might
+conceivably be finite proofs} which cannot
+be stated in ... ''
+%
+%%%%%%%%%%%%%%%%%%%%%%%% 
+%
+\end{quote}
+Some of the issues
+that troubled 
+G\"{o}del 
+in the 
+statement 
+$\bullet$
+can 
+perhaps 
+be 
+partially
+resolved if one
+compares the reflection principles of sentences
+\eq{reflect} 
+and \eq{T-reflect}.
+This is because 
+\eq{reflect} is probably
+unnecessary to explain how thinking beings can
+appreciate their
+ $~\Pi_1^\xi~$
+ theorems 
+%$~$---$~$ 
+when 
+\phx{ppp6}'s specialized
+logics can,
+instead,
+ use the fact that
+its $\Pi_1^\xi$ sentences  satisfy at least
+\eq{T-reflect}'s
+modified
+ reflection principle. 
+%% 
+%% (with
+%% $~\Psi^T~$ being required to be both 
+%% $~\Pi_1^\xi~$ and equivalent to $~\Psi~$ in the
+%% Standard-M model). 
+%% 
+%%
+%
+%In other words, 
+Thus,
+some of the mystery surrounding the
+Second Incompleteness
+Effect can be
+clarified
+ when one notices that 
+%the validity of
+\eq{T-reflect}'s translational reflection princible 
+is a useful precept, that was
+shown by \phx{ppp6} to be
+technically
+%essentially
+ unrelated to G\"{o}del's observation
+that no
+reasonable formalism  
+can prove
+%proves 
+\eq{reflect}'s 
+{\it purist}
+%a reflection 
+principle
+ for all 
+ $~\Pi_1^\xi~$
+sentences
+simultaneously.
+\end{remark}
+
+
+%%% 
+%%% 
+%%% Some of the issues 
+%%% G\"{o}del 
+%%% had raised in statement $~\bullet~$
+%%% can be
+%%% partially
+%%%  resolved when one
+%%% compares the reflection principles of sentences
+%%% \eq{reflect} 
+%%% and \eq{T-reflect}.
+%%% This is because 
+%%% \eq{reflect}'s reflection principle is probably
+%%% unnecessary to explain how thinking beings can
+%%% appreciate their
+%%%  $~\Pi_1^\xi~$
+%%%  theorems $~$---$~$ 
+%%% when some logics can,
+%%% instead,
+%%%  use the fact that
+%%% every $~\Pi_1^\xi~$ statement  satisfies at least
+%%% \eq{T-reflect}'s
+%%% modified
+%%%  reflection principle. 
+%%% %% 
+%%% %% (with
+%%% %% $~\Psi^T~$ being required to be both 
+%%% %% $~\Pi_1^\xi~$ and equivalent to $~\Psi~$ in the
+%%% %% Standard-M model). 
+%%% %% 
+%%% %%
+%%% In other words, some of the mystery surrounding the
+%%% Second Incompleteness
+%%% Effect can be
+%%% resolved
+%%%  when one notices that the validity of
+%%% \eq{T-reflect}'s translational reflection princible 
+%%% (for some {\it specialized} 
+%%% logics)
+%%% is unrelated to G\"{o}del's observation
+%%% that no
+%%% reasonable formalism can feasibly prove 
+%%% \eq{reflect}'s 
+%%% {\it purist form} of
+%%% a
+%%% reflection principle
+%%%  for all 
+%%%  $~\Pi_1^\xi~$
+%%% sentences
+%%% simultaneously.
+%%% %% \end{remark}
+
+%% 
+%% ) $~$---$~$
+%% {\it while at the same time} \phx{ppp6}
+%% permits a 
+%%  formalism
+%% to,
+%% nevertheless, 
+%% support  \eq{T-reflect}'s modified reflection principle and
+%% simultaneously
+%%  prove all Peano Arithmetic's  $~\Pi_1^\xi~$
+%% theorems.
+
+
+
+\bigskip
+
+
+
+\medskip
+
+\begin{remark}
+\label{remhappy}
+\rm
+Theorem \ref{ppp6} is
+also
+significant
+because it explains how
+its  specialized logics 
+ can grapple with a 
+$\, \Pi_1^\xi \, $ encoded
+G\"{o}del sentence
+$~\mho~$ 
+which asserts {\it ``There is no proof of me''}.
+This issue is challenging because
+routine constructions, such as
+\cite{ww1}'s Theorem 7.2,  
+demonstrate that no natural logic
+can 
+verify
+statement
+  \eq{reflect}'s validity 
+for all $~\Pi_1^\xi~$ 
+sentences $~\Psi~$
+(on account of the
+well-known
+syllogism
+posed by 
+$~\mho$'s  G\"{o}del sentence).
+%% 
+%% 
+%% on account of the challenge posed by 
+%% $~\mho\,$'s available  
+%%  $~\Pi_1^\xi~$ encoding. 
+%% 
+%% 
+Theorem \ref{ppp6} 
+constructs,
+%demonstrates,
+however,
+a 
+%method to 
+%serious
+reply to this challenge. 
+%% 
+%%% 
+%%%  that this 
+%%% challenge
+%%% %difficulty
+%%% is not quite as disturbing
+%%% as  might first 
+%%% appear
+%%% %be anticipated.
+%%% 
+This is
+because 
+%%%Theorem \ref{ppp6}'s 
+its self-justifying systems do
+%%% can
+surprisingly
+%{\it at least,}
+prove,
+without difficulty \footnote{Since 
+$\Psi$ and $\Psi^T~$ are equivalent under
+the Standard-M Model {\it but not also
+equivalent}  
+from the perspective of the 
+%axiom 
+system 
+$~\alpha~=~   B^\xi \, +  \, $SelfCons$^1(\,B^\xi \,)$,
+the conventional contradictions, produced by
+% sentence 
+\eq{reflect}'s reflection principle, disappear when it
+is replaced by 
+\eq{T-reflect}.
+%'s
+% theoretical
+% alternative principle. 
+(Thus, there is no danger that
+$\alpha$ could
+use \eq{T-reflect}'s reflection principle to 
+prove 
+an analog of $\,\mho \,$'s 
+forbidden G\"{o}del sentence.) \fend
+%the forbidden statement
+%{\it ``There is no proof of me''} 
+%using
+%%
+%% 
+%% The 
+%% ``difficulty'' that conventionally arises is that an
+%% axiom system 
+%% $~\alpha~$
+%% becomes inconsistent when it proves
+%% $\mbox{Reflect}_{\alpha,d}(~\mho~)~$'s sentence because this causes
+%% it to prove $~\mho\,$'s sentence, which asserts {\it ``There is no
+%% proof of me''.}  
+%% This difficulty vanishes,
+%% however, 
+%%  when $~\alpha~$
+%% proves the translational reflection sentence of 
+%% $\mbox{Reflect}^T_{\alpha,d}(~\mho~)$.
+%% This is because the latter replaces
+%% $~\mho~$ in  \ep{reflect}
+%% with \eq{T-reflect}'s sentence $~\mho^T~$.
+%% This second sentence  is equivalent to $~\mho~$
+%% under the Standard-M
+%% model.
+%% However,
+%% $~\alpha~$ is unable to recognize
+%% this equivalence (and  its
+%% thus escapes the syllogism of
+%% a G\"{o}del-like diagonalization argument).
+%% 
+%% 
+}, the validity of
+\eq{T-reflect}'s 
+{\it translated modification}
+of  \eq{reflect}'s
+% formally 
+unobtainable
+$\Pi_1^\xi$ 
+styled
+variant of
+a reflection principle.
+\end{remark}
+
+
+\bigskip
+
+\begin{remark}
+\label{new14}
+\rm
+We encourage the reader to examine the work of
+Beklemishev, Kreisel-Takeuti
+ and Verbrugge-Visser
+\cite{Be95,Be97,Be3,KT74,VV94,Vi5}
+to see
+alternative
+reflection principles
+and 
+their uses.
+(The  constraint on proof-length,
+by Verbrugge-Visser,
+ is certainly
+one 
+alternative to
+\thx{ppp6}'s machinery.
+Likewise, 
+%\cite{KT74}'s 
+Kreisel-Takeuti's
+Second Order
+Logic {\it CFA}
+reflection is another alternative,
+although it will not \footnote{Kreisel-Takeuti indicate on
+page 25 of \cite{KT74} that CFA's reflection principle for
+a first order formula ``A'' infers the validity of
+{\it only the relativized formula} ``$~A^N~$'' from $A$'s proof.
+Also, their proof predicate is similarily relativized. 
+Thus CFA's second-order logic
+reflection principle, while fascinating, does
+not generalize to first-order logic environments. }
+generalize to first-order 
+logics.) 
+One 
+complicating aspect of
+%drawback to
+our  \thx{ppp6}'s
+%translational 
+reflection
+%principle
+method
+ is
+%  quite 
+that
+Appendix E 
+% formally 
+proves it becomes
+%%% unfortunately
+% fully
+ inoperable
+when an axiom system is 
+% simply
+sufficiently conventional
+to satisfy G\"{o}del's Second Incompleteness Theorem.
+In essence, \thx{ppp6}'s translational reflection
+principle is a
+specialized
+methodology,
+% an attractive type of
+%machinery,
+%reflection methodology, 
+% that is
+intended for logics using
+Definition \ref{xd+1x7}'s
+Level(1) style
+of self-justification. 
+% ``SelfCons''
+% self-justifying axiom.
+\end{remark}
+
+
+%%%% cccccccccc
+%%%% Some readers may wish to examine the work of
+%%%% Beklemishev, Verbrugge and Visser
+%%%% \cite{Be95,Be97,Be3,VV94,Vi5}
+%%%% to see
+%%%% some  different 
+%%%% perspectives about 
+%%%% reflection principles.
+%%%% (The Verbrugge-Visser mechanism,
+%%%% with its  constraints  \cite{VV94,Vi5} on proof-length,
+%%%%  is 
+%%%% a
+%%%% % an especially
+%%%% fascinating alternative to
+%%%% \thx{ppp6}'s machinery.) 
+%%%% One 
+%%%% complicating aspect of
+%%%% %drawback to
+%%%% our  \thx{ppp6}'s
+%%%% %translational 
+%%%% reflection
+%%%% principle is
+%%%% %  quite 
+%%%% that
+%%%% Appendix E 
+%%%% % formally 
+%%%% proves it becomes
+%%%% %%% unfortunately
+%%%% % fully
+%%%%  inoperable
+%%%% when an axiom system is 
+%%%% % simply
+%%%% sufficiently conventional
+%%%% to satisfy G\"{o}del's Second Incompleteness Theorem.
+%%%% Thus, self-justifying systems support a type
+%%%% of 
+%%%% %surprising 
+%%%% reflection principle that is 
+%%%% uniquely intended 
+%%%%  for their particular
+%%%% % special types of
+%%%% applications.
+%%%% %inapplicable in conventional settings.
+
+
+\bigskip
+
+%\smallskip
+
+\begin{remark}
+\label{recc1}
+\rm
+The preceding discussion clearly shows 
+%that 
+self-justifying logics are 
+% very 
+% quite 
+tempting.
+At the same time,
+it is necessary 
+to be 
+very
+cautious
+because there are also two 
+fundamental 
+barriers
+limiting 
+such results:
+\bed
+\topsep -4pt
+\itemsep +2pt
+\item[   a  ] 
+The
+first is the
+ Theorem 2.1 arising from the joint work of
+Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris \cite{Ne86,Pu85,So94,WP87}.
+It
+showed
+no 
+reasonable 
+system recognizing successor
+as a total function can verify its own Hilbert consistency.
+%Moreover,
+Also,
+Willard \cite{ww2,ww7} established 
+%% ww0
+analogous results 
+under semantic tableaux consistency 
+for 
+% most
+systems recognizing multiplication
+as a total function. 
+%These results show that every effort to evade
+Thus, each effort to evade
+the  Second Incompleteness Theorem 
+must
+encounter 
+%inevitable
+ robust
+barriers.
+\item[   b  ]
+A second issue is that Definition \ref{xd+1x7}'s
+``SelfCons''
+{\it ``I am consistent''} 
+axiom
+sentence is less than ideal because 
+it causes axiom systems to produce
+essentially
+a
+1-line proof of their
+own 
+ consistency.
+Such an excessively compressed proof corresponds more
+closely to an axiom system formulating
+ an
+{\it instinctive faith} in its own consistency
+(rather than it  supporting 
+a 
+% more elaborate 
+full-length proof-justification
+of this fact).
+\ennd
+Part of the reason 
+self-justifying systems are of
+interest, despite these limitations, is that they illustrate
+how some formalisms {\it are compatible} with at least an 
+{\it instinctive faith} in their own self-consistency.
+(This compatibility issue is non-trivial
+because Item (a) 
+implies
+there are
+many circumstances where a generalization of the Second Incompleteness
+Theorem will make it infeasible for a formalism to satisfy
+{\it both} 
+Parts (i) and (ii) of Section 1's definition
+of Self-Justification.)
+Moreover, three of Appendix D's four sample self-justifying
+configurations,
+called $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$,
+ will be Type-A systems that
+ recognize
+addition as a total function. These configurations will
+thus
+possess
+the following three 
+significant
+finitized 
+ features:
+\bee
+\topsep -4pt
+\itemsep +2pt
+\item
+They will be able to construct the entire infinite set of integers
+{\it by finite means} because they recognize addition as a total function.
+\item
+For any r.e. logical
+configuration $  (   \nal{B}  ,  D  )  $, 
+it will be possible to develop a 1-sentence
+ {\it finitized} 
+extension for the base axiom systems of any of  
+the configurations of  $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~,~$
+which deploy
+ \eq{globsim}'s
+ Global Simulation Sentence
+to simulate the 
+ $~\Pi^\xi_1~$ knowledge of  $  (   \nal{B}  ,  D  )  $.
+This means 
+that some fully
+finite-sized
+extensions of the base-formalisms of 
+  $~\xi^*~$, $~\xi^{**}~$ and $~\xi^R~$ will contain a
+non-trivial amount of 
+ $~\Pi^\xi_1~$ styled
+knowledge,
+since  $  (   \nal{B}  ,  D  )  $ can 
+correspond to, say,
+%represent for example
+Peano Arithmetic. 
+\item
+The 
+key point is
+ that a 1-sentence extension of an
+axiom system  containing  features (1) and (2)
+can formalize 
+how a
+logic
+ can possess
+an instinctive faith in its own
+consistency
+via \thx{pqq5}'s explicitly
+{\it finitized} structure.
+(Moreover, \thx{ppp6}'s 
+Translational  Reflection Principle
+is applicable to 
+Appendix D's
+ generic configurations of
+ $~\xi^*~$ and $~\xi^{**}~$.
+It will 
+thus imply that  their single 
+{\it finitized}
+ Level-1 self-justifying
+axioms enable 
+% each of
+them to prove
+an  {\it infinite number} of incarnations
+of \eq{T-reflect}'s
+translational 
+reflection principle, where  each
+$\Pi_1^*$ sentence  
+$\Psi$ 
+is mapped onto one such 
+%well-definded unique 
+unique instance.)
+%%% 
+%%% 
+%%% Appendix G's
+%%%  more 
+%%% sophisticated
+%%%  Theorems G.2 and G.3,
+%%% %%% 
+%%% %%% accompanied by 
+%%% %%% Appendix F's
+%%% %%% epistemological  
+%%% %%% discussion,
+%%% %%% in{re
+%%% will
+%%% further strengthens   
+%%% these results.)
+%%% 
+%%% 
+\ene
+The contrast between Items 1-3's positive remarks
+about ``finitized'' cogitation with
+Items (a) and (b)'s 
+opposing comments 
+is 
+obviously
+formidable. 
+It is clearly preferable
+to
+view these
+positive results 
+cautiously and treat them
+as being no more than
+boundary-case  exceptions  to
+the Second Incompleteness Theorem. 
+The essential reason why 
+these exceptions are of
+interest
+is that G\"{o}del's famous centennial paper has
+implicitly raised
+the following 
+%unresolved 
+puzzling issue:
+\begin{quote}
+ \#  How is it that Human Beings
+ manage 
+to muster 
+the physical 
+drive
+to think (and prove theorems) when the many
+generalizations of 
+G\"{o}del's Second 
+Incompleteness Theorem assert 
+conventional logics
+lack knowledge of
+their own consistency?  
+\end{quote}
+There will, of course, never be any perfect answer to the 
+puzzle
+posed by
+$~\# ~$ 
+because philosophical paradoxes and ironical dilemmas never yield
+perfect answers. 
+However, 
+ part of an
+imperfect
+ answer to 
+$~\# ~$ is that 
+Items 1-3 reply to
+ Challenges (a) and (b) by formalizing how
+a thinking being can muster an
+% at least
+approximate partial
+{\it instinctive faith} in its own self-consistency.
+(Moreover, the
+tight
+ contrast 
+between various 
+generalizations of the Second
+Incompleteness Theorem
+\cite{Ad2,AZ1,BS76,BI95,HP91,HB39,Ko6,Lo55,Pu85,Sa11,So94,Sv7,WP87,ww2,wwlogos,ww7}
+with the
+self-justifying systems appearing
+in Appendixes D and G
+suggests 
+that these
+come close to
+being
+maximal forms
+of
+feasible   
+results.)
+\end{remark}
+
+%\smallskip
+
+Our remaining discussion 
+will consist of four optional sections, called
+ Appendixes D, E,  F and G,
+% consists of three 
+% optional
+% sections,
+which
+can be skimmed,
+omitted or  examined
+in 
+any
+% whichever 
+order 
+ the reader 
+%so 
+prefers.
+A 
+%brief 
+summary of 
+their contents 
+is given 
+%provided
+below:
+\bed
+\itemsep +4pt
+\item[   I  ] 
+The Appendix D provides four examples of
+generic configurations
+ that
+utilize
+Theorems \ref{ppp1}, \ref{ppp2},
+\ref{pqq3}, \ref{pqq4} 
+and \ref{pqq5}.
+Its most prominent
+examples involve 
+\ep{totdefxa}'s
+Type-A axiom systems
+where the deduction method is
+either semantic tableaux
+or a modified version of tableaux that permits
+a modus ponens rule for $\Pi_1^\xi$ and $\Sigma_1^\xi$ sentences.
+\item[ II  ]
+The Appendix E introduces a generalization of
+the  Second Incompleteness
+Theorem which shows that \thx{ppp6}'s
+Translational Reflection Principle
+applies {\it only to} 
+self-justifying logics. (It is thus  fully inoperative
+for conventional logics.
+This may explain why 
+\thx{ppp6}'s
+self-justifying systems are an interesting topic.)
+\item[ III  ]
+The Appendix F differs from the rest of
+this paper
+by having a 
+philosophical
+slant.
+It will 
+offer a 3-page summary about why we suspect 
+\thx{pqq3}'s 
+self-justification
+formalism and Remark \ref{recc1}'s
+notion of ``instinctive faith'' are useful.
+%and related to Theorem \ref{ppp}'s reflection principle.
+\item[  IV  ]
+The Appendix G 
+introduces 
+a ``Braced$^\xi(  \Phi  ,j  )$''
+construct
+ and two new theorems that hybridize
+the methodologies of
+ Theorems \ref{pqq4} 
+and \ref{pqq5}.
+These  results will
+improve upon
+ Theorem \ref{pqq4} because 
+their self-justifying  systems 
+contain only a finite 
+number of axiom-sentences beyond those lying in
+$~\xi \,$'s base formalism
+of  $B^\xi $.
+They 
+will
+improve upon 
+ Theorem \ref{pqq5} because
+they can prove the important  
+ Braced$^\xi(  \Phi  ,j  )$  
+subset of
+ $ (   \nal{B} , D )  $'s $\Pi_1^\xi$ theorems
+in a full sense
+ (rather than in
+Remark  \ref{re4-2}'s 
+weaker 
+simulated respect).
+Appendix G's results are useful
+because {\it for arbitrary $k$}
+and 
+for
+any of Appendix D's four sample configurations,
+every
+$\Pi_1^\xi$ theorem
+of  $  (   \nal{B}  ,  D  )  $
+ containing $~k~$ or
+fewer bounded and unbounded quantifiers will 
+be proven 
+by its Theorem G.3 to be self-justifying
+in an undiluted pure sense. (This is because each
+such $\Pi_1^\xi$ sentence, with fewer than $~k~$
+quantifiers, will 
+lie in
+ some
+ fixed  Braced$^\xi(  \Phi  ,j  )$  
+set $~$---$~$  where solely 
+the value of
+$~k~$ determines the values for
+ $~\Phi~$ and $~j ~$. )
+% \nop
+\ennd 
+
+%\njp
+
+\cvmew
+\cvs
+
+
+
+It is 
+probably
+desirable
+ to 
+ concentrate 
+primarily
+ on
+ Theorems \ref{ppp1}, \ref{ppp2},
+\ref{pqq3}, \ref{pqq4}, \ref{pqq5}
+and  \ref{ppp6}
+during 
+one's first reading of this paper.
+This is because
+Appendixes A-G
+are
+less central than these 
+% five 
+core theorems,
+although their material
+does
+add 
+several
+useful
+further
+perspectives to this subject.
+
+
+%%notre
+  
+\medskip
+
+
+\cvb
+\parskip 3pt
+
+
+%%notre-only  
+%%notre-only    {\bf Reminder to the Referee and Editor:} The cover
+%%notre-only    page of this article indicated any (or all) of 
+%%notre-only    Appendixes A-F can be removed from this article if it
+%%notre-only    is deemed too long. In that case, the manuscript could
+%%notre-only    cite the Cornell Archive paper 
+%%notre-only    \cite{ww11}
+%%notre-only    containing these appendixes.
+
+
+%% 
+%%   {\bf Reminder to the Referee:}
+%% % and Editor:} 
+%% The cover
+%%   page of this article indicated any (or all) of 
+%%   Appendixes A-F can be removed from this article if it
+%%   is deemed too long. 
+%% (This is because 
+%% these results
+%% are stored in \cite{ww11}'s archives.)
+
+
+%% In that case, the manuscript could
+%%   cite the Cornell Archive paper 
+%%   \cite{ww11}
+%%   containing these appendixes.
+
+
+
+
+\section*{7.  Concluding Remarks}
+
+
+The research in 
+this article 
+has been
+ a continuation of our prior
+research \cite{ww93}-\cite{ww9} that simultaneously
+has simplified, unified
+and extended the prior results. 
+It
+has explored self-justification
+with a 3-part approach
+where:
+\bee
+% \itemsep +3pt
+\item
+Sections \ref{3uuuu2} and \ref{3uuuu3}
+introduced
+three different
+  stem components
+that can be used to generate
+ self-justifying
+ systems. (These are
+     the relatively simple Lemma  \ref{lemex4}
+and the mathematically more sophisticated
+Theorems  \ref{ppp1}  and \ref{ppp2}.)
+\item
+Section \ref{sect64} and Appendix G then
+generalized our initial stem-like theorems
+in the
+six 
+% five 
+different directions 
+formalized
+by
+Theorem  \ref{pqq3},  \ref{pqq4},
+  \ref{pqq5}, 
+  \ref{ppp6}, 
+G.2 and G.3
+\item
+Appendix D 
+subsequently
+provided four examples of generic configurations
+that are applications of 
+Section \ref{sect64}'s results.
+\ene
+This 3-part approach is
+very
+different from the methods
+used
+in our  prior articles
+\cite{ww93,ww5,wwapal,ww9}. The latter
+examined
+ particular isolated applications
+in 
+thorough
+detail (rather than
+compartmentalize
+and separate
+ the analysis
+into three stages). The virtue of this 3-stage analysis
+is 
+it leads to many new theorems, in addition to
+unifying our prior results.
+
+%%cc \bigskip
+
+It is desirable to 
+categorize
+the maximal
+generality and 
+strongest form of
+boundary-case exceptions
+for the Second Incompleteness Theorem
+that are feasible because
+G\"{o}del's centennial discovery 
+ beckons the 
+scholarly community
+to sharpen their understanding of his 1931 
+landmark discovery,
+that
+has
+fundamentally
+ reshaped mathematics.
+
+
+
+
+%%cc \bigskip
+
+
+It should be emphasized that
+our over-all 
+research
+in \cite{ww93} -- \cite{ww9}
+ has spent an approximately
+equal 
+effort
+in
+ exploring generalizations of
+the Second Incompleteness Theorem
+\cite{ww2,wwlogos,wwapal,ww7} 
+and 
+in
+examining
+its 
+viable 
+boundary-case exceptions
+\cite{ww93,ww1,ww5,ww6,wwapal,ww9}
+(although the current article
+focused on
+the latter topic).
+ This is because
+the
+Second 
+Incompleteness Theorem
+is 
+a starkly 
+robust result that
+ imposes 
+sharp
+limits
+on how strong self-justifying systems may become.
+
+%%cc \bigskip
+
+
+Finally, we encourage the reader to take another brief glance
+at Remarks \ref{f88} -- \ref{recc1}.
+They
+offer a brief
+summary
+of both the
+strengths and limitations
+of our
+chief
+ results.
+They also explain how Theorem \ref{ppp6}'s reflection
+principle for
+$\Pi_1^\xi$ styled theorems is a  
+very  unexpected
+ result.
+
+%quite unexpected
+%and pleasing result.
+
+% implications for  
+% Remark \ref{remhappy} additionally explains
+% what types of
+
+%reflection principles. 
+
+%%%%% that are associated with 
+%%%%boundary-case
+
+
+
+
+  
+%%%%%  \bigskip
+%  
+
+
+%%cccorn-ff  
+%%cccorn-cc  
+%%cccorn-cc  
+
+
+  \bigskip
+  \bigskip
+    
+  
+  
+    
+    {\bf ACKNOWLEDGMENTS: }
+    I thank
+    Bradley Armour-Garb,
+    Seth Chaiken and
+    Kenneth W. Regan
+    for many useful suggestions about how to
+    % significantly
+     improve
+    the presentation 
+    style
+    of some earlier drafts of this article.
+    As was noted in the text of this article, some
+    of my research
+    during the last 17 years was influenced by some private 
+    communications
+    that I had with Robert M. Solovay 
+    in 1994 and
+     Leszek 
+    Ko{\l}odziejczy  in 2005.
+
+
+
+\newpage
+
+\parskip 2pt
+
+\cvt
+
+\section*{Appendix A: The
+$~\Pi_1^\xi~$ encoding for 
+SelfCons$^k(\beta,d)$}
+
+
+
+
+
+
+This appendix will  
+summarize how to 
+formalize
+a $\Pi_1^\xi$ encoding for
+Definition \ref{xd+1x7}'s 
+SelfCons$^k(\beta,d)$ predicate.
+It
+will
+use
+the following notation:
+\bee
+\itemsep +1pt
+\baselineskip = 1.0 \normalbaselineskip
+\topsep -3pt
+\itemsep 4pt
+\item
+Neg$^k(x,y)$,
+will denote 
+a
+$\Delta_0^{\xi}$
+formula 
+indicating that $~x~$ is the G\"{o}del
+number of a 
+$\Pi_k^\xi$ sentence and
+that $~y~$
+represents the  
+$\Sigma_k^\xi$ sentence
+which is its logical negation.
+\item
+$\mbox{Prf} \, _\beta^d~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that
+ $~p~$ is  a proof of theorem $~t~$ from the axiom
+system  $~\beta~$ using the deduction method $~d.~$
+\item
+$\mbox{ExPrf} \, _\beta^d\,( \, h \, , \, t \, , \, p \,)$  
+will denote that
+$ \, p \, $ is a proof
+(using $d$'s deduction method)
+ of
+a theorem $t$ 
+from the union 
+of the axiom system
+$\beta$ with the added
+sentence whose G\"{o}del number equals
+$\, h \,$. 
+\item
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ encodes a sentence
+that replaces all occurrence of free variables in $g \,$ with
+a term of $\bar{g}$ (that specifies $g$'s
+G\"{o}del number). 
+\item
+$\mbox{SubstPrf} \, _\beta^d~( \, g \, , \, t \, , \, p \,)$  
+will denote the 
+hybridization of Items 3 and 4
+that yields a Boolean value of TRUE 
+when there
+exists an integer $~h~$ 
+satisfying 
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf} \, _\beta^d~( \, h \, , \, t \, , \, p \,)$.
+\ene
+
+
+It is easy to apply \cite{ww1}'s methodologies
+to confirm 
+Items 1-5 
+can be encoded as $\Delta_0^\xi$ formulae.
+Thus,  
+Appendixes C and D of  \cite{ww1}
+explained how  the theory of
+LinH functions 
+\cite{HP91,Kr95,Wr78}
+implied  there existed 
+ $\Delta_0$ encodings 
+for  formulae 1-4,
+and
+these 
+ $\Delta_0$ encodings can be
+easily rewritten
+ \footnote{  \b55 This rewriting
+of conventional $\Delta_0$ formulae into
+a  $\Delta_0^\xi$ format
+ is possible because  
+Part 2  of Definition \ref{def3.3}
+indicated that
+two 3-way predicates 
+of
+Add$(x,y,z)$
+and Mult$(x,y,z)$ do 
+encode  addition and multiplication
+in a $\Delta_0^\xi$ styled form.}
+as
+ $ ~ \Delta_0^\xi ~$ expressions.
+\ep{encode} 
+ uses this information to formulate a
+$\Delta_0^{\xi}$ encoding for 
+$\mbox{SubstPrf} \, _{~\beta}^d \,(   g   ,   t   ,   p  )$'s 
+graph. It is
+equivalent to 
+$~$``$~\exists ~h~ \{ ~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf} \, _{\beta}^d(    h    ,    t    ,    p   )\,  \}  \, \,$''$,~$
+  but \ep{encode} is written in
+ a $\Delta_0^{\xi}$ format --- {\it unlike} the quoted
+expression.
+\begin{equation}
+\label{encode}
+\mbox{Prf} \, _{\beta}^d~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~ \{ ~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf} \, _{\beta}^d~( \, h \, , \, t \, , \, p \,)~ \}   
+\end{equation}
+
+Using \eq{encode}'s
+  $\Delta_0^{\xi}$ encoding for 
+$\mbox{SubstPrf} \, _{\beta}^d(   g   ,   t   ,   p  )$, it is
+easy to encode 
+SelfCons$^k(\beta,d)$
+ as  a $\Pi_1^{\xi}$ 
+axiom-sentence.
+Thus,  let
+$ \, \Gamma^k(g) \, $ 
+denote  \eq{encode12}'s formula, and
+let  $ \, \bar{n} \, $ denote $ \, \Gamma(g)$'s
+G\"{o}del number.
+\begin{equation}
+\label{encode12}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^k(x,y) ~ \wedge~ 
+\mbox{SubstPrf} \, _{\beta}^d  \, (  g    ,    x    ,    p   ) ~
+\wedge  ~
+\mbox{SubstPrf} \, _{\beta}^d  \,    (    g    ,    y    ,    q   )  \, \}
+\end{equation}
+Then  $~$``$~\Gamma^k(~ \bar{n}~)~$''$~$ 
+is a $\Pi_1^{\xi}$ 
+encoding 
+for SelfCons$^k(\beta,d)$'s formalization of
+the statement $+$ from Definition
+\ref{xd+1x7}.
+Thus,  $~\Gamma^k(~ \bar{n}~)~$
+is 
+encoded is as follows:
+\begin{equation}
+\label{encode12n}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^k(x,y) ~ \wedge~ 
+\mbox{SubstPrf} \, _{\beta}^d  \, (  \bar{n}     ,    x    ,    p   ) ~
+\wedge  ~
+\mbox{SubstPrf} \, _{\beta}^d  \,    (     \bar{n}   ,    y    ,    q   )  \, \}
+\end{equation}
+
+
+
+{\bf Reminder about \ep{encode12n} :}
+This sentence's
+definition for 
+SelfCons$^k(\beta,d)$ 
+does not assure 
+\eq{encode12n}
+is
+true under the Standard-M model. Indeed for nearly
+all $(\beta,d)$, it will be false
+when $k \geq 2$. This
+is the reason that the study of
+SelfCons$^k(\beta,d)$,
+under Theorems \ref{ppp1} and \ref{ppp2},
+has focused on the cases where $~k~$ equals 0
+or 1.
+Moreover,
+the preceding
+construction did assure that 
+SelfCons$^k(\beta,d)$
+had a
+$~\Pi_1^\xi~$ encoding
+because 
+such
+an
+{\it ``I am consistent''} axiom 
+carries 
+ more meaning
+than  a 
+$\Pi_2^{\xi}$ encoded axiom.
+
+
+
+
+
+
+\parskip 0 pt
+
+
+\section*{Appendix B: The Proof of Lemma \ref{llpp3}}
+
+
+\cvt
+
+Lemma \ref{llpp3} is
+a crucial interim step used to verify
+each of
+Theorems \ref{pqq3}, \ref{pqq4} and \ref{pqq5}.
+Its proof will employ
+the following three 
+straightforward observations:
+\bed
+\item[Fact B.1 ]
+Lemma  \ref{llpp3}'s 
+hypothesis
+implies that
+ $\mxi$ 
+is a
+generic configuration. (This is because 
+it specified that
+ $\xi$ 
+was a generic configuration  
+and that
+all the $\Pi^\xi_1$ sentences of
+$~\mheta~$ were
+true
+in the
+ Standard-M model.
+Thus,
+ $\mxi$ 
+must also be a
+generic configuration.)
+\item[Fact B.2 ]
+The
+associative identity of 
+ $ \, \theta \, \cup \, (\mheta \,  \cup B^\xi) \,  \, = \,  \, 
+( \, \theta \, \cup \, \mheta \, ) \cup B^\xi \, $ 
+obviously holds. It 
+implies
+a sentence
+$ \, \Uxp \, $
+is a theorem of
+ $ \, \theta \, \cup \, (\mheta \,  \cup B^\xi) \, $ 
+if and only if 
+it is a theorem of 
+ $ \, ( \, \theta \, \cup \, \mheta \, ) \cup B^\xi \, $.
+
+
+
+\item[Fact B.3 ]
+Lemma \ref{llpp3}'s hypothesis 
+directly
+\footnote{\s55 The identity of 
+$~ \zzz \mheta ~)~=~\infty~$
+must be true
+because the 
+hypothesis 
+of Lemma  \ref{llpp3}
+ indicated that 
+all the $\Pi^\xi_1$ sentences in
+$~\mheta~$ are 
+true  under the Standard-M model.
+By  Definition \ref{chg}, 
+this
+ implies $~ \zzz \theta ~)~= ~ \zzz \mheta \cup \theta ~)~$.}
+implies $ \zzz \theta )=  \zzz \mheta \cup \theta )$.
+\ennd
+
+\smallskip
+
+The justification of
+claims (i)-(iv)
+are
+ consequences of 
+Facts B.1 through B.3.
+We
+will
+ provide a detailed proof of only Claim (i) here
+ because all
+four claims have 
+similar proofs.
+
+
+
+
+
+
+
+\medskip
+
+{\bf Proof of Claim (i) :}
+The hypothesis of Claim (i) indicated that $~\xi~$ 
+was 0-stable. Therefore, it
+satisfies
+ Definition \ref{ostab}'s
+invariant of $~***~$. 
+In a context where
+$~\phi ~$ is a variable
+designating a r.e. set of $\Pi_1^\xi$ sentences 
+and  $ \, \Uxp \, $ is a variable corresponding to a 
+$\Delta_0^\xi$ sentence, 
+ the invariant  $~***~$
+can be rewritten in a  quasi-rigorous form as :
+\beq
+\label{whatshit1}
+\forall ~ \phi ~~ \forall ~ \Uxp 
+ ~~~~~  \mbox{\it the below
+statement, called $~ \Psi_1(\phi,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $ \, \phi \,  \cup B^\xi \, $ 
+via a proof $ \, p \, , \, $ 
+whose length satisfies
+ Log$(p) \, \leq \zzz \phi \, ) \, +1 \, , \,  $
+ then  $ \, \Uxp \, $ 
+is true under the Standard-M model.
+\end{quote}
+Since \eq{whatshit1}'s
+ universally quantified variable 
+$~\phi~$
+can designate any r.e. set of $\Pi_1^\xi$ sentences, it 
+may
+designate the object  ``$~\theta ~\cup~\mheta~$'', where
+$~\mheta~$ is the r.e. set of  $\Pi_1^\xi$ sentences
+defined by Lemma \ref{llpp3}'s hypothesis
+(and $~\theta~$ is any second r.e. set of sentences).
+Thus,  \eq{whatshit1} 
+directly
+implies :
+\beq
+\label{whatshit2}
+\forall ~ \theta ~~ \forall  \Uxp
+ ~~~~~  \mbox{\it the below
+statement, called $~ \Psi_2(\theta,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$~\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from the
+axiom system $~(~\theta~\cup~\mheta~)~ \cup B^\xi~$ 
+via a proof $~p~,~$ 
+whose length satisfies
+ Log$(p) \, \leq \zzz  \, \theta\cup\mheta)~   +1, $
+ then  $ \, \Uxp \, $ 
+is true  under  Standard-M.
+\end{quote}
+Facts B.2 and B.3 enable one to 
+simplify 
+\eq{whatshit2}'s
+ terms of  $( \, \theta \, \cup \, \mheta \, ) \,  \cup B^\xi$ 
+and  $\zzz ( \, \theta \, \cup \, \mheta \, ) \, ~    )$  
+and thus to derive \eq{whatshit3} as a consequence.
+\beq
+\label{whatshit3}
+\forall ~ \theta ~~ \forall  \Uxp ~~~~
+   \mbox{\it the below
+statement, called $~ \Psi_3(\theta,\Uxp)$,  is true  }
+\end{equation}
+\begin{quote}
+$~\forall ~p~~$ If $ \, \Uxp \, $ is a
+$\Delta_0^\xi$ theorem 
+ derived
+from 
+axiom system $\theta~\cup~(~\mheta~ \cup B^\xi~)$ 
+via a proof $~p$, 
+whose length satisfies
+ Log$(p) \, \leq \zzz \theta) \, + \,1 $,
+ then  $  \Uxp  $ 
+is true under the Standard-M model.
+\end{quote}
+We will now use Fact B.1's observation that
+$\mxi$ 
+is a generic configuration.
+The sentence
+\eq{whatshit3} 
+indicates
+this configuration
+satisfies  
+Definition \ref{ostab}'s invariant of
+$***$.
+Hence, 
+$\mxi $
+ is
+0-stable.
+$\Box$
+
+%%cccorn-ff
+
+
+
+  
+  \medskip
+  
+  \cvs
+  
+  {\bf Brief Comments about the Justifications of Claims (ii)-(iv):}
+  This appendix has  omitted 
+  proving 
+  (ii)-(iv) 
+  for the sake of brevity.
+  Their proofs are   similar
+   to Claim (i)'s proof.
+  For instance,   Claim (ii)'s proof differs
+  from  Claim (i)'s proof  by having
+  the 0-stability invariant in $***$
+   replaced
+  by the 
+  A-stability invariant of $~*~$.
+  This will  cause the analogs of
+  \eq{whatshit1} -- \eq{whatshit3} 
+  to undergo the 
+  following two 
+  simple
+  changes under Claim (ii)'s proof :
+  \bee
+  \topsep -10pt
+  \itemsep +2pt
+  \item 
+  $ \, \Uxp \, $ 
+  will  represent a 
+  $\Pi_1^\xi$ 
+  (rather than a
+  $\Delta_0^\xi$ ) theorem-statement under the 
+  revised versions of $\Psi_1$, 
+  $\Psi_2$ and $\Psi_3$  
+   used in Claim (ii)'s proof
+  \item 
+  The requirement
+  (in 
+  sentences \eq{whatshit1}-\eq{whatshit3} of Claim i's proof)
+   that
+  $  \Uxp  $  be
+  true in the Standard model
+  is
+   changed 
+  to the stipulation that 
+  $  \Uxp  $ 
+     satisfies 
+   the Good$\{ \, \tftt  \zzz \phi) \, \}$
+  and
+  Good$\{  \tftt  \zzz \theta)  \}$
+  conditions
+   under the revised forms of
+   $\Psi_1$, 
+  $\Psi_2$ and $\Psi_3$  
+  used   to prove Claim (ii).
+  \ene
+  Other
+  minor adjustments in Claim (i)'s proof
+  shall verify Claims iii and iv.
+  
+  
+  
+
+\nop
+
+
+\section*{Appendix C: The Proof of \phx{ppp2} }
+
+
+
+Our  proof of  \phx{ppp2}
+is a straightforward modification of
+\phx{ppp1}'s 
+ proof. 
+It
+will be divided into
+two lemmas.
+
+
+\medskip
+
+{\bf Lemma C.\ref{B1-lem}.}
+{\it 
+Every generic configuration that is either E-stable or
+A-stable will automatically satisfy
+Definition \ref{ostab}'s 0-stability condition.}
+
+\medskip
+
+
+
+{\bf Proof.} $~$
+Lemma C.\ref{B1-lem} is 
+a 
+consequence of
+the 
+``Special Note''  appearing at the end of 
+Definition \ref{xd+1x4}.
+For every  $N\geq 0$,
+it
+ indicated that
+if
+$\Uxp$ is a 
+$\Delta_0^\xi~$ 
+sentence then
+Scope$_E$($\Uxp,N)$
+is  equivalent to  $~\Uxp~$.
+This implies 
+(via Definition \ref{xd+1x5})
+that if 
+$\Uxp$ is a  
+Good(N)
+$\Delta_0^\xi~$ formula
+then Scope$_E$($\Uxp,N)$ is automatically
+true under the Standard-M model.
+
+
+
+
+
+
+The latter observation makes it easy to confirm
+Lemma C.\ref{B1-lem}. This is 
+because  every
+$\Delta_0^\xi$ sentence
+is a
+$\Pi_1^\xi$ and
+$\Sigma_1^\xi$ statement.
+Hence, the application of the invariants
+$~*~$  and $~**~$
+from Definitions 
+\ref{astab} and \ref{estab}, in the degenerate case
+where
+$\Uxp$ is a
+ $\Delta_0^\xi$ theorem,
+corroborates 
+Lemma C.\ref{B1-lem}'s claim
+(by showing that
+ Definition \ref{ostab}'s 
+ invariant of
+ $\,***\,$ does hold).
+$~~\Box$
+
+\medskip
+
+The remainder of this appendix will  focus on
+Definition \ref{ostab}'s 0-stability condition.
+(This is sufficient to justify \thx{ppp2}
+because Lemma C.\ref{B1-lem}
+showed all 
+ E-stable and A-stable
+configurations are
+ 0-stable.)
+
+\smallskip
+
+{\bf Lemma C.\ref{B2-lem}.}
+{\it 
+Let $~\xi~$
+denote a
+generic configuration
+ \gggen 
+that is
+0-Stable.  Then 
+the axiom system of
+$B^\xi+$SelfCons$^0(B^\xi,d)$
+will be consistent (and 
+hence self-justifying).}
+
+\smallskip
+
+
+
+\smallskip
+
+{\bf Proof:}
+It will be awkward 
+to write repeatedly
+the expression   ``$~B^\xi+$SelfCons$^0(B^\xi,d)~$''
+during our proof.
+Therefore,  $\zhz $ will
+be an abbreviated
+name for this 
+system. 
+Our justification
+ of Lemma C.\ref{B2-lem},
+is similar to \phx{ppp1}'s proof,
+$\,$except  it replaces a 
+Level$(1^\xi)$  form of self-justification
+with  
+a 
+Level($0^\xi)$.
+It
+will thus
+  be
+ abbreviated and 
+use the following notation:
+\bee
+\itemsep +4pt
+\item
+$~\mbox{Prf}_\zhz (t,p)~$  is a
+$\Delta_0^\xi$ formula 
+specifying
+$~p~$ 
+is
+ a proof of the theorem $~t~$ from
+the  axiom system $~\zhz ~$
+(using 
+$~\xi \,$'s
+deduction method of $d$ ).
+\item
+$~\mbox{Neg}^0(x,y)~$ 
+ is a
+$\Delta_0^\xi$ formula 
+ indicating
+$~x~$ is
+the G\"{o}del encoding of a
+ $\Delta_0^\xi$ 
+sentence and
+$~y~$ is a   $\Delta_0^\xi$ 
+sentence  representing
+$~x \,$'s negation.
+\ene 
+Expression \eq{qpencode} 
+denotes
+ $~\zhz \,$'s 
+ Level$(0^\xi)$
+self-justification axiom.
+It is encoded using
+Appendix A's
+methodology,
+similar to 
+its counterpart 
+used in \phx{ppp1}'s proof (i.e.  
+\ep{pencode} ).
+\begin{equation}
+\label{qpencode}
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q 
+~~~  \neg  ~~
+ \{ \,\mbox{Neg}^0(x,y) ~ \wedge~ 
+\mbox{Prf}_\zhz   \, (     x    ,    p   ) ~
+\wedge  ~
+\mbox{Prf}_\zhz  \,    (       y    ,    q   )  \, \}
+\end{equation}
+
+
+Our proof of Lemma C.\ref{B2-lem}
+will be a proof by contradiction.
+It will thus
+ begin with
+the 
+contrary
+assumption that $~\zhz ~$ is inconsistent
+and have  $~\Phi~$ denote
+\eq{qpencode}'s sentence.
+The inconsistency of $~\zhz ~$ 
+ implies that 
+$~\Phi~$ is
+false under the Standard-M model.
+Hence via
+Definition \ref{chg}, we get:
+\begin{equation}
+\label{qpunk}
+ \zzz \Phi~) 
+   ~~ < ~~ \infty
+\end{equation}
+\noindent
+\ep{qpunk} 
+implies there  exists
+a tuple $(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying \eq{qpencodepunk}.
+(This is because such a 
+$(\bar{p},\bar{q},\bar{x},\bar{y})$ 
+corroborates \eq{qpunk}'s implication that a 
+counter-example to
+\eq{qpencode}'s sentence 
+does
+exist.)
+\begin{equation}
+\label{qpencodepunk}
+  \,\mbox{Neg}^0(~ \bar{x}~, ~ \bar{y}~) ~ \wedge~ 
+\mbox{Prf}_\zhz   \, (     ~ \bar{x}~    ,    ~ \bar{p}~   ) ~
+\wedge  ~
+\mbox{Prf}_\zhz  \,    (       ~ \bar{y}~    ,    ~ \bar{q}~   )  
+\end{equation}
+% Furthermore, the  
+The
+$(p,q,x,y)$
+ satisfying \eq{qpencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{\, \mbox{Max}[p,q,x,y] \, \}$ 
+will
+additionally
+%also
+satisfy
+\eq{qpunkless}.
+(This  observation follows
+from the analog of the
+Footnote \ref{footp1}
+appearing 
+in \phx{ppp1}'s proof. 
+Thus, Section \ref{3uuuu3}'s Equations
+of
+\eq{pencodepunk}
+and
+\eq{punkless}
+are 
+the
+analogs of the current
+ Equations
+\eq{qpencodepunk}
+and
+\eq{qpunkless}.
+The 
+Footnote \ref{footp1} 
+showed
+the particular
+$(p,q,x,y)$
+ satisfying \eq{pencodepunk}
+with  minimum value for 
+$ \mbox{Log}\{~\mbox{Max}[p,q,x,y] ~\}$ 
+satisfied
+\eq{punkless}.
+By the same reasoning,  
+the minimal 
+$(\bar{p},\bar{q},\bar{x},\bar{y})$
+satisfying
+\eq{qpencodepunk} will     satisfy
+\eq{qpunkless}.)
+\begin{equation}
+\label{qpunkless}
+ \mbox{Log}~\{~\mbox{Max}[~\bar{p}~,~\bar{q}~,~\bar{x}~,~\bar{y}~] ~~\}
+   ~~~ =~~~\zzz \Phi~) ~~+~~1
+\end{equation}
+
+
+
+
+Equations 
+\eq{qpencodepunk} and \eq{qpunkless} 
+shall
+bring 
+Lemma C.\ref{B2-lem}'s
+ proof-by-contradiction to its 
+sought-after
+ end. Thus,
+let $~\Upsilon~$ denote the
+ $\Delta_0^\xi$ 
+sentence specified
+by $~\bar{x}~$. Then
+$~\neg ~ \Upsilon~$ 
+corresponds to the 
+ $\Delta_0^\xi$ 
+ sentence denoted
+by $~\bar{y}~$. 
+\ep{qpunkless}
+indicates
+that
+both
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+have  proofs 
+such that the logarithms of
+their G\"{o}del numbers are bounded
+by
+$~\zzz \Phi~) ~+~1~$.
+Using
+Definition \ref{ostab}'s
+invariant of $~***~$,
+these facts establish  
+\footnote{\s55 To  
+apply Definition \ref{ostab}'s
+ invariant $~***~$ in the present setting,
+one simply sets $~\theta\,$'s R-View 
+equal to $~\Phi \,$'s
+1-sentence statement. 
+Then $~***~$ implies that both 
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+must
+be 
+true under the Standard-M model
+because both  their proofs had lengths 
+$\, \leq ~\zzz \Phi~) ~+~1~$.
+} that both 
+ $~\Upsilon~$ and
+$~\neg ~ \Upsilon~$
+are
+true under the Standard-M model.
+
+ 
+But it is impossible for
+  a sentence and its
+negation to be both
+true.
+This finishes
+Lemma C.\ref{B2-lem}'s
+proof
+because the
+temporary
+ assumption that 
+$\zhz $ was inconsistent
+has
+led to a
+contradiction.
+$~~\Box$
+
+\medskip
+
+\phx{ppp2} is a
+ consequence of 
+the  Lemmas C.\ref{B1-lem} and
+ C.\ref{B2-lem} because
+Lemma
+ C.\ref{B2-lem}'s formalism 
+generalizes
+to all
+E-stable and A-stable configurations
+via Lemma C.\ref{B1-lem}'s reduction methodology. 
+
+
+\nop
+
+
+\section*{Appendix D: Applications and Examples}
+
+
+
+This appendix will illustrate
+four examples of generic configurations 
+that satisfy
+Theorems \ref{ppp1} and \ref{ppp2} 
+(and 
+which
+therefore  are self-justifying).
+It will be divided into three parts.
+Section D-1 will define our first example of
+self-justifying configuration, called $~\xi^*~$.
+It will use
+semantic tableaux
+ deduction.
+Section D-2 will 
+prove that          $~\xi^*~$ is  EA-stable. 
+Section D-3 will 
+briefly sketch
+        three additional examples of
+   stable generic      configurations
+
+
+It is likely preferable to examine
+Sections \ref{3uuuu1} -- 
+\ref{sect64} before  this appendix.
+However, Section
+ D-1's 
+ short 2-page discussion 
+can be
+ read
+quite easily
+either before or
+ after 
+Section \ref{sect64}.
+
+
+
+
+
+
+
+\subsection*{D-1. $~$Definition of
+the EA-Stable Configuration  
+ $~\xi^*~$ }
+
+
+Our first example of an 
+EA-Stable configuration,
+ called \xxi,
+will be defined in
+this section.
+Its deduction method will be
+semantic tableaux.
+Its 
+ base axiom 
+system $\, B^* \,$ will be a
+ Type-A  formalism,
+which 
+treats addition but not
+multiplication as a total function
+(i.e. see  \ep{totdefxa} $~).$
+
+
+
+
+The closest analog of  $B^*$
+and  \xxi 
+ in our prior work
+% had
+appeared in Section 5 of \cite{ww5}.
+%(Its formalism was
+(It 
+differed
+from  \xxi 
+partly because
+it
+%\cite{ww5}
+did not 
+use Definition  \ref{def3.3}'s
+unifying
+notation.)
+In 
+\cite{ww5},
+% \cite{ww5}'s  discussion,
+a function 
+ $\, F \, $ 
+was     called
+ {\bf Non-Growth} 
+iff  
+$ F(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$. Six  examples of  
+non-growth functions are:
+\begin{enumerate}
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {\it the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0,~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lceil ~x^{1/y}~ \rceil$ when $~y\geq 1,~$
+and
+it equals $~x~$ when $~y=0.$
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+and  zero otherwise. 
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\end{enumerate}
+These operations were called
+{\bf Grounding Functions}
+in \cite{ww5}.
+The term
+{\bf U-Grounding Function}  
+referred 
+to a
+set of
+functions that included the
+Grounding  operations
+ plus 
+the further 
+primitives  of addition and
+{\it Double$(x)=x+x$.} 
+(The Double operation  
+is helpful because it significantly
+ enhances
+\cite{ww5}'s linguistic efficiency \footnote{\s55 The symbol
+{\it Double$(x)$}
+was technically unnecessary in \cite{ww5}'s formalism
+because
+$x+x$ can encode
+ Double$(x)$.
+However,
+its notation
+adds expressive power to
+\cite{ww5}'s language because, for example,
+Double(Double(Double(Double(x))) 
+requires less memory space to encode than
+than
+$~x~$ added to itself 16 times.
+} .)
+
+ 
+
+\medskip
+
+The symbol $~\Delta_0^*~$ 
+will be the
+analog of Definition \ref{xd+1x1}'s 
+ $\Delta_0^\xi$
+construct under
+the U-Grounding function language.
+(It will be defined to be any formula in a U-Grounding 
+language where all its quantifiers are bounded.)
+It is easy in this context to encode
+a $\Delta_0^{    *}$ 
+formula $Mult(x,y,z)$ 
+for 
+representing
+  multiplication's graph.
+For instance,
+\ep{neweq1} 
+is one such $\, \Delta_0^{    *} \, $ formula
+(which actually does not employ any bounded quantifiers):
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+Expression
+\eq{neweq1} 
+is significant because Part 2
+of Definition \ref{def3.3}
+indicated every generic configuration must have 
+ available some 
+method to represent the graphs of
+addition and multiplication 
+in a
+ $\Delta_0^\xi$ 
+styled
+format, similar to     
+\eq{neweq1}'s paradigm.
+ (Addition can be
+treated trivially
+because the 
+U-grounding language
+           possesses 
+an addition function symbol.)
+The
+footnote
+ \footnote{\s55  Section \ref{3uuuu1} 
+explained
+during
+its discussion
+of  
+Lemma \ref{lex22}
+ that
+every generic configuration $  \xi  $
+must have a means to encode 
+the graphs of
+addition and multiplication
+as $  \Delta_0^{\xi}  $ formulae
+(visavis Parts 1 and 2 
+of 
+ Definition \ref{def3.3}).
+This enabled Lemma \ref{lex22}'s
+procedure to translate
+all of conventional arithmetic's 
+$\Sigma_j$ and $\Pi_j$
+formulae 
+into
+equivalent 
+$\Sigma_j^{\xi}$ and $\Pi_j^\xi$
+expressions. \fend }
+% in $ \, \xi \,$'s language. 
+serves as a reminder about why
+these $\Delta_0^*$ encodings are 
+needed. 
+Our next goal is to define 
+the generic configuration 
+ \xxi that
+ Section D-2 will prove  
+is EA-stable.
+
+%\hgskip
+
+\bigskip
+
+
+{\bf Definition
+D.\ref{D1-def}.}
+The 
+language
+ $~L^{*}~$ of the generic configuration 
+ \xxi 
+will be built in a natural manner
+out of 
+the eight 
+U-grounding  function operations,
+the usual
+atomic predicate
+symbols of ``$~=~$'' and 
+ ``$~\leq~$'',
+and the
+three constant symbols 
+ $~K_0,~ K_1\,$ and $\,K_2\,$ (that define the
+integers of 0, 1 and 2).
+The
+other components
+of  \xxi 's configuration
+  are defined
+below:
+\bed
+\cvh
+\itemsep 4pt
+\item[    i  ]
+As previously noted,
+  $~\Delta_0^{*}~$
+is defined to
+represent the set of all
+formulae in $ \, L^* \,$'s 
+language, 
+whose  quantifiers are bounded 
+in an
+arbitrary
+ manner by terms employing 
+the U-Grounding function
+symbols.
+(It will thus generate via Definition \ref{xd+1x1}
+the  $\Pi_n^*$ and $\Sigma_n^*$ sentences of
+$ \, L^* \,$.)
+\item[   ii  ]
+The base axiom system 
+ $~B^*~$ 
+for \xxi 
+will be allowed to be any
+consistent
+set of   $\Pi_1^{\xi^*}~$ sentences
+that is capable of proving every
+ $\Delta_0^*$ sentence that is valid
+in Standard-M.
+It will 
+also
+include   sentence
+ \eq{d1a}'s 
+{\it very precise} \footnote{\g55 \label{fodii}
+The two appearances of the term ``$~x+y~$'' in sentence
+\eq{d1a} 
+may at first appear to be redundant. (This
+statement is 
+equivalent to sentence  \eq{totdefsymba}'s
+declaration that addition is a total function,
+which had avoided such redundancy.) The virtue of
+\eq{d1a}'s format is that it is a    
+ $\Pi_1^*$ styled 
+statement, unlike \eq{totdefsymba}'s
+ $\Pi_2^*$ styled format. 
+This sharpened
+ $\Pi_1^*$ perspective
+will help simplify some of 
+our proofs. }
+$\Pi_1^*$  styled
+declaration that addition is a total function.
+\beq 
+\label{d1a}
+\small
+\forall x ~~~~~~\forall y~~~~~~ \exists~ z \leq  x+y ~~:~~~ ~~~
+  \{~~~z=x+y~~~\}
+\enq
+\item[  iii  ]
+\xxi's deduction method will be the semantic tableaux
+method.
+\item[   iv  ]
+\xxi' s 
+G\"{o}delized
+method $~g~$
+for encoding
+a semantic tableaux proof
+ can be essentially
+any natural method 
+ that satisfies 
+the 
+minor stipulation
+that
+ at least $~5 \, J~$ bits
+are required
+ to encode a semantic tableaux
+proof that has $~J~$ function symbols. 
+This stipulation is called the 
+{\bf ``Conventional Tableaux Encoding Requirement''}.
+It is trivial 
+\footnote{\g55 The Conventional Tableaux Encoding Criteria
+requires that the 
+ G\"{o}del 
+number of a semantic tableaux proof,
+with  $~J~$ function symbols,
+ must be 
+least as large as $~32^J~$.
+It is clear that
+all
+the usual methods for 
+generating the G\"{o}del codes
+satisfy this criteria.
+This is because 
+any proof that has  $~J~$ function symbols will contain at least
+ $~2  \, J~$ logical symbols and thus employ 
+at least
+ $~5  \, J~$ bits. \fend
+} to corroborate   
+ that all the
+usual methods for encoding semantic tableaux proofs
+satisfy this 
+criteria.
+(The  Appendix A of \cite{ww5} provides
+one 
+example of 
+a possible tableaux encoding method.
+Any other natural mechanism for encoding 
+tableaux proofs is equally suitable.)
+\ennd
+
+\smallskip
+
+
+Section D-2
+will,
+interestingly,
+prove  that   $\xi^*$ is EA-stable.
+This 
+will imply (via \phx{ppp1})
+that
+$~B^*~\cup~\mbox{SelfCons}^1\{~B^*~,d~\}$ 
+is 
+self-justifying.
+Theorems \ref{pqq4},  \ref{pqq5}, G.2 and G.3 will,
+formalize, in this context,
+four different methods in which 
+$~B^*~$ can be extended to 
+construct self-justifying formalisms that are able to prove
+Peano Arithmetic's 
+$~\Pi_1^*$ theorems.
+
+\nskip 
+
+\medskip
+
+ Thus while  self-justifying axiom systems
+contain unavoidable weaknesses, they  also possess
+the nice feature that they are able to prove many
+of the useful theorems of mathematics.
+
+
+\nskip
+
+\subsection*{D-2. $~$Proof of the  EA-Stability of
+ $~\xi^*~$ }
+
+\nskip
+
+This section
+ will prove \xxi 
+is EA-stable and thus
+satisfies the paradigms of 
+Theorems \ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5}
+and  \ref{ppp6}.
+Our proof will be based on modifying some of the methodologies
+from \cite{ww5}, so that they
+become applicable to \xxi .
+Many readers may prefer to omit examining
+both this part of Appendix D and Section 
+ D-3 because they 
+are unnecessary for understanding
+the 
+% further
+ material in
+Section \ref{sect64} and
+Appendixes E and F. (Our recommendation is that the latter
+material be read first.)
+
+
+
+
+\nskip
+\smallskip
+
+
+Our notation for defining a
+  semantic tableaux
+proof
+        in the next paragraph
+will be similar to the 
+conventional
+      definitions
+appearing in 
+Fitting's and Smullyan's textbooks  \cite{Fi90,Sm68}.
+It will employ
+\cite{ww5}'s
+notation
+so that we can employ
+two of its
+lemmas during our analysis of semantic tableaux proofs.
+
+\nskip
+
+%\cvmew
+
+{
+%notre \cvh
+In our discussion,
+$~\Phi~$
+will be called
+a {\bf  Prenex-Level($\,$m$^* \,$)}
+  sentence
+ iff it is a 
+ $\Pi_m^*$ or $\Sigma_m^*$ 
+expression
+that satisfies the usual prenex requirement (that
+all its unbounded 
+quantifiers   lie  
+in its leftmost part).
+If $\Phi$ is  Prenex-Level$(m^* )$
+then {\bf Reverse($\Phi$)} 
+shall 
+denote a second
+ Prenex-Level$( \, m^* \,)$
+sentence that is equivalent to
+$~\neg~\Phi~$  rewritten 
+\footnote{\s55  For example,
+if $\Phi$ denotes ``$~\forall \, x ~ \exists \,y ~~\psi(x,y)~$''
+then Reverse$(\Phi)$ 
+would be written  as
+ ``$~ \exists \, x ~ \forall \,y ~~\neg~\psi(x,y)~$''.}
+%% \footnote{\s55 Thus  
+%% % For example,
+%% if $\Phi$ denotes ``$~\forall \, x ~ \exists \,y ~~\psi(x,y)~$''
+%% then Reverse$(\Phi)$ is 
+%% %would be written  as
+%%  ``$~ \exists \, x ~ \forall \,y ~~\neg~\psi(x,y)~$''.}
+in a
+ Prenex Level$( \, m^* \, )$ form.
+For a fixed axiom system $ \, \alpha ,  \, $ its
+{\bf $\Phi$-Based Candidate Tree}
+will  be defined
+to be a  tree structure 
+whose root is
+the sentence 
+Reverse$(\Phi)$ 
+ and whose all other nodes are
+either axioms of $ \, \alpha \, $ or deductions from higher
+nodes of the tree,
+via the rules 1--8 
+given
+below. (The symbol
+``{\bf $ \,  $A$  ~  \Longrightarrow  ~  $B$  \, $}''
+in  rules 1-8
+will mean
+that {\bf $ \,   $B$  \, $} is a valid deduction
+from its ancestor {\bf $ \,  $A$  \, $}
+in the germane 
+ deduction tree.)
+\begin{enumerate}
+\itemsep 5pt
+%notre \small
+%%corbl \baselineskip = 1.05 \normalbaselineskip
+\item $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Gamma ~$ .
+$~~~$
+\item $ \,  \neg  \,\neg \, \Upsilon  \,  \Longrightarrow  \,  \Upsilon. \, $  
+Other rules for
+the ``$ \, \neg \,$'' symbol  are:
+$ \, \neg ( \Upsilon \vee \Gamma )  \,  \Longrightarrow  \,  \neg \Upsilon
+\wedge \neg \Gamma$,
+%\newline
+$ \, \neg ( \Upsilon \rightarrow \Gamma )  \,  \Longrightarrow  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \Longrightarrow  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $
+ and $ ~~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed
+when their ancestor is
+$~\Upsilon \, \vee \, \Gamma.~$
+\item A pair of sibling nodes $ \,  \neg \Upsilon  \, $ and $ \,  \Gamma  \, $ is
+allowed
+when their ancestor is
+$ \, \Upsilon \, \rightarrow \, \Gamma$.
+\item $~ \exists v \, \Upsilon  (v) ~ \Longrightarrow ~ \, \Upsilon(u) ~$
+where $\,u \,$ is a newly introduced ``Parameter Symbol''. 
+\item 
+ $~ \exists v \leq s ~ \, \Upsilon  (v) ~~ \Longrightarrow ~ ~
+u \leq s ~ \wedge~ \Upsilon(u) ~$
+is the
+ variation of Rule 5 for  bounded existential quantifiers
+of the form $~$``$~ \exists v \leq s ~$''.
+\item $\forall v \, \Upsilon  (v)  \,  \Longrightarrow    \, \Upsilon(t)  \, $
+where $t$ denotes a U-Grounded 
+term. These terms may be any one of
+a constant symbol,
+a  parameter symbol (defined  by a 
+prior application of  Rules 5 or 6 to some
+ some ancestor of the current node),
+or a U-Grounding function-symbol with  
+recursively defined
+inputs.
+\item 
+ $\forall v \leq s  \, \Upsilon  (v) ~~ \Longrightarrow ~~
+t \leq s \, \rightarrow \, \Upsilon(t) $ is 
+the variation of Rule 7 
+for a
+ bounded  quantifier such as
+ ``$~ \forall v \leq s ~ $''
+\end{enumerate} }
+
+%%corbl  \baselineskip = 1.1 \normalbaselineskip 
+\noindent
+Let us say a leaf-to-root branch (in a candidate
+tree) is {\bf Closed} iff it contains both some sentence
+$ \,  \Upsilon  \, $ and its negation ``$ \,  \neg \, \Upsilon  \, $''.  
+Then a
+ {\bf  Semantic
+Tableaux Proof} of $  \Phi  $,
+from the axiom system  $  \alpha $,
+is defined to be a 
+$\Phi$-Based Candidate Tree
+whose every
+ leaf-to-root branch is closed.
+
+\cvt
+
+\hgskip
+
+%\cvwew
+
+\nskip
+
+
+%bbbbb {
+
+It is next helpful to define
+the notion of a 
+{\bf Z-Based Deduction Tree},
+in a context where $~Z~$ represents 
+an
+ axiom system,
+typically different from 
+the prior paragraph's
+$~\alpha~$.
+This object 
+will be
+defined to be
+identical to
+a semantic tableaux proof, except for the following
+changes:
+\bed
+%%corbl  \baselineskip = 1.1 \normalbaselineskip 
+\itemsep 5pt
+\item[   i   ]
+{\it Every node} in a $Z-$Based 
+deduction tree
+must be either an axiom of  $~Z~$
+or a deduction from a higher node of the tree via
+the 
+rules 1-8.
+(This
+applies
+also to the root of
+a $Z-$Based  deduction tree.
+It will  store an 
+axiom of  $~Z~$
+ in its root, unlike a semantic tableaux proof
+ which had
+stored
+ Reverse$(\Phi)$ 
+in its
+root.)
+\item[  ii   ]
+There will be no requirement that each 
+ leaf-to-root branch
+be
+ closed
+in a 
+$Z-$Based  deduction tree.
+(Indeed, 
+some branch will 
+automatically
+not be closed
+if  $~Z~$ is  consistent.) 
+\ennd
+Items (i) and (ii) make it 
+apparent
+ that
+$Z-$Based  deduction trees
+are different 
+% properties
+from 
+semantic tableaux proofs.
+It will
+turn out,
+nevertheless, 
+ that the study of
+$Z-$Based  deduction trees
+will clarify the nature of 
+ semantic tableaux proofs.
+
+
+\nskip
+\nskip
+
+\medskip
+
+
+
+{\bf Definition 
+D.\ref{D2-def}.}
+Let $ ~   a ~   $ and $ ~   b ~   $ denote two
+integers that are powers of $\,  2 \,  $ 
+satisfying
+$  a \, > b \,
+\geq \, 2 \, $
+Then
+ an axiom system $ \, Z \, $
+(employing $ \, L^* \,$'s language)
+will be called
+a {\bf Normed(a,b)} formalism  iff:
+\begin{enumerate}
+\item All  $Z$'s 
+axioms  are 
+either $\Pi_1^{*}$
+or  $\Sigma_1^{*}$ sentences.
+\item 
+Each  $\Pi_1^{*}$ axiom of $~Z~$
+will satisfy Definition \ref{xd+1x5}'s
+Good($~$Log$_2a~)$ 
+criteria, and
+ each
+ $\Sigma_1^{*}$ axiom of $~Z~$
+will
+likewise
+  satisfy Good($~$Log$_2b~)$. 
+\ene
+
+\nskip
+\nskip
+
+
+{\bf  Clarification about Definition D.\ref{D2-def}
+: } The
+``Normed(a,b)''  
+concept (above)
+is obviously 
+equivalent to the same-named notion
+appearing in
+Definition 4 of \cite{ww5}.
+It uses,
+however,
+ a 
+different notation
+to make it compatible 
+with
+Section \ref{3uuuu2}'s formalism.
+Thus, Item 2's assertion
+that the
+  $\Pi_1^{*}$ axiom 
+$ \, \forall  \,  v_1 \,  \, \forall  \,  v_2 \,  \,  ...
+\forall  v_k   \,  \,  \,  \, \phi(v_1,v_2,...v_k) \,  \, $
+satisfies
+Good($~$Log$_2a~)$  is equivalent to 
+\eq{normscopede}'s statement.
+The
+Good($\,$Log$_2b\,)$ property of 
+$\exists \,v_1\,\exists \,v_2\, ...\,\exists \,v_k\, ~~\phi(v_1,v_2,...v_k)~$
+is,
+likewise,
+ equivalent to \eq{normscopedx}.
+
+\nskip
+
+\beq
+\label{normscopede}
+\forall ~ v_1~ < ~a~~\forall ~ v_2~ < ~a~~ ...
+\forall ~ v_k~ < ~a
+   ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~~.
+\enq
+
+\nskip
+
+\beq
+\label{normscopedx}
+\exists ~ v_1~ < ~b~~\exists ~ v_2~ < ~b~~ ...
+\exists ~ v_k~ < ~b
+ ~~~~~ : ~~~~~
+\phi(v_1,v_2,...v_k)~~.
+\enq
+
+\parskip 5pt
+
+\nskip
+
+Our interests in
+this notation
+will center around Fact D.3's invariant:
+
+
+
+{
+%%corbl  \baselineskip = 1.17 \normalbaselineskip 
+
+\medskip
+
+\nop
+
+{\bf Fact D.3$~.$}
+Let
+$ \, \xi^* \, $ 
+denote 
+ Definition D.\ref{D1-def}'s
+generic configuration, and 
+$  Z $ be an extension of $ \, \xi^* \,$'s
+base axiom
+system  $ B^*  $ 
+which
+satisfies Definition D.\ref{D2-def}'s
+Normed$(a,b)$ constraint.
+Then any
+$Z-$Based  deduction tree
+$\, T \, $that
+has a G\"{o}del
+number 
+smaller than
+ $ \, (a/b)^4 \,$
+must 
+contain
+at least one 
+root-to-leaf branch, called $ \, \peta \,  \, , \, $
+that is not
+``closed''. {\rm
+(In other words, this path   $ \, \peta \, $
+will be contradiction-free, insofar as it
+does not contain  both  some
+sentence $~\Psi~$ and its formal
+negation).}
+}
+
+%%notre
+  
+%%notre-only  {\bf Proof:} $~$ The  justification of
+%%notre-only  Fact D.3 is essentially 
+%%notre-only  an
+%%notre-only  %a direct and
+%%notre-only  immediate
+%%notre-only   consequence
+%%notre-only  of the
+%%notre-only  %% no  
+%%notre-only  %% no \newpage
+%%notre-only  %% no \noindent
+%%notre-only  %% no 
+%%notre-only  Lemmas 1 and 2 from \cite{ww5}
+%%notre-only  after one maps these two lemmas's notation into the
+%%notre-only  context of Fact D.3's
+%%notre-only  hypothesis. (We provide a detailed justification
+%%notre-only  of this 
+%%notre-only  technical
+%%notre-only  fact in the Footnote 23 of the slightly longer
+%%notre-only  unabridged version of this article 
+%%notre-only  that
+%%notre-only  resides in the Cornell Archives \cite{ww11}.) 
+  
+
+  
+  \smallskip
+  
+\nskip
+
+%%cccorn-ff
+
+%%cccorn-cc  
+%%cccorn-cc  
+   {\bf Proof:}
+   $~$ The 
+   justification of
+   Fact D.3 is
+% essentially 
+%   an
+   a direct 
+% and    immediate
+    consequence
+   of 
+   %% no  
+   %% no \newpage
+   %% no \noindent
+   %% no 
+the   Lemmas 1 and 2
+appearing in article \cite{ww5} 
+   (see footnote 
+   \footnote{\label{newtry}
+   \baselineskip = 1.3 \normalbaselineskip  
+   A 
+   %formal 
+   proof of Fact D.3 from first principles would
+   be 
+   quite
+   complicated because there are eight elimination rules employed
+   by semantic tableaux deduction, $~$each
+   of which needs to
+   be 
+   % simultaneously 
+   examined by such a proof's umbrella
+   formalism. Fortunately, we do not need provide such a
+   complicated analysis here 
+   because  
+   a 4-page proof of
+   the Lemmas 1 and 2 
+   in Section 5.2
+   of \cite{ww5}
+   had 
+    already
+   visited these issues. 
+   Thus,
+    Fact D.3 
+   turns out to be an easy consequence 
+   of these two lemmas
+   %the Lemmas 1 and 2 of \cite{ww5}
+   after the following two straightforward issues are addressed: 
+   \bee
+   \topsep -4pt
+   \itemsep -2pt
+   \item Section 5.2 of \cite{ww5} 
+   had
+   defined the $\,$``U-Height''$\,$ of a deduction tree to
+   be the 
+   largest
+   number of 
+   U-Grounding function symbols
+   that appear in any of its root-to-leaf
+   branches. Its Lemma 1
+   proved that
+   every deduction tree with a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b \, $
+   will contain at least one branch
+   satisfying a condition,
+   which \cite{ww5}
+    called ``Positive(a,b)''.
+   The Lemma 2 in \cite{ww5} then showed that this
+   Positive(a,b) property  implies that the germane 
+   deduction tree must contain some branch that is 
+   contradiction-free.
+   The combination of these two lemmas
+   thus amounts to the establishing of
+   the following 
+   %slightly 
+   rephrased 
+   hybridized statement:
+   %invariant:
+   \begin{quote}
+   $\bullet~~~$
+   If a Z-based deduction
+   tree  has 
+                   a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b \, $,
+   then some branch of it 
+   is contradiction-free
+   (i.e.  this branch cannot
+   contain  both  some
+   % $\Delta_0^*$
+   sentence $~\Psi~$ and its
+   negation).
+   \end{quote}
+   \item
+   %% 
+   %% Item $\, \bullet \,$
+   %% applies to Fact D.3 's  
+   %% deduction trees
+   %% because 
+   %% 
+   Fact D.3 's   hypothesis indicated the
+   G\"{o}del number $g$ for its deduction tree
+   satisfied the following 
+   conditions:
+   \bed
+   \topsep -7pt
+   \itemsep -1pt
+   \item[   I.   ]
+   $~~g~\leq ~ (a/b)^4~$
+   \item[  II.   ]
+   The   U-Height of $~g \,$'s deduction tree
+   is less than $~\frac{1}{5}~$Log$_2~g~$. (This
+   is
+   simply
+    because
+    Fact D.3 presumes that
+   the ``Conventional Tableaux Encoding'' methodology
+   from Part-iv of 
+   Definition D.\ref{D1-def}
+   was used to encode $~g$'s
+   G\"{o}del number.)
+   \ennd
+   Items
+   I and II imply 
+   $g \,$'s  tree has  
+    a U-Height  $\, \leq \, Log_2a  \, - \, Log_2b  $.
+   The invariant  $\, \bullet \,$ 
+   %then, in turn, implies
+   %that this deduction tree
+   then implies
+   this deduction tree
+   has at least one branch
+   that is contradiction-free (as  Fact D.3
+   claimed). $\Box$
+   \ene
+   We emphasize that the 
+   %The
+   above 
+   % 2-part 
+   justification
+    of Fact D.3 is 
+   {\it much simpler}
+   than a proof from first principles.
+   %
+   %This is  because the 
+   %
+   The
+   latter would
+   require examining 
+    eight different 
+   tableaux 
+   elimination rules,
+   as the detailed
+   proofs of \cite{ww5}'s
+    Lemmas 1 and 2 
+   actually
+   did do.}
+   %
+   %\noindent
+   %-----------------------------------------------------------------}
+for more details).  $~~\Box$
+
+
+%%%% 
+%%%% provides   some
+%%%%    %some
+%%%% %   precise 
+%%%% % details 
+%%%%    details about how 
+%%%%    the   Lemmas 1 and 2
+%%%% of \cite{ww5} do
+%%%%  imply Fact D.3).
+%%%% 
+
+
+\nop
+
+
+
+We will now apply Fact D.3 to prove 
+Theorem D.\ref{D.4-theorx}.
+Its invariant will, 
+interestingly,
+collapse entirely
+\footnote{\h55 $~$The
+difficulty posed by 
+%the 
+multiplication 
+%operation
+can be easily understood when one compares
+two 
+integers sequences  
+   $ x_0, x_1, x_2,  ... $ 
+and  $ y_0, y_1, y_2,  ... $,
+defined 
+as follows:
+\begin{center}
+$x_{i}~~=~~x_{i-1}+x_{i-1}~~~~~$
+$~~$ AND  $~~$ 
+$~~~~~y_{i}~~=~~y_{i-1}*y_{i-1}$
+\end{center}
+It turns out that the faster growth rate of  multiplication
+under
+the series  $~ y_0, y_1, y_2,  ...~ $
+enables
+one to to construct  tiny 
+$Z-$Based  deduction trees
+$\, T$ that
+violate
+the analog of Fact D.3 's paradigm.
+(This is 
+because 
+such trees
+ can have
+G\"{o}del
+numbers 
+smaller than
+ $ \, (a/b)^4 \,$,
+while 
+all their 
+root-to-leaf branches 
+can be simultaneously   
+``closed'' via contradictions.)
+This 
+property of
+            multiplication is 
+analogous
+to Example \ref{ex3-3}'s
+observations about
+          how        the
+differing growth rates of  
+$ x_0, x_1, x_2,  ... $ 
+and  $ y_0, y_1, y_2,  ... $
+are related to the
+threshold where the semantic tableaux version of 
+ Second Incompleteness Theorem can be evaded. \fend }
+%% 
+%% It
+%% explains
+%% intuitively 
+%%  why   Fact D.3 and 
+%% Theorem D.\ref{D.4-theorx} 
+%% use
+%% paradigms
+%% that recognize addition but not multiplication
+%% as total functions
+%% during their evasions of
+%% the semantic tableaux version of
+%% the Second Incompleteness Theorem.  \fend }
+%% 
+ if one were to
+merely  add
+a multiplication function symbol to 
+the
+    U-Grounding language.
+This is why
+our 
+boundary-case exceptions
+to the semantic tableaux version of the Second Incompleteness
+allow 
+a Type-A axiom system 
+to
+recognize addition
+as a total function (but
+suppress a similar 
+ treatment of
+ multiplication).  
+
+
+
+
+\hgskip
+
+
+{\bf Theorem
+D.\ref{D.4-theorx}.}
+{\it The generic configuration $\xi^*$ 
+is both A-stable and
+E-stable.} {\rm (This 
+implies  many different 
+self-justifying formalisms
+exist
+via Theorems 
+\ref{ppp1}, \ref{pqq3}, \ref{pqq4}, \ref{pqq5},
+ \ref{ppp6},
+ G.2 and G.3.)}
+
+\cvs
+
+\hgskip
+
+Our proof of
+Theorem D.\ref{D.4-theorx} will 
+separately
+show 
+ $\xi^*$  is 
+A-stable and E-stable. 
+
+\hgskip 
+
+\parskip 3pt
+
+{\bf Proof of 
+ $ \, \xi^* \,$'s A-stability :}
+Suppose for the sake of establishing a
+proof by
+ contradiction
+that  $ \, \xi^* \, $ was not A-stable.
+Then the constraint $ \, * \, $ 
+of Definition \ref{astab}
+would  be violated
+by at least
+%%%%%  bbbbb
+some
+$\theta \, \in \,$RE-Class$(\xi)$.
+This violation
+ will cause the statement $ \, + \, $ to be true
+for 
+such a
+ $\theta~$:
+\begin{description}
+\item[  +   ]
+There exists a
+semantic tableaux
+ proof $~p~$
+of a
+ $~\Pi_1^*~ $ theorem,
+called say 
+$ \, \Uxp ~,~ $ 
+from
+ the
+axiom system of $~\theta \cup B^\xi~$
+such that
+ Log$(p)~\leq \zzz \theta)~+1~ $
+and where 
+ $ \, \Uxp \, $ also  
+fails to
+satisfy 
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\end{description}
+Let us recall that
+if  $\Uxp$ is $\Pi_1^*\,$ then
+Reverse$(\, \Uxp \,)~$ 
+is a 
+$\Sigma_1^*$ sentence equivalent to $~\neg \Uxp~$.
+Thus, 
+Reverse$(\, \Uxp \,)~$ will satisfy
+Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$
+criteria (simply because
+it
+% Reverse$(\, \Uxp \,)~$
+  has the opposite 
+goodness
+property as
+$\,\Uxp \,$ ).
+Also, if
+$~Z~$ 
+denotes the axiom system of
+ $~\theta \cup B^\xi \, + \, $Reverse$(\, \Uxp \,)$,
+it is easy to verify
+\footnote{\g55    \label{jnorm}
+The axiom system
+ $Z$  must satisfy 
+\nor1
+because:
+\bee
+\item
+The
+quantity
+$~2^{ \zzz \theta  \, )}$
+is a valid
+first component
+for $Z$'s   norming constraint 
+because
+all the axioms of $~B^\xi~$ are 
+true in the Standard-M model
+and
+because
+Definition \ref{chg}
+implies
+all of
+$ \theta \,$'s axioms satisfy 
+Good $\{\,\zzz \theta) \,\}$.
+\item
+The
+quantity      $ ~\sqrt{~2^{ \zzz \theta \,   )}} ~$
+is a valid
+second component
+for $Z$'s   norming constraint 
+because
+ Reverse$(\, \Uxp \,)~$ 
+is the only $\Sigma_1^*$ sentence belonging to
+Z, and  because  Reverse$(\, \Uxp \,)~$ 
+satisfies 
+ Good$\{~ \, \tftt  \zzz \theta) \, ~\}~$.
+\ene}
+that $~Z\,$'s axioms 
+will
+satisfy the 
+\nor1
+criteria.
+
+
+
+\smallskip
+
+It is next helpful to  observe that what is
+a proof from one perspective corresponds to
+being a
+deduction tree from 
+a different
+perspective.
+Thus, Item $~\, + \,$'s proof
+ $~p~$
+of the 
+theorem
+$ \, \Uxp \,$ 
+from
+ the
+axiom system of $~\theta \cup B^\xi~$
+corresponds to being  a Z-based deduction tree,
+with $~Z~$ representing the
+axiom system of
+$~\theta \cup B^\xi \, + \, $Reverse$(\, \Uxp \,)$.
+In this context,
+Item  $\, + \,$'s 
+inequality of 
+ Log$(p)~\leq \zzz \theta  \, )~+1~ $
+implies 
+\footnote{ \g55 \label{fd7} 
+Without loss of generality, we 
+may assume that 
+every non-trivial proof $~p~$ satisfies
+  Log$(p)~\geq ~64~$ (since a 
+string with fewer than 64 bits is too short to be a
+proof).
+Then the footnoted paragraph's 
+ Log$(p)~\leq \zzz \theta)~+1~ $ inequality 
+trivially implies 
+$~p~<~3^{ \zzz \theta  \, )}~$.
+In a context where 
+ $~Z~$ 
+is a
+\nor1
+axiom system,
+the
+ latter inequality 
+certainly
+implies
+$~p~$, viewed as a deduction tree for $~Z~,~$
+has a small enough G\"{o}del number to
+ satisfy the
+hypothesis for Fact D.3. 
+(This is because if one sets
+$~a~=~2^{ \zzz \theta  \, )} ~$ 
+and 
+$~b~=~\sqrt{2^{\zzz \theta  \, )}} ~$ 
+then obviously 
+$ ~~p~<~3^{ \zzz \theta  \, )}~ <~4^{ \zzz \theta  \, )}~ =~
+(a/b)^4~~~).$  \fend } 
+$\,$that 
+$~p~$, viewed as a deduction tree for $~Z~,~$
+ satisfies the
+hypothesis of Fact D.3 . 
+Hence, Fact D.3 establishes
+that $~p~$ must contain
+at least
+ one contradiction-free
+root-to-leaf branch.
+
+\smallskip
+
+
+This last observation 
+is all that is needed
+to confirm
+ $~\xi^* \,$'s A-stability,
+via a proof-by-contradiction.
+This is because the 
+definition of a semantic tableaux proof implies
+every 
+one of its
+root-to-leaf branches 
+must
+end with a pair of
+contradicting nodes.
+However, the last paragraph showed 
+$~p~$
+will not satisfy this
+required
+ property,
+if $~\xi^*~$ is not A-stable.
+Hence 
+our construction has proven the A-stability of
+ $~\xi^*~$ by showing that otherwise an infeasible
+circumstance will arise.
+ $~~\Box$ 
+
+
+
+
+
+\hgskip
+
+{\bf  Proof of 
+ $  \xi^*  $'s E-stability :}
+A
+proof-by-contradiction
+will verify
+ $  \xi^*  $ is
+E-stable, analogous
+ to  the
+ proof of
+its
+    A-stability.
+Thus if
+ $  \xi^*  $ was not
+E-stable, then
+statement $++$ would be true
+for some  $\theta$. (This is because 
+at least one
+$\theta \, \in \,$RE-Class$(\xi)$
+would then
+ violate
+Definition \ref{estab}'s
+requirement of  $~**~~$.)
+\begin{description}
+\item[ ++   ]
+There exists a
+semantic tableaux
+ proof $~p~$
+of a $\Sigma_1^\xi$ theorem 
+$ \, \Uxp \, $ from the
+axiom 
+system $\theta \cup B^\xi$
+such that
+ Log$(p) \,\leq \zzz \theta) +1 $
+and  
+ $  \Uxp  $  also
+fails to
+satisfy 
+Good$\{  \tftt  \zzz \theta) \,\}.$
+\end{description}
+Item $++  $ 
+ implies 
+Reverse$( \Uxp )$ 
+satisfies
+ Good$\{  \tftt  \zzz \theta) \}$
+(because Reverse$( \Uxp )$ again
+   has
+ the opposite 
+goodness property
+as 
+$ \Uxp $ ).
+Let
+$Z$ now
+denote the formal axiom system of  
+ $\, \theta \cup B^\xi  \, +  \, $Reverse$( \Uxp )$.
+The footnote \footnote{ \sm55   
+The axiom system
+ $~Z~$ must satisfy 
+\xor2
+because:
+\bee
+\item The first component of its norming constraint 
+can be set equal to
+ $ ~\sqrt{~2^{ \zzz \theta   \, )}} ~$
+because Reverse$(\, \Uxp \,)~$ 
+is a 
+ Good$\{~ \, \tftt  \zzz \theta  \, ) \, ~\}~$
+$\Pi_1^*$ sentence, and all $~Z\,$'s other 
+$\Pi_1^*$ sentences satisfy more relaxed constraints.
+\item The second component of $Z$'s norming constraint 
+is satisfied by the constant of
+2 because Definition D.\ref{D2-def} implies  this 
+quantity is always permissible
+when $~Z~$ 
+contains no $\Sigma_1^*$ axiom sentences.
+\ene   \fend }
+then
+   uses
+reasoning 
+similar
+to footnote    \ref{jnorm} 
+to
+show 
+$Z$ 
+satisfies 
+\xor2
+
+\parskip 3pt
+
+\smallskip
+
+As before 
+via a simple change in notation, 
+ $~p\,$'s
+semantic tableaux proof of
+$ \, \Uxp \,$ 
+can be viewed 
+as 
+ a deduction tree
+using $~Z\,$'s 
+axioms.
+Also as before, we may 
+use the combination of the facts that 
+ $~Z~$ is a
+\xor2 system
+and that  Item ++ 
+indicated 
+ Log$(p) \,\leq \zzz \theta) +1 $ to 
+deduce\footnote{\sm55 The proof that 
+ $~p~$ is
+ small enough to satisfy 
+Fact D.3 's
+hypothesis in the current E-stable case 
+is almost
+identical to Footnote  \ref{fd7}'s analysis of
+the A-stable case. 
+Thus as in the earlier case,
+Item ++'s inequality of
+ Log$(p)~\leq \zzz \theta)~+1~ $ 
+trivially implies 
+$~p~<~3^{ \zzz \theta  \, )}~$.
+Also, we may again assume that
+  Log$(p)~\geq ~64~$ (since a sequence with fewer than 64 
+bits cannot amount to a proof of any interesting fact under
+all normal coding conventions). 
+An analog of Footnote  \ref{fd7}'s chain of inequalities
+will then allow us to conclude that 
+$~p~$
+is
+ small enough 
+proof from a 
+\xor2
+system
+to
+ satisfy the
+hypothesis for 
+Fact D.3.  \fend }  that
+ $~p~$ is small enough to satisfy 
+Fact D.3 's
+hypothesis.
+ Hence once again,
+Fact D.3  implies that
+$~Z~$ must contain
+at least
+ one contradiction-free
+root-to-leaf branch.
+As before, the existence 
+ of this
+contradiction-free
+path 
+violates the definition of a semantic tableaux proof
+and 
+enables
+ our proof-by-contradiction to
+reach its
+desired end. $~\Box$ 
+
+\hgskip
+
+
+
+{\bf Remark D.5} {\it  (about 
+Theorem  D.\ref{D.4-theorx}'s
+significance) :}
+Part-ii of
+ Definition
+D.\ref{D1-def}
+indicated  $~\xi^* \,$'s base axiom of
+$~B^*~$ was a Type-A formalism that recognized addition
+as a total function. This is significant because
+ \cite{ww0,ww2,ww7,ww9} 
+showed
+nearly all Type-M formalisms, including 
+all the common axiomatizations for
+I$\Sigma_0$,
+are unable to recognize their 
+semantic tableaux consistency.
+Thus, the declaration that multiplication is a total
+function
+is {\it the trigger-point} causing
+\footnote{\sm55 
+We formally proved in \cite{ww0,ww2,ww7,ww9} 
+that
+multiplication's totality property
+causes
+ the semantic tableaux version of the Second
+Incompleteness Theorem to become active.
+The  
+ Example \ref{ex3-3} 
+summarizes the main
+%%%  
+%%%  offers a 
+%%%  % nice brief 
+%%%  summary of the
+%%%  % underlying
+%%%  
+ intuition behind these
+results.}
+the semantic tableaux version of the
+Second Incompleteness Theorem
+to become active. 
+This threshold effect is 
+significant
+% quite tight 
+because
+Theorem D.4, combined with
+Theorems 
+ \ref{pqq4}, \ref{pqq5}, G.2 and G.3,
+formalize       {\it four different respects} in which 
+Type-A
+self-justifying 
+formalisms can
+prove all 
+Peano Arithmetic's
+ $\Pi_1^*$ theorems
+{\it ( after} multiplication's totality axiom is suppressed).
+
+\cvt
+
+
+
+\subsection*{D-3. $~$Three Further Examples of
+Stable Generic Configurations}
+
+
+
+
+Our second 
+example
+of an
+ EA-stable
+        configuration
+is
+called $~\xi^{**}~$. 
+It will be identical to 
+ $~\xi^*~$ except that it will replace
+semantic tableaux with a stronger
+deduction method,
+which 
+\cite{ww5} 
+    called  Tab$-U_1^*$.
+The latter 
+is          a
+revised version of 
+semantic tableaux
+that permits
+ a modus ponens
+rule to perform
+deductive cut 
+ operations on
+$\Pi_1^*$ 
+and $\Sigma_1^*$ sentences.
+(The formal
+definition of 
+ Tab$-U_1^*$ 
+deduction 
+had appeared in \cite{ww5}.
+It will  be
+unnecessary to
+ repeat here.)
+
+
+
+
+The Section 5.3 of 
+\cite{ww5} 
+noted
+Tab$-U_1^*$ 
+has
+ similar self-justification properties
+as conventional semantic tableaux.
+%It is thus unnecessary to discuss
+%Tab$-U_1^*$ in detail here.
+All the results that
+Section D-2 
+proved about
+ $~\xi^{*}~$
+% ,
+% however,
+apply also  to  $~\xi^{**}~$,
+via their natural generalization
+under
+\cite{ww5}'s
+Tab$-U_1^*$
+deduction
+method.
+Thus,
+ $~\xi^{**}~$
+is also EA-stable.
+
+
+%%%%% 
+%%%%% 
+%%%%% Our second 
+%%%%% example
+%%%%% of an
+%%%%%  EA-stable
+%%%%%         configuration
+%%%%% is
+%%%%% called $~\xi^{**}~$. 
+%%%%% It will be identical to 
+%%%%%  $~\xi^*~$ except that it will replace
+%%%%% semantic tableaux with a stronger
+%%%%% deduction method,
+%%%%% which 
+%%%%% \cite{ww5} 
+%%%%%     called  Tab$-U_1^*$.
+%%%%% The latter 
+%%%%% is          a
+%%%%% revised version of 
+%%%%% semantic tableaux
+%%%%% that permits
+%%%%%  a modus ponens
+%%%%% rule to perform
+%%%%% deductive cut 
+%%%%%  operations on
+%%%%% $\Pi_1^*$ 
+%%%%% and $\Sigma_1^*$ sentences.
+%%%%% (The 
+%%%%% definition of 
+%%%%%  Tab$-U_1^*$ 
+%%%%% deduction 
+%%%%% appeared in \cite{ww5}.
+%%%%% It
+%%%%% is unimportant to
+%%%%%  repeat here.)
+%%%%% 
+%%%%% 
+%%%%% 
+%%%%% 
+%%%%% The Section 5.3 of 
+%%%%% \cite{ww5} 
+%%%%% noted
+%%%%% Tab$-U_1^*$ 
+%%%%% has
+%%%%%  similar self-justification properties
+%%%%% as conventional semantic tableaux.
+%%%%% It is thus unnecessary to discuss
+%%%%% Tab$-U_1^*$ in detail here.
+%%%%% %%  
+%%%%% %% 
+%%%%% %% We will not
+%%%%% %%  go  into the full details
+%%%%% %% again,
+%%%%% %% for the sake of brevity.
+%%%%% %% 
+%%%%% %% 
+%%%%% All the results that
+%%%%% Section D-2 had
+%%%%% proved about
+%%%%%  $~\xi^{*}~$
+%%%%% % ,
+%%%%% % however,
+%%%%% apply also  to  $~\xi^{**}~$,
+%%%%% via their natural generalization
+%%%%% under
+%%%%% \cite{ww5}'s
+%%%%% Tab$-U_1^*$
+%%%%% deduction
+%%%%% method.
+%%%%% Thus,
+%%%%%  $~\xi^{**}~$
+%%%%% is also EA-stable.
+
+
+ 
+A key point is that 
+ there is a
+non-trivial
+distinction between  
+$  \xi^{*}  $ 
+and  $  \xi^{**}  $, despite
+the fact that they
+have 
+similar technical qualities.
+This is because 
+ $  \xi^{**}  $ contains  a
+Level-1
+modus ponens rule
+(unlike $  \xi^{*}  $ ).
+ If 
+it were infeasible to expand  $  \xi^{*}  $ 
+into 
+a broader 
+ $  \xi^{**}  $, $  $then both formalisms could,
+perhaps,
+ be easily
+dismissed as having 
+negligible pragmatic 
+significance (since
+modus ponens is central
+to
+cogitation). 
+However in a context where  $  \xi^{**}  $ does permit
+a  Level-1
+modus ponens rule, 
+it is a tempting formalism
+(despite its limited modus ponens rule).
+
+
+
+
+Unlike
+ $~\xi^{*}~$ 
+and  $~\xi^{**}~$,
+our third 
+example of an EA-stable
+configuration,
+called $~\xi^{-}~$,
+will support            an unlimited modus ponens
+rule.
+This will be possible because 
+ $~\xi^-\,$'s
+language of $~L^-~$ 
+will 
+be weaker than the languages of
+ $~\xi^{*}~$ 
+and  $~\xi^{**}~$.
+Thus $~L^-$  will
+include 
+ the six Grounding functions, but
+not the Growth functions of addition and doubling.
+It will thus treat addition and multiplication as
+3-way atomic predicates,
+ Add$(x,y,z)$ 
+and Mult$(x,y,z)$, rather than as
+formal functions.
+
+
+\cvt
+\parskip 2pt
+
+
+
+This
+perspective
+enabled  $~\xi^-~$   to
+support an evasion of 
+the Second Incompleteness Theorem with
+an unlimited  modus ponens rule
+present,
+in a context where
+the other four parts of 
+its
+generic configuration are defined below:
+\bee
+\item
+The 
+  $~\Delta_0^{-}~$
+class for $~\xi^-~$
+will 
+be built in an
+essentially natural
+ manner from
+the Grounding function
+set.
+It will 
+thus 
+include
+all formulae in $~L^- \,$'s language, 
+whose 
+quantifiers
+are bounded
+ in any arbitrary manner 
+using
+the Grounding function
+primitives.
+\item
+The
+base axiom system
+$~B^-~$ of $~\xi^-~$
+ will
+employ  an infinite number of constant symbols,
+denoted as 
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+where 
+ $K_1=1$ 
+and where 
+ $K_{i+1}$ is
+a power of 2
+ defined by the axiom of:
+\beq
+\label{addc}
+\mbox{Add}(~K_{i}~,~K_{i}~,~K_{i+1}~)
+\enq
+Thus, the combination of 
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+with the
+Grounding function  
+of subtraction allows
+the language  $L^-$
+to encode the value of any
+arbitrary natural
+number (as 
+Part 1 
+of Definition \ref{def3.3} 
+had required).
+Essentially,
+$~\xi^- \,$'s 
+base axiom system
+of $~B^-~$ 
+can
+be any 
+consistent
+r.e. set of
+$~\Pi_1^-~$ sentences 
+that includes \eq{addc}'s axiom schema
+and is
+able to
+prove every 
+ $\Delta_0^{-}~$ sentence which is valid
+in the Standard-M model.
+\item
+$~\xi^- \, $'s
+deduction method can be any version of a classic
+Hilbert-style proof methodology. (Thus, it  will  include
+a modus ponens rule with no restrictions.)
+\item
+$~\xi^- \, $'s
+G\"{o}delization method can be essentially
+any natural technique.
+\ene
+An interesting aspect of  $\xi^-$
+is it can be 
+proven to be
+EA-stable
+via an analog of
+Section D-2's treatment of  $\xi^*$.
+Thus,
+Theorem \ref{pqq4}
+implies
+every axiom system
+$\alpha$,
+whose
+ $\Pi_1^-$ theorems 
+hold true
+in the Standard-M model,
+can be mapped
+onto an extension of  $\xi^-\,$'s base
+axiom system
+that can 
+ recognize
+its own 
+Hilbert
+ consistency 
+and
+prove 
+$\alpha $'s  $\Pi_1^-$ theorems.
+Except for
+minor changes in notation,
+this  
+result
+represents a new way of proving
+\cite{wwapal}'s Theorem$\, 3.~$
+
+
+
+
+
+
+\medskip
+
+
+
+The self-justifying features
+of
+ $ \xi^{*} $,  $ \xi^{**} $  and  $ \xi^{-} $
+are of interest
+primarily 
+ because
+the 
+Second 
+Incompleteness Theorem implies 
+that they
+cannot be 
+improved
+much 
+further.
+This tight fit is
+summarized by  Items 1-4.
+\begin{enumerate}
+\topsep -15pt
+\item 
+The Theorem 2.1 
+(due to  the combined work of 
+Nelson,
+Pudl\'{a}k,
+Solovay and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} )
+implies no natural axiom system can
+prove Successor is a total function and recognize its own
+Hilbert consistency. This theorem 
+thus explains why
+the presence of
+growth functions 
+must be omitted from
+ $\xi^{-}\,$'s
+base axiom system of $B^-$.
+\item
+Moreover, \cite{wwapal}
+  proved 
+$~\xi^{-} \,$'s method 
+for evading  
+the Second Incompleteness
+Theorem
+will 
+collapse
+if one replaces  
+\ep{addc}'s  ``addition-based named sequence'' of
+constant symbols
+$    K_1   ,   K_2   ,   K_3   , \,   ...   $ 
+with a faster 
+growing
+``multiplicative 
+convention'',
+where 
+the
+constant symbols 
+$    C_1   ,   C_2   ,   C_3   , \,   ...   $
+are formally defined via 
+\eq{multc}'s
+ schema.
+\beq
+\label{multc}
+\mbox{Mult}(~C_{i}~,~C_{i}~,~C_{i+1}~)
+\enq
+Thus,  \cite{wwapal} showed that there
+exists a 
+ $\Pi_1^{-}$ sentence $ \,W \,$ 
+(provable from Peano Arithmetic)
+such that no consistent  system can
+simultaneously prove  $W$,
+contain \eq{multc}'s axiom schema and prove the non-existence
+of proof of $0=1$ from itself.
+There is no space to prove it here, but 
+a generalization of the Second Incompleteness
+Theorem implies  the modification
+of $\xi^-$ that replaces
+\eq{addc}'s  axiom schema with 
+\eq{multc}'s schema 
+{\it is not even 0-stable.}
+\item
+Similarly,
+\cite{ww2,ww7} proved that if 
+ $ \, \xi^{*} \, $'s and  $ \, \xi^{**} \, $'s 
+ base 
+ axiom system of 
+$ \, B^* \, $ 
+was strengthened
+to include the assumption that
+multiplication was a total function then \cite{ww5}'s
+two
+semantic tableaux  evasions of
+the Second Incompleteness Theorem would 
+both collapse.
+\item
+Also, \cite{wwlogos} 
+proved
+that  an analog of $\xi^{**} \,$'s
+evasion of the Second Incompleteness Theorem
+will collapse if its 
+modus ponens rule was expanded to apply to
+either $\Pi_2^*$ or $\Sigma_2^*$ sentences.
+\ene
+The Item 3
+is especially interesting because
+\cite{ww6} 
+proved \cite{ww5}'s evasion of the Second Incompleteness
+Theorem 
+was compatible with its formalism recognizing
+an infinitized generalization of a
+computer's floating point multiplication
+as a total function.
+Thus
+ while the semantic tableaux formalisms
+of 
+ $\xi^{*}$ or  $\xi^{**}$ 
+are provably unable \cite{ww2,ww7} to
+ recognize integer
+multiplication as a total function,
+their relationship to floating point multiplication
+is 
+more subtle.
+
+
+
+
+
+
+\smallskip
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Our fourth example of an
+application of Section \ref{sect64}'s theorems
+was stimulated by
+some
+insightful 
+email  we
+received from
+L. A. Ko{\l}odziejczyk
+\cite{Ko5} in 2005.
+It noted
+there existed a
+potential
+ exponential gap between the lengths
+of semantic tableaux and Herbrand-style proofs 
+under some 
+circumstances.
+Our earlier research \cite{ww2} 
+addressed a 
+1981 Paris-Wilkie open question \cite{PW81} 
+by
+generalizing some 
+Adamowicz-Zbierski techniques 
+\cite{Ad2,AZ1} 
+to show
+a natural  axiomatization of I$\Sigma_0$ 
+ satisfied the semantic tableaux version of the Second
+Incompleteness Theorem.
+In this context,
+Ko{\l}odziejczyk
+asked whether this would
+apply to       all
+plausible axiomatizations for I$\Sigma_0$ ?
+
+
+
+
+
+
+
+We replied in
+\cite{ww9}
+to
+Ko{\l}odziejczyk's
+stimulating question
+ by 
+distinguishing between Example \ref{ex3-1}'s 
+$\Delta_0^A$ 
+and $\Delta_0^R$  formulae
+and by using
+the Paris-Dimitracopoulos \cite{PD82}
+translation algorithm for 
+$\Delta_0$ formulae. (The latter procedure was summarized
+earlier by
+ Lemma \ref{lex22}. It
+demonstrated
+ how to
+ map
+classic arithmetic's
+$\Delta_0^A$ formulae
+onto  equivalent   
+$\Delta_0^R$  formulae
+in  the Standard-M
+model.) Our reply to 
+Ko{\l}odziejczyk's
+question, 
+thus,
+employed this
+translation methodology to show that there existed an axiom system,
+called Ax-3, which proved the identical set of theorems as
+the more common Ax-1 and Ax-2 encodings of 
+ I$\Sigma_0$ and which possessed the following 
+pair of 
+quite
+fascinating 
+  contrasting properties:
+\bed
+\topsep -3pt
+\item[   A  ]
+No
+consistent
+ superset $~\beta~$ of Ax-3's set of axioms is capable of
+proving its
+own
+ semantic tableaux consistency \cite{ww9}.
+\item[   B  ]
+In contrast,  
+if  ``Herb'' denotes
+the next paragraph's
+ Herbrand-styled deduction
+and if ``SelfRef'' denotes 
+the sentence $~\bullet~$ from
+Section \ref{secc1},
+then 
+Ax3$\, + \,$SelfRef(Ax-3,Herb) will
+be a self-justifying axiom system.
+%%% 
+%%% 
+%%% In contrast,  
+%%% if  ``Herb'' denotes
+%%% the next paragraph's
+%%%  Herbrand-styled deduction
+%%% method
+%%% then 
+%%% Ax3$\, + \,$SelfRef(Ax-3,Herb) will
+%%% satisfy both Parts (i) and (ii) of 
+%%% Section \ref{secc1}'s 
+%%%  definition of
+%%% self-justification.
+%%% 
+%%% 
+\ennd
+
+The intuition behind \cite{ww9}'s proof of Items A and B can be 
+easily summarized if 
+we define a
+ ``Herbrandized-style''  proof of a 
+theorem $ \,\Phi \,$ from an axiom system $ \, \alpha \,$
+as being an essentially 
+2-part
+structure where:
+\bee
+\topsep -3pt
+\item Each  of $\alpha$'s axioms and
+also the sentence $~ \neg \Phi~$ are
+first written as  Skolemized
+expressions.
+\item
+A propositional calculus proof 
+is then used to show
+ that some formal conjunction
+of instances of Item 1's Skolemization schema has no satisfying
+truth assignment.
+\ene
+Such a
+ formalism is 
+different from the definition
+of a semantic tableaux proof (appearing in for example Fitting's
+textbook \cite{Fi90} ).
+This is because the latter replaces the use
+of Skolemization in Items 1 and 2 with an existential quantifier
+elimination rule. It turns out that this distinction enables
+some semantic tableaux proofs to be exponentially more compressed
+than their Herbrandized counterparts,
+as Ko{\l}odziejczyk observed 
+\cite{Ko5,Ko6}. This fact
+ enabled \cite{ww9}
+to prove that Herbrandized and semantic tableaux proofs
+have the divergent properties summarized by Items A and B.
+
+
+One reason
+Ax-3's evasion of the Second Incompleteness Theorem
+is of interest is that I$\Sigma_0$ supports many more generalizations
+of the Second Incompleteness Theorem
+than evasions 
+of it.
+ Thus,
+Willard  \cite{ww2,ww7,ww9}
+proved that the semantic tableaux version of
+the Second Incompleteness Theorem 
+was valid for three different encodings of
+I$\Sigma_0$, and
+Adamowicz, Salehi and  Zbierski
+have discussed in great detail
+\cite{Ad2,AZ1,Sa11}
+ various Herbrandized generalizations
+of the Second Incompleteness Theorem for
+particular encodings of  
+I$\Sigma_0$ 
+and I$\Sigma_0+\Omega_i$. 
+Moreover, an added 
+facet of \cite{ww9}'s 
+Ax-3 encoding for I$\Sigma_0$ is that most automated
+theorem provers use a 
+particular
+variant of the Resolution method
+that causes \cite{ww9}'s 
+unusual 
+methodology
+to apply
+also to them \footnote{
+\b55 
+The main theorems in \cite{ww9} generalize for resolution because
+Resolution-based theorem provers
+employ skolemization analogously to Herbrand deduction. \fend }.
+
+
+The reason for our 
+%particular 
+interest in
+\cite{ww9}'s results 
+% in the current article
+is that it represents a fourth example 
+%% of
+where the 
+ meta-theorems from Sections 
+\ref{3uuuu3} and  \ref{sect64} 
+can be useful.
+Thus,
+the footnote
+\footnote{\b55  The 
+discussion in \cite{ww9}
+ did not
+technically  
+use
+ Definition  \ref{estab}'s machinery
+to establish 
+%that 
+there existed an extension of
+its ``Ax-3'' encoding for 
+I$\Sigma_0$ that could recognize its own Herbrand consistency.
+Its formalism,
+however,
+ could be
+easily couched in terms of 
+ Definition  \ref{estab}'s machinery, if one uses a 
+generic 
+configuration $~\xi^R~$ where
+\bee
+\item  $~\xi^R\,$'s 
+base language is the same as the usual language of
+arithmetic, 
+\item  $~\xi^R\,$'s 
+$~\Delta_0^R~$ sub-class is defined by
+Item (b) in Example \ref{ex3-1}, 
+\item  $~\xi^R\,$'s 
+base axiom system 
+is \cite{ww9}'s ``Ax-3'' system, 
+\item  $~\xi^R\,$'s 
+deduction method
+is either a Herbrandized styled-method
+or a Resolution system that relies upon
+Skolemizatin in a similar manner.
+\item  $~\xi^R\,$'s G\"{o}del encoding scheme
+may be any such 
+natural method.
+\ene
+This approach 
+supports a 
+% somewhat
+  stronger form of
+self-justification result
+than
+had 
+appeared in \cite{ww9}.
+This is
+because $~\xi^R~$ can be proven to be E-stable (by a 
+% straightforward 
+generalization of \cite{ww9}'s analysis
+techniques). 
+Thus,
+ \phx{ppp2} 
+implies
+that
+Ax-3 has a well-defined self-justifying extension
+that
+can  recognizes its own
+formalized
+ Level$(0^R)$ consistency.
+(This 
+self-justification result
+is  stronger than
+\cite{ww9}'s main theorem.
+The latter
+merely established
+that some extension of Ax-3
+recognized the non-existence of a
+Herbrandized deduction of $0=1$ from itself.) \fend }
+summarizes how  a fourth 
+type of
+generic configuration,
+called $~\xi^R~$,
+can be defined
+ that both duplicates
+\cite{ww9}'s
+main
+ self-justification results 
+under the above definition of
+Herb-deduction,
+as well as  strengthens
+them. (In particular, 
+  $~\xi^R~$ meets
+Theorem \ref{ppp2}'s
+requirements,
+and  self-justifying 
+extensions of its Ax-3 system thus
+recognize their 
+%own
+ Level$(0^R)$ consistency.)
+
+
+
+
+\parskip 2pt
+
+
+
+
+
+The properties of our
+four generic configurations of $~\xi^R$,
+  $~\xi^*~$,
+ $~\xi^{**}~$ and $~\xi^-~$ are  summarized by Table I.
+These configurations are
+listed in ascending order according to the strength
+of their deduction methods $~d~$. As their deduction methods
+increase in strength, 
+these configurations 
+% the associated configurations 
+have
+their ability
+reduced
+%weakened
+ to recognize 
+the totality of 
+the addition and multiplication operations.
+%% 
+%% 
+%% (This is because the corresponding generic configurations will
+%% violate the requirements of the Second Incompleteness Theorem,
+%% if they simultaneously use too strong a deduction method
+%% and possess too strong an understanding of their
+%% own consistency.)
+
+
+
+
+
+$~\xi^R\,$ is thus a Type
+Almost-M 
+system
+that
+can prove multiplication is a total function
+(but 
+which
+does not contain 
+\ep{totdefsymbm}'s
+ totality statement {\it as an axiom)}.
+On the other hand, $~\xi^-~$ 
+uses a stronger Hilbert-styled
+ deduction methodology,
+which is  incompatible with treating
+the totality of addition or multiplication as either
+axioms {\it or as derived theorems.} 
+%% (This incompatibility is unavoidable because the
+%% Theorem 2.1, due to the joint work of
+%% Nelson,
+%% Pudl\'{a}k,
+%% Solovay and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87}, 
+%% implies that self-justifying systems cannot simultaneously
+%% recognize successor as a total function and prove their
+%% own Hilbert consistency.)
+
+
+
+
+
+
+Each of Table I's
+rows
+ $ \, \xi^R$,
+ $ \, \xi^{**} \, $ and $ \, \xi^- \, $  
+are  maximal (in that an alternate row
+improves upon one   column's measurement 
+only when it is 
+weaker from the perspective of
+another 
+column).
+Only  $ \, \xi^* \, $ 
+is
+an exception to this rule: It is
+strictly
+weaker
+\footnote{\b55 \label{f28} $  \xi^{**}  $ employs a stronger
+deduction method 
+than   $  \xi^*  $
+because it allows a modus ponens rule for $\Pi_1^*$ and
+$\Sigma_1^*$ sentences to be added to semantic tableaux 
+deduction
+(see \cite{ww5} for the precise definition of this  
+``Tab$-U_1^*~$''
+ modification of the  semantic tableaux 
+deductive method).  \fend }
+     than
+     $ \,  \xi^{**}  $. 
+    This appendix has discussed 
+      $ \, \xi^* \, $ 
+%% despite its sub-optimality, 
+%%    %primarily 
+%%    partly 
+because it makes
+    Theorem D.\ref{D.4-theorx}'s proof simpler
+    (and also because 
+     semantic tableaux 
+    % deduction 
+    is a 
+    % very
+    frequent topic in the logic literature).
+
+    %%%footnnnnnnnnn 
+
+
+
+%\smallskip
+
+\begin{center}
+{\bf Table I}
+\end{center}
+
+
+\bigskip
+
+\small
+\baselineskip = 1.1 \normalbaselineskip 
+\noindent
+\begin{tabular}{|c|c|c|c|c|c|c|} \hline
+Name & Deduction Method  & Type & Almost   & Type  & {\bf Axiom} 
+& {Self-Just }  \\
+ &     & {\bf A} 
+& {\bf M} & {\bf M} &  { Format} &  { Level}
+\\ \hline\hline 
+ &  Resolution and/or  & &  &  &  & \\
+ $\xi^R$ & Herbrandized analogs & Yes$^{35}$
+  & Yes  & No &  E-stable &Level $(0^R)$ \\ \hline
+ &  & &  &  &  & \\
+ $\xi^*$ & Semantic Tableaux & Yes  & No  & No & EA-stable & Level $(1^*)$  \\ \hline
+ &  & &  &  &  & \\
+ $\xi^{**}$ & Tab$-U_1^*$ Deduction$\, ^{34}$ & Yes  & No  & No &  EA-stable &Level $(1^*)$   \\ \hline
+ &  & &  &  &  & \\
+ $\xi^{-}$ & Hilbert Deduction & No  & No  & No &  EA-stable &
+ Level $(\, \infty^{-} \,)$   \\ \hline
+\end{tabular}
+
+%\njp
+
+\bigskip
+\normalsize
+\cvl
+
+
+%\cvt
+
+\parskip 0pt
+
+%\nskip
+%\nskip
+
+\bigskip
+The footnote
+\footnote{\b55    For the sake of simplicity,
+    the Ax-3 system of \cite{ww9}  did not
+    use either 
+    Equations
+    \eq{totdefxa} or  
+    \eq{totdefsymba}'s
+  as axiom statements
+(since  they were
+provable as  theorems).
+ All
+    \cite{ww9}'s 
+     results 
+    do,
+however,  generalize
+%    readily
+    when 
+\eq{totdefxa}'s
+statement  about
+addition's totality 
+is included as 
+%added 
+an axiom.
+Thus, it is appropriate to attach the designation
+of ``Yes'' with a caveat to  the ``Type-A'' entry
+in Table I's first row. (This row is called 
+``Resolution and/or Herbrandized analogs'' 
+because it applies
+to 
+essentially 
+any deduction scheme that relies upon Skolemization
+as an alternative to  \cite{Fi90}'s
+semantic tableaux
+existential quantifier elimination rule.)} ,
+ attached to Table I's first row,
+explains why  a caveat is attached to its first
+``Yes'' entry. 
+The theme of Table I is 
+%thus 
+that self-justifying
+axiom systems have some nice redeeming features,
+although
+the
+ Second
+Incompleteness Theorem
+clearly also
+imposes
+severe
+ limits on their abilities.
+This point  will be reinforced 
+when
+Appendix E introduces a generalization of the
+Second Incompleteness Theorem,
+that shows 
+\thx{ppp6}'s translational reflection principle is close
+to being a maximal feasible result, and when Appendix F 
+discusses the epistemological  significance
+of self justification.
+
+
+%% 
+%% the next two sections discuss
+%% hybridizations of
+%% Theorems  \ref{pqq4}
+%% and \ref{pqq5}
+%% and the epistemological  implications
+%% of self justifying systems.
+
+
+
+%% 
+%% an alternate system
+%% and when 
+%% Appendix F discusses the
+%% epistemological  implications
+%% of self justification.
+
+
+%% 
+%% the next two sections discuss
+%% hybridizations of
+%% Theorems  \ref{pqq4}
+%% and \ref{pqq5}
+%% and the epistemological  implications
+%% of self justifying systems.
+
+\nskip
+\nskip
+
+
+%% 
+%% an alternate system
+%% and when 
+%% Appendix F discusses the
+%% epistemological  implications
+%% of self justification.
+
+
+\nop
+
+\section*{Appendix E: A Clarification of
+Theorem \ref{ppp6}'s Significance}
+
+% An Anti-Reflection Principle for
+%\LARGE
+% \baselineskip = 1.8 \normalbaselineskip 
+
+
+It has been known since the time of G\"{o}del that
+most conventional 
+arithmetic 
+axiom systems will satisfy the following two 
+invariants:
+\bee
+\item
+They are {\it physically unable} to prove
+their own consistency
+\item
+They are  $\Sigma_1$ 
+complete. This means they can
+formally prove any  $\Sigma_1$
+arithmetic sentence that holds
+true in the Standard-M model, and
+they can likewise refute any
+$\Pi_1^\xi$ sentence 
+that is false.
+\ene
+Let $~\xi~$ denote any generic configuration of the
+form \gggcp . This appendix will use the
+term {\bf $\xi -$Conventional} to describe any axiom
+system that satisfies analogs of the
+preceding 
+%two 
+conditions for generic configurations.
+Thus
+$~\alpha~$ is
+$\xi -$Conventional 
+iff it satisfies the following two
+criteria:
+\bed
+\item[ a.  ]
+The axiom system $~\alpha~$
+will be {\it  unable} to verify
+its own consistency under $~\xi\,$'s
+deduction
+    method  of $~d~$.
+\item[ b.  ]
+The axiom system $~\alpha~$
+will be an extension of $~\xi \,$'s base axiom of $~B^\xi~$.
+Part-3 of Definition \ref{def3.3} will thus imply it
+is 
+$\Sigma^\xi_1$ 
+complete. (Hence, $~\alpha~$ can
+formally prove any  $\Sigma^\xi_1$ sentence that holds
+true in the Standard-M model, and
+it can likewise refute any
+$\Pi^\xi_1$ sentence 
+that is false.)
+\ennd
+
+This section will prove no analog of \ep{T-reflect}'s
+translational reflection principle is feasible for 
+$\xi -$Conventional axiom systems. Thus,
+Theorem \ref{ppp6} must be close to being a maximal result,
+since it cannot 
+plausibly
+be further extended to hold under conventional axiom systems.
+
+\medskip
+
+{\bf Theorem E.1}  (A New Type of Version of the Second Incompleteness
+Theorem): 
+{\it
+There exists no 
+$\xi -$Conventional 
+axiom
+system $~\alpha~$ 
+that can prove the validity of 
+\eq{G-reflect}'s
+Translational Reflection Principle
+for any translation-mapping T.}
+(In other words,
+there exists no
+ algorithm  $~T~$ that 
+maps 
+$~\Pi_1^\xi$ sentences $~\Psi~$  
+onto alternate  $~\Pi_1^\xi$ 
+sentences
+$~\Psi^T~,~$ 
+which 
+ are equivalent 
+to $~\Psi~$
+in the
+Standard-M model
+and where $~\alpha~$ can
+verify
+\eq{G-reflect}'s reflection principle for every
+$~\Pi_1^\xi$ 
+ sentence $~\Psi .~)$
+%
+%simultaneously
+%for 
+% all  $~\Pi_1^\xi$ sentences
+% $~\Psi~$.) 
+\beq
+\label{G-reflect}
+\forall ~p~~~[~~ \mbox{Prf}_{\alpha,d}(~\lceil \, \Psi \, \rceil~,~p~)
+  ~~~ \Rightarrow ~~~ \Psi^T~~]
+\enq
+
+
+{\bf Proof:} $~$It is  easy to prove Theorem E.1 via a
+proof-by-contradiction. Thus consider the possibility
+that Theorem E.1's translational mapping $~T~$ did exist.
+One can then easily select a  $~\Pi_1^\xi$ sentences $~\Psi~$  
+that is false in the 
+Standard-M model. 
+Then 
+$~\Psi^T~$ is also false under the 
+Standard-M model 
+(since 
+ $~\Psi~$  
+and
+$~\Psi^T~$ 
+are equivalent in this model).
+
+
+%% Moreover, $T$'s definition requires 
+%% $\Psi^T$ 
+%% to have a $\Pi_1^\xi$ format.
+%% Thus,
+
+Hence
+Part-b of the definition of 
+$\xi -$Conventionality implies
+$~\alpha~$ must prove 
+$~\neg~\Psi^T$
+ (on account of $~\Psi^T\,$'s 
+$\Pi_1^\xi$ format).
+
+
+It is at this juncture that our proof-by-contradiction will
+reach its end. This is because if $~\alpha~$ can prove
+\eq{G-reflect}'s statement
+and
+{\it also prove}
+ the sentence $~\neg~\Psi^T~$,
+ then it
+certainly can combine these two facts to prove the
+non-existence of a proof of $~\Psi~$.
+The latter 
+contradicts 
+Part-a of the definition of 
+$\xi -$Conventionality (because it  shows $~\alpha~$
+can verify its own consistency). $~~\Box$ 
+
+
+%%  The latter would
+%% contradict 
+%% Part-a of the definition of 
+%% $\xi -$Conventionality (because it would show $~\alpha~$
+%% can verify its own consistency). $~~\Box$ 
+
+\bigskip
+
+{\bf Remark E.2.} 
+We remind the reader that 
+Footnote \ref{imper} 
+pointed out that
+$T$'s translational mapping 
+would lose its main functionality, if 
+it did not require $\Psi^T$ 
+to have a  $~\Pi_1^\xi~$ format,
+similar to $\Psi$.
+In essence,
+Theorem E.1 is
+of interest
+because it shows that 
+Theorem \ref{ppp6}'s evasion of the Second Incompleteness Theorem 
+is close to
+being a maximal result. 
+%(because 
+(It thus shows that \eq{G-reflect}'s
+translational reflection principle does not generalize 
+to conventional 
+axiom systems.)
+%settings.)
+This 
+dichotomy
+may explain why self-justifying axiom systems,
+along with    Theorem \ref{ppp6}'s
+particular
+ invariant,
+are 
+potentially useful results.
+
+%surprising topics.
+
+
+ % {\bf Remark E.2.} 
+% The 
+% preceding proof 
+% of Theorem E.1 makes it
+% clear
+% that its generalization of the
+% Second Incompleteness Theorem
+% is a 
+% straightforward
+% result. It is 
+
+
+ 
+\cvl
+
+
+
+
+
+\section*{Appendix F: Epistemological 
+Perspective and Speculations}
+
+
+
+\cvs
+
+\parskip 6pt
+
+It is desirable to include a short
+purely
+epistemological
+discussion 
+within this 
+mostly mathematical article so that 
+the more subtle nature of our
+ results 
+cannot be
+misconstrued.
+
+Part of the reason 
+Self Justification
+can lend itself 
+to
+easy
+misinterpretations is that 
+the
+First Incompleteness Theorem 
+demonstrates 
+the
+impossibility of 
+constructing an ideally
+optimal axiomatization of number theory. 
+For
+any initial
+r.e$. \,$axiom system $~\alpha~$
+and deduction
+method$~d$, 
+G\"{o}del 
+thus
+noted
+it is
+easy
+\footnote{  \b55 
+Let $\mho(a,d)$ the
+classic
+ G\"{o}del
+sentence 
+that  asserts:
+{\it ``There is no proof of {\it this sentence} from 
+$\alpha$'s
+axiom system 
+under $\,d$'s deduction method.''}
+G\"{o}del
+\cite{Go31}
+ noted 
+$~\alpha+\mho(\alpha,d)~$ 
+always proves  more theorems than  $\alpha$.}
+ to
+develop
+ an extension of  $~\alpha~$
+that can prove 
+strictly more
+theorems than $~\alpha~$ under 
+ $\,d$'s
+deduction method. 
+Moreover, a large number of generalizations of the Second Incompleteness
+Theorem, starting with its 1939  
+Hilbert-Bernays version
+\cite{HB39}, are known to be 
+%very 
+robust results.
+
+Such considerations 
+naturally lead to  questions
+about whether
+any
+ r.e$. \,$axiom system can encompass the workings
+of the human mind. It may surprise some readers to learn
+that this author 
+shares 
+such
+ skepticism.
+That is, 
+we
+doubt
+{\it any single ISOLATED} self-justifying
+r.e.$\,$logic 
+can {\it  fully} approximate
+the complex workings of the human mind.
+
+
+In this short appendix,
+let us instead
+view  cogitation
+as 
+{\it roughly} a
+process wondering
+though some universe $ \,   \nal{U}$, 
+comprised of
+{\it both} consistent and
+inconsistent axiom systems,
+with a
+trial-and-error evolutionary method 
+focusing 
+ its attention
+over time  
+increasingly  
+onto
+ the members of this universe 
+ $~  \nal{U}~$ that are found to be
+ consistent.
+It is 
+%
+%%notre
+%
+%
+%%notre-only straightforward
+%
+%easy
+%
+%%cccorn-ff  
+%%cccorn-cc  
+  straightforward\footnote{  \b55 It 
+   is trivial from a theoretical perspective
+   to design a 
+   learning heuristic that
+   will utilize all 
+   consistent axiom systems
+   from
+   its available  universe $  \nal{U}$ 
+    eventually, and
+   it will
+   spend only an infinitesimal fraction of its effort on
+   inconsistent systems as time runs to infinity.
+   (This because
+   there exists only
+   a countable number of distinct r.e. sets
+   belonging to the universe
+   $  \nal{U}.~)~$
+   Also, 
+   this
+   learning
+    process can 
+   presumably be made to
+    employ 
+   some type of
+   smart souped-up
+   AI heuristics to enhance its efficiency,
+   whose details will not concern us 
+   % here
+   in this abbreviated 3-page appendix.
+   What is 
+   central to the current discussion
+    is that some type of formally
+   {\it non-recursive} 
+   and presumably trial-and-error
+   method must 
+   obviously
+   be used 
+   by this learning process
+    to find
+   the consistent elements of
+   $~  \nal{U}~,~$ 
+   on account of G\"{o}del's
+   undecidability results.}
+   %%
+ %%
+ to define
+many 
+universes  $~  \nal{U}~$  and
+ evolutionary processes that 
+fall into this gendre.
+Our 
+goal 
+in this section
+will be to
+examine
+ Section \ref{3uuuu3}'s
+``R-View'' $\theta$ and its RE-Class$(\xi)$.
+
+ 
+
+Thus,  $\theta$ will denote an 
+R-View
+that consists of
+an arbitrary
+r.e$. \,$set of
+$\Pi_1^\xi$ sentences.
+Also, 
+RE-Class$(\xi)$ will
+again
+denote the
+set of all
+$~\theta~$ which 
+can be built under $~\xi \,$'s language
+of
+$\,L^\xi$.
+(Section \ref{3uuuu3}
+had
+allowed
+ both valid and invalid
+R-Views  $~\theta$ 
+ to appear in
+RE-Class$(\xi)$ because 
+no recursive 
+decision
+procedure can 
+identify
+all  
+the Standard-M model's true  $\Pi_1^\xi$ sentences.)
+
+%% \nop
+
+The epistemological purpose of this notation was revealed
+in Section \ref{sect64}.
+For the cases where $k  =  0$
+or 1,
+Section \ref{sect64}
+defined 
+$G^\xi_k(\, \theta \,)$
+to be  the 
+axiom system:
+\begin{equation}
+\label{f4gedef}
+G^\xi_k(~ \theta ~)~~= ~~
+\theta~\cup~B^\xi~\cup~\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}
+\end{equation}
+Also, Definition \ref{dap4-1} indicated that
+  the function 
+$ \, G^\xi_k ~ $
+(which maps $ \, \theta \, $ onto 
+$G^\xi_k(\, \theta \,)~~~)$ would be
+called 
+{\bf Consistency Preserving} iff 
+$ \, G^\xi_k(\, \theta \,) \, $is 
+assured to be consistent whenever 
+all the sentences in $~\theta ~$
+are  
+true  under the Standard-M model.
+\thx{pqq3} 
+indicated,
+in this context,
+ that
+$~G^\xi_1~$ 
+satisfies 
+this 
+property
+whenever $~\xi~$ is
+EA-stable.
+Likewise,
+$~G^\xi_0~$ 
+is      consistency preserving whenever $~\xi~$ is
+one of A-stable, E-stable or 0-stable.
+
+\nskip
+
+
+These results indicate 
+a trial-and-error experimental process
+can, indeed, walk 
+{\it in an 
+unusually
+orderly manner} through an universe
+of self-reflecting 
+candidate
+formalisms, when
+RE-Class$(\xi)$  denotes 
+$~  \nal{U}\,$'s 
+universe  
+and
+ $~\xi~$ satisfies any of
+the  EA-stable, E-stable, 
+A-stable or 0-stable conditions.
+This is because if 
+$~\theta~$ designates
+a 
+set of
+$\Pi_1^\xi$ sentences
+holding true 
+in
+ the Standard-M model, 
+then  
+$~G^\xi_k(\theta)~$ 
+will
+{\bf automatically}
+satisfy both Parts (i) and (ii)
+of Section 1's definition of Self Justification,
+according to \thx{pqq3}.
+
+Such 
+consistency preservation is
+surprising because 
+it is simply inapplicable to
+%does not apply to
+the $\,G^\xi_k \,$ 
+% functions of
+ functions for
+most 
+pairs $~(\xi,k).$
+\thx{pqq3}'s  first
+ contribution 
+is,
+ thus,  
+that it formalizes
+how  $G^\xi_k\,$'s  mapping function 
+can
+represent
+ a type of approximation
+for
+instinctive faith,
+under certain well-defined circumstances.
+
+
+This notion of instinctive faith is, of course, 
+less
+robust than a conventional proof.
+One
+obvious
+  difficulty
+is that a 1-sentence proof,
+using an {\it ``I am consistent''} axiom,
+is 
+less convincing
+than a full-length proof from first principles. Also, if
+the initial formalism $~\theta~$ contains a false
+$~\Pi_1^\xi~$ sentence then  $~B^\xi+\theta~$
+and $~G^\xi_k(\theta)~$ 
+will be  both inconsistent.
+
+
+
+\nop
+
+\cvl
+
+Nevertheless
+for $\,k\,$ equals 0 or 1,
+ if   $~\theta~$ is comprised of the true sentences
+in the Standard-M model, then 
+\thx{pqq3} 
+will
+assure that
+$~G^\xi_k(\theta)~$ is 
+a consistent system that has an ability
+to
+use its {\it ``I am consistent''}
+axiom sentence
+to 
+formalize
+its
+own consistency.
+Moreover, the axiom system
+$~G^\xi_k(\theta)~$ is helpful because
+G\"{o}del's famous centennial paper
+% has certainly 
+implicitly
+raised
+the following
+bedeviling 
+issue:
+%dilemma:
+\begin{quote}
+$\#~~$   How is it that Human Beings
+ manage 
+to muster 
+the physical 
+drive
+to think (and prove theorems) when the many
+generalizations of 
+G\"{o}del's Second 
+Incompleteness Theorem 
+demonstrate
+conventional logics
+lack knowledge of
+their own consistency?  
+\end{quote}
+While  philosophical paradoxes and ironical 
+dilemmas,
+similar to 
+$~\# ~,~$ 
+ never yield
+perfect answers, the preceding discussion is helpful
+because it 
+explores
+a
+certain
+syllogism
+% paradigms
+whereby a logic
+can 
+formalize
+ at least some
+fragmented
+operational
+ appreciation of its own consistency.
+
+Moreover, 
+Part-3 of Appendix D
+ indicated that its 
+four self-justifying configurations were
+ close to being maximal results
+that cannot  be 
+much
+improved,
+on account of various
+barriers imposed by
+ the Second Incompleteness Theorem. 
+Thus, these
+particular
+ positive results,
+combined with Theorems \ref{ppp1}
+\ref{pqq3},   \ref{pqq4},  
+\ref{pqq5},
+ \ref{ppp6},
+D.4, E.1,
+ G.2, G.3 and Remarks
+\ref{re4-1} and
+ \ref{recc1},  
+come 
+close to formalizing the
+maximal variants of instinctive faith that a 
+first-order logic can 
+bolster.
+
+
+
+The theme of the last two paragraphs
+is
+thus
+ that our approximation of 
+ {\it ``instinctive faith''} may be imperfect, but it
+is still a useful partial reply to 
+$~\# \,$'s puzzling
+dilemma
+{\it in a context where}
+unambiguous 
+full 
+resolutions to $\, \# \,$
+{\it are not permitted by}
+the Second Incompleteness
+Theorem.
+Furthermore,
+\ep{T-reflect}'s
+translational reflection principle,
+together with Theorem \ref{ppp6}
+and the Remarks \ref{f88} and \ref{remhappy},
+illustrate how the notion of 
+ an instinctive faith 
+about the usefulness of $\Pi_1^\xi$ theorems
+ can
+be
+almost physically
+{\it hard-wired} into self-justifying formalisms.
+
+
+%%% instinctive faith 
+%%% about the validity of $\Pi_1^\xi$ theorems
+%%%  can
+%%% be
+%%% almost physically
+%%% {\it hard-wired} into self-justifying formalisms.
+
+
+
+
+
+
+%%cccorn-ff 
+
+  \bigskip        
+     {\bf A Yet Further
+      Facet
+       of this Unusual Epistemological Interpretation: } 
+      Let
+      % us use 
+      the term 
+      {\it Epistemological Bundle Theory} 
+      refer to
+       the underlying
+      theory, advanced in this appendix, which 
+      speculates about a 
+       Thinking Agent
+      % as 
+      walking
+      through   
+      RE-Class$(\xi)$'s
+      bundled universe of valid and invalid
+      collections of $\Pi_1^\xi$ sentences
+      and 
+      then applying  some heuristic  to
+      attempt to
+      identify
+      %locate
+      % locate (via  heuristics)
+       those
+      % particular elements
+       $\theta \, \in \,$RE-Class$(\xi)$
+      whose sentences are true under the Standard-M model.
+      
+      Such a 
+      theory has a second virtue, aside from 
+      % its
+      addressing  $ \# \,$'s
+       paradoxical question
+      about the nature  
+      of {\it ``instinctive faith''. }
+      It also clarifies
+      the meaning of 
+      our main theorems
+      %%  
+      %%   
+      %%   and simplifies
+      %%  the 
+      %%  % mathematical
+      %%   structure of 
+      %%  Sections
+      %%  \ref{3uuuu1}-\ref{sect64}'s theorems  
+      %%  % formal
+      %%  
+      %%  
+      and the related
+           E-stability, A-stability,
+       EA-stability and 
+      RE-Class$(\xi)$ constructs.
+      
+      
+      %%
+      %%help 
+      %%analyze 
+      %%%prove their theorems about 
+      %%self-justification.
+      %%It turns out that
+      %%%A nice aspect about the  
+      %% epistemological bundling
+      %%can explain the motivation behind these
+      %%theorems.
+      %%
+      %%
+      %%*is that it can explain much of the intuition behind
+      %%* these theorems.
+      
+      
+      This is because the
+      Items $\, * \,$ and  $\, ** \,$
+      from the
+      definitions of 
+      A-stability and E-stability
+      in Section \ref{3uuuu3} 
+      formalize
+      how a thinking agent $~T~$ can view short
+      proofs from a {\it technically inconsistent} axiom system
+      of $~B^\xi \cup \theta~$ as containing
+      % some 
+      pragmatically
+      useful information
+      {\it under the assumption} that the
+      lengths 
+      %   FIXED ALREADY length's length's length's
+      of $~T\,$'s proofs {\it are shorter}
+      than  
+      the errors in
+      $~\theta\,$'s $\Pi_1^\xi$ styled-statements.
+      The pleasing aspect 
+       about this
+      observation, illustrated by 
+      % for example
+      Remark \ref{re3-1},
+      %
+      % epistemological bundling
+      is that those same invariants,
+      $\, * \,$ and  $\, ** \,$, 
+      which 
+%%%%%      can 
+      tempt a
+       thinking agent $~T~$ 
+      to engage in a trial-and-error walk through
+      RE-Class$(\xi)$'s bundled universe, 
+      %% are 
+      also 
+      %% the invariants that 
+      make
+      viable
+      %% the operating prerequisites for making
+       \thx{ppp1}'s self-justifying formalisms. 
+      
+      
+      %active.
+      
+      Thus aside from 
+      addressing
+       $\, \# \,$'s dilemma about the nature of 
+      instinctive faith, 
+      the 
+      % mathematical
+      meta-formalism in this appendix
+      is
+      %was 
+      % also
+      useful 
+      %in 
+      %venting
+      % for
+      in  explaining
+      the 
+      % underlying
+       motivation behind the 
+      %very
+      %quite
+      % fairly
+      elaborate
+      network 
+      % labyrinth
+      of
+      theorems, proofs and definitions 
+      %raised
+      that were introduced 
+      % had 
+      %appeared 
+      in this 
+      paper.
+In summary,
+EA-stable logics are thus
+ interesting   both
+in their own right
+(as a vehicle 
+ enabling a Thinking Being to partially tolerate
+its own errors), and 
+because
+they are  useful in explaining
+how a Thinking Being 
+can possess a type of instinctive
+faith in its own consistency
+(via the 
+reflection
+principles of
+ Theorem \ref{ppp6}
+and 
+of  Remarks \ref{f88} and \ref{remhappy}).
+
+
+
+
+
+
+\cvl
+
+\section*{Appendix G:  Improvements upon Theorems  \ref{pqq4} 
+and \ref{pqq5}  }
+
+Let us recall that 
+Remark \ref{re4-n} indicated that there was a
+subtle
+trade-off between 
+Theorems  \ref{pqq4} 
+and \ref{pqq5},
+where neither result was 
+strictly
+ better than the other.
+This section
+will introduce
+two hybrid 
+methodologies, using 
+Definition G.1's
+formalism, that 
+improve upon \thx{pqq5}
+while  retaining a large part of \thx{pqq4}'s
+nice features.
+
+\hgskip
+
+% \cvt
+\parskip 2pt
+
+
+{\bf Definition G.1}
+Let $~\xi~$ denote 
+the  generic configuration, 
+whose base axiom system is again denoted as 
+$ \, B^\xi \,$,
+$\,~ \Phi~$ denote any          $\Pi_1^\xi$ 
+sentence 
+that   is true in    the Standard-M model
+and 
+$~j~$ denote an index that represents some
+predicate 
+ Test$^\xi_j \,$ 
+lying 
+in Definition \ref{gsim}'s
+ TestList$^\xi$ sequence. Then
+a
+$\,\Pi_1^\xi \,$ sentences $\Psi $ 
+ will be said to be a 
+{\bf Braced}$^\xi( \, \Phi \, , \, j \, )$ expression when
+$~ B^\xi \, + \, \Phi~$ can prove:
+\beq
+\label{punch}
+\{~~~ \forall ~x~~~
+\mbox{Test}_j^\xi(~\lceil~\Psi~\rceil~,~x~) ~~~\}
+ ~~~~\longrightarrow ~~~~ \Psi 
+\enq
+
+
+\medskip
+
+
+
+
+
+{\bf Theorem G.2}
+{\it $~$Let  $ \,\xi \,$ again 
+denote an arbitrary  generic configuration  
+  \gggcp, 
+and let
+ $(  \nal{B},D)$ again denote any second axiom system and deduction
+method whose $\Pi_1^\xi$ theorems are true under the
+Standard-M model.
+Then for any 
+integer $~j~$ and for any 
+  $\Pi_1^\xi$ sentence
+$~\Phi~$ that is  true  in the
+Standard-M model,
+ the following  invariants
+do hold:}
+\bed
+\item[   i ] 
+{\it If $ \, \xi \, $ is EA-stable\sss  
+then there 
+will exist a self-justifying
+$~\beta_j~\supset~B^\xi$ that 
+can recognize its 
+Level$(1^\xi$)
+ consistency,
+contains
+only {\bf a finite number} of additional axioms 
+beyond those appearing in 
+$~B^\xi$,
+ and which
+can 
+prove
+all  of $(  \nal{B},D)$'s $\Pi_1^\xi$ theorems that 
+are Braced$^\xi(  \Phi  ,j)$ expressions.}
+\item[  ii ] 
+{\it Likewise, 
+if $\xi$ is 
+  E-stable, A-stable  or 0-stable 
+then 
+ a
+self-justifying
+$\beta_j \supset B^\xi$ 
+will exist
+with the same properties except 
+that it
+recognizes its own
+Level$(0^\xi$)
+ consistency.}
+\ennd
+
+\medskip
+
+
+
+
+{\bf Proof.}
+To justify
+Theorem G.2, we must first define
+the axiom system  $~\beta_j~,~$ 
+whose existence is claimed by
+Items (i) and (ii).
+It will be defined to
+consist of the union of
+the initial base axiom system
+ $B^\xi$ 
+with the following three added axiom-sentences.
+\bed
+\item[   1  ]
+The $\Pi_1^\xi$ sentence $~\Phi~$
+used
+by  Definition G.1's
+ Braced$^\xi(  \Phi  ,j  )$ formula.
+
+\smallskip
+
+\item[   2  ]
+A GlobSim$^D_{  \nal{B}} \,(\xi,j)$ sentence whose indexing
+integer $ \, j \,$ 
+is defined by  Definition G.1.
+This global simulation sentence is
+thus the statement:
+\beq
+\label{glob2}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{Prf}_{  \nal{B}}^D \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}^\xi(t)~~]~~~
+\longrightarrow ~~~ 
+\mbox{Test}_j^\xi(t,x)  ~~~\}
+\enq
+\item[   3  ]
+A  $\Pi_1^\xi$ sentence
+ of the form
+$\mbox{SelfCons}^k\{~[~\theta \,\cup \,B^\xi~]~,d~\}~$
+where:
+\bed
+\item[   a   ] 
+$\theta~$ is an R-view
+consisting of the two
+ $\Pi_1^\xi$ sentences defined by
+Items 1 and 2.
+\item[   b   ] 
+$B^\xi\,$ is $~\xi \,$'s base axiom system,
+and
+\item[   c   ] 
+$k~$ equals respectively 
+ 1  and 0 under formalisms  (i) and (ii).
+\ennd
+\ennd
+Thus,
+the  system $\beta_j$
+uses identical definitions
+ under
+formalisms (i) and (ii),      
+except 
+that
+its third sentence will use a different value for
+$~k~$.
+Our proof of 
+Theorem G.2
+will require        first confirming
+the following fact:
+\begin{description}
+\item[  Claim *  ]
+The axiom system
+$\,\beta_j \,$ (which consists of the union of
+$B^\xi$  with the
+  sentences 1-3)
+will have a capacity
+to prove every 
+Braced$^\xi(  \Phi  ,j)$ 
+sentence $~\Psi~$
+that is a
+ $\Pi_1^\xi$
+  theorem of
+$(  \nal{B},D)$.
+\end{description}
+The proof of Claim * is quite simple.
+It will rest on
+the following
+three
+observations:
+\bed
+\item[   a  ]
+For each  $\Pi_1^\xi$ sentence $\Psi$,
+the  system $~\beta_j~$
+must certainly
+have a capacity to prove 
+\eq{glob21}'s sentence (which 
+states 
+that $~\Psi\,$'s
+G\"{o}del number 
+formally
+encodes
+a $\Pi_1^\xi \,$  statement). This  
+is because 
+\eq{glob21}
+ is true
+ in the Standard-M
+model
+ and
+because Part 3 of Definition \ref{def3.3}
+indicated
+that
+ the 
+$~B^\xi~$ 
+sub-component of $~\beta_j~$
+has a capacity to prove 
+ every
+$\Delta_0^\xi$ sentence
+ that is true.
+\beq
+\label{glob21}
+\mbox{Check}^\xi( ~ \lceil \, \Psi \, \rceil ~)
+\enq
+\item[   b  ]
+Since
+Claim $\,* \,$ specifies
+ $~\Psi~$ is a theorem of 
+ $(  \nal{B},D)$, 
+there must certainly exist some integer
+$~N~$ that is the G\"{o}del number of its proof from
+ $(  \nal{B},D)$. This implies that 
+\eq{glob22} must be a true
+$\Delta_0^\xi$ sentence
+ under the Standard-M model.
+As was the case with \ep{glob21}, this implies
+that
+it must be provable from
+$~B^\xi~$ (because it is a valid 
+$\Delta_0^\xi$ sentence).
+\beq
+\label{glob22}
+\mbox{Prf}_{  \nal{B}}^D \,(   ~ \lceil \, \Psi \, \rceil ~  , ~  N ~ )
+\enq
+\item[   c  ]
+It is apparent that 
+Equations \eq{glob2}, \eq{glob21} and \eq{glob22}
+imply the validity of \eq{glob23}.
+Moreover, Part 4 of Definition \ref{def3.3} indicated that
+the generic configuration $~\xi\,$'s deduction method
+does satisfy  G\"{o}del's Completeness Theorem.
+This fact assures that 
+$~\beta_j~$ must be able to prove 
+\eq{glob23} because it contains
+\eq{glob2}
+as an axiom and 
+ \eq{glob21} and \eq{glob22} 
+as derived theorems \footnote{\label{fcomp} \sm55 
+Every deduction method $\,d$,
+$\,$satisfying G\"{o}del's Completeness Theorem,
+will be automatically able to prove
+a theorem $~Z~$
+when it contains
+$X$, $Y$ and $~(X \wedge Y)~\rightarrow ~Z~$ as
+theorems, irregardless of whether or not it contains
+an explicit built-in
+modus
+ponens rule.
+Thus $~d~$ can prove
+\eq{glob23} because of its knowledge about 
+\eq{glob2}--\eq{glob22}'s validity.}.
+\ennd
+\vspace*{- 0.1 em}
+\beq
+\label{glob23}
+\forall ~x~~~
+\mbox{Test}_j^\xi(   ~ \lceil \, \Psi \, \rceil ~  , ~  x ~ )
+\vspace*{- 0.1 em}
+\enq
+Claim $*$ is a
+consequence of 
+Observations a-c. This is  because
+$ \, \Phi \, $ 
+is one of $ \, \beta_j \,$'s defined axioms,
+and
+Definition G.1 
+indicated 
+ $ \, B^\xi \, + \, \Phi \, $ was capable of proving
+\eq{punch}'s statement
+ for every  Braced$^\xi(  \Phi  ,j  )$ sentence $ \, \Psi \, $.
+These facts corroborate Claim $*$ 
+because they imply
+that $ \, \beta_j \, $
+must be able to verify
+Claim $\,* \,$'s
+ sentence
+ $ \, \Psi \, $ (because
+ $ \, \beta_j \, $
+can verify statements
+\eq{punch} and \eq{glob23}).
+
+
+
+
+
+
+
+\medskip
+
+The remainder of 
+Theorem G.2's proof is 
+analogous
+to
+\phx{pqq5}'s proof.
+This is because the prior paragraph 
+established
+that $~\beta_j~$  can prove 
+every 
+ Braced$^\xi(  \Phi  ,j  )$ 
+theorem of
+ $(  \nal{B},D)$
+(as was required by
+ Claims i and ii ). 
+The only 
+remaining task is to
+show that
+ $~\beta_j~$  is a self-justifying formalism that can
+recognize its 
+Level($1^\xi$) and Level($0^\xi$)
+consistencies,
+as specified by
+ Claims i and ii.  
+This part of 
+Theorem G.2's
+ verification
+is identical
+to
+the methods used
+to prove
+Theorems \ref{pqq3} and \ref{pqq5}.
+It
+will
+thus
+ not be repeated
+here.
+  $~~\Box$
+
+
+
+\medskip
+
+
+
+The last part of this appendix will
+require the
+following additional
+ notation to formalize
+the main intended application
+of
+Theorem G.2's 
+formalism.
+\bee
+\item
+\topsep -7pt
+ Count$(  \Psi )$ will denote
+the number of 
+quantifiers appearing in the  sentence $\Psi$
+(including both its bounded and unbounded quantifiers).
+\item 
+Size$^\xi(c)$ 
+will
+denote the  set of $\Pi_1^\xi$ sentences
+$\Psi$ where  Count$(  \Psi ) \, \leq \, c \,$.
+\ene
+Our next theorem will be   a specialized
+variant of
+Theorem G.2, using the 
+Size$^\xi(c)$  construct.
+It  will explain the 
+ intended application
+of
+this
+formalism:
+
+
+
+
+\medskip
+
+
+{\bf Theorem G.3.}
+{\it 
+$~$Let $~\xi~$ denote any one of Appendix D's four sample
+generic configurations of $~\xi^*~$, 
+$~\xi^{**}~$,  $~\xi^-~$  or   $~\xi^R~$.
+Then 
+ for any  $~c>0~$, 
+Theorem G.2's axiom systems
+of $~\beta_j~$
+can be arranged so that 
+they can prove all of 
+  $  (    \nal{B} ,  D  ) $'s
+Size$^\xi(c)$ 
+ $\Pi_1^\xi$ 
+theorems while
+simultaneously also
+ recognizing their:
+\bee
+\item Level(1)
+consistency  for the cases
+when  $ \, \xi \, $ is one of   $ \, \xi^* \, $,
+$ \, \xi^{**}\,   $ or   $ \, \xi^-$.
+\item
+ Level(0) consistency when $~\xi~$ is
+$~\xi^R~$.
+\ene}
+
+\cvmew
+
+{\bf Proof Sketch:}
+%% 
+%%  There is 
+%% insufficient
+%%  space to prove
+%% Theorem G.3 here, but its intuition 
+%% is easy to summarize.
+%% 
+%% 
+ The  intuition behind
+Theorem G.3's proof is 
+quite easy to summarize.
+For 
+arbitrary $     c>0      $
+and any
+of Appendix D's configurations of
+ $   ~  \xi^*  ~   $, 
+$   ~  \xi^{**}  ~   $,  $  ~   \xi^-  ~   $  and   $  ~   \xi^R  ~   $, 
+it is 
+routine to
+ construct an ordered pair $  ~   (\Phi,j) ~    $ where every
+$\Pi_1^\xi$ sentence of
+Size$^\xi(c)$ is a 
+ Braced$^\xi(  \Phi  ,j  )$ expression.
+Theorem G.3's 
+first claim is,
+thus, a 
+consequence of 
+Part (i)
+of Theorem G.2
+ and 
+the fact that each of
+ $   ~  \xi^*  ~   $, 
+$   ~  \xi^{**}  ~   $ and  $  ~   \xi^-  ~   $  
+are EA-stable.
+Likewise, Theorem G.3's 
+second claim follows from
+Part (ii)
+of Theorem G.2
+ and the fact that 
+$ ~  \xi^R  ~ $ is E-stable,
+$~~\Box$  
+
+\bigskip
+\medskip
+%\parskip 0pt
+
+{\bf Remark G.4.}
+The  Theorems G.2 and G.3 are
+of interest  because the set of 
+$\Pi_1^\xi$ sentences of
+Size$^\xi(c)$ is a natural class to examine.
+It is,
+thus, tempting to consider
+a system that 
+recognizes
+its own
+formal 
+ consistency, uses only a finite
+number of axiom sentences beyond those in $~B^\xi~,~$ and 
+which
+can
+ prove all of
+  $  (    \nal{B} ,  D  ) $'s
+ $\Pi_1^\xi$ theorems 
+of Size$^\xi(c)$.
+Such a system 
+replies to 
+Remark 
+\ref{re4-n}'s challenge by
+% nicely
+ hybridizing the properties of
+Theorems  \ref{pqq4} 
+and \ref{pqq5},
+%%  .
+in a seemingly pragmatic manner.
+
+
+
+
+
+
+\newpage
+
+\begin{thebibliography}{99}
+
+
+\baselineskip =  1.15 \normalbaselineskip
+
+\parskip 6 pt
+
+
+
+
+\bibitem{Ad2}
+Z. Adamowicz,
+``Herbrand Consistency and Bounded
+Arithmetic'', 
+{\it Fundamenta Mathematicae}
+% {\it Fund. Math.}
+171 (2002) pp. 279-292.
+
+
+ 
+ \bibitem{AB1}
+ Z. Adamowicz and T. Bigorajska,
+ ``Existentially Closed Structures and G\"{o}del's Second Incompleteness
+ Theorem'', {\it Journal of Symbolic Logic}
+ 66 (2001), pp. 349-356.
+
+
+
+
+
+
+
+
+\bibitem{AZ1}
+Z. Adamowicz and P. Zbierski,
+``On Herbrand consistency in weak theories'', 
+{\it Archives for Mathematical Logic}
+40(2001)  399-413
+
+
+
+\bibitem{Ar90}
+T. Arai, ``Derivability Conditions on Rosser's Proof Predicates'',
+{\it Notre Dame Journal of Formal Logic} 30(1990) 487-490
+
+
+
+\bibitem{Be95}
+L Berkemishev,
+``Iterated Local Reflection Versus Iterated Consistency'',
+{\it Annals of Pure and Applied Logic}
+75 (1995) pp. 25-48.
+
+
+
+
+
+
+\bibitem{Be97}
+L Berkemishev,
+``Induction Rules, Reflection Principles and Provably
+Recursive Functions'',
+{\it Annals of Pure and Applied Logic}
+85 (1997) pp. 193-242.
+
+
+
+\bibitem{Be3}
+L Berkemishev,
+``Proof Theoretic Analysis
+by Iterated Reflection'',
+{\it Archives on Mathematical Logic}
+42 (2003) pp. 515-552.
+
+
+
+
+\bibitem{BS76}
+A. Bezboruah and J. C. 
+ Shepherdson,
+``G\"{o}del's Second Incompleteness Theorem for Q'',
+{\it Journal of  Symbolic Logic} 41 (1976) pp.  503-512
+
+
+\bibitem{Bu86}
+S. R. Buss, {\it Bounded Arithmetic,} (Ph D Thesis)
+Proof Theory 
+Notes \#3,  Bibliopolis 1986.
+
+
+
+\bibitem{BI95}
+S. R. Buss and A. Ignjatovic,
+``Unprovability of Consistency Statements in Fragments of
+Bounded Arithmetic'', 
+{\it Annals of Pure and Applied Logic} 74 (1995) pp. 221-244.
+
+
+
+\bibitem{Fe60}
+S. Feferman, ``Arithmetization of Mathematics in a General Setting'',
+{\it Fundamenta Mathematicae}
+% {\it Fund. Math.} 
+49  (1960) pp. 35-92.
+
+
+
+
+\bibitem{Fi90}
+M. Fitting, First Order Logic and Automated Theorem Proving,
+Springer-Verlag, 1996.
+ 
+
+
+
+
+
+\bibitem{Fr79b}
+H. M. 
+Friedman, ``Translatability and Relative Consistency'',
+ Ohio State  Tech Report, 1979.
+
+
+
+
+
+
+\bibitem{Go31}
+K. G\"{o}del,
+`` \"{U}ber 
+formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandte Systeme I'',
+{\it Monatshefte f\"{u}r Mathematik und Physik} 37 (1931) pp. 349-360.
+
+
+
+
+
+
+
+
+\bibitem{HP91}
+P. H\'{a}jek and P. Pudl\'{a}k,
+{\it Metamathematics of First Order Arithmetic,}
+Springer Verlag 1991. 
+
+
+
+\bibitem{HB39}
+D. Hilbert and P. Bernays,
+{\it Grundlagen der Mathematic}, Springer 1939.
+
+
+
+
+
+\bibitem{Je71}
+R. 
+ Jeroslow, ``Consistency Statements in 
+Mathematics '', 
+{\it Fundamenta Math}
+ 72 (1971)  pp.17-40.
+ 
+
+
+
+
+\bibitem{Ka91}
+R. Kaye,
+{\it Models of Peano Arithmetic}, Oxford University
+ Press, 1991. 
+
+
+
+
+
+
+
+
+
+
+\bibitem{Kl38}
+S. C.
+Kleene,
+``On the Notation of Ordinal Numbers'',
+{\it Journal of   Symbolic Logic}
+3 (1938), 
+150-156.
+
+
+
+
+
+
+
+\bibitem{Ko5}
+L. A. Ko{\l}odziejczyk, Private Email Communications during November
+of 2005.
+
+
+
+\bibitem{Ko6}
+ L. A. Ko{\l}odziejczyk,
+``On the Herbrand Notion of Consistency for Finitely
+Axiomatizable Fragments 
+of Bound Arithmetic'',
+{\it Journal of Symbolic Logic}
+71 (2006) pp. 624-638.
+
+
+
+
+\bibitem{Kr87} J. Kraj\'{i}cek,
+``A Note on Proofs of Falsehood'', 
+{\it Archives on Math Logic} 26 (1987) pp. 169-176.
+
+
+
+
+ 
+ \bibitem{Kr95} J. Kraj\'{i}cek,
+ {\it Bounded Propositional Logic and Complexity Theory,} Cambridge 
+University
+ Press 1995.
+
+
+\bibitem{KT74}
+G. Kreisel and G. Takeuti,
+``Formally Self-Referential Propositions  for Cut-Free Classical
+Analysis'',
+{\it Dissertationes Mathematicae}
+118
+(1974) pp. 1--55
+
+
+
+
+\bibitem{Lo55} M. H. L\"{o}b, A Solution to a Problem by Leon Henkin,
+{\it Journal of Symbolic Logic}
+20 (1955)  115-118.
+
+
+
+\bibitem{Ne86}
+E. Nelson, {\it  Predicative Arithmetic,}
+Mathematics Notes
+ Princeton U Press,
+  1986.
+
+
+    \bibitem{Pa71}
+  R. Parikh, ``Existence and Feasibility in Arithmetic'',
+  {\it Journal of  Symbolic Logic}
+   36 (1971), pp. 494-508
+
+
+\bibitem{PD82}
+J. B. Paris and C. Dimitracopoulos,
+``Truth Definition for $\Delta_0$
+Formula'',
+{\it Monagraphie de L'Ensignment Mathematique}
+30 (1982) pp. 317-329. (See especially the first two pages
+of this article to
+find the counterpart of   
+Lemma \ref{lex22} in it.)
+
+
+
+\bibitem{PD83}
+J. B.  Paris and C. Dimitracopoulos,
+``A Note on  
+Undefinability of Cuts'', 
+{\it Journal Symbolic  Logic}
+48 (1983)
+  564-569
+
+
+
+
+\bibitem{PW81}
+J. B.  Paris and A. J.  Wilkie,
+``$\Delta_0$ Sets and Induction'', 
+{\it 1981 Jadswin  Conference Proceedings,} 
+pp. 237-248.
+
+
+
+\bibitem{Pa72}
+C. Parsons, 
+On $n-$Quantifier Elimination,
+{\it Journal of Symbolic Logic}
+37 (1972), pp. 466-482.
+
+
+
+
+ \bibitem{Pu84}
+ P. Pudl\'{a}k,
+ ``On Lengths of Proofs of Finistic Consistency Statements in
+ First order Theories'',
+ {\it Logic Colloquium 84}, North Holland (1986)
+ pp. 165-196.
+
+
+
+\bibitem{Pu85}
+P. Pudl\'{a}k,
+``Cuts, Consistency Statements and Interpretations'',
+{\it Journal of Symbolic Logic}
+50 (1985)  423-442
+
+
+
+
+\bibitem{Pu96}
+P. Pudl\'{a}k,
+``On the Lengths of Proofs of Consistency'',
+in {\it Collegium Logicum: 1996 Annals of the Kurt G\"{o}del
+Society} ( Volume 2),  Springer-Wien-NewYork,
+pp 65-86.
+
+
+
+
+
+
+
+
+
+
+
+\bibitem{Ro67}
+H. Rogers, {\it 
+Recursive Functions and Effective 
+Compatibility,} McGrawHill 1967.
+
+
+
+\bibitem{Ro36}
+J. B. Rosser,
+``Extensions of Some
+Theorems by
+G\"{o}del
+and Church'',
+{\it Journal of  Symbolic  Logic}
+1 (1936) pp.  87-91.
+
+
+\bibitem{Sa11}
+S. Salehi,  Herbrandized Consistency of Some 
+Finite Fragments of Bounded
+Arithmetic Theories (CoRR 2011)
+{\rm http://arxiv.org/abs/1110.1848}. 
+
+
+
+ 
+ \bibitem{Sm83}
+ C. A.  Smory\'{n}ski, 
+ The Incompleteness Theorem, 
+ {\it Handbook on Mathematical Logic,}
+ pp. 821--866, 1977.
+ 
+
+
+
+
+
+
+
+\bibitem{Sm85}
+C. A. Smory\'{n}ski, ``Non-standard Models and Related Developments
+in the Work of Harvey Friedman'', in {\it Harvey Friedman's
+Research in the  Foundations of Mathematics},
+NH 1985, pp 179-229.
+
+
+
+
+
+\bibitem{Sm68} 
+R. Smullyan, {\it First Order Logic,} Springer-Verlag, 1968.
+
+
+
+
+\bibitem{So76}
+R. M. 
+Solovay,  ``Interpretability in set theories''.
+Unpublished
+letter to Petr 
+H\'{a}jek
+dated
+ August 17, 1976 and
+available on Petr    H\'{a}jek's home page.
+
+
+
+\bibitem{So76b}
+R. M. 
+Solovay,  ``Provability Interpretations in Modal Logics'',
+{\it Israel Journal of Mathematics} 25 (1976) pp. 278-304.
+
+
+
+\bibitem{So88}
+R. M. 
+Solovay,  ``Injecting Inconsistencies into Models of PA'',
+{\it Annals Pure and Applied Logic''} 44(1988) 102-132
+
+{\baselineskip =  1.1 \normalbaselineskip
+
+
+\bibitem{So94}
+R. M.  Solovay,  Private 
+%telephone  
+communications
+in  1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using the additional formalisms of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+Solovay 
+never published 
+this result (called Theorem 2.1 in Section 2)
+or
+his other
+observations 
+that
+several logicians \cite{HP91,Ne86,PD83,PW81,Pu85,Pu96,So76,Sv7,WP87}
+have 
+attributed to his
+private communications.
+The Appendix A of 
+\cite{ww1} offers our 
+4-page interpretation of
+Solovay's basic suggestion.
+See also
+reference \cite{So76} for a pdf file consisting of
+Solovay's 
+earlier unpublished
+1976
+ letter to Petr 
+H\'{a}jek.}
+
+
+
+
+\bibitem{Sv83}
+V. \v{S}vejdar, ''Modal Analysis of 
+Rosser Sentences'',
+{\it Journal of Symbolic Logic} 48 (1983)  986-999.
+
+
+\bibitem{Sv7}
+V. \v{S}vejdar, ''An Interpretation of Robinson Arithmetic in its
+Grzegorczjk's Weaker Variant''
+{\it Fundamenta Informaticae} 81 (2007)
+pp. 347-354.
+
+
+\bibitem{Ta53} 
+G. Takeuti,
+``On a Generalized Logical Calculi'',
+{\it Japan Journal of Mathematics}
+32 (1953) pp. 39-96. 
+
+
+
+ 
+ \bibitem{Ta87}
+ G. Takeuti, {\it Proof Theory,} 
+ Studies in Logic Volume 81, 
+ North Holland, 1987.
+
+
+
+\bibitem{Ta0} 
+G. Takeuti,
+``G\"{o}del Sentences of Bounded Arithmetic'',
+{\it Journal of Symbolic Logic} 65 (2000)  1338-1346.
+
+
+
+
+
+\bibitem{Ta36}
+A. Tarski,
+``Der Wahrheitsbegriff in den 
+formalisisierten 
+Sprachen'',
+{\it Studia Philosphie.} 1 (1936) 261-405
+
+
+
+
+\bibitem{TMR53}
+A. Tarski, A. Mostowski and R. M. Robinson,
+{\it Undecidable Theories}, North Holland, 1953.
+
+
+\bibitem{VV94}
+R, Verbrugge and A. Visser,
+``A Small Reflection Principle for Bounded Arithmetic'',
+{\it Journal of Symbolic Logic} 59 (1994) pp. 785-812.
+
+
+
+\bibitem{Vi89}
+A. Visser,
+``Peano's Smart Children: A Provability Logical Study of
+Systems with Built-in Consistency'',
+{\it Notre Dame Journal of Formal Logic},
+30 (1987) pp. 161-196.
+
+
+
+
+\bibitem{Vi92}
+A. Visser,
+``An Inside View of Exp'',
+{\it Journal of Symbolic Logic} 57 (1992)
+131--165.
+
+
+
+ 
+ \bibitem{Vi93}
+ A. Visser,
+ ``The Unprovability of Small Inconsistency'',
+ {\it Archive for Math Logic} 32 (1993) pp. 275-298.
+ 
+ 
+
+\bibitem{Vi5}
+A. Visser,
+ ``Faith and Falsity'',
+ {\it Annals of Pure and Applied Logic}
+131 (2005) pp. 103-131.
+
+
+\bibitem{VH73}
+P. Vop\v{e}nka and P. H\'{a}jek,
+``Existence of  a Generalized Semantic Model of
+G\"{o}del-Bernays Set Theory'',
+{\it Bulletin de l'Academie Polonaise des
+Sciences,
+Math, 
+Astromiques et Physiques}
+12 (1973) 
+1079-1086.
+
+
+
+\bibitem{WP87}  
+A. J. Wilkie and J. B. Paris,  
+``On the Scheme of Induction for Bounded
+ Arithmetic'', {\it
+Annals of  Pure and  Applied Logic} (35) 1987, 261-302.
+
+
+
+
+
+ 
+\bibitem{ww93}
+D. Willard,
+``Self-Verifying Axiom Systems'', {\it
+Computational Logic and Proof Theory:$~$
+The   Third Kurt G\"{o}del
+Colloquium}
+(1993),
+Springer-Verlag LNCS\#713,  325-336.
+
+
+  
+  
+  \bibitem{ww0}
+  D. Willard, 
+  ``The Semantic Tableaux Version of the Second
+  Incompleteness Theorem Extends Almost to 
+  Robinson's 
+  Arithmetic 
+  Q'',
+  {\it Automated Reasoning with Analytic Tableaux
+  and Related
+  Methods}, 
+  Springer Verlag (2000) LNCS\#1847, 
+   pp. 415-430. 
+
+
+\bibitem{ww1}
+D. Willard, ``Self-Verifying  Systems, the Incompleteness
+Theorem and Related Reflection
+Principles'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+
+
+\bibitem{ww2}
+D. Willard, 
+``How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem 
+Almost 
+to 
+Robinson's Arithmetic'',
+{\it Journal of Symbolic Logic}
+$~\,67~ (2002)~$pp. 465--496.
+
+
+
+\bibitem{wwlogos}
+D. Willard, 
+``A Version of the
+Second Incompleteness Theorem For Axiom
+Systems that Recognize Addition 
+But Not Multiplication as a Total Function'',
+{\it First Order Logic Revisited,}
+Logos Verlag (Berlin) 2004, pp. 337--368.
+
+\bibitem{ww5}
+D. Willard, 
+``An Exploration of the Partial Respects in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency'', 
+{\it Journal of Symbolic Logic} 70  (2005) pp. 1171-1209. 
+
+\bibitem{ww6}
+D. Willard, 
+``On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency'', 
+{\it Journal of Symbolic Logic} 71 (2006)
+pp. 1189-1199.
+
+\bibitem{wwapal}
+D. Willard,  
+``A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+
+
+
+\bibitem{ww7}
+D. Willard,  
+``Passive induction and a solution to a Paris-Wilkie 
+question'',
+{\it Annals of Pure and Applied Logic}
+146(2007) 
+pp. 124-149.
+
+\bibitem{ww9}
+D. Willard,  
+``Some Specially Formulated Axiomizations for
+I$\Sigma_0$ 
+Manage to
+Evade
+the Herbrandized Version of the Second Incompleteness Theorem'',
+{\it Information and Computation}
+207(2009)  1078-1093
+
+\bibitem{Wr78}
+C. Wrathall, ``Rudimentary Predicates and Relative Computation'',
+{\it Siam Journal on Computing} 7 (1978), pp 194-209
+\end{thebibliography}
+
+
+
+\end{document}
+
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--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2015-aug10.bak
@@ -0,0 +1,7575 @@
+% 2015 upstairs aug 10 4.2 pm SUNY
+
+%% 2015 home % august 2 8.1 pm while listenihng to BobbyLee DoWap
+
+%% 12.45 appointment nicoloson
+
+ % 2015 july 4 3.4 am after spell 10.1 am after sinatra
+
+ % 2015 july 2 3.15 pm 
+
+%% 2015 july 2 2.50 pm upstairs 
+
+%% 2015 july 1  10.30  am downstairs
+
+
+
+%% notarized notes 2015 april 2 6.3  am april 4 notarize again 
+
+%% home 2014 feb 8 1.15am (new email address)
+
+%% home 2015 feb6 4.3 am   suny 2.40 pm home 6.15 pm
+
+
+%% gmail dan.willard.albany and Prof.DanEdwardWillard
+%% gmail password cpZ9ar48s
+
+
+%%% SUNY JAN 11 Brad Copy 8.4  pm
+
+%%  SUNY   jan11 5/30pm spell check
+
+%% 2015 HOME jan 10  9.4  pm  pm New Abstratct
+
+%% 512 6932
+
+%% Towards a Restructuring of Hilbert's Consistency Program
+
+% www.cs.albany.edu/~dew/algor/
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[12pt]{article}
+\documentclass[11pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+
+
+
+% \addtolength{\oddsidemargin}{-0.95in}
+\addtolength{\oddsidemargin}{-1.0in}
+
+
+
+\setlength{\textheight}{9.6 in}
+\setlength{\textheight}{8.8 in}
+\setlength{\textheight}{9.3 in}
+\setlength{\textheight}{9.35 in}
+%above too short
+
+
+\setlength{\textwidth}{6.3 in}
+%% PRINT
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{5.4 in}
+
+\setlength{\textwidth}{6.5 in}
+% \setlength{\textwidth}{7.0 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+% Above IDeall
+
+
+%% \setlength{\textwidth}{6.4 in}
+%%%% above brad with 11 point
+
+%\setlength{\textwidth}{6.0 in}
+%\setlength{\textwidth}{5.7 in}
+
+%\setlength{\textwidth}{6.4 in}
+
+%\setlength{\textwidth}{5.5 in}
+
+%\addtolength{\topmargin}{-1.0in}
+%\addtolength{\topmargin}{-0.9in}
+\addtolength{\topmargin}{-0.8in}
+%\addtolength{\topmargin}{1.2in}
+
+%\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+
+\title{ A 2-Part Conjecture about
+How  a Much-Diluted but Non-Trivial
+%Variant
+Fragment
+ of
+Hilbert's Consistency Program 
+Is
+Likely
+%Plausible 
+Feasible
+for the
+% Even the Challenging
+Case of
+Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+% \title{\large \bf On the Revival of a Modified and Diluted Version of
+% Hilbert's Consistency Program (Extended Abstract)} 
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+%Email = dew@cs.albany.edu.}}
+%\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+%\begin{abstract}
+\baselineskip = 1.17 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+
+
+
+\begin{abstract}
+
+%\Large
+
+% \baselineskip = 1.8 \normalbaselineskip 
+%aaaaaaaaaaa
+
+\large
+\LARGE
+ \normalsize
+\large
+
+\baselineskip = 1.25 \normalbaselineskip 
+
+It is well known that
+the combined work of Pudl\'{a}k and Solovay \cite{Pu85,So94},
+enhanced by some added techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87}, implies
+ no reasonable axiom system
+can verify its own Hilbert consistency, when it recognizes
+Successor as a total function and treats addition and multiplication
+as 3-way relations (as Example \ref{ex-2.3} will explain).
+These considerations will lead us to examine
+unconventional
+ axiomatizations
+for arithmetic that continue to view addition and multiplication as
+3-way relations, but
+which  replace the successor function symbol with an
+entirely new operator, called the ``$~\theta~$'' primitive.
+
+\medskip
+
+% Our Propositions 
+% \ref{th-3.3} and \ref{th-6.1}
+% will show this $~\theta~$ operator
+% allow us 
+
+This $~\theta~$ operator
+will
+allow us 
+to encode any integer $~n~$ by a term $~T_n~$
+whose length will exceed the $O[~$Log$(n)~]$ length of a
+binary encoding 
+only by the
+relatively
+ small magnitudes formalized by
+Propositions 
+\ref{th-3.3} and \ref{th-6.1}.
+This 
+paradigm will be
+% issue is
+%will be 
+significant because the combination of
+our Appendix A and prior published results in \cite{wwapal}
+will provide good reasons for conjecturing  the
+ $~\theta~$ primitive can be used to construct logics that
+can  verify their own consistency under 
+the  desired setting of
+a Hilbert-styled 
+deductive methodology.
+
+\medskip
+
+The relationship between our new system and the aspirations of
+Hilbert's consistency program will be discussed in detail.  There can,
+obviously, never exist a self-justifying logic that can  evade the force
+of G\"{o}del's Second Incompleteness Theorem in a fully ubiquitous respect.
+Our conjecture will be, however, that  
+our methodology 
+implies a 
+{\it well-defined}
+fragment of the goals of Hilbert's Consistency
+program can be
+% positively 
+achieved in a
+{\it  diluted
+% but non-trivial} 
+but interesting} 
+respect.
+
+
+\end{abstract}
+
+\bigskip
+\bigskip
+
+\large
+{\bf Keywords:}
+\small
+G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+Open Question,
+Hilbert-styled Deduction (and its Frege-like analogs).
+
+
+
+
+% \bigskip
+% 
+% 
+% 
+% {\bf Mathematics Subject Classification:}
+% 03B52; 03F25; 03F45; 03H13 
+% 
+% 
+% 
+% \bigskip
+% \bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+%\begin{abstract}
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+\newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+\def\gvs{ }
+
+
+\def\gvs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+\def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 0pt}
+
+\def\gvs{ \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \large \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+
+\noindent
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\gvs
+
+
+\section{Introduction}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+\gvs
+
+\parskip 4pt
+
+%  hhhh
+Two historic results
+were established
+by
+%%% in
+ G\"{o}del's
+millennial
+%centennial
+%  1931 
+ paper \cite{Go31}.  The First Incompleteness
+Theorem 
+showed  there existed no decision procedure
+for identifying the true statements of Arithmetic.
+%G\"{o}del's
+Its Theorem XI,
+later known as the ``Second  Incompleteness
+Theorem'',
+%,
+%appearing in G\"{o}del's millennial paper \cite{Go31}. 
+demonstrated
+that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+ Russell-Whitehead Principia Mathematicae formalism
+% $\, P \,$
+can
+% could
+ verify
+its own consistency.
+ G\"{o}del's
+two
+observations
+% Theorem XI 
+were  historic
+mainly  because they
+ demonstrated, unequivocally,
+that the 
+initial 
+objectives of Hilbert's Consistency
+Program were too far-reaching.
+Thus at best,  only a
+% very  
+sharply curtailed
+form of Hilbert's goals 
+would be
+% was 
+plausible.
+This fact was further reinforced by a new version of the
+Second Incompleteness Theorem, due to
+the combined work of Pudl\'{a}k and Solovay \cite{Pu85,So94},
+enhanced by some added techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87}.
+
+%plausibly feasible
+%from Theorem XI's perspective.
+
+Within these curtailed limits, we have published since 1993
+a series of articles 
+\cite{ww93}-\cite{ww14},
+outlining generalizations of the Second Incompleteness
+Theorem and
+its 
+sometimes-feasible
+% plausible 
+ boundary-case exceptions.
+%  that were formally feasible.
+Pavel Pudl\'{a}k
+examined
+%in great detail,
+% the preprints 
+an early preprint
+ of  
+our
+article \cite{wwapal}
+and suggested 
+\cite{Pupriv} 
+%that 
+we
+consider  attempting to hybridize 
+its formalism with some of
+Ajtai's observations about 
+ about
+the Pigeon Hole effects
+\cite{Aj94}. 
+The Section 6 of 
+  \cite{wwapal}
+did subsequently
+formalize one type of response to 
+Pudl\'{a}k's insightful observation.
+Our new results in this current paper will examine
+this topic 
+from yet another perspective.
+
+The
+particular
+ goals in the current paper will transgress 
+significantly
+ beyond our earlier research in
+\cite{ww93}-\cite{ww14}.  
+(The latter focused on examining generalizations and boundary-case
+exceptions for the Second Incompleteness Theorem
+that
+% were formally proven feasible). 
+viewed
+roughly, at least, addition from a traditional perspective.)
+Our new 
+Propositions 
+\ref{th-3.3} and \ref{th-6.1}
+will explore the properties of an
+entirely unconventional methodology for constructing
+the foundations of integer arithmetic (which does not use
+any of
+the
+ traditional 
+function symbols 
+% of
+for 
+%formalizing the 
+%conventional primitive operations of 
+successor, addition and
+multiplication
+as primitive
+ operations). 
+
+This topic 
+will 
+have interesting 
+philosophical implications and 
+% also 
+lead to
+our conjecture 
+% 
+% be
+% interesting unto itself,
+% from a 
+% solely
+% %purely
+% philosophical perspective.
+% We will also conjecture
+% that
+% it 
+% will
+% provide a machinery for reviving a
+% % {\it  VERY VERY  
+% 
+that a 
+{\it  much-diluted} but
+non-trivial fragment of Hilbert's Consistency Program
+is plausible. 
+
+%More precisely during our discussion of axiomatizations for
+%integer arithmetics, 
+
+During our discussion.
+%% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+ will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us say 
+%that 
+an
+axiom 
+system
+%basis
+ $\, \alpha \,$
+{\bf recognizes}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+can prove
+\eq{totxtefs} - \eq{totxtefm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+%\small
+\beq 
+\label{totxtefs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totxtefa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totxtefm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+\noindent
+Our axiomatizations for integer arithmetics 
+%in the current article 
+will differ from 
+their conventional counterparts
+{\it by neither 
+possessing} an ability to prove the totality statements
+in Lines 
+\eq{totxtefs} - \eq{totxtefm}, 
+{\it nor
+containing} function symbols for
+formalizing
+the traditional addition, multiplication and successor operations.
+%%% nor recognizing the validity of the identities
+% 
+% which state that these primitive
+% are ``total functions''. 
+% 
+Instead, we will rely upon an entirely
+different type of primitive function symbol, called the ``$~\theta~$''
+operator, to construct the endless sequence of integers
+$~3,4,5, ~...~$ from the
+three  initial starting constants of 0, 1 and 2.
+
+Its unorthodox
+% This alternative 
+means for constructing the set of non-negative integers
+%% in an unorthodox manner 
+will be a curious philosophical development,
+unto itself,
+apart from 
+its
+%  the
+surprising  implications
+% it will have 
+about a
+possible
+ conjectured
+revival of a {\it much-diluted} but non-trivial 
+variant
+%form of cousin
+of Hilbert's Consistency Program. 
+
+The reader should be forewarned
+that this article will not provide a  proof of its main
+2-part
+conjecture.
+% A later section of this article
+We
+ will 
+% later
+explain
+why 
+% how 
+our
+conjecture is likely correct,
+%We consider it 99 \%
+%probable that Section ???'s conjecture is correct, 
+but its proof
+is
+almost certainly
+% be 
+% much
+too
+long to fit into  
+%%
+%%for it to be practical to provide
+%%within the space of 
+%%
+a
+% one
+single
+% short 
+%abbreviated 
+paper.
+%                                %
+Our 
+% This
+conjecture
+%It 
+will suggest
+% that
+  a
+%%
+%%(or even 50) pages
+%%The goal of this article
+%%will be to stimulate  interest
+%%in 
+%%
+perhaps
+5-10 \% 
+fragment of the goals of Hilbert's consistency program
+can be positively achieved,
+%but 
+in a context where
+the
+Incompleteness Theorem
+precludes, definitively, 
+a more 
+ambitious project.
+
+
+
+% in a context where
+% the
+% % G\"{o}del's 
+% Incompleteness Theorem
+% %definitively establishes that 
+% % clearly 
+% precludes
+% a more ambitious project.
+
+
+
+
+%The discourse in t
+
+This article 
+%has been carefully composed
+%so that it 
+can be
+read 
+ without 
+% first
+examining any of our previous
+papers. If a reader
+does wish to skim
+one of our earlier
+articles,
+% before the current  article,
+we recommend 
+Sections 3 
+\& 4
+ of \cite{wwapal} 
+be skimmed
+(in a context where the 
+subsequent
+sections
+% portions 
+of \cite{wwapal}
+%will be
+are
+%are fully
+unrelated to our
+% chief 
+conjecture).
+
+% 
+%  our recommendation is that
+% Section 3 of \cite{wwapal} 
+% be skimmed 
+% $~---~$
+% in a context
+% where the subsequent sections of \cite{wwapal}
+% are unrelated to our main theorem and 
+% can 
+% be 
+% entirely
+% omitted.
+
+
+
+
+%% 
+%% In a context where I will be submitting this paper for publication
+%% shortly before my 67-th birthday, the reader should be forewarned
+%% that this article will not provide a formal proof of the main
+%% conjecture that it will 
+%% propose.
+%% We consider it 99 \%
+%% probable that Section ???'s conjecture is correct, 
+%% but its proof
+%% looks too tedious and long for it to be practical to provide
+%% within the scope of this 25-page paper.
+%% The goal of this article
+%% will be to stimulate the interest
+%% of a new generation of researchers,
+%% who consider our new formalism and its relationship to
+%% Hilbert's Year-1900 Second Open Question to be of interest.
+%% 
+
+\section{Returning to the 1931-1939 Period}
+\label{ss2}
+ G\"{o}del's 
+Second Incompleteness Theorem 
+was published in two 
+quite different 
+ forms during the
+1931-1939 period.
+Its initial 1931 variant, formalized by Theorem XI
+in  G\"{o}del's millineal paper \cite{Go31},
+% 
+% 
+% Its Theorem XI,
+% later known as the ``Second  Incompleteness
+% Theorem'',
+% %,
+% %appearing in G\"{o}del's millennial paper \cite{Go31}. 
+% 
+demonstrated
+that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+ Russell-Whitehead Principia Mathematicae formalism
+% $\, P \,$
+could
+% could
+ verify
+its own consistency.
+The widely quoted more general
+result, that 
+every consistent r.e. extension
+ of Peano Arithmetic must
+be unable to prove a theorem affirming its
+own consistency, 
+was 
+first
+published 
+%% 
+%% (see \footnote{ Boolos states in \cite{Bool}
+%% that it has been open to scholarly debate
+%% whether or not the 1939
+%% Hilbert-Bernays generalization of the Second Incompleteness Theorem
+%% is or (is not) a straightforward generalization of
+%% G\"{o}del's initial result} ) 
+%% 
+in the 1939 edition of the 
+the Hilbert-Bernays
+textbook \cite{HB39}. It has been considered
+to be the definitive demonstration of the broad reach of
+the Second Incompleteness Effect. 
+It also established, beyond any reasonable doubt, that any type
+of formalism possessing a conventional knowledge of its own consistency,
+must rely upon a 
+foundational structure
+ fundamentally different from Peano Arithmetic. 
+(This is because the 
+Hilbert-Bernays 
+textbook formalized the forerunner of
+what has now been known as the 
+Hilbert-Bernays Derivability Conditions \cite{HB39,HP91,Lo55,Mend},
+as a mechanism for
+% foreseeing 
+envisioning
+the 
+astonishing
+broad generality of the
+Second Incompleteness Effect.) 
+ 
+
+It is, thus, fascinating that Hilbert,
+as the co-author of 
+an important 
+%very 
+% historic
+generalization of the Second Incompleteness Theorem,
+never withdrew the
+% chose to  never fully withdraw his 
+1926 justification
+ \cite{Hil26}
+for his consistency program:
+\begin{quote}
+$*~$~ 
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+Indeed, 
+%Instead,
+Hilbert 
+always insisted that some
+special new formalism would at
+least partially vindicate the
+prior
+%initial 
+ goals of his
+consistency program.
+He thus arranged to 
+have  its often-quoted 
+motto
+({\it ``Wir m\"{u}ssen wissen---Wir werden wissen''} )  
+%of his 
+%nevertheless, 
+engraved on his tombstone.
+
+
+%Moreover, it 
+
+It is 
+also
+known
+\cite{Da97,Go5,Yo5}
+that G\"{o}del
+was 
+doubtful about the generality of the Second Incompleteness
+Theorem for at least two years after its publication.
+He thus inserted the following
+cautious caveat into
+his famous
+1931 
+millennial
+paper  \cite{Go31}:
+% whose closing section 
+%%% 
+%%% One of the closing paragraphs of
+%%%  \cite{Go31} 
+%%% thus
+%%% included
+%
+%
+%%% contained the following cautious disclaimer:
+%caveat:
+\newpage
+\begin{quote}
+\it
+%\baselineskip = 1.0 \normalbaselineskip 
+$~**~~$ 
+``It must be expressly noted that
+Theorem XI
+%'s incompleteness result
+(e.g. the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and {\it there might
+conceivably be finite proofs} which cannot
+be stated in P or in ... ''
+\end{quote}
+
+
+The 
+% above 1931
+statement $**$ has 
+had
+%been subject to 
+numerous
+%many
+different
+interpretations
+\footnote{
+Some 
+scholars
+have interpreted
+$\,**\,$
+as
+%as, possibly,'
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.}.
+All
+ G\"{o}del's 
+biographers
+\cite{Da97,Go5,Yo5}
+%%%have
+noted
+his
+% that G\"{o}del's
+initial intention
+was
+to
+establish
+%achieve
+Hilbert's proposed objectives, before 
+he proved
+%proving
+% G\"{o}del proved
+a result 
+% 
+% however,
+% %%%%%his
+% G\"{o}del
+% did  originally 
+% seek 
+% % goal was 
+% to
+% establish
+% %achieve
+% Hilbert's proposed objectives before 
+% proving
+% % G\"{o}del proved
+% a result 
+% 
+leading
+%that led 
+in the opposite direction.
+Yourgrau \cite{Yo5}
+records
+%furthermore,
+ how
+von Neumann 
+surprisingly
+%did
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive 
+%%%  achievement of a'
+ termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+It is known 
+ G\"{o}del 
+began to 
+more fully
+endorse 
+the Second Incompleteness
+Theorem
+during a 1933
+%% Vienna
+lecture  \cite{Go33},
+and he 
+% told biographers he
+completely embraced it
+after learning about Turing's work
+\cite{Tur36}.
+
+
+Our research
+in \cite{ww93}-\cite{ww14}
+%has been 
+is
+related to issues 
+%analogous
+ similar 
+to those
+%that were 
+raised by Hilbert and 
+G\"{o}del
+in
+ statements  $*$ and $**$.
+This is because it is counter-intuitive and awkward to
+%presume that 
+explain how
+human beings can maintain the
+psychological drive and 
+needed energy-desire to cogitate, without
+being stimulated by an instinctive faith in their own
+consistency (under a definition of 
+% formal  consistency
+such
+% this concept 
+that is suitably 
+gentle and
+% delicate
+soft
+ to 
+be consistent with the 
+Incompleteness Theorem's requirements).
+
+% preclude a violation of the 
+% restrictions imposed by the Incompleteness Theorem).
+
+Accordingly, our research in
+\cite{ww93}-\cite{ww14} 
+has explored both generalizations and
+boundary-case exceptions of the Incompleteness Effect, so as
+to determine what type of boundary-case evasions are permitted.
+Our prior research in \cite{ww93}-\cite{ww14} 
+had used mostly cut-free forms of deduction to
+evade the 
+restrictions imposed by the
+Second Incompleteness Effect. The current article will instead
+focus on more pristine Hilbert-Frege methods of deduction.
+They are likely to support an evasion of the Second Incompleteness
+Effect when our axiom systems replace the traditional  
+growth properties of the addition, multiplication and successor 
+function symbols with our new $~\theta~$ primitive.
+
+The motivation for this replacement will be
+explained during the next section of this article.
+It is needed
+essentially
+because
+a
+ version of the Second Incompleteness Theorem,
+due to the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+%will show 
+demonstrates
+that 
+if an axiom system $~\alpha~$
+proves any 
+of \eq{totxtefs} - \eq{totxtefm}'s totality statements
+then it is incapable of confirming its own consistency
+under a Hilbert-style deductive method.
+
+
+
+
+
+\smallskip
+
+Our results will suggest  it is possible to obtain a
+{\it part-way 5-10 \%
+positive} interpretation
+for what
+Hilbert and G\"{o}del 
+were
+seeking 
+% a Consistency Program
+to establish
+%%  seeking to accomplish
+% contemplating 
+in their 
+statements
+$*$ and $**$, within a context where 
+it is  known that the 
+Second Incompleteness Effect precludes a full achievement of
+%these 
+Hilbert's
+objectives
+from ever transpiring.
+The last five minutes of
+a recent 60-minute YouTube 
+presentation
+ by Harvey Friedman
+\cite{Fr14},
+entitled the 
+{\it ``Blessing and Curse of Kurt  G\"{o}del''},
+suggested that it is
+%the themes of Hilbert and G\"{o}del's
+%remarks $*$ and $**$ by indicating that 
+%it is 
+interesting to explore futuristic partial 
+boundary-like evasions of the Second Incompleteness Theorem,
+despite the stunning strength of 
+G\"{o}del's result.
+It is within this context where our proposed use of a new
+$~\theta~$ primitive 
+symbol to replace the growth properties
+of the traditional addition, multiplication and successor function
+symbols may be of potential interest.
+
+The development of our $~\theta~$ primitive 
+was partially influenced by a private email
+communication 
+we had received
+from  
+Pavel Pudl\'{a}k \cite{Pupriv},
+as Sections 3-4  shall explain.
+We also emphasize that the conventional interpretation of
+the Second Incompleteness Theorem, as precluding
+Hilbert's Consistency Program from ever achieving its initially
+specified objectives, is certainly correct.
+Our only caveat is that the latter should not lead  one
+to
+ignoring the role that a
+human's instinctive faith in his/her's internal
+consistency 
+%crucially stimulates and motivates
+plays in stimulating and motivating
+human cognition.
+% humans to cogitate.
+%
+% crucially
+% gain the
+% motivation for stimulating cogitation. 
+It is
+from this 
+special
+perspective where our prior research and
+new 2-part conjecture 
+will
+% does 
+suggest that 
+an approximate
+%at least a
+5-10 \%
+fragment 
+% fraction
+of what Hilbert and G\"{o}del 
+%suggested in 
+had
+sought
+in $*$ and $**$ 
+%could
+should be
+ plausibly 
+%is
+%% be formally 
+ feasible.
+
+
+
+\gvs
+
+\section{Motivation for Research and Background Notation}
+%  222222}
+\label{ss3}
+
+
+%%! 
+%%! This article will be written in a style so that its
+%%! overall theme (if not full details)
+%%! should become 
+%%! {\it  quickly} comprehensible to a reader who has
+%%! examined 
+%%! only
+%%!  one of the
+%%! % introductory 
+%%! logic textbooks by say Enderton,
+%%! Fitting, H\'{a}jek-Pudl\'{a}k,
+%%!  Mendelson
+%%! or Papadimitriou \cite{End,Fi96,HP91,Mend,Papa}.
+%%! %% 
+%%! %% We will rely mostly upon the
+%%! %% precise
+%%! %% deductive calculi notation employed
+%%! %% in Section 2.4 of Enderton's textbook,
+%%! %% but any of 
+%%! %% the 
+%%! %% similar
+%%! %% Hilbert-style deductive calculi of 
+%%! %%  H\'{a}jek-Pudl\'{a}k,
+%%! %%  Mendelson
+%%! %% or Papadimitriou \cite{HP91,Mend,Papa}.
+%%! %% will also be suitable for achieving our results.
+%%! 
+%%! %% 
+%%! %%  
+%%! %% ( Papadimitriyou's textbook
+%%! %% generously states it employs a deductive notation
+%%! %% that 
+%%! %% has
+%%! %% stemmed from its predecessor in 
+%%! %% Enderton's textbook.)
+%%! 
+%%! 
+
+
+
+%% In order to make our 
+%% results
+%% %research 
+%% apply to the 
+%% formalism
+
+It is helpful to employ a flexible vocabulary so
+% that our
+our 
+results
+will apply 
+%research 
+%to the 
+%formalisms
+to any of the
+%% 
+%% accessible to
+%% some 
+%% readers who are acquainted with only one of the 
+%% 
+textbook 
+formalisms of
+% settings outlined by 
+%  
+say 
+Enderton,
+Fitting, H\'{a}jek-Pudl\'{a}k,
+or Mendelson
+\cite{End,Fit,HP91,Mend}.
+% 
+% ,
+% %% or
+% %%  Papadimitriou 
+% %% \cite{End,Fit,HP91,Mend,Papa},
+% %% 
+% %% 
+% %the widest
+% % possible  
+% %audience, 
+% it is 
+% helpful
+% % useful to 
+% use
+% %employ 
+% a
+% % very 
+% flexible
+%  vocabulary.
+% %%
+% %%that 
+% %%allows a reader to 
+% %%%quickly 
+% %%%translate results 
+% %%traverse
+% %%from
+% %%one textbook to another.
+% % Therefore, let us define a
+Let us 
+% thereby
+call an
+ordered pair $(\alpha,d)$ a
+    {\bf ``Generalized Arithmetic''}
+% therefore
+iff its 
+% first and second 
+two components 
+%% 
+%%  Each of the 
+%% textbooks \cite{End,Fi96,Mend,Papa,HP96} have
+%% employed
+%% substantially
+%%  different variants of Basis and 
+%%  Deductive-Apparatus structures. 
+%% 
+are 
+% described
+formalized
+% defined 
+as below:
+%% 
+%% Their
+%% definitions in
+%%  Items (1) and (2)
+%% %simple 
+%% %%%definitions of these two notions
+%% %given below,
+%%     allow one to easily translate 
+%% %theorems 
+%% formalisms
+%% % methodologies
+%% from
+%% one textbook 
+%% % source
+%% to another:
+%% 
+%% %\njp
+%% % \newpage
+%% \parskip 2pt
+\bee
+\item
+The {\bf ``Axiom Basis''} $~\alpha~$ 
+for an arbitrary arithmetic
+shall be defined as
+the set of 
+{\it proper axioms} employed by the 
+formalism  $(  \alpha  , d  )$.
+\item
+An arithmetic's {\bf ``Deductive Apparatus''} $~d~$ 
+is defined as
+the 
+{\it combination} of its formal rules for inference
+and 
+%its
+the
+ built-in
+ logical axioms ``$~L_d~$''
+%  (that are 
+% implicitly 
+employed by these rules.
+\ene
+
+%%%\item
+%%%The term {\bf ``Deductive Apparatus''} $~d~$ will
+%%%refer to the 
+%%%{\it combination} of the rules of inference
+%%%used by an arithmetic and its
+%%%the logical axioms ``$~L_d~$'' that 
+%%%render meaning to
+%%%%are an automatic part of
+%%%$~d\,$'s machinery.
+
+
+\begin{exx}
+\label{ex-2.1}
+%\label{ex-basis}
+\rm
+This notation 
+allows one to
+ conveniently separate  the logical axioms
+$~L_d~,~$ associated
+with  $(  \alpha  , d  )~$, from 
+ $\, \alpha \,$'s
+ ``basis axioms''.
+%basis axioms
+It also allows one to  isolate
+and compare
+%  , conveniently, 
+various
+apparatus techniques,
+%technique,
+% employed in the exact formalisms 
+including the
+ $~d_E~$,  
+ $~d_M~$, 
+ $~d_H~$,  
+and  $~d_F~$
+methods
+%that we will now define:
+defined below:
+%% 
+%%  Three
+%% examples of this are illustrated below,
+%% in a context where
+%% are the deductive apparatus machineries defined
+%% in  Enderton's, Mendelson's and Fitting's textbooks
+%% \cite{End,Fi96,Mend}.
+%% 
+\bed
+\item[   i. ]
+The  $~d_E~$ apparatus,
+formalized in 
+\textsection
+ 2.4 of  Enderton's textbook, 
+% will
+uses only  modus ponens
+as a rule of inference.
+The latter will be accompanied
+by 
+a
+4-part 
+system of
+ logical axioms,
+called $~L_{d_E}~$, $\,$ to endow
+ $~d_E~$ 
+with an
+ability to  support
+% apparatus
+% agility so that it  supports
+%can satisfy
+%the analog of 
+G\"{o}del's Completeness Theorem.
+%% '
+%% (similar to  other'
+%% % full-scale '
+%% deductive methodologies).'
+%% 
+%%%% 
+%%%% (Papadimitriyou's
+%%%% % in-depth  exploration
+%%%% textbook \cite{Papa}  about
+%%%% % examination  of 
+%%%% the Logic-Computer interface
+%%%% relies
+%%%% explicitly 
+%%%%  upon
+%%%% % uses  
+%%%% Enderton's  
+%%%% % underlying 
+%%%% apparatus mechanism.)
+%%%% 
+%% %uses
+%% relies upon
+%%  Enderton's 
+%% approach   $d_E$.)  
+
+%% %relies 
+%% does rely
+%% upon 
+%% Enderton's apparatus
+\item[  ii. ]
+The  $~d_M~$ 
+apparatus in
+\textsection 2.3 
+of  Mendelson's textbook
+and  the $d_H$ 
+ apparatus
+in  \textsection 0.10
+of the H\'{a}jek-Pudl\'{a}k's
+ textbook
+employ a more compressed set of logical axioms
+than $\, d_E \,$,
+but
+they use
+two rules of inference
+(formalizing
+separately
+ modus ponens and generalization).
+%% plus a smaller set of logical axioms, which Mendelson
+%% has called A1-A5.
+%% Also, the $d_H$ 
+%%  apparatus 
+%% on pages ???? 
+%% of the 
+%%  H\'{a}jek-Pudl\'{a}k textbook
+%% uses a slightly different variation of a generalization. 
+%% (In the end, 
+In the end, 
+% both
+ $~d_M~$ 
+and  $~d_H~$ 
+prove the same set of theorems
+as   $~d_E~$ with 
+only minor and unimportant changes in
+proof length.
+\item[ iii. ]
+The 
+``semantic tableaux''
+ $~d_F~$ 
+apparatus in
+Fitting's 
+and Smullyan's 
+textbooks 
+\cite{Fit,Smul}
+was
+ the main focus of our 
+investigations in \cite{ww93,ww1,wwlogos,ww5,ww6,ww14}.
+It will be rarely used
+in the current article,
+however.
+Unlike
+ $~d_E~$,  $~d_M~$  and  $~d_H~$, it
+employs no logical axioms.
+It instead
+ uses a more complicated rule of inference.
+This tableaux apparatus 
+% and also Resolution, have  been 
+ has 
+% been found to  have many 
+a wide array of
+applications
+for automated deduction,
+although it is 
+less efficient than
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+in
+% under
+% extremal
+worst-case 
+environments.
+% settings.
+%circumstances.
+\ennd
+\end{exx}
+
+
+\begin{dff}
+\label{def-2.2}
+\rm
+Each of the
+% deductive 
+methods of
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+have the property that if a theorem $\, \Psi \,$
+has a proof
+with  length $~L~$ 
+ from an arbitrary 
+axiom basis $~\alpha~$ 
+under one of these deductive systems,
+then it will have a proof from these other formalisms
+with lengths bounded by Polynomial$(L)$.
+The term 
+{\bf ``Hilbert-style''} deductive method will,
+thus, refer to any  deductive 
+% apparatus will refer to any other
+apparatus $~d~$ that
+employs
+% similarity has its 
+proof lengths
+% being 
+equivalent to within a polynomial magnitude
+to
+%of
+the comparable proof lengths from $d_E$,  $d_M$  and  $d_H$
+and which 
+also
+assures 
+that the proofs of any
+two theorems $~\Phi~$ 
+and $~\Psi~$ 
+(under $d$ from any
+axiom basis $~\alpha~$) 
+will
+always 
+have
+% by more than a constant factor 
+the sum of the lengths of the proofs
+of $~\Phi \rightarrow \Psi ~$ and $~\Phi~$ 
+% under $~d~$ from $\alpha$ always 
+formally
+bound the length of
+$~\Psi\,$'s proof. 
+\end{dff}
+
+
+\begin{exx}
+\label{ex-2.3}
+%\label{ex-basis}
+\rm
+Some added notation is
+ needed to
+explain why
+% help outline
+% an important distinction between 
+a Hilbert style
+deductive apparatus, such as $\,d_E\,$,  $\,d_H\,$
+ or $\,d_M\,$, should be distinguished from
+ $d_F$'s
+``tableaux'' apparatus.
+Let
+% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$
+once again 
+% will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us recall 
+that 
+an
+axiom basis
+ ``$\, \alpha \,$''
+is said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+%\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+
+\noindent
+%Then an 
+In this context, an
+``axiom basis''
+$\alpha$
+will be called 
+{\bf Type-M} if it contains
+\eq{totdefxs}-\eq{totdefxm}
+% \ref{totdefxs}-\ref{totdefxm}
+as theorems,  
+{\bf Type-A} if it contains 
+%only
+\eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+$\alpha$ 
+%will be 
+is
+called 
+{\bf Type-NS} if it  can prove
+none of these theorems.
+%In this context,
+%The
+%Items (a) and (b) illustrate 
+%%%Below are illustrated several
+%%%implications of this notation:
+The implications of this notation
+are formalized by Items a and b:
+
+%% 
+%% 
+%% , below, 
+%% %will 
+%% illustrate how
+%% a
+%% %% 
+%% %%  the
+%% %% prior
+%% %%  literature has
+%% %% 
+%% %% 
+%% ``Hilbert-style''
+%% deductive apparatus, such as $\,d_E\,$
+%%  or $\,d_M\,$, supports very different generalizations
+%% of the Second Incompleteness Theorem
+%% than  $\,d_F\,$'s
+%% ``tableaux-style'' apparatus:  
+%% 
+%%  the prior literature most germane
+%% to our current article is summarized as follows:
+%% 
+%% 
+%% The relationship of these constructs to 
+%% self-justification 
+%% is explained by
+%% items (a) and (b):
+\bed
+\item[   a.  ]
+The
+%%  
+%% above 
+%% evasions of the Second Incompleteness
+%% Theorem are known to be near-maximal in a mathematical sense.
+%% This is because
+%% the
+%% 
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as formalized by statement $\, ++ \,$,
+%has  implied  
+implies
+no
+natural 
+Type-S system can recognize  its
+own  consistency
+under
+any of $\, d_E \,$'s, $\, d_H \,$'s  
+or
+ $\, d_M \,$'s
+  Hilbert-style 
+versions
+of deduction:
+\begin{quote}
+{\bf ++ }
+%\footnotesize
+ \baselineskip = 1.05 \normalbaselineskip 
+{\it 
+(Solovay's  
+modification
+%Generalization 
+\cite{So94}
+%1994 Generalization \cite{So94}
+%of a 1985 theorem 
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+with
+%using 
+%some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let $ \, \alpha \, $ denote 
+%%! any consistent
+% a logic
+an axiom  
+basis  
+% axiom system 
+%basis  system 
+supporting
+% which contains 
+\eq{totdefxs}'s
+Type-S statement
+and 
+assuring
+%which assures 
+%that 
+the successor operation
+%always 
+does satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+cannot verify is
+%will be unable to recognize its
+%own  
+consistency
+under any Hilbert-style 
+%deductive 
+apparatus $d$,
+%whenever
+if  it
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+the usual % identity,
+associative, commutative 
+ distributive 
+and identity axioms.
+%   -axiom
+% properties.
+\end{quote}
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of $++$'s result. 
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+ by  Buss-Ignjatovic,
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k initial 1985 article  \cite{Pu85} 
+% implicitly
+captured 
+the majority 
+%most
+%%%  much 
+of $++$'s 
+main
+% underlying
+formalism,
+and
+Friedman did some
+related work
+% in
+\cite{Fr79a}.
+
+\item[   b.  ]
+Part of what makes
+the Pudl\'{a}k-Solovay discovery in
+ $++$ interesting is that 
+\cite{ww1,ww5,wwapal}
+%Willard
+developed two 
+% separate
+methods for 
+basis systems
+%%% $\alpha$ 
+to confirm their own consistency, whose
+natural hybridizations is precluded by $++$. These results involve
+either a Type-NS
+% basis 
+system 
+\cite{ww1,wwapal}
+ verifying its own consistency
+under 
+any  of the 
+ $d_E$ or  $d_H$
+ or $d_M$'s
+Hilbert-style methods,
+or   a Type-A 
+%basis 
+system \cite{ww93,ww1,ww5,ww6,ww14} 
+verifying
+ its
+% own 
+self-consistency
+under $d_F$'s tableaux 
+%deductive 
+apparatus.
+Also, Willard \cite{ww2,ww7} observed how one could
+refine $++$ with Adamowicz-Zbierski's
+methodology \cite{AZ1} to show 
+ Type-M  systems
+cannot recognize their semantic tableaux consistency.
+\ennd
+\end{exx}
+
+The roles of
+% Observations
+Items (a) and (b)
+in our research
+%from the current example, 
+will become more evident
+as this article progresses. 
+Essentially, our
+prior research
+% ,
+% best summarized in \cite{ww14},
+%  has 
+had focused mostly on Type-A arithmetics
+that could verify their consistency under either semantic
+tableaux deduction or some near-cousin of this concept
+(as was explained in \cite{ww14}'s short 16-page summary
+of \cite{ww93}-\cite{ww9}'s results).
+The 
+new
+$~\theta~$ operator, defined in the next section, will
+raise the question about whether a
+surprisingly powerful class of new Type-NS systems may 
+satisfy an analogous
+property in the context of
+Definition \ref{def-2.2}'s more pristine
+Hilbert-style methodology for deduction. 
+
+
+% have
+% a similar property.(The article \cite{ww14} offers a nice 16-page summary
+% of our prior results 
+% \cite{ww93}-\cite{ww9}
+% about Type-A arithmetics, but 
+% none of these results will
+% be
+%  needed
+% %  to be examined 
+% during our current article's exploration of 
+% the properties of the new % 
+% $~\theta~$ operator.)
+
+% It will be unnecessary for a reader to examine any of our
+
+ 
+%% % year-2014 
+%% Wollic-2014 paper \cite{ww14}
+%% summarized and extended our
+%% results about
+%% semantic tableaux consistency.
+%% % and this 
+%% The current 
+%% new 
+%% year-2015
+%% paper 
+%% will, now,
+%% explore whether
+%%  systems  can
+%% also corroborate
+%% their Hilbert-styled consistency
+%% under certain well-defined circumstances.
+
+%% (and seek to explore the restrictions $++$ imposes upon
+%% Hilbert-styled deduction).
+
+% 
+% (The latter topic
+% % is 
+% %very 
+% %entirely
+% %different 
+% differs
+% from the former
+% because
+% constraint  $++$ 
+% applies only 
+% to its
+% particular
+%  domain.)
+
+%in the second context.)
+
+
+%% The constraints imposed by $++$ 
+%% are challenging 
+%% because Type-NS arithmetics 
+
+This
+topic
+% subject 
+is challenging because 
+essentially all
+Type-S arithmetics
+are forbidden by $++$ from
+verifying the consistency
+of their own Hilbert-styled deductions
+(and 
+conventional forms of
+Type-NS formalisms are
+typically 
+% usually 
+quite weak).
+%% 
+%% (Thus, our efforts
+%% to design
+%% self-verifying systems
+%%  must focus on
+%% Type-NS arithmetics).
+%% 
+Our new Propositions \ref{th-3.3} and \ref{th-6.1}
+will suggest a 
+plausible partial
+solution to this 
+problem by
+% daunting challenge by
+illustrating how
+an {\it unusual class} of 
+Type-NS arithmetics will be able to construct the
+full set of integers $~0,1,2,3,...~$ by finite means 
+{\it without using}
+any of the successor, addition or multiplication
+function symbols. 
+As a result, we will suggest 
+a
+%  a {\it part-way} 
+% that an interesting
+non-trivial 
+(although diluted)
+fragment
+of what Hilbert
+and G\"{o}del
+% sought 
+% referred to
+sought
+in 
+statements 
+$*$ and $**$ 
+% will
+%  be formally  achieved 
+% become tempting 
+is likely
+% be 
+viable
+under Definition \ref{def-2.4}'s formalism.
+
+
+
+% 
+% This
+% topic
+% % subject 
+% is challenging because 
+% $++$'s 
+% Type-NS arithmetics 
+% %  obviously 
+% have sharply circumscribed powers
+% (demonstrating the 
+% broad reach
+% % ubiquitous  nature 
+% of
+% %the Second Incompleteness Theorem's reach).
+% G\"{o}del's 
+% second theorem).
+% %%  
+% %% The current article will
+% %% show, however,  that 
+% %% some Type-NS arithmetics are 
+% %% substantially
+% %%  stronger than previously 
+% %% anticipated
+% %% (and they will have useful applications 
+% %% in
+% %% computer science settings).
+% %% 
+% %% Thus in a context where 
+% %% the power of both G\"{o}del's initial
+% %% Second Incompleteness Theorem and $++$'s strengthening of it
+% %% are stunning
+% %% and
+% %% have pervasive implications,
+% %% we will show that a 
+% %% {\it partial-and-much-less-than-full}
+% %% fragment 
+% %% of what Hilbert
+% %% desired
+% %%  in statements $*$ and $**$ an be
+% %% positively  achieved.
+% %% 
+% The current article will
+% %show, however, 
+% suggest,
+% however, 
+% % that 
+% some Type-NS arithmetics are
+% % , however,  
+% % significantly 
+% %% substantially
+% more far-reaching
+% than 
+% previously 
+% anticipated. Thus,  a 
+% % well-defined 
+%   {\it
+% partial but non-trivial} fragment of what Hilbert
+% and G\"{o}del
+% % sought 
+% % referred to
+%  anticipated
+% in 
+% statements 
+% $*$ and $**$ will
+% %  be formally  achieved 
+% % become tempting 
+% look
+% % be 
+% viable
+% under Definition \ref{def-2.4}'s formalism.
+
+
+\begin{deff}
+\label{def-2.4}
+\rm
+Let
+$~\alpha~$ again 
+denote an axiom basis 
+and $~d~$ 
+designate
+ a
+deduction apparatus.
+Then the  ordered pair
+ $~(  \alpha  , d  )$
+will
+be called  {\bf Self Justifying} when:
+\begin{description}
+% \xxitch
+% \small
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+(or at least one of its axioms)
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+\end{deff}
+
+
+\begin{exx}
+\label{ex-2.5}
+\rm
+Using 
+Definition \ref{def-2.4}'s
+ notation, our research  
+in
+\cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+developed
+%\cite{ww93}-\cite{ww14}
+%has  consisted of
+% developing
+ordered pairs  $~(  \alpha  , d  )$
+that 
+were
+%are
+``Self Justifying''.
+It 
+%  has 
+also explored
+how the Second Incompleteness Theorem formalizes
+limits beyond which such formalisms cannot transgress.
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+% second
+%%% axiom 
+system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition.
+%of 
+% this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+%%%
+%%% the following 
+%%% %%% added
+%%% further
+%%% sentence,  
+%%% called
+%%% %%% that we call
+%%% {\bf SelfRef$(\alpha,d)~$}:
+\begin{quote} 
+%\xxitch
+$\oplus~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of this
+{\bf $\,$``SelfRef$(\alpha,d)$''$\,$}
+statement.
+%%% SelfRef$(\alpha,d)$'s
+%%% self-referential statement.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition.
+This is because if the 
+%  ordered
+ pair $(\alpha,d)$ is too strong
+then a classic G\"{o}del-style diagonalization argument can
+be applied to the axiom system 
+$\alpha^d~~=~~ \alpha \, + \, $ SelfRef$(\alpha,d)$,
+where the added presence of the statement 
+SelfRef$(\alpha,d)$ 
+will cause this extended version of 
+$\, \alpha\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, the machinery of the sentence
+``SelfRef$(\alpha,d)$'' is relatively easy to 
+encode,
+%make well-defined 
+via an application of the Fixed Point Theorem,
+but it
+is
+ironically 
+%%%%%{\it most often  
+{\it 
+typically
+%usually
+useless! }
+\end{exx}
+
+%\newpage
+
+
+Unlike our earlier work, which focused 
+ mostly around a 
+semantic
+tableaux apparatus for deduction,
+the current paper
+will
+explore
+%paper will explore
+\dfx{def-2.2}'s
+more pristine Hilbert-style methodologies.
+%% 
+%% analogous to 
+%% Example
+%% \ref{ex-2.1}'s 
+%% textbook
+%% methods.
+%% 
+% of 
+% $d_E$,  $d_M$
+% and  $d_H$.
+%%! 
+%%! in
+%%! the textbooks by 
+%%!  Enderton,  H\'{a}jek-Pudl\'{a}k,
+%%! Mendelson and Papadamiriyou \cite{End,HP91,Mend,Papa}.
+There are, of course, many types of generalizations
+of the Second Incompleteness Theorem known to
+arise in  Hilbert-like  settings
+\cite{BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Lo55,Kr87,Kr95,Pa71,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,wwapal}.
+Each such
+generalization
+formalizes
+a paradigm where 
+self-justification is infeasible
+under a Hilbert-style apparatus.
+
+\smallskip
+
+Our 
+main
+prior research about
+%%! 
+%%! main work about arithmetics displaying knowledge
+%%! about their 
+%%! 
+Hilbert consistency appeared in \cite{wwapal}.
+Its ISCE$(\beta)$ 
+formalism could recognize its own 
+Hilbert consistency and 
+%%could 
+prove analogs of any 
+r.e. 
+extension $~\beta$ of
+Peano Arithmetic's $\Pi_1$ theorems.
+It unfortunately required
+the use of an infinite number of constant symbols, with
+ ISCE$(\beta)$ 
+using one built-in constant
+symbol 
+$~C_i~$,
+for each power of 2. An alternative in  \cite{wwapal},
+called ISINF$(\beta)$, required use of only three constant symbols,
+but its proof lengths were impractically long. 
+
+
+%%! required
+%%! % in excess of
+%%! an impractical
+%%!  $O(N)$ length proof to construct an integer $N$.
+
+\smallskip
+
+Prior to \cite{wwapal}'s publication 
+Pavel Pudl\'{a}k
+\cite{Pupriv}
+examined this article and asked 
+us
+the crucial question about 
+whether
+we could improve 
+upon ISINF's properties
+ by using Ajtai's observations
+\cite{Aj94}
+ about
+the Pigeon Hole principle.
+%%  
+%% Sam Buss \cite{Bupriv} also asked us
+%% a 
+%% %similar 
+%% related
+%% question
+%% (during a more 
+%% informal 
+%% %abbreviated
+%% conversation).
+%% %(in a more informal manner).
+%% 
+Our prior
+partial
+answer to  Pudl\'{a}k's
+question
+was offered  
+%issue
+%appeared
+%in Sections 6 and 7 of \cite{wwapal}.
+in Sections 6 of \cite{wwapal}.
+A very different type of reply will be offered
+% 
+% We will offer 
+% % an alternate much  
+% a 
+% % much
+% more sophisticated
+% and different 
+% type of reply
+% % analysis
+% %  formalism 
+% 
+in
+the current 
+paper.
+%article.
+
+
+%%! an abbreviated version of a similar
+%%! question after we verbally summarized to him \cite{wwapal}'s
+%%! planned results.
+
+% in 1997.
+
+
+\begin{deff}
+\label{def-2.6}
+\rm
+Throughout our discussion, a
+% A
+primitive $~F~$ will be called a 
+{\bf Q-Function} 
+iff is is 
+sufficiently ambiguous
+for there to exist an UNCOUNTABLY infinite number of
+different 
+{\it 
+plausible  sequences} of
+ordered pairs in expression \eq{wow} where
+$~F(i)=a_i~$  is allowed as a
+% logically 
+permissible
+%plausible 
+%formalization 
+representation of $F$
+under some fixed axiom system $~\gamma~$. 
+\begin{equation}
+\label{wow}
+  (0,a_0) 
+ ~,~  (1,a_1)   ~,~  (2,a_2)   ~,~  (3,a_3)   ~,~  (4,a_4)~ ...
+\end{equation} 
+\end{deff}
+
+\gvs
+
+%It turns out most
+ 
+Most
+Q-Function symbols are
+unsuitable for 
+analyzing
+%producing a positive resolution to
+Hilbert's  Second Open Question or most 
+issues in
+% other prominent
+% % mathematical 
+% questions within 
+mathematics.
+ This is because the
+presence of an 
+ uncountably
+ infinite
+  number of
+different 
+plausible  sequences,
+formalized by Line
+\eq{wow} for solving
+$~F(i)=a_i~,~$  is 
+typically more of a burden than a benefit.
+An exception to this general rule
+of thumb
+ will be
+provided by
+ the next
+section's $~\theta~$ operator.
+It will be germane to 
+ $\, ++ \,$'s generalization of the Second Incompleteness
+and suggest a mechanism whereby an efficient form of
+``Type-NS''self-justifying 
+arithmetic
+can recognize its own Hilbert consistency,
+without
+viewing
+% recognizing 
+%%%%%% any of  addition, multiplication and
+even
+successor  as
+a total function.
+
+
+%% 
+%% It will enable us to develop ground terms for formulating
+%% any integer $~N~$ using
+%%  $O\{~$Log$(N)~\}~$ 
+%% logical symbols,
+%% in a context where
+%% {\it  none of the} addition, multiplication or
+%% successor  function symbols are employed
+%% by $~\theta \,$'s analog of an 
+%%  $O\{~$Log$(N)~\}~$ 
+%% lengthed 
+%% binary-like 
+%% encoding 
+%% for integers.
+%% % of an integer as a binary number. 
+%% This  alternate
+%%  $O\{~$Log$(N)~\}~$ 
+%% format
+%% for encoding an integer $~N~$ is 
+%% potneitlally useful
+%% %fascinating
+%% because  
+%% Item $\, ++ \,$'s generalization of the Second Incompleteness
+%% Theorem, due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% does not preclude evasions of its invariant when a
+%% ``Type-NS''
+%% axiom system ceases to recognize 
+%%  addition, multiplication and
+%% successor  as total functions.
+
+
+ 
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question.
+% A special class of Q-Functions
+% %  , however,
+% will
+% walk through
+%   Cantor's 
+% % the
+% world of the Uncountably Infinite
+% %% in a more 
+% in an
+% enticing manner,
+% however.
+% %% however.
+% %It 
+% They
+% will suggest a
+% %%% much
+%  {\it  diluted but non-trivial}
+% variant
+% of the aspirations
+% which 
+% %that 
+% Hilbert and G\"{o}del 
+% expressed
+%  in
+% $*$ and $**$ 
+% are 
+% %applicable to
+% feasible under
+% % plausible in the context of 
+% Hilbert Deduction.
+% (This
+% % Q-function a
+% analysis will be 
+% %%%%%%%%%%%%%% quite
+% % entirely 
+% different
+% from 
+% \cite{ww14}'s 
+% examination of
+% %formalisms
+% %the formalisms appearing in our Wollic-2014 paper
+% %%% because 
+% %it will replace 
+% semantic tableaux deduction
+% because it will
+% apply  
+% uniquely 
+% to 
+% Definition \ref{def-2.2}'s 
+% ``Hilbert-styled'' deduction methods.)
+
+ 
+%%% is replaced by the more efficient
+%%% %with the more pristine 
+%%% Hilbert-style deductive methodology.)
+
+%can be achieved.
+
+
+%%! This will suggest
+%%! a {\it limited}  
+%%! and very-much {\it down-sized} version of the formalism that
+%%!  Hilbert
+%%! advocated
+%%! is
+%%! likely
+%%! %probably
+%%! feasible
+%%! and 
+%%! germane to 
+%%! the
+%%! future
+%%! %  computational 
+%%! needs of automated
+%%! theorem provers.
+%%! Our 
+%%! exploration 
+%%!  will also provide a
+%%! % quite
+%%!  new interpretation of the
+%%! meaning of the statements $*$, $**$ and
+%%!  $***$.
+
+% by Hilbert and G\"{o}del.
+
+\vspace*{- 0.6 em} 
+
+% \section{Revisiting a World which Hilbert called
+% {\it ``Cantor's Paradise''}}
+
+\section{Main Formalism}
+\label{ss4}
+\label{seee3}
+
+%333333333333333333333333333
+\vspace*{- 0.4 em} 
+
+% OLD Title was {\it Notation and Basic Concepts}
+
+Throughout this 
+paper,
+%article,
+a function 
+ $\, H \, $ 
+will be     called
+ {\bf Non-Growth} 
+iff  
+$ H(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$.  Six  examples of  
+ non-growth functions are:
+\bee
+\parskip 0pt
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lfloor ~x^{1/y}~ \rfloor$ when $~y\geq 1~$
+%% 
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% 
+(and zero otherwise),
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+and
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\ene
+%% 
+%% 
+%% \bee
+%% \baselineskip = 0.8 \normalbaselineskip 
+%% 
+%% \item
+%% {\it Integer Subtraction} 
+%% where ``$~x-y~$'' is defined to equal zero 
+%% in {\it the special case} where
+%%  $~x \leq y,$
+%% \item
+%% {\it Integer 
+%% Division}
+%% where ``$~x \div y~$'' equals
+%% $~x~$ when $~y=0,~$ and
+%% it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+%% \item
+%% $Root(x,y)~$ which equals $~ \ulcorner ~x^{1/y}~ \urcorner$ when $~y\geq 1,~$
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% \item
+%% $Maximum(x,y),~~$
+%% \item
+%% $ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+%% and  zero otherwise. 
+%% \item
+%% $Count(x,j)~~=~~$the number of ``1'' bits
+%% among $~x$'s rightmost $~j~$ bits.
+%% \end{enumerate}
+%% 
+These operations were called
+either the 
+{\bf ``Grounding''} 
+or {\bf ``Ground-Level''} 
+ functions
+in our articles \cite{ww1,wwlogos,ww5,ww14}.
+We will
+use the latter nomenclature in the current article
+ because the notion of a  ``Ground-Level''
+function should not be confused with the very different notion
+of a ``Grounded Term'' employed by Definition
+\ref{def-3.4}. 
+
+%  TWO DEFS or ONE ???????? 
+
+
+
+Our
+starting  language $L^G$  shall also contain
+the
+two atomic 
+symbols
+%% relations 
+of ``$~=~$'' and ``$~\leq~$'' and three
+built in constants symbols, $~C_0~$, $~C_1~$ and $~C_2~$, 
+for representing
+the values of 0, 1 and 2. 
+Within this context, expressions 
+\eq{newadd} and \eq{newmult} formalize how addition and multiplication
+can be encoded as two 3-way predicates,
+%%  in  $L^G$,
+ denoted as
+Add$(x,y,z)$ and Mult$(x,y,z)$.
+
+\vspace*{- 0.6 em} 
+\beq 
+\label{newadd}
+z ~ -~x~~=~~ y~~~~ \wedge ~~~~ z~\geq~x
+\end{equation}
+
+\vspace*{- 0.6 em} 
+\begin{equation}
+\label{newmult}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+Also, 
+our constant symbols,
+ $~C_1~$ and $~C_2~,~$ for representing
+the quantities 1 and 2, 
+ allow us to
+formalize the following further
+%useful 
+%%%%%%%% function 
+operations:
+\bed
+\small
+\topsep -4pt
+\itemsep -1pt
+\it
+\item[   $~$a.  ] 
+Pred$(x)~~=~~x-1$
+(in a context where 
+%Item 1's
+%the prior paragraph's
+our prior
+definition for Subtraction implies
+Pred(0)$=0$.$~)$   
+\item[   $~$b.  ] 
+Half$(x)~~=~~x \div 2$
+(in a context 
+where 
+``$ ~x \div 2~$'' equals technically 
+$~\lfloor \, \frac{x}{2} \, \rfloor~$ 
+under 
+%our
+the 
+%preceding 
+prior
+% 
+paragraph's 
+notation).
+% of Item 2's definition for Division).  
+\item[   $~$c.  ] 
+Pred$^n(x)$ defined to be $~n~$ iterations of the
+Predecessor operation
+\item[   $~$d.  ] 
+Half$^{ \,n}(x)$ defined to be $~n~$ iterations of the
+halving operation.
+%%
+%%\item[   $~$e.  ] 
+%%Bit$(x,j)~=~$ $Count(x,j)~-~Count(x,j-1)~$
+%%(e.g. the value of $~x \,$'s rightmost
+%% $j-$th bit)
+%%\item[   $~$f.  ] 
+%%Min$(x,y)~~=~~x~-~(y-x)~~$
+%%(e.g. a quantity that represents the minimum of $x$ and $y$
+%%because our definition of ``integer subtraction'' 
+%%implies  $~y-x \,= \, 0 ~$ whenever $\, x \geq y$.$~~)$  
+%%
+\ennd
+
+
+
+Let us say
+that
+ a function symbol $H(x_1,x_2...x_j)$
+is
+ {\bf Growth Permitting}
+iff for each integer $~k \geq 2~$ there exists a 
+``growth-tuple''
+$(a_1,a_2...a_j)$ 
+satisfying
+  $~H(a_1,a_2...a_j) \, >\, k~$
+and 
+also
+% simultaneously 
+having
+each $ \, a_i \leq k$ \, 
+It will  be necessary 
+%% for  us 
+to employ
+ either an infinite number
+of constant symbols or some Growth-Permitting function 
+so that an extension of the 
+language $L^G$ can construct the 
+full
+%infinite
+collection of
+ integers of $~3,4,5,6~....~$.
+
+
+
+
+%% One  awkward aspect of this notation is that 
+%% it provides
+%%  no guarantee
+%% that    integers larger than 2 will exist         without the 
+%% presence of some 
+%% further
+%% methodology for producing larger integers.
+
+
+% 
+\smallskip
+
+
+One method for resolving this problem was presented in \cite{wwapal}.
+% 
+% It employed  an infinite number of further constant symbols. The
+% latter's
+%  ISCE$(\beta)$ 
+%  system was
+% %   shown to be 
+% compatible with                     self-justification,
+% but such an infinite number of constant symbols clearly trespassed on
+% Hilbert's goal of using a 
+% %strictly 
+% finite-sized formalism.
+% 
+Its  ISCE$(\beta)$ axiom basis
+ employed  an infinite number of further constant symbols. It
+was
+compatible with                     self-justification,
+but deviated from 
+%{\it very sharply from} 
+Hilbert's
+intended
+ goals 
+because it employed
+% by employing 
+%an  
+a {\it highly awkward}
+infinite number of 
+distinct
+ constant symbols.
+
+\smallskip
+
+The 
+% self-justifying
+``ISINF'' formalism 
+% in
+of 
+\cite{wwapal} 
+offered an alternate method for resolving this difficulty.
+%% in the context of a self-justifying logic. 
+It
+% required the use of
+used
+ {\it only
+three} constant symbols.  It   could prove analogs of all
+of Peano Arithmetic's
+$\Pi_1$ theorems, but almost all
+of
+ its proofs
+unfortunately
+ had lengths longer 
+than the number of atoms in the universe.
+Most other approaches, for resolving this dilemma, 
+% are 
+were
+also problematic
+because the
+invariant
+ $~++~$,
+which Example \ref{ex-2.3}
+attributed to the joint work of 
+  Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris,
+showed that essentially every Type-S arithmetic is unable to
+recognize its
+own consistency under a Hilbert-style deductive apparatus.
+
+\smallskip
+
+
+The challenge posed by $++$ 
+is, thus, certainly formidable.
+Our goal in 
+%the current 
+this
+article
+will be to suggest 
+how
+%that 
+a Q-function primitive $F(x)$,
+that has
+% an extraordinarily 
+a deliberately
+ambiguous
+ function definition,
+can help overcome the constraints that $++$ imposes. 
+Such an
+% ultra-ambiguous defined 
+unusual
+primitive $~F~$ will have
+an  uncountable number
+of vectors, analogous to Line \eq{wow}, that are permitted 
+solutions to
+$F$'s
+definition.
+Our basic goal
+% in this article 
+will be to outline 
+how this unusual concept is
+likely  germane to the
+self-justifying
+%% 
+%% why it is
+%% likely such Q-functions will enable one to build
+%% surprisingly efficient self-justifying logics that
+%% partially (but not fully) achieve the 
+%% %{\it  diluted portion} of the
+%% 
+aspirations  
+that
+Hilbert and G\"{o}del
+expressed in 
+%statements 
+$*$ and
+$**~$.
+
+\smallskip
+
+%\normalsize \baselineskip = 0.98 \normalbaselineskip
+%\gvs
+
+
+%% \el{wow}'s 
+%% dizzying
+%% $\aleph_1$ distinct solutions.
+%% We will traverse in the opposite direction in this article
+%% because 
+%% Definition \ref{def-2.6}
+%% %\eq{wow} 
+%% formalizes 
+%% % fascinating 
+%% %  a possible
+%% an 
+%% avenue
+%% available
+%% for evading $++$'s generalization of the 
+%% %Second 
+%% Incompleteness
+%% Theorem.
+
+% {\it in at least a partial sense.}
+
+
+%%  
+%% %They 
+%% This level of multiplicity will be
+%% %These will turn out to be 
+%% useful in the present setting
+%% because
+%% %% 
+%% %% Our hypothesis is that such
+%% %% solutions, while awkward and  disadvantageous
+%% %% in many 
+%% %% evident
+%% %% %%%%%obvious 
+%% %% respects,
+%% %% should not be discarded.
+%% %% This is 
+%% %%  because 
+%% %% 
+%% it
+%% can
+%% formalize a type of 
+%% allowed
+%% growth-permitting function,
+%% that is
+%% not prohibited by $++$'s generalization of the Second Incompleteness
+%% Theorem. 
+%% 
+%% 
+
+%\smallskip
+
+\begin{deff}
+\label{def-3.1}
+\rm
+Let us say 
+% an integer-valued
+a function symbol
+$F(x_1,x_2,...x_j)$ is {\bf `` 1-Definitive ''} iff it has only one
+solution under its definition by an axiom system $\gamma$. 
+Let us call $~F~$  {\bf ``Indeterminate''} otherwise.
+%(The remainder of this article 
+(Mathematicians
+obviously typically avoid using 
+function definitions
+%%%%%%%%%, $~F~$ 
+with 
+%that have 
+%%%%  having
+even two solutions, not to
+speak of
+what will be
+\el{wow}'s 
+dizzying
+quantity
+of possibly
+%potential
+%possible
+$\aleph_1$ distinct 
+% number of 
+solutions,
+considered in the current article.
+Our 
+main
+conjecture will be that this unconventional
+approach is 
+germane to the challenge posed
+by  $++$'s 
+broad-scale
+generalization of the 
+%Second 
+Incompleteness
+Theorem.
+This is because our conjecture will be that
+our proposed new $~\theta~$ 
+primitive
+ will represent
+an efficient Indeterminate 
+function
+that eschews  $++$'s prohibitions.) 
+\end{deff}
+
+% helpful when addressing 
+% the challenge 
+% We will traverse in 
+% an unconventional
+% %  the opposite 
+% direction in this article
+% because
+% our conjecture will be that
+% Indeterminate functions 
+% with $\aleph_1$ different 
+% allowed solutions for
+% \el{wow} 
+% formalize
+% %%  
+% %% Definition \ref{def-3.1}
+% %% % {def-2.6}
+% %% %\eq{wow} 
+% %% will formalize 
+% %% a 
+% %% possible
+% %% %feasible
+% %% %plausible
+% %% 
+% an
+% avenue
+% % available
+% for evading $++$'s 
+% broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem.)
+% 
+% (Our conjecture in this article will be that
+% %%
+% %%The next two sections
+% %%will explore
+% %%how 
+% %%
+% \el{wow}'s indeterminate ``Q-Function'' symbol $F$
+% %%%can clarify
+% is germane to
+% the
+% aspirations 
+% that 
+% Hilbert and G\"{o}del
+% % expressed
+% stated
+%  in 
+% % statements 
+% $*$ and
+% $** \, $, $\,$when
+% one employs an 
+% operator
+% $ \, F \, $ 
+% that
+% owns
+% % has
+% $\aleph_1$ distinct 
+% available
+% solutions). 
+% \end{deff}
+
+
+%%are viable,
+%%especially in an automated theorem proving setting,
+%%when Ambiguous function operatives are judiciously
+%%employed.
+%%
+
+%%!  
+%%! can
+%%! %%and germane about automated
+%%! %%deduction, will 
+%%! % theorem proving,
+%%! %computational deduction,
+%%! be satisfied by a self-justifying logic that employs
+%%! one 
+%%! %single $\aleph_1$ 
+%%!   Growth-Permitting function $F(x)$,
+%%! that is
+%%! % inherently 
+%%! ``ambiguous'',
+%%!  {\it accompanied 
+%%! by} a finite number of non-growth primitives
+%%! % {\it  non-growth}
+%%!  $G_1,~G_1,~... G_k$
+%%! that are ``unambiguous''.
+%%! 
+%%! It will also be explained how such results should have useful 
+%%! applications for automated theorem proving, even when they
+%%! employ 
+%%! only
+%%! diluted forms of  self-justifying logics.
+
+%% 
+%% \vspace*{- 1.0 em}
+%% 
+%% \subsection{Main Notation Conventions}
+%% % about Cantor's Paradise}
+%% % \large
+%% % \baselineskip = 1.8 \normalbaselineskip 
+%% 
+%% %\vspace*{- 0.7 em}
+
+{\bf More Notation:}
+$~$Let us say
+an axiom system $~\alpha~$ 
+has 
+{\bf Infinite Far Reach} iff
+it relies upon
+{\it only a  finite number} of
+distinct constant symbols
+(and/or axiom sentences) to 
+% but still can 
+prove 
+for each $n$
+the
+\el{farreach}'s invariant.
+
+%for each particular integer $n$.
+
+\vspace*{- 0.8 em} 
+
+\beq
+%% \small
+\label{farreach}
+\exists ~~x~~~ \mbox{Pred}^n(x)~\geq ~1
+\enq
+The ISINF axiomatic       framework 
+from
+   \cite{wwapal} 
+was
+    a self-justifying
+system with Infinite Far Reach.
+%% 
+%% The opening paragraph of
+%% \cite{wwapal}'s Section 6
+%% %%% quite 
+%% %was frank 
+%% warned the reader
+%% about ISINF's limitations. 
+%% These arose because
+%% 
+%% 
+%% It
+%% used the word ``unnatural'' to describe 
+%% the ISINF system.
+%% Such caution
+%% % deliberately
+%% % self-deprecating term 
+%% was appropriate because
+%% 
+Unfortunately, this result was mostly useless because
+nearly all
+   theorem-proofs
+%of trivial theorems0000000 
+from ISINF 
+were
+longer than the number of atoms in the universe.
+
+The reason
+\cite{wwapal} defined ISINF,  
+%ISINF was worthy of mention,
+despite 
+such
+%%% these
+% plainly 
+%%%  obvious 
+limitations,
+was 
+because
+ISINF 
+demonstrated some self-justifying logics,
+knowledgeable about their own Hilbert consistency,
+were
+% technically 
+able to
+prove all of Peano Arithmetic's $\Pi_1$ theorems
+together with the
+existence of
+the infinite set of integers $ \, 1,2,3,... \,  \, $.
+This result 
+% is interesting because it casts 
+did cast
+% casts 
+a
+new 
+perspective
+%light 
+on  $\,++\,$'s  
+invariant
+% $++$  (appearing on \pag2)
+by showing how some 
+Type-NS
+forms of self-justifying arithmetics
+escape $++$'s almost-ubiquitous 
+ reach
+by managing to possess infinite far reach
+ without taking
+% {\it without recognizing} 
+Successor as a total function.
+
+
+%%  of the current article.
+%% The latter result indicated that Type-S arithmetics, recognizing merely
+%% Successor as a total function, are unable to confirm their own
+%% Hilbert consistency. 
+%% Yet,
+%% %%  despite this fact, 
+%% ISINF was able to produce an 
+%% {\it  eye-squinting} caveat because it 
+%% supported the above ``Infinite Far Reach'' property
+%% without 
+%% needing
+%% %being able to prove
+%%  Line
+%% \eq{totdefxs}'s declaration that successor is a total function.
+
+%\smallskip
+
+We sent an advanced copy of \cite{wwapal}
+to
+Pudl\'{a}k.
+He
+appreciated the nature of the challenge we faced,
+concerning the delicate nature of self-justifying 
+arithmetics that are
+able to prove
+% satisfy 
+\eq{farreach}'s invariant
+{\it for each fixed $~n~$} while 
+being prohibited 
+by $++$
+from
+recognizing  successor as a total
+function.
+% (due to $++$'s restrictions).
+Pudl\'{a}k  
+%private 
+%His
+emailed communications
+\cite{Pupriv}
+suggested 
+that we look at 
+Ajtai's
+work 
+\cite{Aj94}
+about a
+%the 
+Pigeon-Hole function
+ $~ \glamb(x)~$ defined by the identities
+\eq{zm1} and \eq{zm2}.
+
+\newpage
+\vspace*{- 1.2 em} 
+\beq
+%% \small
+\label{zm1}
+\forall ~~x~~~~~ \glamb(x)~ \neq ~ 0
+\enq
+
+\vspace*{- 1.2 em} 
+
+\beq
+\label{zm2}
+%% \small
+\forall ~~x~~~ \forall ~~y~~~~ x ~ \neq~ y ~~ \Rightarrow ~~
+\glamb(x)~ \neq ~\glamb(y)
+\enq
+The relevance of 
+$~\glamb~$ 
+% Pigeon-Hole functions
+can be
+best 
+%readily 
+appreciated
+% if 
+when
+%we let
+$~\glamb^n(x)~$ 
+ denotes
+% the 
+a
+term
+ $~\glamb(~\glamb(~ ... \glamb(x)))~$
+consisting of $~n~$ iterations of the $~\glamb~$ operator.
+Then 
+% the
+\el{DUMB1}'s
+composite
+term $~S_n~$
+% , defined below, shall
+will
+% then 
+satisfy
+Pred$^n(~S_n~)~\geq ~1.~$ 
+%% 
+%% An axiom system, employing the primitive 
+%% operation
+%%  $~ \glamb~,~$ 
+%% can thus 
+%% can easily
+%% prove
+%% Line \eq{farreach}'s 
+%% assertion.
+%% %claim.
+%% %under almost all conventional logics.
+%% 
+%% 
+\beq
+\label{DUMB1}
+S_n~~~=~~~\mbox{Max}[~\glamb(0)~,~\glamb^2(0)~,~\glamb^3(0)~,~...~~\glamb^n(0)~]
+\enq
+Pudl\'{a}k
+observed
+that
+%the
+% Pigeon-Hole function 
+ $~ \glamb(x)~$  
+will
+grow too slowly (in the worst case)
+for
+one to be able to 
+deduce
+successor is a total function
+from its properties  
+% further observed that it is known 
+ \footnote{ \baselineskip = 0.94 \normalbaselineskip
+ The operation $\glamb(x)$ will grow
+at a slower rate than Successor, 
+if it equals $x+1$ for all standard
+numbers $~x~$ and if $\glamb(x)=x-1$ 
+ when $~x~$ is
+a non-standard integer. This seemingly minute detail
+implies  one cannot infer 
+Successor is a total function from
+ $\glamb$'s behavior, since the latter is contradicted by a
+ model  where
+ all  non-standard
+numbers have 
+%their 
+sizes bounded by some fixed  
+% non-standard 
+number B.
+(This 
+subtle detail, 
+raised by
+Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+it 
+%shows that
+raised the question about whether
+ a partial exception to
+Example \ref{ex-2.3}'s
+invariant $++$
+%% on \pag2,  
+might plausibly exist.)  }.
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+His insightful email \cite{Pupriv} asked
+whether 
+the inequality 
+Pred$^n(~S_n~)~\geq ~1~$
+might
+%would,
+thus, 
+% still
+enable a formalism,
+% based around
+utilizing the
+ $\, \glamb \,$ operative, 
+to 
+somehow
+improve upon \cite{wwapal}'s results ?
+
+
+% our
+% formalisms could be 
+% revised
+% %modified 
+% so that 
+% % the  Pigeon-Hole function 
+%  $~ \glamb(x)~$ 
+%  could improve upon \cite{wwapal}'s results.
+
+%%
+%%(possibly using Ajtai's methodologies \cite{Aj-focs}).
+%%Sam Buss raised, interestingly,  a 
+%%partially
+%%similar 
+%%issue during an informal conversation
+%% \cite{Bu-priv} in 1977.
+%%
+%%\smallskip
+%%
+%%These questions
+%%% by
+%%%Pudl\'{a}k and Buss 
+%%were insightful because they isolated 
+%%an
+%%important juncture where $++$'s underlying methodology does not apply.
+%%A partial answer to these questions appeared in 
+%%\cite{wwapal}'s closing section, but a more comprehensive full
+%%answer  has always eluded us.
+
+%This is  because there always seemed to appear
+%one wrinkle of details that precluded a full proof.
+
+
+\smallskip
+
+
+It  was
+initially
+ unclear 
+%%%%% to us
+whether a positive answer to 
+Pudl\'{a}k's
+ probing
+ question would resolve ISINF's main difficulties.
+This is
+because
+% Expression 
+\eq{DUMB1}'s
+term
+$~S_n~$  requires $O(~n^2~)$ logic symbols to encode
+% essentially 
+an integer quantity
+greater than
+ $~n~$
+(since its term
+$~\glamb^j(0)~$ uses $O(j)$ logic symbols).
+%an integer quantity that exceeds the quantity $~n~$ in size.
+Thus once again, the quantity $~2^{100}~,~$ whose binary encoding
+requires 100 bits,  would require in excess of 
+ $~2^{100}~$ bits to encode. 
+Such quantities, exceeding the number of atoms in the universe,
+were troubling because our 
+general
+goal has been to 
+construct self-justifying arithmetics that 
+ possessed, at least,
+some
+partial  facets of
+ pragmatic value.
+
+
+% 
+% find a partial
+% answer to Hilbert's
+% Year-1900 Second
+% Problem  
+%  that would 
+%  possess, at least,
+% some
+% partial  facets of
+%  pragmatic value.
+% 
+
+\bigskip
+\gvs
+
+The remainder of this section will outline how a different type of
+Q-Function operator will 
+be much better than
+ $~ \glamb~$ for meeting our needs.  
+During our discussion,
+Power$(x)$ will denote 
+a primitive specifying
+% that
+ $~x~$ is
+a power of
+$~2~$.  It is 
+%formally 
+encoded
+by 
+\eq{wep2} 
+because
+%under
+our Grounding language
+has
+``Logarithm$(x) \,$''$ ~ = ~ \lfloor \,$Log$_2(x) \, \rfloor \,$.
+\beq
+\vspace*{- 0.6 em}
+\label{wep2} 
+%\small
+x=1 ~~~\vee ~~~ \mbox{Logarithm}(~x~)~\neq~\mbox{Logarithm}(~x-1~)
+\enq
+In this context,
+ $\zzthe(x)$ 
+will denote the analog of
+the $\glamb(x)$ function  
+%% haphazard
+that walks among the powers of 2 in a manner 
+similar to 
+$\glamb(x)$'s
+%  haphazard
+ walk through conventional
+integers. 
+It is 
+formally
+defined by \eq{walk1}-\eq{walk4}.
+% 
+% It will thus satisfy
+% the axiomatic constraints below (which are 
+% $\zzthe(x)$'s analog of the more modest constraints given in
+% % sentences 
+% \eq{zm1} and \eq{zm2}).
+% The most important difference between these two constructs
+% is that axiom \eq{walk1} requires that 
+%  $\zzthe(x)$ maps power of 2 onto powers of 2.
+
+% {\small
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+
+\vspace*{- 1.2 em}
+\noindent
+{\it It needs to be emphasized} that 
+ \eq{walk1} -- \eq{walk4} will be the
+{\it only vehicle} our
+self-justifying axioms 
+%will 
+have available to construct
+integers $\geq \, 3$. $~$These
+axioms will be called
+%They will be henceforth called
+the {\bf $~$Up-Walking$~$} axioms. 
+(The axiom \eq{walk4} 
+is,
+% does not,
+technically, 
+unnecessary
+ to construct any
+integer  $\geq \, 1\, $, but it is helpful because
+it 
+% allows us to 
+formalizes how our methodology will treat  integers
+which are not powers of 2.)
+
+Both
+the 
+Q-functions 
+% the operators 
+$\glamb$ 
+ and $\zzthe$ are 
+challenging
+%daunting
+because there are a  
+ dizzying
+$\aleph_1$ distinct 
+vectors, analogous to 
+ Line \eq{wow}, 
+that are
+%where their definitions permits 
+representations of these functions.
+%% 
+%% Also, we may combine either operation with our 
+%% language $L^G$'s grounding function-primitives to formulate a term
+%% $~T_n~$ that defines any arbitrary integer $~n~$.
+%% 
+We will soon see that
+there is, however, a
+distinction
+%  major difference 
+between these two concepts
+from a computational complexity perspective.
+
+\begin{definition}
+\label{defx-3.2}
+\rm
+Let $~L^Q~$ 
+and $~L^{Q^*}~$ 
+denote the 
+extensions
+of $~L^G\,$'s Grounding language that contain the
+respective
+additional 
+function symbols of  
+  $\zzthe$
+ and
+$\glamb$. Then 
+$~~L^Q~$ shall be called the {\bf  Q-Grounding} language, and 
+ $~~L^{Q^*}~$ 
+will be called the  {\bf  Q* Grounding} language. 
+\end{definition}
+
+\begin{propp}
+\label{th-3.3}
+In contrast to the
+Q* Grounding language 
+that requires $O(~n^2~)$ function symbols
+for defining a term $~T^*_n~$ for representing the integer
+$~n,~$ the    Q-Grounding language
+%% will need no more than 
+needs
+% uses 
+only
+$O(~$Log$^{ \, 3\,} \,n~)$ symbols to 
+encode
+%formalize 
+a term 
+$~T_n~$ representing
+$~n$.
+\end{propp}
+
+\vspace*{- 1.0 em}
+
+\begin{center}
+% \small
+% Our proof of \phx{th-3.3} 
+\phx{th-3.3}'s 
+proof
+will rely upon the following notation  convention:
+\end{center}
+
+\vspace*{- 0.8 em}
+
+\begin{definition}
+\label{def-3.3}
+\rm
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+% Throughout this article,
+Then
+%for any  $~j \, \geq 1~,~$ 
+%the symbol 
+$~E_j~$ 
+will
+% shall
+% will 
+denote
+the quantity produced by
+\eq{ej-def}'s division operation: 
+
+\vspace*{- 0.6 em}
+
+\beq
+\small
+\label{ej-def}
+ \frac{~\mbox{Max}
+~[~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~~] }
+{~~\mbox{Half}^{\,j\,} ~ \{ ~\mbox{Max}~[
+ ~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~]~ 
+ ~ \}~~ } 
+%
+% \mbox{Max}(~\zzthe^j(1),~\zzthe^{j-1}(1),~... ~\zzthe^1(1)~ 
+\enq
+It is each to see
+$  E_j  =  2^j  $ for every
+$j   \geq 1$.
+This is
+ because \el{ej-def}'s
+twice-repeating term
+object
+ of
+$  \mbox{Max}
+~[~\zzthe^j(1),  \zzthe^{j-1}(1),...\zzthe(1) \,]$
+% is at least as large as $\, 2^j\,$.
+is a power of 2 exceeding  $\, 2^j\,$.
+%% 
+%% The definitions of the
+%% % Q-Grounding 
+%% functions of ``Half'', ``Max'' and
+%% ``$~\zzthe~$''   imply 
+%% $~E_j~=~2^j~$ for each
+%% $j \, \geq 1$. 
+%% 
+For the additional case where $~j=0~,~$
+we will 
+% formally
+define  $~E_0~=~1~$ (by 
+using the 
+%%
+%%setting it equal to 
+%%our
+%%%the
+%%
+built-in constant symbol
+of $~C_1~$).
+\end{definition}
+
+%% , which 
+%% is intended to
+%% %formally 
+%% represent the integer of ``1'').
+
+
+
+{\bf Proof of  \phx{th-3.3}:}
+%The justification of   \phx{th-3.3} is an 
+Easy consequence of
+\dfx{def-3.3}'s machinery. Thus if $~n~$ is a power of
+2 of the form $~2^j~$ then
+% the preceding
+% definition's
+expression $~E_j~$ is a term representing $~n \,$'s value
+that employs
+  $O(~$Log$^{ \, 2\,} \,n~)$ 
+logical
+symbols. On the other hand, if  
+ $~n~$ is not a power of
+2 then it can be defined 
+with   $O(~$Log$^{ \, 3\,} \,n~)$ symbols by 
+setting
+$~E_j~$ equal to the least power of 2 greater than $~n~$ and
+subtracting from $~E_j~$ those powers of 2 that are needed to
+produce $\,n\,$'s  value. 
+For example since $76~=~128~-~32~-~16~-~4~,\,$ it can
+be formalized as a term $T_{76}$ defined by
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$.
+
+% $~~~~\Box$
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\begin{definition}
+\label{def-3.4}
+\rm
+A term in mathematical logic
+is defined to be a syntactic object,  built
+out of
+solely
+ symbols for representing 
+functions,
+constants and variables.
+% 
+% The nomenclature in
+% % classical
+% logic has 
+% %formally 
+% defined
+% a {\it ``term''} to be a syntactic object,  built
+% out of symbols for representing 
+% functions,
+% constants and variables.
+% 
+% 
+Such an object is called
+% either 
+a {\bf ``Ground Term''} 
+%% (or for precision a
+%%% {\bf ``Tree-Oriented Ground Term''} )
+when it  is built {\it out of solely}
+function and
+ constant symbols.
+For example in our Q-Grounding language (which 
+uses
+%owns only
+$  C_0  $, $  C_1  $ and $  C_2  $ 
+as
+built-in
+ constants)
+% symbols), 
+the expression
+%of 
+``$\, C_2-  C_1\,$'' 
+is a 
+% such 
+a Ground term.
+Two more complex 
+examples of 
+Ground terms are
+``Max$(   C_2 ,  C_1 - C_0)$'' 
+and ``Max$( ~\zzthe(C_1)~,~C_2 ~)$''.
+Also,
+expression $~E_j~$ 
+in Line \eq{ej-def}
+should be viewed as 
+a Ground term (when one 
+views
+its
+use of the
+symbol
+ ``1'' as an %informal 
+abbreviation
+for the
+constant
+ ``$~C_1~$''). 
+\end{definition}
+
+
+\begin{remm}
+\label{rem-def-3.4}.
+\rm
+Section \textsection \ref{ss6} 
+and its   Proposition \ref{th-6.1} 
+will technically distinguish between two
+kinds of Ground terms, that it calls the
+%
+%{\bf Comment of Definition \ref{def-3.4}'s Notation:} 
+%We will  distinguish  between two
+%kinds of Ground terms in Section \textsection \ref{ss6},
+%called its 
+{\bf ``Tree-Oriented''} and 
+{\bf ``Dag-Oriented''} formats.
+The latter will differ from a more 
+conventional tree structure
+by having a 
+Directed Acyclic Graph structure replace 
+a logic's
+usual 
+ tree format for defining its quantitative values.
+Our discussion in the next two sections will be
+simplified if we use the shorter phrase of
+{\it ``Ground Term''} 
+to refer to what Section \textsection \ref{ss6} will more
+accurately call a 
+{\it ``Tree-Oriented Ground Term''}.
+(It will turn out
+% that our   
+Proposition \ref{th-6.1} 
+will later explain how Dag-Oriented Ground Terms
+differ from
+their
+ tree-oriented 
+counterparts
+% Ground Terms by allowing us
+%to reduce the 
+by reducing the
+$O(~$Log$^{ \, 3\,} \,n~)$ length of a
+tree-oriented term to a more compact
+$O(~$Log$\, \,n~)$ size.)
+ \end{remm}
+
+
+\begin{definition}
+\label{def-3.5}
+\rm
+A ground term $~T~$ will be called an 
+{\bf ``Observable''} 
+object iff there is
+%{\it only one} 
+an
+unique
+interpretation of its 
+quantified value in the
+%meaning  in our
+Q-Grounding language.
+It 
+%will be
+ is
+called an
+{\bf ``Unobservable''} iff it has multiple 
+%plausible 
+such
+interpretations
+due to $\zzthe$'s ``indeterminate'' definition
+(e.g. see Definition \ref{def-3.1}). 
+\end{definition}
+
+%%% (due to the 
+%%% %uncountably 
+%%% ambiguous nature of 
+%%% % our built-in function 
+%%% $~\zzthe~~$).
+%%% \end{definition}
+
+\begin{exx}
+\label{ex-3.6}
+\rm
+The previously mentioned ground term
+Max$( ~\zzthe(C_1)~,~C_2 ~)$ is an ``unobservable''
+because it can assume any of the plausible integer values
+of $~2 \, , \, 4 \, , \,8  \,  , \,16  \, 
+ \, ... ~$.
+On the other hand, 
+%% expression 
+\eq{xoo} 
+is an ``observable''
+that
+ represents
+ the integer value of ``3''.
+(This is because 
+its 
+twice-repeating
+term 
+``$~\mbox{Max}[ ~C_2~,  ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~$'' is bounded
+below by 4, causing the left and right sides of its subtraction
+operation to differ by 
+% an amount of 
+exactly 3.) 
+\beq
+%% \small
+\label{xoo}
+\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~] ~~-~~
+\mbox{Pred}^{\, 3 \,} \{~\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~\} 
+\enq
+Our notation 
+%thus 
+also
+implies that Line \eq{ej-def}'s
+ expression $~E_j~$ 
+is an
+ ``observable''. This implies, in turn, that 
+ \phx{th-3.3}'s term $~T_n~$ is an  ``observable''
+ employing no more than 
+ $O(~$Log$^{ \, 3\,} \,n~)$ 
+logical
+symbols.
+For example since $76~=~128~-~32~-~16~-~4~,\,$ 
+it follows that $~ T_{76}~$
+corresponds to the term
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$,
+$~$where each $~E_j~$ employs only 
+ $O(~$Log$^{ \, 2\,} \,j~)$ symbols.
+%%%%%\end{exx}
+\end{exx}
+
+
+
+Thus, \dfx{def-3.5} and Example \ref{ex-3.6} have illustrated
+%that 
+how
+the realm of ``observable'' objects is a 
+% very 
+broad and accessible world,
+of 
+non-trivial
+%% pragmatic
+ significance.
+It allows every integer $~n~$ to be represented by a
+% reasonably small 
+term $~T_n~$ with
+% an
+a tight
+ $O(~$Log$^{ \, 3\,} \,n~)$ length
+(in a context where Section \textsection \ref{ss6}'s more elaborate formalism
+will allow us to reduce this length to a yet more
+attractive  $O(~$Log$ \,n~)$ size).
+
+%% 
+%%  We will soon see
+%% how  Arithmetic's conventional $\Pi_1$ sentences can 
+%% also have
+%% % pleasingly
+%%  terse encodings.
+%% 
+
+
+%% Also, all
+%% Arithmetic's conventional logical sentences do have
+%% likewise
+%% % pleasingly
+%%  terse encodings.
+%% % under our notation convention.
+
+
+The distinction between
+``Observables'' and ``Unobservables''
+% ground terms 
+will 
+%also 
+%% 
+%% cast a
+%% delightful
+%% 
+offer a
+ new perspective on
+the aspirations
+% that 
+which
+Hilbert and G\"{o}del 
+expressed
+in 
+their
+statements $*$ and $**$
+under our proposed
+\newline
+2-part conjecture.
+It
+ will suggest
+how the Second Incompleteness Theorem
+can
+%  remain to 
+be seen as a majestic result
+from a purist perspective
+, while
+a {\it well-defined fragment} of
+%their 
+what
+Hilbert and G\"{o}del
+sought in 
+ $*$ and $**$
+%aspirations
+%% in  statements $*$ and $**$
+can 
+likely
+%almost certainly
+be 
+%part-way
+satisfied (in at least a 
+% well-defined 
+limited sense).
+
+
+\begin{remm}
+\label{rem-3.7}
+{\bf (explaining the goals of this paper):$~$}
+\rm
+Let us say 
+that
+a basis axiom system $~\alpha~$ owns
+a {\it ``Finitized Perspective''}  of the Natural Numbers
+if it requires only a 
+{\it finite number} of proper axioms
+to construct the full set of integers
+$~0,1,2,3~... ~$. All conventional arithmetics have this property.
+%It is useful to divide such 
+Such
+logics
+%arithmetics
+fall into two categories,
+called {\it Single} and {\it Double-Formatted} systems.
+%as defined below: 
+They are defined below: 
+ %These constructs are defined below:
+\bed
+\item[   a.  ]
+%An axiom basis $~\alpha~$
+%will be called
+{\bf Single-Formatted Arithmetics} consist of
+axiomatic basis systems
+% $~\alpha~$
+%all of 
+whose 
+%iff all its 
+ground terms are 
+all 
+Observables.
+(Most conventional arithmetics
+%%%% will 
+%fall into this 
+lie in this
+category
+%when the 
+because they
+employ the
+ growth
+%  function
+properties 
+ of
+the  Successor
+operation
+%function
+ in
+% a straightforward 
+%the
+%% a  conventional
+the traditional
+manner.)
+%% since the 
+%% the simple  growth function of Successor
+%% easily
+%% generates
+%% all the natural numbers). 
+%are {\it ``Single-Formatted Formatted''} logics. 
+\item[   b.  ]
+{\bf Double-Formatted Arithmetics} 
+% representing 
+represent
+systems
+%%%consisting of
+%%%%axiomatic 
+%%%logics
+%%%%%basis systems
+whose ground terms 
+may be either
+ Observables
+or Unobservables.
+(Axiomizations
+for Q-Grounded logics 
+%%  of
+%% the
+%% % our
+%% Q-Grounding language
+are
+%%% will 
+%obviously 
+%%% be
+``Double-Formatted''
+because they
+allow $\theta$'s analog of 
+\el{wow}'s function symbol $F$ 
+to have
+an uncountable number of 
+different allowed
+representations).
+% 
+% (Our
+% Q-Grounding language 
+% gives support to such a system.
+% This is because it
+% can  have its function primitives
+% defined by a finite number of 
+% proper axioms,)
+% %axiom-sentences.) 
+\ennd
+The distinction between 
+categories
+%Items 
+(a) and (b) is
+significant
+% important
+because 
+Example \ref{ex-2.3}
+%%%  \pag2 
+%%% had 
+already explained how
+ statement $++$'s generalization
+of the Second Incompleteness Theorem applies to 
+any formalism recognizing Successor as a total function.
+Thus, Item (b)'s Double-Formatted logics 
+are useful, if one wishes to consider alternatives
+to
+%formalism that do not recognize
+successor as a total function.
+%More precisely, 
+In this context,
+Hilbert's 
+%  famous 
+%Year-1900 
+Second 
+Open
+Problem
+can be viewed
+ as a  {\it  2-part question}, 
+composed of sub-queries Q-1 and Q-2:
+%%%%%
+%%%%% {\it  2-part question}.
+%%%%%The separation of Hilbert's question into two parts,
+%%%%%called Q-1 and Q-2, will allow 
+%%%%%%% 
+%%%%%%% This 
+%%%%%%% bipartite
+%%%%%%% distinction 
+%%%%%%% is useful because it
+%%%%%%% can enable 
+%%%%%%% 
+%%%%%the academic community to better
+%%%%%  with
+%%%%% what Hilbert and G\"{o}del were
+%%%%%seeking to accomplish
+%%%%%in 
+%%%%%their
+%%%%%statements
+%%%%%of $*$, $**$ and $***$.
+\bed
+\small
+\item[ {\bf Question Q-1$~~$}] {\it Are any axiom systems
+able to
+ prove 
+theorems 
+verifying
+ their own consistency in a robust sense?$~~$}
+The answer to Q-1 is clearly ``No'' because the combination
+ G\"{o}del's initial 1931 result \cite{Go31} with 
+%the 
+%further
+Hilbert-Bernays's result 
+\cite{HB39} 
+and the Pudl\'{a}k-Solovay invariant $++$
+(from Example \ref{ex-2.3})
+%% \pag2) 
+imply 
+arithmetics of ordinary strength cannot prove
+their own consistency in a robust sense.
+\item[ {\bf Question Q-2$~~$}]
+ {\it Can 
+logic systems
+%arithmetic logics
+%axiomatizations of Arithmetic
+%  , at least, 
+%somehow 
+``appreciate'' 
+% (not formally ``prove'')
+ their
+own consistency in some 
+{\bf REDUCED} sense, that is diluted
+but not fully immaterial?}
+$~~\,$The answer to 
+%question 
+Q-2 is 
+complex
+%%% more complex than Q-1
+%less clear-cut
+because 
+%several types of 
+some
+arithmetics,
+such as \cite{ww93,ww1,ww5,wwapal,ww9,ww14}'s paradigms,
+ can
+formalize
+% ``recognize''
+their 
+own consistency 
+using Example \ref{ex-2.5}'s
+% a 
+Fixed-Point {\it ``I am consistent''}
+axiom.
+\ennd
+
+
+%% %  sentence $\,\oplus\,$.
+%% %%Using
+%% %%%%%%%%%Under
+%% % Using the notation from 
+%% Under
+%% Lines
+%% \eq{totdefxs}--\eq{totdefxm}'s notation,
+%% these paradigms include:
+%% % both:
+%% \bee
+%% \small
+%% \baselineskip = .86 \normalbaselineskip 
+%% \item
+%%  Type-A arithmetics
+%% \cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+%% %capable of 
+%% recognizing their self-consistencies under
+%% either the deductive mechanics of semantic tableaux or one
+%% of its cousins. (See especially \cite{ww14}'s
+%% recent Wollic-2014 paper.)
+%% \item
+%%  Type-NS arithmetics recognizing their Hilbert consistency,
+%% such as the formalisms of \cite{ww1,wwapal}
+%% %further 
+%% improved, possibly,
+%% with the added techniques introduced in
+%% this  article.
+%% \ene
+%% 
+%% \ennd
+
+
+A theme of this article will be that
+% distinction 
+the distinguishing
+between questions Q-1 and Q-2 and between
+Single and Double-Formatted Logics 
+is 
+related
+% likely central 
+to the mystery
+% that has enshrouded 
+enshrouding
+the Second Incompleteness Theorem.
+This is
+%is germane to the aspirations of automated theorem proving
+%will be germane to this article 
+because there 
+%is no doubt
+can be no doubt that
+% can be no question 
+%%%%%%%% that 
+the Second
+Incompleteness Theorem is  fully
+robust
+% result 
+from a purist 
+%pristine 
+mathematical perspective.
+Yet,
+it is still problematic to fully
+% 
+%  simultaneously
+% % at the same time,
+% it is 
+% hard to 
+% entirely
+% 
+dismiss
+ Hilbert's 1926
+suggestion that 
+ some 
+specialized forms of logics should
+%declaration 
+%% 
+%% concerns
+%% in $\,*\,$ 
+%% that 
+%% {\it ``the honor of human understanding''}
+%% requires
+%% examining
+%% % explaining 
+%% % considering
+%% how  logic systems can
+%% 
+possess
+a type of well-defined 
+ knowledge about their
+own 
+internal
+consistency.
+(This is because it is
+highly
+ awkward to explain how and why
+human beings 
+are able to
+%can 
+%manage to 
+motivate 
+their 
+%cogitations,
+cognitive process,
+% themselves to think,
+ if they do not own
+some type of 
+% instinctive
+internal
+knowledge about their own
+ consistency.)
+
+% sufficient
+% % enough 
+%  knowledge about their
+% % own 
+% internal
+% consistency
+% to motivate 
+% cognition.
+
+% Bad change above
+%cogitation.
+% themselves to cogitate. 
+
+%%It is also
+%%especially 
+%%% very 
+%%tempting 
+%%to divide Hilbert's Year-1900
+%%Open Question into its Q-1 and Q-2 separate parts
+%% during the 21st century,
+%%as computers share with humans cogitative abilities.
+%%
+%%Maybe DELETE above sentence ??? 
+\end{remm}
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+The next 
+two
+sections will 
+describe our 
+2-part
+conjecture about how 
+%a
+Double-Formatted Logics are 
+likely to 
+%produce some 
+cast
+new perspectives
+on 
+this topic.
+%the nature of the Second Incompleteness Theorem.
+Before starting this subject, it should be mentioned
+that other unusual interpretations of the Second Incompleteness
+Theorem have followed
+from Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and also
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a preferable Hilbert-style deductive apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspective  
+should
+% certainly,
+ be examined by researchers
+interested in the
+Second 
+Incompleteness Theorem,
+although 
+%but
+it is
+%% 
+% they are 
+unrelated to 
+our particular
+% the next section's 
+%specific analysis of
+type of
+Hilbert-styled self-justifying effects,
+studied in the current article.
+
+
+%% systems 
+%% formalizing
+%% %verifying 
+%% their
+%% own consistency
+%% %%%%%Definition \ref{def-2.2}'s
+%% %%% approximate 
+%% under
+%% Hilbert-styled 
+%% deduction.
+
+
+%deduction.
+%  Hilbert deduction.
+
+%methods.
+%formalism.
+
+
+%% It is,
+%% % They
+%% %are, 
+%% however,  not germane to the next section's
+%% perspective.
+
+%methodology.
+%main formalisms.
+%methods.
+%results.
+
+ % \baselineskip = 1.8 \normalbaselineskip 
+
+%\section{
+%\small 
+%Improving \cite{wwapal}'s Results with a 
+%``Double-Formatted'' Logic  }
+
+\section{First Half of our 2-Part Conjecture}
+
+% \label{ss32}
+\label{ss5}
+
+
+The only aspect of our prior research that will be directly
+related
+to our 2-part conjecture is the ISCE axiom system,
+defined in \cite{wwapal}'s Sections 3 \& 4.
+The next several paragraphs will review 
+\cite{wwapal}'s results, for the reader's convenience.
+%% 
+%% This section will
+%% review \cite{wwapal}'s results in sufficient detail
+%% so that a reader need not examine \cite{wwapal}'s formal 
+%% text,
+%% 
+%% %%%%%definition of the ISCE axiom system. 
+%% 
+%% During our discussion, 
+%% 
+
+
+During our 
+discussion,
+%review of \cite{wwapal}'s results,
+$~L^G~$ will once again denote
+our Grounding-level language built out of our six
+non-growth functions,
+consisting
+ of 
+the
+Subtraction, Division,
+Maximum, Logarithm, Root and Count operations.
+Also, $\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ will denote
+three constant symbols designating the
+integers values of 
+``0'', ``1'' and  ``2''.
+In a context where Pred$(x)$ is an abbreviation for
+``$\,x \,- \, 1\,$''
+(or more precisely  
+``$\,x \,- \, C_1\,$'' ), 
+the ISCE axiom system from \cite{wwapal}
+used
+\eq{start}'s axiom
+ statement 
+to define
+ $\,C_0\,$, $\,C_1\,$ and $\,C_2~$:
+% these three constants:  
+\begin{equation}
+\label{start}
+\mbox{Pred}(    C_0    )  =  C_0~ \, \wedge ~ \,
+C_1 \neq C_0~ \, \wedge ~ \,
+\mbox{Pred}(    C_1    )  =  C_0 ~ \, \wedge ~ \,
+\mbox{Pred}(    C_2    )  =  C_1 
+\end{equation}
+%Also,
+The  challenge 
+\cite{wwapal} 
+faced was its formalism could
+not use any of the
+%  conventional 
+function-operations of
+successor, addition or multiplication to infer the existence
+of larger integers from the initial constants of 
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$. This was because
+the Pudl\'{a}k-Solovay result $++$ 
+indicated
+  the 
+presumption  successor is a total function 
+precludes   
+most
+%axiom 
+systems
+from recognizing their
+own Hilbert
+consistency.
+
+Our article
+\cite{wwapal} 
+considered two alternatives
+%to a conventional Successor 
+%function symbol 
+for overcoming these difficulties,
+ called
+the  {\bf Additive} and  {\bf Multiplicative Naming}
+conventions.
+They defined 
+some
+further constant symbols $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+where 
+%respectively
+$~C_j~=~2^{j-1}~$ and $~C^*_j~=~2^{\,  2^{ \,j-2}}~$.
+
+The definition of these
+% new 
+constants
+%  symbols
+is 
+easy
+%straightforward
+under $L^G\,$'s
+% Grounding-level 
+language.
+% called $L^G~,~$
+%all of whose function objects are non-growth primitives. 
+This is because
+Lines
+\eq{newadd}  and \eq{newmult}
+%had 
+specify how
+% that
+two 3-way predicates, called
+Add$(,x,y,z)$  and  Mult$(,x,y,z)\,$, 
+%can 
+%do
+encode the identities of
+% can be encoded to specify,  respectively,
+$x=y+z$ and $x*y=z$.
+Our additive and multiplicative 
+% naming 
+conventions
+can,
+%will,
+ then, define 
+ $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+%by  using
+via
+an infinite number of instances  of
+%utilizing respectively
+ \eq{addcov} and
+\eq{multcov}'s
+%{\it infinitely long}
+axiom schemas:
+% two infinite schemas of  axiom-sentences:
+%% 
+%% that belong to our 
+%%  ``Additive'' and  ``Multiplicative
+%%  Naming Conventions'',
+%% then the values for 
+%% $~C_j~$ and 
+%% $~C^*_j~$ can be easily derived from $j-2$ instances of
+%% %respectively \eq{addcov} and \eq{multcov}'s 
+%% these
+%% schema:
+%% 
+
+
+{
+%\small
+\beq
+\label{addcov}
+\mbox{Add}(~C_{j-1}~,~C_{j-1}~,~C_{j}~)
+\enq
+\beq
+\label{multcov}
+\mbox{Mult}(~C^*_{j-1}~,~C^*_{j-1}~,~C^*_{j}~)
+\enq}
+The methodology in
+ \cite{wwapal} 
+%% employed \eq{addcov}  and  \eq{multcov}'s schema in a context where it 
+presumed
+% assured
+%the Y of 
+the ``names'' for its constants $ C_j $ 
+and $ C^*_j $ 
+had nice compact encodings using $O(~Log(j)~)$ bits.  
+Its formalism calculated
+%, thereby, 
+the values of ``unnamed'' integers from 
+named entities via the {\it non-growth} Subtraction and
+Division primitives. For instance since $~20~=~32-8-4~,~$
+the quantity 20 
+can be encoded as $~C_6-C_4-C_3$.
+%%%%%%%%%%%%%%%% under \eq{addcov}'s naming convention.
+
+
+%% required
+%% $O(~Log(j)~)$ bits.  
+%% Thus, the length of these encodings was 
+%% much 
+%% smaller
+%% than the respective
+%% numbers
+%% % magnitudes of 
+%%  $2^{j-1}~$ and $2^{2^{j-2}}$ 
+%% %that 
+%% these constants represent.
+
+The challenge \cite{wwapal} 
+faced was to determine whether 
+%it was possible to formulate 
+self-justification
+was possible
+under 
+either
+\eq{addcov}'s
+% ``Additive'' 
+or  \eq{multcov}'s 
+% ``Multiplicative'' 
+%% naming
+schema.
+It found 
+%that 
+ \eq{multcov}'s 
+multiplicative
+% naming 
+convention was  incompatible 
+with self-justification (due to its 
+%%very 
+speedy growth rate),
+but
+%In contrast,
+\eq{addcov}'s additive 
+% naming 
+schema did,
+conveniently,
+ permit self-justification. 
+
+\medskip
+
+Our new proposed Double-Formatted form of a self-justifying
+axiom system is easiest to describe, if we first
+review \cite{wwapal}'s Single-Formatted formalism
+and then incrementally refine it.
+
+The extension of our base-language $~L^G~$
+that includes the Additive Naming Convention (ANC)'s
+additional constants 
+ $~C_3,~C_4,~ C_5,~ ...~$ will be called
+an {\bf ANC-Based Language} and be denoted
+as  $~L^{ANC}~$. 
+Also if
+ $\, t \,$ denotes   any term in $\, L^{ANC} \,$'s   
+language, then 
+the quantifiers in 
+the two wffs of
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and
+$\exists ~ v \leq t~~ \Psi (v)$
+will be  called $\, L^{ANC} \,$'s   
+{\bf ``Bounded 
+Quantifiers''}.
+
+
+\begin{deff}
+\label{def-3.8}
+\rm
+The analogs of a conventional arithmetic's
+$\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+formulae
+in the
+language $L^{ANC}$ will be denoted as
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$
+ and $\Sigma^{ANC}_n$.
+Thus,
+a formula will be defined to be
+$\Delta^{ANC}_0$  iff all its quantifiers are bounded.  
+The
+%%%%%%%%%  below 
+definitions 
+of  $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$
+formulae are also  quite conventional:
+\bee
+%\small
+\parskip -2 pt
+\baselineskip = 0.8 \normalbaselineskip 
+\item
+Every  
+$\Delta_0^{ANC}$ formula is considered to
+be 
+also 
+a
+$\Pi_0^{ANC}$  and 
+an
+$\Sigma_0^{ANC}  $ expression.
+%% 
+%% ``$~\Pi_0^{ANC}~ \,$''  and 
+%% %  also 
+%%  ``$~\Sigma_0^{ANC}~ \, $''. 
+%% 
+\item
+A
+formula
+is  called
+ $ \,\Pi_n^{ANC} \,$
+when it
+% is 
+can be
+encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  
+where
+%with
+$\Phi$ is  $\Sigma_{n-1}^{ANC}$
+\item
+A formula
+is  called
+ $\Sigma_n^{ANC}$
+when it can be encoded as 
+$\exists v_1~ ...~ \exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{ANC}$.
+\ene
+\end{deff}
+
+
+
+%%\begin{deff}
+%%\label{def3.9}
+%%\rm
+
+ \parskip 1pt
+
+
+Given an initial axiom system $\beta,$
+the Theorem 3 of \cite{wwapal} defined a 
+self-justifying logic, called
+ISCE$(\beta)$ 
+that could prove all 
+$~\beta\,$'s $\Pi_1^{ANC}$ theorems and 
+verify its own consistency under a Hilbert-style deductive
+apparatus. It consisted of the following four
+groups of axioms:
+%
+%  \newpage
+\begin{description}
+ \parskip 0pt
+\item
+{\bf GROUP-ZERO:} 
+This 
+schema
+%  axiom group 
+will
+use \el{start}'s axiom to define the constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ 
+and 
+%employed
+%an infinite number of instances of
+\el{addcov}'s Additive Naming 
+schema
+%convention
+to define 
+ the further constants
+ $    C_3,  C_4,  C_5,  ...   $  
+\item
+{\bf GROUP-1:}
+It is convenient  to
+define
+ ISCE's Group-1 and Group-2
+axioms using a notation that 
+will support \cite{wwapal}'s Theorem 3
+in a 
+% slightly
+ more 
+general sense than
+appeared in \cite{wwapal},
+%%% under a slightly different notation convention,
+% is transparently equivalent
+% (but slightly different) from \cite{wwapal}'s counterpart,
+so that
+% a 
+the
+new
+% proposed
+%%%% second
+ ``IQFS''
+formalism
+(appearing later in this section)
+% shall
+will
+%proposal shall
+% framework will 
+be easier to describe.
+Let us
+therefore
+ say  a $\Pi_1^{ANC} $  sentence is {\bf Simple}
+iff the only built-in constants it employs are
+$\,C_0\,$, $\,C_1\,$ and $\,C_2$.
+Then ISCE's Group-1 scheme will
+allowed to
+ be any finite set of 
+simple $\Pi_1^{ANC} $  axioms, called $~S~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following properties:
+\bee
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+axioms
+will be capable of proving all $\Delta^{ANC}_0 $  
+statements which
+are true.
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+scheme
+will also be capable of proving 
+that
+the ``=" and ``$\leq$" predicates
+% own 
+support
+their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+Any finite set 
+$\Pi_1^{ANC} $  axioms
+with the above properties can be used to define $~S~$
+and 
+support
+%prove 
+an analog of
+\cite{wwapal}'s Theorem 3,
+by a trivial generalization
+\footnote{A formal proof of this generalization of
+\cite{wwapal}'s results is 
+%absolutely 
+entirely
+routine.
+%  and omitted here for the sake of brevity.}
+For the sake of brevity, it  is
+omitted.} 
+%
+%here,}
+ of
+the methodologies from Sections 3 and 4 of
+\cite{wwapal}. (Thus,
+any such finite set $~S~$ supporting Conditions (1) and (2)
+can  be employed
+by
+ISCE's
+Group-1 part.)
+%%% 
+%%% and it is unimportant which
+%%% particular defining
+%%%  set is used.
+% 
+% BBB111
+%  
+% This
+% % schema
+% axiom group 
+% consisted of a finite
+% set of 
+% $\Pi_1^{ANC} $ axioms
+% %, CALLED $F$,
+%  defining ISCE's
+% Grounding  function primitives. 
+% %This means that
+% For each  such function $G$ and set of numbers 
+% $    {k},   {k_1},    {k_2}, ...    {k_m}$,
+% %the combination of 
+% the Group-Zero and Group-1 axioms 
+% %must
+% will
+%  imply 
+% $ G(    {k_1},    {k_2}, ...    {k_m}) \,=\,    {k}   $ when
+% this sentence is true
+% \footnote{ \f55 
+% Our
+%  $\Pi^{ANC}_1$
+% encoding for the
+% Group-1 scheme  needs,
+% technically, 
+% % employ
+%  only
+% employ
+%  the three constant symbol $C_0$, $C_1$ and $C_2$ for the
+% union of all 
+% the
+%  Group-Zero and Group-1 axioms
+% to satisfy 
+% their
+% %its
+% %the 
+% above requirements.} .
+% The Group-1 schema
+% of \cite{wwapal} 
+% will also
+% assign the ``=" and ``$<$" predicates
+% their conventional
+% %  logical
+% properties.
+% %footnoted property.}
+% %%
+% %%(Any finite 
+% %%set of  $\Pi_1^{ANC} $
+% %%sentences  meeting these conditions is
+% %%suitable.)
+% %%
+\item
+{\bf GROUP-2:}
+Let
+$\ulcorner \, \Phi \, \urcorner$ denote $\Phi$'s G\"{o}del number, and
+$\mbox{HilbPrf}_{ \beta  }(x,y)$
+denote a 
+%%%%%%%%%%%  $\Delta _0^{ANC+}$ 
+$\Delta _0^{ANC}$ 
+formula indicating  $y$ is a
+Hilbert-styled
+proof
+from axiom system $\beta  $ of the theorem 
+$x$.
+%
+% Suppose that
+%$~\beta~$ uses the same Grounding function symbols as
+%ISCE$^{ANC}(\beta)$,
+%and it therefore generates
+%a set of 
+%% $\Pi_1^{ANC+} $ theorems.
+% $\Pi_1^{ANC} $ 
+%theorems.
+%
+For each 
+%$\Pi_1^{ANC+} $
+$\Pi_1^{ANC} $
+ sentence  $\Phi$,
+the Group-2 schema
+for ISCE$(\beta)$ 
+%
+%was defined in  \cite{wwapal} 
+%did 
+will 
+contain
+% an 
+one
+axiom of the form:
+%% 
+%% \begin{equation}
+%% \small
+%% \label{group2nold}
+%% \forall ~x~\forall ~y~
+%% ~~\{~~[~~ \sigma_{~ \ulcorner \, \Phi \, \urcorner
+%%  ~}(x)~\wedge~
+%% \{~ \mbox{HilbPrf}~_\beta
+%% ~(~ x ~,~y~)~~]~~
+%% \Rightarrow ~~ \Phi~~ \}
+%% \end{equation}
+%% % {\bf IMPORTANT CLARIFICATION:} 
+%% %{\small 
+%% %%{{\bf DECIPHERING LINE \eq{group2nold}:$~$}
+%% {{\bf Clarification:$~$ }
+%%  \el{group2nold} is {\it helpful}
+%% because   ISCE(\beta)$  can             infer
+%% \eq{group2old}'s {\it simpler statement}
+%% directly
+%% from the combination of
+%% \eq{group2nold},
+%% % it,
+%% the Group-1 schema and \el{deltf}'s definition of
+%% ``$~\sigma~$''.}
+%% 
+\begin{equation}
+% \small
+\label{group2old}
+\forall ~y~~~\{~ \mbox{HilbPrf}~_\beta
+~(~ \ulcorner \Phi \urcorner ~,~y~)~~
+\Rightarrow ~~ \Phi~~\}
+\end{equation}
+\item
+{\bf GROUP-3:}
+This last part of
+%%%%%%%%%%%%%%%  \cite{wwapal}'s
+ISCE$(\beta)$
+formalism was
+ a single 
+self-referencing
+$\Pi_1^{ANC}$
+sentence  
+stating:
+ %% essentially declaring:
+\begin{quote}
+% \small
+%%%%%%%%%%%%% $ \oplus ~ \oplus ~~~$
+$ \oplus  \oplus ~~~$
+ ``There 
+%is 
+exists
+no
+Hilbert-style proof of 0=1 from the union of the Group-0, 1 and 2
+axioms  with {\it THIS  SENTENCE} (referring to itself)''.
+\end{quote}
+\end{description}
+%{\bf CLARIFICATION:}
+{\bf Clarifying $ \oplus  \oplus$'s Meaning:}
+ $~$Several of our articles
+\cite{ww1,ww5,wwapal,ww9}
+employed
+self-referential
+  $\Pi_1^{ANC}$ constructions,
+similar 
+to 
+%%%%%%%%%the sentence
+ $ \oplus  \oplus \,$
+as Example \ref{ex-2.5} had mentioned.
+%% 
+%% whose 
+%% % precise implications were outlined in
+%% significance was explained by
+%% %formalized by 
+%% Example \ref{ex-2.5}.
+%% 
+A reader can find
+several
+%detailed
+slightly different
+ illustrations about how 
+$~ \oplus  \oplus ~ $
+%  $\, \oplus  \oplus $'s 
+%   self-referential statement 
+is encoded in  these articles.
+
+
+% 
+% Each of these articles provide examples of
+% how analogs for
+% $\, \oplus  \oplus $'s 
+% self-referential
+% statement
+% are encoded.
+
+
+ 
+% If the reader wishes to see
+% a formal encoding  for
+% $\, \oplus  \oplus $'s 
+% %self-referential
+% % Fixed-Point 
+% statement,
+% %it 
+% one such example
+% is provided by  
+% \cite{wwapal}'s 
+%  Lemma 1.
+% 
+
+
+\begin{deff}
+\label{def-3.9x10}
+\rm
+Let $~I(~\bullet~)~$ denote
+an operation that maps
+an initial axiom basis $\, \beta \,$ onto an alternate
+system  $\,I(\beta)\, $.
+(One example of
+such an operation is the
+  ISCE$( \, \bullet \, )$ 
+framework,
+that maps 
+an initial axiom basis of 
+  $~\beta~$ onto 
+the alternate formalism of
+ ISCE$(\beta).~)~$ 
+Such an operation  $~I(~\bullet~)~$
+is  called {\bf Consistency Preserving}
+iff  $\,I(\beta)\, $ is consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemas is
+consistent.
+\end{deff}
+
+
+%Most of our research in 
+% \cite{ww93}-\cite{ww14}
+% has 
+
+Several of our research projects 
+%centered around
+had employed
+ \dfx{def-3.9x10}'s
+framework.
+For instance, 
+%% 
+%% the 
+%% 
+%% 
+%% Its
+%% %%%  main
+%% % central 
+%% focus in
+\cite{wwapal} 
+demonstrated
+%consisted of showing
+ the   ISCE$( \, \bullet \, )$ 
+mapping was consistency preserving.
+Thus if PA+ denotes the extension of
+Peano Arithmetic that 
+includes
+PA's traditional  Addition and Multiplication
+functions
+%% 
+%% 1n addition to the conventional
+%% functions of addition and multiplication
+%% contains 
+%% 
+%% 
+plus $L^G\,$'s six
+added
+%previously mentions
+ Grounding-level function 
+primitives,
+%functions,
+then
+ ISCE$( \, $PA+$ \, )$
+will 
+be automatically
+%be
+ consistent
+(because  PA+ was consistent).
+% consistent whenever PA+ is consistent.
+Hence while Peano Arithmetic is unable to
+verify its own consistency (on account of G\"{o}del's
+seminal 1931 discovery), it is sufficiently agile to
+prove the following relative-consistency statement:
+\begin{center}
+%% \small
+$\#~~~$ If PA is consistent then 
+ ISCE$( \, $PA+$ \, )$ is  
+ self-justifying.
+ \end{center}
+This
+%The above
+% statement
+ relative-consistency statement
+%does offer 
+provides
+a partial 
+positive
+answer to
+the
+Q-2 version of Hilbert's Second  Question.
+%This is because it formalizes one respect 
+It 
+captures
+% Brad change encapsulizes 
+one
+% positive 
+respect
+in which
+%such as  
+ISCE$( \, $PA+$ \, )$
+can {\it appreciate} its own consistency.
+This respect is, obviously,
+only
+of a limited nature
+because $++$'s generalization of the Second
+Incompleteness Theorem indicates 
+that
+no Type-S arithmetic
+can
+% simultaneously 
+recognize 
+% {\it both} 
+its Hilbert consistency and
+take
+successor 
+to be
+ a total function.
+%The consistency-preservation property of 
+% ISCE$( \, \bullet \, )$
+%dies, however,
+It does,  however,
+ raise the following
+enticing
+ question:
+\begin{quote}
+$\# \, \#~ $
+Can the infinite number of
+distinct
+ constant symbols, employed by
+ISCE's Group-Zero schema, be reduced to a finite size
+by a Type-NS Self-Justifying Logic,
+without resorting to \cite{wwapal}'s inefficient
+``ISINF'' 
+methodology (which requires 
+a proof 
+having an expensive
+ $\Omega(N)$ length for constructing integers $N$
+whose binary encoding uses $O(~$Log$(N)~)$ bits) ?
+\end{quote}
+The remainder of this section will outline how an encouraging
+answer to 
+$\, \# \, \# \, $'s query
+is likely to
+%%%should,
+% conveniently
+arrive,
+%be plausible
+when one 
+% carefully
+%delicately 
+modifies ISCE's formalism
+with  the Q-function operative of $~\zzthe~$.
+
+\begin{deff}
+\label{def-3.10}
+\rm
+Let $L^Q$ 
+% once 
+again denote the extension of
+$~L^G\,$'s Grounding language that includes
+the
+% further
+ Q-function symbol of $\, \theta $.
+Then
+$\Delta^Q_0$,
+ $\Pi^Q_n$ and $\Sigma^Q_n$
+will,
+intuitively,
+%similarly
+ denote the
+%  1-to-1 
+analogs of
+\dfx{def-3.8}'s
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$'s
+formulae
+in $~L^G\,$'s  language.
+In particular, if $~\Phi~$ 
+is one of an
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+formula,
+then
+% the formula 
+$~\Phi^Q~$
+will be called
+% respectively
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+when
+%if 
+it
+differs from $~\Phi~$ 
+only
+by
+ replacing each constant $~C_J~$
+from the set $~C_3,C_4,C_5...~$ 
+with Line \eq{ej-def}'s 
+% mathematically equivalent term of
+term  $~E_{J-1}~$. 
+\end{deff}
+
+\parskip 2pt
+
+%% 444444444444444
+
+\begin{example}
+\label{ex-3.11}
+\rm
+Suppose $~\Phi$ 
+is one of a 
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+sentence that employs the three constant symbols
+of $C_4$,   $C_6$ and  $C_{10}\,$
+for
+ representing the
+three numbers
+of 8, 32 and 512. 
+Let us recall 
+that
+ $E_3$,   $E_5$ and  $E_9\,$
+%  do
+formulate these three quantities
+under  Line \eq{ej-def}'s notation.
+Then $~\Phi^Q$  will have an 
+identical definition as
+ $~\Phi$ 
+except each $C_j$ is replaced by
+$E_{j-1}$.
+
+
+A formula is,
+moreover,
+ defined to lie in one
+of the 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+classes 
+{\it only if} it is constructed in such a manner.
+This fact
+% brad assures 
+ensures
+that all the terms employed in these
+three classes of sentences are 
+{\it ``Observable''} terms.
+Hence ``Unobservable'' ground terms are allowed in
+$~L^Q\,$'s language, but {\it they are excluded}
+from occurring in the 
+{\it ``end-product''} 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+theorems 
+that 
+%%will now be discussed.
+%it proves !
+do
+encapsulate
+%formalize
+ the {\it intended use of 
+%its
+this 
+formalism.}
+\end{example}
+
+\begin{deff}
+\label{def-3.12}
+\rm
+The term {\bf IQFS($ ~\bullet~$)$~$}  refers
+to the self-justifying analog of
+  ISCE($ ~\bullet~$)$~$
+%that will be employed 
+under  $L^Q\,$'s
+language.
+(The acronym ``IQFS'' stands for 
+``Introspective Q-Function Semantics''.)
+In a context where $~\beta~$ is 
+%some i
+an initial axiom
+system that proves theorems 
+%under
+in 
+the 
+language  $L^Q$, the 
+system
+%formalism 
+ IQFS($ \, \beta\,$)
+%$~$
+will 
+be defined as a
+ 4-part 
+formalism,
+analogous to  ISCE($\beta$),
+except for the following 
+%relatively modest 
+changes: 
+\bed
+\parskip 0pt
+\item[  a.  ]
+The Group-Zero schema of
+ IQFS will
+differ from ISCE's analog
+by replacing 
+\el{addcov}'s ``Additive Naming'' schema with  
+the
+Up-Walking axioms,
+given in Lines  \eq{walk1}--\eq{walk4}.
+(This is because
+the language  $L^Q$ differs from
+ $L^{ANC}$ by 
+having the 
+ Q-function operator of $~\zzthe~$
+define the formal quantities that are represented by
+the constant symbols
+of $~C_3,C_4,C_5~~....~$ 
+under  $L^{ANC}.~~)$ 
+Otherwise both
+these
+Group-Zero 
+schemes will be 
+identical.
+Thus,
+ they 
+will 
+both 
+use \el{start}'s axiom to define the 
+three initial constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,~$.  
+\item[  b.  ]
+The Group-1 scheme of IQFS will be identical to ISCE's counterpart
+except it will reflect Item a's modification of the Group-Zero
+scheme for $\, L^Q\,$'s revised language
+(e.g. 
+the footnote \footnote{\f55
+In a context where a $\Pi_1^Q $  sentence is
+called  ``Simple''
+when it contains no $~E_j~$ term with $~j \geq 2~,~$ 
+the Group-1 scheme of IQFS will be analogous to
+ISCE's counterpart by consisting of
+ any finite set of 
+simple $\Pi_1^Q $  axioms, called $~S^*~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following properties:
+\bee
+\item
+  The union of $~S^*~$ with IQFS's Group-Zero
+axioms
+will be capable of proving all $\Delta^Q_0 $ 
+statements which  are true.
+\item
+  The union of $~S^*~$ with IQFS's Group-Zero
+scheme
+will also be capable of proving 
+that
+the ``=" and
+ ``$\leq$" predicates
+support their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+The above two properties are the 1-to-1 analogs of
+their counterparts used by ISCE's Group-1 scheme.
+As was the case with ISCE's formalism,
+any finite set
+of simple 
+$\Pi_1^Q $  axioms
+with the above properties can be used to define $~S^*~.~$
+Once again,
+it is unimportant which 
+particular
+finite-sized
+definition for $S^*$
+% such set 
+is used.} describes 
+such a
+%the
+resulting
+%% quite
+ straightforward revision of
+the Group-1 scheme.)
+\item[  c.  ]
+All the $\Pi_1^Q$ axioms lying in IQFS's
+%Group-1 and 
+Group-2 scheme will be identical to their counterparts
+under ISCE, except  they
+will
+ employ 
+\dfx{def-3.10}'s machinery for translating 
+ $ \,\Pi_1^{ANC} \,$ 
+sentences into their equivalent
+ $ \,\Pi_1^Q \,$ counterparts. 
+\item[  d.  ]
+The Group-3 axiom of  IQFS
+will be similar to ISCE's Group-3 
+{\it ``I am consistent''} 
+axiom-statement, except 
+the latter's notion of ``I'' will reflect the above
+changes in the Groups 0, 1 and 2 schemes. 
+It
+%Thus, the new
+%Group-3 axiom 
+will,
+thus,
+be a $\Pi_1^Q$ sentence declaring that
+{\it ``There is no 
+Hilbert-style
+proof of 0=1 from the union of the preceding axioms
+with THIS SENTENCE (looking at itself)''.}
+\ennd
+\end{deff}
+
+The properties of IQFS
+ are
+interesting
+% especially intriguing 
+from a 
+% Computer Science 
+Complexity perspective
+because
+\phx{th-3.3} showed 
+% that 
+every integer $\,n\,$ 
+can
+%could 
+be encoded
+%under it by
+by
+%via
+a
+ term $T_n$ that has an $O\{~[~$Log$(n)~]^3~\}$ length.
+This is
+unlike the 
+%much 
+%%%%%%%%%% far worse 
+asymptote  $\Omega(n^2)$ that results
+when the 
+$~\zzthe$ primitive 
+(from Lines  \eq{walk1}-\eq{walk4})
+is replaced by 
+Lines \ref{zm1} and  \ref{zm2}'s
+less efficient
+primitive of 
+$~\glamb~$.  
+
+
+% function
+
+
+%% It
+%% is
+%% also
+%%  almost
+%% as good as the 
+%% %encoded
+%%  $O\{~$Log$(n)~\cdot ~ $LogLog$(n)~\}$ length
+%% that ISCE produces.
+%% % while 
+%% Moreover
+%% IQFS,$~$ {\it unlike ISCE,$~$}
+%% requires only a finite number of constant symbols
+%% to define any 
+%% %starting 
+%% integer $N$. 
+
+%In other words, the 
+
+
+An open question of fundamental interest is whether or not
+IQFS framework is
+% fundamentally 
+analogous to ISCE by also
+satisfying 
+Definition \ref{def-3.9x10}'s Consistency Preservation property.
+The
+% close 
+similarity between the definitions of these two
+axiomatic frameworks
+ strongly 
+suggests that
+\cite{wwapal}'s proof of ISCE's consistency preservation
+property 
+%should 
+ought to
+be extensible to IQFS.
+
+The first half of our 2-part conjecture is,
+%thus,
+the presumption
+ that IQFS
+does have this 
+desirable
+characteristic. (Unfortunately, we suspect a
+formally
+rigorous proof of 
+this consistency preservation property
+ will be 
+% much
+%significantly
+ more 
+complex
+%complicated 
+than 
+it was for
+ISCE's
+analog in Theorem 3 of \cite{wwapal}
+because a more complicated mathematical infrastructure will
+be needed to overcome certain bottle necks that would
+otherwise arise during the
+proof.) 
+
+
+%% More will be said about this conjecture
+%% in
+%% 
+
+If 
+our conjecture
+does hold then IQFS will be 
+% much 
+substantially
+more germane to the
+goals that Hilbert  sought in statement $*$ than
+was ISCE.
+This is
+because the former framework
+differs from the latter by
+ needing only three starting constants,
+representing the integers of 0, 1 and 2, in order to
+construct the 
+full
+infinite span of integers $~3,4,5,~...~$.
+
+%\bigskip
+
+\smallskip
+
+Moreover,
+our conjecture that  IQFS will 
+duplicate ISCE's consistency-preservation property is based on
+more than the analogous definitions between these two frameworks.
+It is also because  Item  $++$'s
+ Pudl\'{a}k-Solovay
+generalization of the Second Incompleteness Theorem
+% 
+% (from the Item  $++$ given 
+% in Example
+% \ref {ex-2.3} )
+% is
+%   proven 
+% by Appendix A 
+% 
+is demonstrated by the Proposition A.1
+(in Appendix A)
+to be inapplicable to 
+ IQFS's formalism.
+It is within this combined context that IQFS formalism
+seems to
+become especially enticing, when it 
+differs from ISCE by needing only three starting constant
+symbols to construct the full infinite range of integers.
+
+%from a purely finite basis.
+
+
+\parskip 2 pt
+
+\section{Second Half of our 2-Part Conjecture}
+
+\label{ss6}
+
+Both the virtues and drawbacks of \textsection \ref{ss5}'s
+conjecture
+are highlighted by Proposition \ref{th-3.3}'s 
+characterization of
+the  $O\{~[~$Log$(n)~]^3~\}$ 
+%invariant for 
+quantity of logical symbols 
+used for encoding
+% needed to to encode 
+an integer $~n~$ as a
+ grounded term
+ $T_n$. 
+Thus,
+this
+ amount is clearly
+%much
+significantly
+ better than the alternate
+$O(~n^2~)$ 
+length
+that arises when the $~\theta~$ function symbol
+is replaced by the less efficient primitive 
+$~\glamb~,~$ 
+%%%%% operator 
+defined by Lines 
+\eq{zm1} and \eq{zm2}.
+On the other hand, one would ideally 
+prefer our ground terms to resemble the conventional encoding
+of a binary number that uses 
+ $O\{~$Log$(n)~\}$ 
+logical symbols to encode an
+arbitrary  number $~n~$.
+
+It turns out  it is possible to improve 
+Proposition \ref{th-3.3}'s encodings 
+to such a compressed  $O\{~$Log$(n)~\}$ size,
+if one adds only a minor
+wiggle to the logic's notation
+%%%%%%%%%%%%%%%%%% conventions 
+convention.
+This distinction arises because most traditional
+logic
+languages
+% , as typically formalized in textbooks,
+will
+formalize
+% both conventional terms and 
+Definition \ref{def-3.4}'s 
+``Ground terms''
+as tree-like structures.
+An 
+easy
+alternative modification of this 
+%perspective 
+construct
+will
+% would 
+allow these terms to
+own the more general structure of
+a Directed Acyclic Graph (Dag).
+ 
+% Kozen has noted ????.
+%some computer scientists have noted.
+
+This distinction has been
+traditionally viewed
+ as an unimportant 
+wrinkle
+%issue 
+because it can be 
+readily
+%easily
+proven that every Dag-oriented  term can be 
+converted into its Tree-oriented counterpart with 
+merely a
+% only an usually unimportant  
+Polynomial increase in space. 
+The reason for our special interest in an
+alternate Dag-formulated
+%distinction
+%between a Tree-oriented and Dag-oriented
+base-language for logic 
+is that the latter 
+ushers in
+ more efficiently encoded Ground terms. 
+Thus,   Proposition \ref{th-6.1} 
+will
+indicate
+that
+Proposition \ref{th-3.3}'s earlier
+ ground terms can 
+have their lengths
+ compressed from an
+$O\{~[~$Log$(n)~]^3~\}$ 
+to an   $O\{~$Log$(n)~\}$ 
+magnitude in a Dag context.
+
+%in the latter context.
+
+
+\begin{propp}
+\label{th-6.1}
+% \rm
+Let us consider a Dag-analog of 
+\textsection \ref{ss4}'s formalism where one again:
+\bee
+\item
+$\theta$ is the only available growth permitting function symbol,
+\item
+the only built-in constant symbols are the 
+entities 
+ $~C_0~$, $~C_1~$ and $~C_2~$ 
+for representing
+the values of 0, 1 and 2, and
+\item
+three function symbols for
+representing
+ integer-subtraction, integer-division
+and the maximum operation 
+are, once 
+again, available. 
+\ene
+%Within this notation, 
+In this context,
+any integer $~n~$ can be encoded
+by a Dag-oriented Ground term 
+$~T_n^*~$
+using only
+ $O\{~$Log$(n)~\}$ logical 
+symbols. (As the pointers needed to
+separate these  $O\{~$Log$(n)~\}$ logical objects 
+will have encodings using 
+ $O\{~$LogLog$(n)~\}$ bits, the total amount of 
+memory to encode a
+ Dag-oriented Ground term will require
+ $O\{~$Log$(n)~\cdot ~~$LogLog$(n)~\}$ bits.)
+\end{propp}
+
+The proof for Proposition \ref{th-6.1}
+rests essentially on a more elaborate version of the
+argument that   \textsection \ref{ss4}
+used to justify Proposition  \ref{th-3.3}.
+It
+% essentially rests on 
+uses
+the fact that 
+Proposition  \ref{th-3.3}'s ground term $T_n$
+will have
+% has
+many repeating subterms that can be compressed into
+% one 
+single objects under a Dag-style notation.
+%% 
+%% Our
+%% %We supply in the attached 
+%% $\, 1 ~\frac{1}{2} \,$ page
+%% proof in 
+%% 
+Appendix B  provides
+a  $\, 1 ~\frac{1}{2} \,$ page
+formal proof
+about how 
+%the 
+a more
+nicely
+compressed object $T_n^*$ 
+can be encoded
+using only $O\{~$Log$(n)~\}$ logical symbols.
+
+Let  IQFS$^*$ denote the analog of our  
+IQFS framework that has   
+Proposition \ref{th-6.1}'s
+ Dag-oriented Ground Terms replace
+  \textsection \ref{ss4}'s earlier 
+Tree-oriented Ground Terms
+in an accordingly revised language.
+The second half of our 2-part conjecture is that there is no
+difference between 
+ IQFS$^*$  and
+IQFS
+ from the perspective of
+Definition \ref{def-3.9x10}'s
+Consistency Preservation property.
+Both these formalism are thus anticipated to 
+be analogous to \cite{wwapal}'s ISCE 
+framework, insofar as
+  IQFS$(\beta)$  and  IQFS$^*(\beta)$ 
+will be consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemes is
+consistent.
+
+Although
+ IQFS$^*$ and 
+IQFS will 
+possess
+ similar properties under the second half of
+our 2-part conjecture, the distinction between these
+two formalisms should not be viewed as inconsequential.
+This is because the Ground terms in the  
+ IQFS$^*$ use  $O\{~$Log$(n)~\}$ logical symbols,
+similar to the classic encodings of an integer $~n~$ as
+a binary number. Thus, our conjecture about the self-justifying
+properties of  IQFS$^*$ 
+will be more meaningful than its analog
+for IQFS
+(because  IQFS$^*$  possesses 
+%% owns 
+much greater levels of efficiency).
+
+
+%is much more efficient.
+
+%\end{document}
+
+
+
+\section{On the Epistemological Significance of  Our 2-Part Conjecture}
+\label{ss7}
+
+\parskip 2pt
+
+
+All  our published articles
+(since 1993) 
+ about self-justifying arithmetics
+have emphasized that our evasions of the
+Second Incompleteness Effect  rested upon using arithmetics
+that were much weaker than traditional arithmetics in, at least,
+some 
+particular
+well-defined respects. The attached Appendix A 
+has emphasized
+that there are good reasons for conjecturing that IQFS satisfies
+a consistency-preservation property similar to ISCE, but this 
+does not reply to a second question that many 
+researcher
+% will
+may
+ want
+to have
+ addressed.
+
+% skeptics may initially raise.
+
+% will 
+% understandably
+% not, by itself, change many reader's  skepticism
+% about employing  forms of arithmetic that are weaker than its
+% traditional counterparts.
+
+It concerns the fact that our proposed   IQFS formalisms 
+%even more controversial is that they are what
+employs what
+\textsection \ref{ss4} calls a
+``Double-Formatted'' logic, where a 
+ground-term can correspond to either 
+what Definition \ref{def-3.5}
+calls an 
+ ``Observable''
+or an
+ ``Unobservable''
+object.
+ The latter will possess no
+fully
+ evident meaning
+because
+% our mathematical notation allows an 
+Unobservable ground-terms 
+can 
+%possess 
+represent any of
+multiple different 
+allowed values.
+%meanings.
+Many  
+researchers 
+will thus wish
+to
+% have 
+receive
+a reply to
+% thus raise 
+the following 
+%skeptical 
+resulting
+question:
+%
+% to be
+% addressed:
+% answered:
+\begin{quote}
+\it
+ $  \bullet ~~~$ 
+Does 
+% not 
+the presence of   
+``Unobservable'' ground-terms in 
+the language of  IQFS cause it
+to lie fundamentally outside the domain
+of modern mathematical logic
+(thus making its partial evasion of the Second Incompleteness
+Theorem
+mostly
+ irrelevant) ? 
+\end{quote}
+Our 
+% 
+% positive
+reply to  $ \, \bullet \,$'s skeptical inquiry
+has two quite
+positive
+% distinct 
+facets.
+%  
+% will have
+%  two 
+% % equally important 
+% %parts. 
+% very positive
+% aspects.
+% % facets.
+% 
+It is given below: 
+\bee
+\parskip 0pt
+\item
+% Our
+Propositions 
+\ref{th-3.3} and \ref{th-6.1} have 
+illustrated how
+%emphasized that each
+every 
+integer $~n~$ can be represented by a ground term $T_n$
+with two 
+%%%%% 
+distinctly 
+different forms of efficiency.
+% different permissible types of efficiency.
+%Moreover our 
+In this context,
+Definition \ref{def-3.10}
+and Example \ref{ex-3.11}
+have noted that our language $L^Q$ is sufficiently
+agile so that each of $L^{ANC}\,$'s
+$\Pi_n^{ANC}$ 
+sentences can be directly translated into
+their equivalent $\Pi_n^Q$ counterparts. The presence of
+``Unobservable'' objects
+ in $L^Q$ 
+% 
+is 
+thus 
+% has
+no major
+drawback
+% disadvantage
+because
+% when 
+all the conventional arithmetic objects from   
+ $L^{ANC}$ have efficient translations in 
+ $L^Q\,$'s language.
+\item
+Moreover, there is an added sense where it is
+propitious
+% nice
+that  $L^Q$
+utilize
+% possesses
+ {\it both} 
+``Observable'' and ``Unobservable'' ground-terms.
+This is because human
+beings,
+ in their common colloquial verbal languages,
+use
+references to
+ ambiguous linguistic objects in their everyday
+mundane
+ speech habits
+(similar to ``Unobservable'' ground-terms).
+% and logical sentences that possess such ambiguities). 
+Thus assuming our 2-part conjecture is
+correct, our proposed IQFS
+formalism
+ will own the 
+pleasing
+simultaneous abilities to:
+\bed
+% \baselineskip = 1.05  \normalbaselineskip 
+ \baselineskip = 1.1  \normalbaselineskip 
+% \small
+\parskip 0 pt
+\item[  a.  ]
+verify its
+% (his) 
+own consistency, 
+\item[  b.  ]
+prove analogs of all of Peano
+Arithmetic's $\Pi_1$ theorems
+(despite the fact it starts with only
+$C_0~$, $C_1$ 
+and $C_2\,$ as
+its 
+% three
+ initial constant symbols), and
+\item[  c.  ]
+ resemble the ambiguity 
+that humans display when attempting to decipher the meaning
+of human-type 
+language
+objects (in that Unobservable ground-term
+contain a level of built-in ambiguity
+that is
+ analogous to
+many mundane
+colloquial objects, that typically lack full specification
+% inherent levels of ambiguity 
+under an
+ordinary
+conventional
+human language).
+%
+% sentences, that lie outside the domain of formal mathematics
+\ennd
+\ene
+The combination of Items 1, 2a, 2b and 2c thus suggests
+that
+some perhaps
+ 5-10 \%
+fragment of 
+the aspirations that 
+Hilbert and G\"{o}del 
+had
+expressed
+in statements $*$ and $**$ 
+% thus 
+can be positively acheived
+(within the obvious context where
+the initial objectives of Hilbert's Consistency Program
+were definitively shown to 
+ have been over-ambitious by the Incompleteness
+Theorem).
+
+ 
+%% The current paper will be possibly the last paper I publish about
+%% self-justifying logics in a context where it is being submitted
+%% for publication when I am over 66 years old. I would therefore
+%% like to answer two other further questions that a reader may have
+%% about self-justifying logics in the Remarks 7.1 and 7.2 below:
+
+\medskip
+
+It  
+also
+should
+ be mentioned that the infinite
+number of axiom
+sentences,
+ appearing in the Group-2 schemas
+for ISCE$(\beta)$, IQFS$(\beta)$  and IQFS$^*(\beta)$,
+can be
+% should be able to be
+nicely
+ reduced to a 
+purely
+finite size, with almost no loss 
+in
+%of useful
+information. This was done in \cite{ww14} 
+for the Group-2
+scheme
+of its IS$_D(\beta)$ formalism, 
+with the latter
+%%%%%%%%%%%%%%%% still 
+%where the
+% 
+% germane Group-2 scheme was
+% reduced to one
+% single  axiom sentence 
+% while the resulting 
+% 
+% latter
+%formalism still
+% produced
+producing
+isomorphic counterparts
+of all of $~\beta \,$'s
+full set of
+ $\Pi_1$ theorems
+(e.g. see Sections 5 and 6 of    \cite{ww14}).
+The same methods will
+% trivially 
+routinely
+generalize for the
+% 
+% Analogs of the techniques from Sections 5 and 6 of 
+% \cite{ww14} 
+% % will easily 
+% apply to each of the 
+% 
+ISCE,
+ IQFS and IQFS$^*$ frameworks, if our 2-part conjecture
+does hold true.
+
+
+% 777777
+
+
+\section{Summary of Results}
+\label{ss8}
+
+\baselineskip = 1.21 \normalbaselineskip 
+% \parskip 2pt
+
+\parskip 4pt
+
+The  
+% importance 
+% % significance
+% of the 
+Second Incompleteness Theorem's
+%important 
+%underlying
+vital significance 
+% implications
+in establishing a
+90-95 \%
+refutation of the objectives of
+Hilbert's Consistency Program
+% are, 
+is, 
+of course, undeniable. It would, nevertheless, be of interest if
+some 5-10 \%
+fragment of 
+%its
+Hilbert's
+ initially intended 
+ objectives 
+% 
+% that Hilbert and G\"{o}del set forth in
+% their
+% statements $*$ and $**$ 
+% 
+was
+%were
+%could be
+partially
+ achieved.
+
+This is because it is difficult to fathom how human beings
+can
+maintain
+their psychological motive to engage in cognition without owning some
+type of qualified instinctive faith in their own consistency.
+Moreover, the close similarity between the defining structures of the
+ISCE and IQFS frameworks strongly suggests
+\cite{wwapal}'s proof of ISCE's consistency preservation property
+should generalize for both 
+IQFS  and 
+ IQFS$^*$ under a more elaborate
+% and sophisticated
+inductive machinery.
+Such a self-justifying arithmetic
+would be of interest
+%is very satisfying 
+% be very tempting 
+because it would be
+supportive of  each of
+the invariants of
+1, 2a, 2b and 2c 
+listed in Section
+% from
+% \textsection 
+\ref{ss7}. 
+
+
+%More precisely,
+
+This is because
+ if IQFS satisfies an analog of
+ISCE's consistency preservation property (as we conjecture),
+then IQFS(PA+) would be 
+% a
+philosophically 
+%interesting
+curious.
+% 
+%  This is 
+% % formalism
+% % 
+% % response
+% % to the open questions raised by Hilbert and G\"{o}del
+% % in $*$ and $**$
+% % 
+% because IQFS(PA+) would
+% % then 
+% 
+It would then
+be a self-justifying
+% arithmetic
+logic
+which
+% that
+proves
+% 
+all 
+Peano Arithmetic $\Pi_1^Q$ theorems, that
+% which
+is
+% {\bf  NOT MUCH HARMED} 
+ {\it NOT MUCH DILUTED}
+by the presence of unobservable 
+ground terms in $L^Q\,$'s language (because such
+unobservables
+%%%%% do {\it never physically appear} in 
+% do  
+{\it never formally appear} in 
+IQFS(PA+)'s end-product $\Pi_1^Q$ theorems).
+
+
+%. and it would thus be very tempting.
+
+
+The importance of $++$'s generalization of the
+Second Incompleteness Effect, due
+ to the joint work of 
+  Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+% \cite{Ne86,Pu85,So94,WP87},
+has been repeatedly 
+emphasized.
+%mentioned in this article. 
+%
+%has been repeatedly emphasized and
+%cannot
+%  can never
+% be understated.
+It  demonstrated conventional Type-S arithmetics
+cannot verify their own 
+Hilbert
+consistency.
+It is for this reason that
+our examination of self-justifying logics has focused 
+on arithmetics where either:
+%our subsequent research explored 
+%arithmetics with:
+%% 
+%% For instance, 
+%% this vital invariance
+%%  was
+%% %has been 
+%% cited, in detail,
+%% % mentioned 
+%% in all our published papers
+%% except  \cite{ww93},
+%% (The latter article could not cite 
+%% this topic because our 
+%% 1994
+%% telephone conversations with
+%% Robert  Solovay 
+%% stimulated Solovay's observation about
+%% hybridizing \cite{Pu85,Ne86,WP87}'s
+%% formalisms to establish \cite{So94}
+%% that no
+%% %%  
+%% %%  \cite{ww93} was published.)
+%% %%  
+%% %%  had taken
+%% %%   place
+%% %%  that 
+%% %%  %he 
+%% %%  Solovay
+%% %%  \cite{So94}
+%% %%  %began
+%% %%  %in 1994, that Solovay 
+%% %%  %%%%%%%%%noted
+%% %%  %%%%%%%%%1
+%% %%   observed
+%% %%  % in 1994 
+%% %%  % he know 
+%% %%  how
+%% %%  % to extend 
+%% %%  the work of
+%% %%    Pudl\'{a}k  Nelson and Wilkie-Paris,
+%% %%  \cite{Pu85,Ne86,WP87} 
+%% %%  implied
+%% %%  %show  
+%% %%  all
+%% %%  natural
+%% %%  
+%% natural
+%% Type-S axiom systems 
+%% %are 
+%% can corroborate its
+%% % were
+%% % unable to verify their 
+%% own Hilbert consistency.)
+%% It is within this context that our research has explored two
+%% methods for constructing self-justifying logics where
+%% either:
+%% % 
+%% % types of formalisms where an formalism
+%% % $~\alpha ~$
+%% % can partially
+%% % corroborate its own consistency where either:
+%% % 
+%% 
+\bed
+\parskip 1pt
+\baselineskip = 1.2  \normalbaselineskip 
+\item[   I.  ]  
+a system $~\alpha~$ 
+% Can
+does
+corroborate its own consistency
+under
+both
+ semantic tableaux deduction and its
+ Tab(1)
+generalization
+%in a context where
+ when 
+it 
+% formally
+views
+%under
+%\cite{ww14}'s notion of 
+% ``Type-A'' formalisms that view
+%
+%%% addition as a total function and 
+multiplication
+formally as a 3-way relation
+(as
+was
+ encapsulated by  \cite{ww14}'s
+summary of
+ \cite{ww93,ww1,ww5,ww6}'s
+% \cite{ww93,ww1,ww5,ww6,ww9}'s
+results).
+\item[  II.  ]   
+or alternatively
+a system
+ $~\alpha~$  recognizes its
+own Hilbert consistency
+in a context where it 
+is a ``Type-NS'' logic
+that views
+{\it BOTH}
+  addition and  multiplication as 
+being 3-way relations.
+% (instead of being
+% total functions).
+\ennd
+It is within the context of Topic II where our
+IQFS(PA)  and 
+ IQFS$^*$(PA) frameworks are conjectured to be
+self-justifying 
+arithmetics
+% that are
+ capable of proving
+%  isomorphic counterparts of all of
+all Peano Arithmetic's $\Pi_1$ theorems.
+
+\smallskip
+
+% arithmetics, capable of proving analogs of
+% Peano Arithmetics $\Pi_1$ theorems translated into the
+% language of $L^Q$.
+
+Such an approach will be no full remedy
+%%%%  rrremm
+% 
+% Such a perspective will 
+% have only limited advantages 
+% %be no panacea 
+% 
+when the
+traditional growth properties of the  addition,
+multiplication and successor function operations are replaced
+by an
+alternative  $~\theta~$ function symbol,
+with its
+% surprising
+use of
+a surprisingly modified
+% an
+ ``indeterminate'' function definition.
+It does, however, suggest that 
+{\it a well-defined fragment} of what Hilbert and 
+G\"{o}del sought in statements $*$ and $**$ 
+% is, alas,  feasible in some
+should be 
+likely
+ feasible under
+some special  theoretical 
+circumstances, 
+that are 
+%%%% of 
+%%  , at least,
+% quite
+%limited but  
+partially
+tempting
+%  significance 
+on account of
+% our
+ Propositions 
+% \ref{th-3.3} and \ref{th-6.1}.
+\ref{th-3.3}, \ref{th-6.1} and A.1.
+
+
+
+Thus, our 2-part conjecture 
+raises the prospect of
+a 
+logic
+%formalism 
+simultaneously recognizing the
+% 
+% results can simultaneously
+% recognize the
+%  
+% % undeniable
+central 
+and crucial
+role of the Second Incompleteness Theorem
+in 
+modern
+mathematics,
+%mathematical logic, 
+while also postulating how
+%a 
+% thinking 
+human
+beings can
+formalize
+% formally  own 
+enough of
+a {\it fragmented knowledge}
+about 
+their
+%his/her 
+own 
+internal
+consistency
+% for it 
+to gain 
+% for gaining
+%acquiring
+the psychological stamina 
+% to allow it
+% needed
+for
+motivating
+% engage in
+cognition.
+
+% In other words, \textsection \ref{ss4}'s crucial
+
+\smallskip
+
+This is
+%ultimately
+ because \textsection \ref{ss4}'s 
+% notion of 
+discussion about
+ Double-Formatted arithmetics
+illustrates 
+%suggests 
+how
+to separate the
+crucial
+ notions of ``Observable'' and 
+ ``Unobservable'' ground terms, so
+that 
+% one can recognize 
+%postulate how to
+% {\it simultaneously} 
+the historic
+% importance 
+significance
+of the
+Second Incompleteness Theorem
+is 
+potentially
+compatible with
+% and that
+% while also
+%%while
+% also 
+%% appreciating that
+ a {\it partial fragment} of 
+the aspirations
+% 
+% that
+% %the Y that
+% Hilbert and G\"{o}del
+% %% raised
+% % 
+% expressed 
+% in $*$ and $**$.
+% 
+$*$ and $**$ of
+%that
+%the Y that
+Hilbert and G\"{o}del
+being realized.
+
+\medskip
+
+{\bf Acknowledgments:}
+%%%%As  several Sections 1-4, 
+%\textsection \ref{ss2}, 
+I am
+% much
+%very 
+grateful to
+%was influenced by an emailed letter from 
+Pavel Pudl\'{a}k
+for suggesting 
+\cite {Pupriv}
+I investigate how to apply
+% an analog of 
+Ajtai's study
+\cite{Aj94} of Pigeon-Hole effects 
+for
+refining my prior results about self-justifying logics.
+(The combination of
+ Pudl\'{a}k's
+insightful             suggestion
+% \cite {Pupriv}
+and our
+subsequent
+% further 
+        distinguishing
+between the
+$~\glamb   ~$ and $~\theta~$ operators
+has led to the 
+conjectured
+ improvement of 
+\cite{wwapal}'s ISCE formalism.)
+% I am very grateful to Pudl\'{a}k for making this
+% suggestion. 
+I also thank Bradley Armour-Garb for
+%  many
+several
+comments about how to improve 
+% the 
+this paper's
+presentation.
+
+% in this article.
+
+\small
+\parskip 2 pt
+
+\baselineskip =  0.92 \normalbaselineskip 
+\baselineskip =  0.80 \normalbaselineskip 
+\bibliographystyle{abbrv}
+\bibliography{aa}
+
+%\setlength{\textwidth}{5.0 in}
+
+\gvs
+
+\parskip 0 pt
+
+\section*{Appendix A: An Added Justification for Our Conjecture}
+
+\parskip 3 pt
+
+Section \ref{ss5} indicated that
+our conjecture that  IQFS will 
+duplicate ISCE's consistency-preservation property was based on
+more than the analogous definitions between the
+IQFS and ISCE frameworks.
+It was also because the 
+ Pudl\'{a}k-Solovay
+generalization of the Second Incompleteness Effect
+(from the Item  $++$ given 
+in Example
+\ref {ex-2.3} )
+can be
+  proven 
+ to be inapplicable to 
+ IQFS's formalism.
+This is because our Proposition A.1 below
+demonstrates that the union of the Groups 0, 1 and 2
+axioms of IQFS(PA+) are unable to prove that successor is
+a total function.
+
+{\bf Proposition A.1}
+{\it 
+Let $~\beta~$ denote any
+consistent
+ extension of 
+Page 16's axiom system PA+
+% (employing the $L^G$ language)
+that serves as the input argument for the ISCE($\beta$)
+and IQFS($\beta$)  formalisms.
+Then the union of the Groups 0, 1 and 2 axiom schemes for
+IQFS($\beta$)  are unable to prove that any of the operations
+of successor, addition and/or multiplication are total functions.
+Hence, extensions of these three groups of axioms are not precluded
+from verifying their own consistency by $++$'s generalization of
+the Second Incompleteness Effect.}
+
+
+{
+{\it Proof:}  
+Our article
+ \cite{wwapal}
+showed
+ ISCE($\beta$) 
+was a consistent self-justifying axiom
+system whenever  $~\beta~$ satisfies Proposition A.1's hypothesis.
+% Hence, the invariant $++$ implies that 
+% ISCE($\beta$) is unable to confirm that successor is a total function.
+It then easily follows that there exists a 
+non-standard
+model $M_b$ 
+for  ISCE($\beta$) 
+that contains some non-standard
+integer $b$ which represents some power 
+\newline
+of 2.
+
+Let  $M_b^b$ 
+denote the subset of  the model  $M_b$ 
+which consists of the set of integers that are
+no larger than $b$.  
+Then  $M_b^b$  is also a model of 
+ ISCE($\beta$) because the latter axiom system contains no growth functions.
+
+Now let us define a function operator $~\theta~$ so
+that 
+\bee
+\item $~~\theta(x)~=~2x~~$ when $x$ is a standard number which is power of 2. 
+\item $~~\theta(x)~=~\frac{x}{2}~~$ when $x$ is a non-standard number which is power of 2. 
+\item $~~\theta(x)~=~0~~$ whenever $x$ is not a power of 2. 
+\ene
+ Also, let $N_b^b$ 
+denote a model identical to   $M_b^b$ 
+except that $N_b^b$
+ contains the added function symbol of $\theta$.
+It is easy to show that 
+ $N_b^b$
+ is 
+a
+model for the Groups 0, 1 and 2 axioms for 
+ IQFS($\beta$)
+  formalisms because Items 1-3 imply that
+$~\theta~$ will satisfy Lines
+ \eq{walk1}-\eq{walk4}'s four ``Up-Walking'' requirements.
+Moreover since the element $~b~$ owns no successor under the model
+ $N_b^b$, our proof has shown that 
+ Groups 0, 1 and 2 axioms for 
+ IQFS($\beta$) do not imply that successor is a total function.
+Hence $++$'s generalization of the Second Incompleteness Theorem
+does not apply to this set of axioms
+(because 
+these three groups of axioms
+ can be formalized by 
+a model that fails to recognize
+successor as a total function).
+ $~~~\Box$
+
+\smallskip
+
+It is useful to close this appendix with the reminder that 
+Proposition A.1 does not formally prove the consistency of
+ IQFS($\beta$). 
+It merely makes this prospect look likely
+because it demonstrates no
+%  analog
+counterpart
+ of $++$'s machinery applies
+to  IQFS($\beta$). 
+It is for this reason that we conjecture that
+ IQFS will 
+duplicate ISCE's 
+analogous
+consistency-preservation property.
+
+
+
+% \newpage
+
+\section*{Appendix B: Providing Proposition \ref{th-6.1}'s Proof}
+
+ 
+Our proof for Proposition \ref{th-6.1}
+will rest upon
+an  essentially   more elaborate version of the
+\textsection \ref{ss4}'s proof for 
+Proposition  \ref{th-3.3}.
+It will use the fact that 
+Proposition  \ref{th-3.3}'s ground term $T_n$ has
+many repeating subterms that can be compressed into
+single objects under a Dag-style notation.
+
+Throughout our proof of Proposition \ref{th-6.1},
+$~G~$ will denote our Directed Acyclic Graph (Dag),
+and 
+the symbol $~M~$ will be an abbreviation for
+$~\lceil~1\, + \, $Log$_2(n)~\rceil~$.
+This directed graph will consist of approximately
+$~5 \, \cdot \, $Log$(n)~$ nodes.
+The first five of its six groups of nodes
+in $G$'s graph
+are defined below 
+in roughly
+% bottom-to-top order:
+bottom-up order:
+\bee
+\item The bottom nodes in $G$'s
+graph
+will be the three  
+built-in constant symbols of
+ $~C_0~$, $~C_1~$ and $~C_2~$ 
+that represent
+the values of 0, 1 and 2.
+%\ene
+%\end{document}
+\item
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+For each $~j \leq M\,$, the next $j$ levels of
+$G$'s directed graph will define 
+nodes $A_j$ that formalize the quantity
+ $~\zzthe^j(1)~$. In a context where 
+$~A_0~$ 
+is an abbreviation for $~C_1~$,
+the remaining $A_j$ are defined by:
+\beq
+A_j ~~ =~~~ \zzthe(~A_{j-1}~)
+\enq
+\item
+For each $~j \leq M\,$, let $B_j$
+denote the value of Max$(A_0,A_1,A_2,...A_j)$.
+In a context where 
+$~B_0~$ 
+is an abbreviation for the entity $~C_1~$,
+the remaining $B_j$ are defined
+chronologically in $G$'s directed graph by:
+\beq
+B_j ~~ =~~~\mbox{Max}(A_j,B_{j-1})
+\enq
+\item
+For each $~j \leq M\,$, let $D_j$
+denote the value of $~2^{-j} \, \cdot \, B_M~$.
+In a context where  
+$~D_0~$ 
+was defined by the prior entry $B_M$ in
+our directed graph, 
+the remaining $D_j$ nodes in our graph
+will be defined via \eq{usediv}'s 
+Division operation:
+\beq
+\label{usediv}
+D_j ~~ =~~\frac{D_{j-1}}{2} ~~~~~~\mbox{e.g.}~~~~~~ 
+D_j ~~ =~~\frac{D_{j-1}}{C_2}
+\enq
+\item
+For each $~j \leq M\,$, 
+the node
+ $E_j$
+will represent the quantity
+$~2^j~$.
+These nodes in $G$'s graph will 
+be defined by 
+\eq{usediv2}'s 
+Division operation:
+\beq
+\label{usediv2}
+E_j ~~ =~~\frac{D_M}{D_{M-j}}
+\enq
+\ene
+
+
+Some added notation is needed to describe the last part of
+$G$'s graph for formalizing $n$'s representation as a
+Dag-oriented ground term
+employing $O \{ ~$Log$(n)~\}$
+logic symbols.
+ Let $T_n$
+ denote 
+Proposition \ref{th-3.3}'s formulation of $~n~$ 
+as a Tree-oriented ground term, and 
+$T_n^*$
+denote its Dag counterpart.
+Our proof of 
+Proposition \ref{th-3.3} noted 
+$T_n$ could be constructed by setting $E_M$ equal to
+the least power of 2 greater than $~n~$ and
+then  subtracting from it
+those powers of 2 which are needed to produce the quantity $n$.
+
+The exact same methodology will now
+be used to construct
+our $T_n^*$ representation of $~n~,~$ except 
+ we
+will now
+obviously
+use the methodologies from Items 1-5 to
+assure
+%  that 
+no more than $O\{~$Log$(n)~\}$
+graph nodes are used to construct
+all 
+\el{usediv2}'s 
+ $E_j$ terms.
+(For example since $86~=~128-32-8-2$  
+%% $118~=~128-8-2$
+which in turn equals ``$~E_7-E_5-E_3-E_1~$'',
+ the final stage of 
+$G$'s construction of $T^*_{86}$ will
+first
+ set node
+$F_1$ equal to ``$~E_7-E_5~$'',
+then
+ set node
+$F_2$ equal to 
+ ``$~F_1-E_3~$'' 
+and lastly have the desired
+output node
+$F_3$ represent the final answer as the
+quantity of  ``$~F_2-E_1~$''.)
+
+It is easy to see that this methodology will never use more
+than  
+  $O\{~$Log$(n)~\}$ logical symbols to encode
+ $T_n^*$ as a Dag-oriented ground term.
+Moreover, the needed pointers in the Dag graph $G$
+will require no more than LogLog$(n)$ bits to
+%separate its
+distinguish between these
+   $O\{~$Log$(n)~\}$ 
+separate objects.
+%logical symbols.
+Hence
+if one selects to use a pointer methodology to formulate
+$G$'s graph,
+then
+%   our full graph $G$ will need 
+no more than
+ $O\{~$Log$(n)~\cdot~$LogLog$(n)~\}$ bits
+will be needed
+to encode
+all these pointers.  $~~\Box$
+
+
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/2011-2019/2015-aug10.tex b/nachlass/collected_dew_materials/2011-2019/2015-aug10.tex
new file mode 100644
index 0000000..b2f60d0
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2015-aug10.tex
@@ -0,0 +1,7575 @@
+% 2015 upstairs aug 10 4.2 pm SUNY
+
+%% 2015 home % august 2 8.1 pm while listenihng to BobbyLee DoWap
+
+%% 12.45 appointment nicoloson
+
+ % 2015 july 4 3.4 am after spell 10.1 am after sinatra
+
+ % 2015 july 2 3.15 pm 
+
+%% 2015 july 2 2.50 pm upstairs 
+
+%% 2015 july 1  10.30  am downstairs
+
+
+
+%% notarized notes 2015 april 2 6.3  am april 4 notarize again 
+
+%% home 2014 feb 8 1.15am (new email address)
+
+%% home 2015 feb6 4.3 am   suny 2.40 pm home 6.15 pm
+
+
+%% gmail dan.willard.albany and Prof.DanEdwardWillard
+%% gmail password cpZ9ar48s
+
+
+%%% SUNY JAN 11 Brad Copy 8.4  pm
+
+%%  SUNY   jan11 5/30pm spell check
+
+%% 2015 HOME jan 10  9.4  pm  pm New Abstratct
+
+%% 512 6932
+
+%% Towards a Restructuring of Hilbert's Consistency Program
+
+% www.cs.albany.edu/~dew/algor/
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[12pt]{article}
+\documentclass[11pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+
+
+
+% \addtolength{\oddsidemargin}{-0.95in}
+\addtolength{\oddsidemargin}{-1.0in}
+
+
+
+\setlength{\textheight}{9.6 in}
+\setlength{\textheight}{8.8 in}
+\setlength{\textheight}{9.3 in}
+\setlength{\textheight}{9.35 in}
+%above too short
+
+
+\setlength{\textwidth}{6.3 in}
+%% PRINT
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{5.4 in}
+
+\setlength{\textwidth}{6.5 in}
+% \setlength{\textwidth}{7.0 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+% Above IDeall
+
+
+%% \setlength{\textwidth}{6.4 in}
+%%%% above brad with 11 point
+
+%\setlength{\textwidth}{6.0 in}
+%\setlength{\textwidth}{5.7 in}
+
+%\setlength{\textwidth}{6.4 in}
+
+%\setlength{\textwidth}{5.5 in}
+
+%\addtolength{\topmargin}{-1.0in}
+%\addtolength{\topmargin}{-0.9in}
+\addtolength{\topmargin}{-0.8in}
+%\addtolength{\topmargin}{1.2in}
+
+%\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+
+\title{ A 2-Part Conjecture about
+How  a Much-Diluted but Non-Trivial
+%Variant
+Fragment
+ of
+Hilbert's Consistency Program 
+Is
+Likely
+%Plausible 
+Feasible
+for the
+% Even the Challenging
+Case of
+Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+% \title{\large \bf On the Revival of a Modified and Diluted Version of
+% Hilbert's Consistency Program (Extended Abstract)} 
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+%Email = dew@cs.albany.edu.}}
+%\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+%\begin{abstract}
+\baselineskip = 1.17 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+
+
+
+\begin{abstract}
+
+%\Large
+
+% \baselineskip = 1.8 \normalbaselineskip 
+%aaaaaaaaaaa
+
+\large
+\LARGE
+ \normalsize
+\large
+
+\baselineskip = 1.25 \normalbaselineskip 
+
+It is well known that
+the combined work of Pudl\'{a}k and Solovay \cite{Pu85,So94},
+enhanced by some added techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87}, implies
+ no reasonable axiom system
+can verify its own Hilbert consistency, when it recognizes
+Successor as a total function and treats addition and multiplication
+as 3-way relations (as Example \ref{ex-2.3} will explain).
+These considerations will lead us to examine
+unconventional
+ axiomatizations
+for arithmetic that continue to view addition and multiplication as
+3-way relations, but
+which  replace the successor function symbol with an
+entirely new operator, called the ``$~\theta~$'' primitive.
+
+\medskip
+
+% Our Propositions 
+% \ref{th-3.3} and \ref{th-6.1}
+% will show this $~\theta~$ operator
+% allow us 
+
+This $~\theta~$ operator
+will
+allow us 
+to encode any integer $~n~$ by a term $~T_n~$
+whose length will exceed the $O[~$Log$(n)~]$ length of a
+binary encoding 
+only by the
+relatively
+ small magnitudes formalized by
+Propositions 
+\ref{th-3.3} and \ref{th-6.1}.
+This 
+paradigm will be
+% issue is
+%will be 
+significant because the combination of
+our Appendix A and prior published results in \cite{wwapal}
+will provide good reasons for conjecturing  the
+ $~\theta~$ primitive can be used to construct logics that
+can  verify their own consistency under 
+the  desired setting of
+a Hilbert-styled 
+deductive methodology.
+
+\medskip
+
+The relationship between our new system and the aspirations of
+Hilbert's consistency program will be discussed in detail.  There can,
+obviously, never exist a self-justifying logic that can  evade the force
+of G\"{o}del's Second Incompleteness Theorem in a fully ubiquitous respect.
+Our conjecture will be, however, that  
+our methodology 
+implies a 
+{\it well-defined}
+fragment of the goals of Hilbert's Consistency
+program can be
+% positively 
+achieved in a
+{\it  diluted
+% but non-trivial} 
+but interesting} 
+respect.
+
+
+\end{abstract}
+
+\bigskip
+\bigskip
+
+\large
+{\bf Keywords:}
+\small
+G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+Open Question,
+Hilbert-styled Deduction (and its Frege-like analogs).
+
+
+
+
+% \bigskip
+% 
+% 
+% 
+% {\bf Mathematics Subject Classification:}
+% 03B52; 03F25; 03F45; 03H13 
+% 
+% 
+% 
+% \bigskip
+% \bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+%\begin{abstract}
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+\newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+\def\gvs{ }
+
+
+\def\gvs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+\def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 0pt}
+
+\def\gvs{ \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \large \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+
+\noindent
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\gvs
+
+
+\section{Introduction}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+\gvs
+
+\parskip 4pt
+
+%  hhhh
+Two historic results
+were established
+by
+%%% in
+ G\"{o}del's
+millennial
+%centennial
+%  1931 
+ paper \cite{Go31}.  The First Incompleteness
+Theorem 
+showed  there existed no decision procedure
+for identifying the true statements of Arithmetic.
+%G\"{o}del's
+Its Theorem XI,
+later known as the ``Second  Incompleteness
+Theorem'',
+%,
+%appearing in G\"{o}del's millennial paper \cite{Go31}. 
+demonstrated
+that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+ Russell-Whitehead Principia Mathematicae formalism
+% $\, P \,$
+can
+% could
+ verify
+its own consistency.
+ G\"{o}del's
+two
+observations
+% Theorem XI 
+were  historic
+mainly  because they
+ demonstrated, unequivocally,
+that the 
+initial 
+objectives of Hilbert's Consistency
+Program were too far-reaching.
+Thus at best,  only a
+% very  
+sharply curtailed
+form of Hilbert's goals 
+would be
+% was 
+plausible.
+This fact was further reinforced by a new version of the
+Second Incompleteness Theorem, due to
+the combined work of Pudl\'{a}k and Solovay \cite{Pu85,So94},
+enhanced by some added techniques of Nelson and Wilkie-Paris
+\cite{Ne86,WP87}.
+
+%plausibly feasible
+%from Theorem XI's perspective.
+
+Within these curtailed limits, we have published since 1993
+a series of articles 
+\cite{ww93}-\cite{ww14},
+outlining generalizations of the Second Incompleteness
+Theorem and
+its 
+sometimes-feasible
+% plausible 
+ boundary-case exceptions.
+%  that were formally feasible.
+Pavel Pudl\'{a}k
+examined
+%in great detail,
+% the preprints 
+an early preprint
+ of  
+our
+article \cite{wwapal}
+and suggested 
+\cite{Pupriv} 
+%that 
+we
+consider  attempting to hybridize 
+its formalism with some of
+Ajtai's observations about 
+ about
+the Pigeon Hole effects
+\cite{Aj94}. 
+The Section 6 of 
+  \cite{wwapal}
+did subsequently
+formalize one type of response to 
+Pudl\'{a}k's insightful observation.
+Our new results in this current paper will examine
+this topic 
+from yet another perspective.
+
+The
+particular
+ goals in the current paper will transgress 
+significantly
+ beyond our earlier research in
+\cite{ww93}-\cite{ww14}.  
+(The latter focused on examining generalizations and boundary-case
+exceptions for the Second Incompleteness Theorem
+that
+% were formally proven feasible). 
+viewed
+roughly, at least, addition from a traditional perspective.)
+Our new 
+Propositions 
+\ref{th-3.3} and \ref{th-6.1}
+will explore the properties of an
+entirely unconventional methodology for constructing
+the foundations of integer arithmetic (which does not use
+any of
+the
+ traditional 
+function symbols 
+% of
+for 
+%formalizing the 
+%conventional primitive operations of 
+successor, addition and
+multiplication
+as primitive
+ operations). 
+
+This topic 
+will 
+have interesting 
+philosophical implications and 
+% also 
+lead to
+our conjecture 
+% 
+% be
+% interesting unto itself,
+% from a 
+% solely
+% %purely
+% philosophical perspective.
+% We will also conjecture
+% that
+% it 
+% will
+% provide a machinery for reviving a
+% % {\it  VERY VERY  
+% 
+that a 
+{\it  much-diluted} but
+non-trivial fragment of Hilbert's Consistency Program
+is plausible. 
+
+%More precisely during our discussion of axiomatizations for
+%integer arithmetics, 
+
+During our discussion.
+%% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+ will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us say 
+%that 
+an
+axiom 
+system
+%basis
+ $\, \alpha \,$
+{\bf recognizes}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+can prove
+\eq{totxtefs} - \eq{totxtefm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+%\small
+\beq 
+\label{totxtefs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totxtefa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totxtefm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+\noindent
+Our axiomatizations for integer arithmetics 
+%in the current article 
+will differ from 
+their conventional counterparts
+{\it by neither 
+possessing} an ability to prove the totality statements
+in Lines 
+\eq{totxtefs} - \eq{totxtefm}, 
+{\it nor
+containing} function symbols for
+formalizing
+the traditional addition, multiplication and successor operations.
+%%% nor recognizing the validity of the identities
+% 
+% which state that these primitive
+% are ``total functions''. 
+% 
+Instead, we will rely upon an entirely
+different type of primitive function symbol, called the ``$~\theta~$''
+operator, to construct the endless sequence of integers
+$~3,4,5, ~...~$ from the
+three  initial starting constants of 0, 1 and 2.
+
+Its unorthodox
+% This alternative 
+means for constructing the set of non-negative integers
+%% in an unorthodox manner 
+will be a curious philosophical development,
+unto itself,
+apart from 
+its
+%  the
+surprising  implications
+% it will have 
+about a
+possible
+ conjectured
+revival of a {\it much-diluted} but non-trivial 
+variant
+%form of cousin
+of Hilbert's Consistency Program. 
+
+The reader should be forewarned
+that this article will not provide a  proof of its main
+2-part
+conjecture.
+% A later section of this article
+We
+ will 
+% later
+explain
+why 
+% how 
+our
+conjecture is likely correct,
+%We consider it 99 \%
+%probable that Section ???'s conjecture is correct, 
+but its proof
+is
+almost certainly
+% be 
+% much
+too
+long to fit into  
+%%
+%%for it to be practical to provide
+%%within the space of 
+%%
+a
+% one
+single
+% short 
+%abbreviated 
+paper.
+%                                %
+Our 
+% This
+conjecture
+%It 
+will suggest
+% that
+  a
+%%
+%%(or even 50) pages
+%%The goal of this article
+%%will be to stimulate  interest
+%%in 
+%%
+perhaps
+5-10 \% 
+fragment of the goals of Hilbert's consistency program
+can be positively achieved,
+%but 
+in a context where
+the
+Incompleteness Theorem
+precludes, definitively, 
+a more 
+ambitious project.
+
+
+
+% in a context where
+% the
+% % G\"{o}del's 
+% Incompleteness Theorem
+% %definitively establishes that 
+% % clearly 
+% precludes
+% a more ambitious project.
+
+
+
+
+%The discourse in t
+
+This article 
+%has been carefully composed
+%so that it 
+can be
+read 
+ without 
+% first
+examining any of our previous
+papers. If a reader
+does wish to skim
+one of our earlier
+articles,
+% before the current  article,
+we recommend 
+Sections 3 
+\& 4
+ of \cite{wwapal} 
+be skimmed
+(in a context where the 
+subsequent
+sections
+% portions 
+of \cite{wwapal}
+%will be
+are
+%are fully
+unrelated to our
+% chief 
+conjecture).
+
+% 
+%  our recommendation is that
+% Section 3 of \cite{wwapal} 
+% be skimmed 
+% $~---~$
+% in a context
+% where the subsequent sections of \cite{wwapal}
+% are unrelated to our main theorem and 
+% can 
+% be 
+% entirely
+% omitted.
+
+
+
+
+%% 
+%% In a context where I will be submitting this paper for publication
+%% shortly before my 67-th birthday, the reader should be forewarned
+%% that this article will not provide a formal proof of the main
+%% conjecture that it will 
+%% propose.
+%% We consider it 99 \%
+%% probable that Section ???'s conjecture is correct, 
+%% but its proof
+%% looks too tedious and long for it to be practical to provide
+%% within the scope of this 25-page paper.
+%% The goal of this article
+%% will be to stimulate the interest
+%% of a new generation of researchers,
+%% who consider our new formalism and its relationship to
+%% Hilbert's Year-1900 Second Open Question to be of interest.
+%% 
+
+\section{Returning to the 1931-1939 Period}
+\label{ss2}
+ G\"{o}del's 
+Second Incompleteness Theorem 
+was published in two 
+quite different 
+ forms during the
+1931-1939 period.
+Its initial 1931 variant, formalized by Theorem XI
+in  G\"{o}del's millineal paper \cite{Go31},
+% 
+% 
+% Its Theorem XI,
+% later known as the ``Second  Incompleteness
+% Theorem'',
+% %,
+% %appearing in G\"{o}del's millennial paper \cite{Go31}. 
+% 
+demonstrated
+that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+ Russell-Whitehead Principia Mathematicae formalism
+% $\, P \,$
+could
+% could
+ verify
+its own consistency.
+The widely quoted more general
+result, that 
+every consistent r.e. extension
+ of Peano Arithmetic must
+be unable to prove a theorem affirming its
+own consistency, 
+was 
+first
+published 
+%% 
+%% (see \footnote{ Boolos states in \cite{Bool}
+%% that it has been open to scholarly debate
+%% whether or not the 1939
+%% Hilbert-Bernays generalization of the Second Incompleteness Theorem
+%% is or (is not) a straightforward generalization of
+%% G\"{o}del's initial result} ) 
+%% 
+in the 1939 edition of the 
+the Hilbert-Bernays
+textbook \cite{HB39}. It has been considered
+to be the definitive demonstration of the broad reach of
+the Second Incompleteness Effect. 
+It also established, beyond any reasonable doubt, that any type
+of formalism possessing a conventional knowledge of its own consistency,
+must rely upon a 
+foundational structure
+ fundamentally different from Peano Arithmetic. 
+(This is because the 
+Hilbert-Bernays 
+textbook formalized the forerunner of
+what has now been known as the 
+Hilbert-Bernays Derivability Conditions \cite{HB39,HP91,Lo55,Mend},
+as a mechanism for
+% foreseeing 
+envisioning
+the 
+astonishing
+broad generality of the
+Second Incompleteness Effect.) 
+ 
+
+It is, thus, fascinating that Hilbert,
+as the co-author of 
+an important 
+%very 
+% historic
+generalization of the Second Incompleteness Theorem,
+never withdrew the
+% chose to  never fully withdraw his 
+1926 justification
+ \cite{Hil26}
+for his consistency program:
+\begin{quote}
+$*~$~ 
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+Indeed, 
+%Instead,
+Hilbert 
+always insisted that some
+special new formalism would at
+least partially vindicate the
+prior
+%initial 
+ goals of his
+consistency program.
+He thus arranged to 
+have  its often-quoted 
+motto
+({\it ``Wir m\"{u}ssen wissen---Wir werden wissen''} )  
+%of his 
+%nevertheless, 
+engraved on his tombstone.
+
+
+%Moreover, it 
+
+It is 
+also
+known
+\cite{Da97,Go5,Yo5}
+that G\"{o}del
+was 
+doubtful about the generality of the Second Incompleteness
+Theorem for at least two years after its publication.
+He thus inserted the following
+cautious caveat into
+his famous
+1931 
+millennial
+paper  \cite{Go31}:
+% whose closing section 
+%%% 
+%%% One of the closing paragraphs of
+%%%  \cite{Go31} 
+%%% thus
+%%% included
+%
+%
+%%% contained the following cautious disclaimer:
+%caveat:
+\newpage
+\begin{quote}
+\it
+%\baselineskip = 1.0 \normalbaselineskip 
+$~**~~$ 
+``It must be expressly noted that
+Theorem XI
+%'s incompleteness result
+(e.g. the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and {\it there might
+conceivably be finite proofs} which cannot
+be stated in P or in ... ''
+\end{quote}
+
+
+The 
+% above 1931
+statement $**$ has 
+had
+%been subject to 
+numerous
+%many
+different
+interpretations
+\footnote{
+Some 
+scholars
+have interpreted
+$\,**\,$
+as
+%as, possibly,'
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.}.
+All
+ G\"{o}del's 
+biographers
+\cite{Da97,Go5,Yo5}
+%%%have
+noted
+his
+% that G\"{o}del's
+initial intention
+was
+to
+establish
+%achieve
+Hilbert's proposed objectives, before 
+he proved
+%proving
+% G\"{o}del proved
+a result 
+% 
+% however,
+% %%%%%his
+% G\"{o}del
+% did  originally 
+% seek 
+% % goal was 
+% to
+% establish
+% %achieve
+% Hilbert's proposed objectives before 
+% proving
+% % G\"{o}del proved
+% a result 
+% 
+leading
+%that led 
+in the opposite direction.
+Yourgrau \cite{Yo5}
+records
+%furthermore,
+ how
+von Neumann 
+surprisingly
+%did
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive 
+%%%  achievement of a'
+ termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+It is known 
+ G\"{o}del 
+began to 
+more fully
+endorse 
+the Second Incompleteness
+Theorem
+during a 1933
+%% Vienna
+lecture  \cite{Go33},
+and he 
+% told biographers he
+completely embraced it
+after learning about Turing's work
+\cite{Tur36}.
+
+
+Our research
+in \cite{ww93}-\cite{ww14}
+%has been 
+is
+related to issues 
+%analogous
+ similar 
+to those
+%that were 
+raised by Hilbert and 
+G\"{o}del
+in
+ statements  $*$ and $**$.
+This is because it is counter-intuitive and awkward to
+%presume that 
+explain how
+human beings can maintain the
+psychological drive and 
+needed energy-desire to cogitate, without
+being stimulated by an instinctive faith in their own
+consistency (under a definition of 
+% formal  consistency
+such
+% this concept 
+that is suitably 
+gentle and
+% delicate
+soft
+ to 
+be consistent with the 
+Incompleteness Theorem's requirements).
+
+% preclude a violation of the 
+% restrictions imposed by the Incompleteness Theorem).
+
+Accordingly, our research in
+\cite{ww93}-\cite{ww14} 
+has explored both generalizations and
+boundary-case exceptions of the Incompleteness Effect, so as
+to determine what type of boundary-case evasions are permitted.
+Our prior research in \cite{ww93}-\cite{ww14} 
+had used mostly cut-free forms of deduction to
+evade the 
+restrictions imposed by the
+Second Incompleteness Effect. The current article will instead
+focus on more pristine Hilbert-Frege methods of deduction.
+They are likely to support an evasion of the Second Incompleteness
+Effect when our axiom systems replace the traditional  
+growth properties of the addition, multiplication and successor 
+function symbols with our new $~\theta~$ primitive.
+
+The motivation for this replacement will be
+explained during the next section of this article.
+It is needed
+essentially
+because
+a
+ version of the Second Incompleteness Theorem,
+due to the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+%will show 
+demonstrates
+that 
+if an axiom system $~\alpha~$
+proves any 
+of \eq{totxtefs} - \eq{totxtefm}'s totality statements
+then it is incapable of confirming its own consistency
+under a Hilbert-style deductive method.
+
+
+
+
+
+\smallskip
+
+Our results will suggest  it is possible to obtain a
+{\it part-way 5-10 \%
+positive} interpretation
+for what
+Hilbert and G\"{o}del 
+were
+seeking 
+% a Consistency Program
+to establish
+%%  seeking to accomplish
+% contemplating 
+in their 
+statements
+$*$ and $**$, within a context where 
+it is  known that the 
+Second Incompleteness Effect precludes a full achievement of
+%these 
+Hilbert's
+objectives
+from ever transpiring.
+The last five minutes of
+a recent 60-minute YouTube 
+presentation
+ by Harvey Friedman
+\cite{Fr14},
+entitled the 
+{\it ``Blessing and Curse of Kurt  G\"{o}del''},
+suggested that it is
+%the themes of Hilbert and G\"{o}del's
+%remarks $*$ and $**$ by indicating that 
+%it is 
+interesting to explore futuristic partial 
+boundary-like evasions of the Second Incompleteness Theorem,
+despite the stunning strength of 
+G\"{o}del's result.
+It is within this context where our proposed use of a new
+$~\theta~$ primitive 
+symbol to replace the growth properties
+of the traditional addition, multiplication and successor function
+symbols may be of potential interest.
+
+The development of our $~\theta~$ primitive 
+was partially influenced by a private email
+communication 
+we had received
+from  
+Pavel Pudl\'{a}k \cite{Pupriv},
+as Sections 3-4  shall explain.
+We also emphasize that the conventional interpretation of
+the Second Incompleteness Theorem, as precluding
+Hilbert's Consistency Program from ever achieving its initially
+specified objectives, is certainly correct.
+Our only caveat is that the latter should not lead  one
+to
+ignoring the role that a
+human's instinctive faith in his/her's internal
+consistency 
+%crucially stimulates and motivates
+plays in stimulating and motivating
+human cognition.
+% humans to cogitate.
+%
+% crucially
+% gain the
+% motivation for stimulating cogitation. 
+It is
+from this 
+special
+perspective where our prior research and
+new 2-part conjecture 
+will
+% does 
+suggest that 
+an approximate
+%at least a
+5-10 \%
+fragment 
+% fraction
+of what Hilbert and G\"{o}del 
+%suggested in 
+had
+sought
+in $*$ and $**$ 
+%could
+should be
+ plausibly 
+%is
+%% be formally 
+ feasible.
+
+
+
+\gvs
+
+\section{Motivation for Research and Background Notation}
+%  222222}
+\label{ss3}
+
+
+%%! 
+%%! This article will be written in a style so that its
+%%! overall theme (if not full details)
+%%! should become 
+%%! {\it  quickly} comprehensible to a reader who has
+%%! examined 
+%%! only
+%%!  one of the
+%%! % introductory 
+%%! logic textbooks by say Enderton,
+%%! Fitting, H\'{a}jek-Pudl\'{a}k,
+%%!  Mendelson
+%%! or Papadimitriou \cite{End,Fi96,HP91,Mend,Papa}.
+%%! %% 
+%%! %% We will rely mostly upon the
+%%! %% precise
+%%! %% deductive calculi notation employed
+%%! %% in Section 2.4 of Enderton's textbook,
+%%! %% but any of 
+%%! %% the 
+%%! %% similar
+%%! %% Hilbert-style deductive calculi of 
+%%! %%  H\'{a}jek-Pudl\'{a}k,
+%%! %%  Mendelson
+%%! %% or Papadimitriou \cite{HP91,Mend,Papa}.
+%%! %% will also be suitable for achieving our results.
+%%! 
+%%! %% 
+%%! %%  
+%%! %% ( Papadimitriyou's textbook
+%%! %% generously states it employs a deductive notation
+%%! %% that 
+%%! %% has
+%%! %% stemmed from its predecessor in 
+%%! %% Enderton's textbook.)
+%%! 
+%%! 
+
+
+
+%% In order to make our 
+%% results
+%% %research 
+%% apply to the 
+%% formalism
+
+It is helpful to employ a flexible vocabulary so
+% that our
+our 
+results
+will apply 
+%research 
+%to the 
+%formalisms
+to any of the
+%% 
+%% accessible to
+%% some 
+%% readers who are acquainted with only one of the 
+%% 
+textbook 
+formalisms of
+% settings outlined by 
+%  
+say 
+Enderton,
+Fitting, H\'{a}jek-Pudl\'{a}k,
+or Mendelson
+\cite{End,Fit,HP91,Mend}.
+% 
+% ,
+% %% or
+% %%  Papadimitriou 
+% %% \cite{End,Fit,HP91,Mend,Papa},
+% %% 
+% %% 
+% %the widest
+% % possible  
+% %audience, 
+% it is 
+% helpful
+% % useful to 
+% use
+% %employ 
+% a
+% % very 
+% flexible
+%  vocabulary.
+% %%
+% %%that 
+% %%allows a reader to 
+% %%%quickly 
+% %%%translate results 
+% %%traverse
+% %%from
+% %%one textbook to another.
+% % Therefore, let us define a
+Let us 
+% thereby
+call an
+ordered pair $(\alpha,d)$ a
+    {\bf ``Generalized Arithmetic''}
+% therefore
+iff its 
+% first and second 
+two components 
+%% 
+%%  Each of the 
+%% textbooks \cite{End,Fi96,Mend,Papa,HP96} have
+%% employed
+%% substantially
+%%  different variants of Basis and 
+%%  Deductive-Apparatus structures. 
+%% 
+are 
+% described
+formalized
+% defined 
+as below:
+%% 
+%% Their
+%% definitions in
+%%  Items (1) and (2)
+%% %simple 
+%% %%%definitions of these two notions
+%% %given below,
+%%     allow one to easily translate 
+%% %theorems 
+%% formalisms
+%% % methodologies
+%% from
+%% one textbook 
+%% % source
+%% to another:
+%% 
+%% %\njp
+%% % \newpage
+%% \parskip 2pt
+\bee
+\item
+The {\bf ``Axiom Basis''} $~\alpha~$ 
+for an arbitrary arithmetic
+shall be defined as
+the set of 
+{\it proper axioms} employed by the 
+formalism  $(  \alpha  , d  )$.
+\item
+An arithmetic's {\bf ``Deductive Apparatus''} $~d~$ 
+is defined as
+the 
+{\it combination} of its formal rules for inference
+and 
+%its
+the
+ built-in
+ logical axioms ``$~L_d~$''
+%  (that are 
+% implicitly 
+employed by these rules.
+\ene
+
+%%%\item
+%%%The term {\bf ``Deductive Apparatus''} $~d~$ will
+%%%refer to the 
+%%%{\it combination} of the rules of inference
+%%%used by an arithmetic and its
+%%%the logical axioms ``$~L_d~$'' that 
+%%%render meaning to
+%%%%are an automatic part of
+%%%$~d\,$'s machinery.
+
+
+\begin{exx}
+\label{ex-2.1}
+%\label{ex-basis}
+\rm
+This notation 
+allows one to
+ conveniently separate  the logical axioms
+$~L_d~,~$ associated
+with  $(  \alpha  , d  )~$, from 
+ $\, \alpha \,$'s
+ ``basis axioms''.
+%basis axioms
+It also allows one to  isolate
+and compare
+%  , conveniently, 
+various
+apparatus techniques,
+%technique,
+% employed in the exact formalisms 
+including the
+ $~d_E~$,  
+ $~d_M~$, 
+ $~d_H~$,  
+and  $~d_F~$
+methods
+%that we will now define:
+defined below:
+%% 
+%%  Three
+%% examples of this are illustrated below,
+%% in a context where
+%% are the deductive apparatus machineries defined
+%% in  Enderton's, Mendelson's and Fitting's textbooks
+%% \cite{End,Fi96,Mend}.
+%% 
+\bed
+\item[   i. ]
+The  $~d_E~$ apparatus,
+formalized in 
+\textsection
+ 2.4 of  Enderton's textbook, 
+% will
+uses only  modus ponens
+as a rule of inference.
+The latter will be accompanied
+by 
+a
+4-part 
+system of
+ logical axioms,
+called $~L_{d_E}~$, $\,$ to endow
+ $~d_E~$ 
+with an
+ability to  support
+% apparatus
+% agility so that it  supports
+%can satisfy
+%the analog of 
+G\"{o}del's Completeness Theorem.
+%% '
+%% (similar to  other'
+%% % full-scale '
+%% deductive methodologies).'
+%% 
+%%%% 
+%%%% (Papadimitriyou's
+%%%% % in-depth  exploration
+%%%% textbook \cite{Papa}  about
+%%%% % examination  of 
+%%%% the Logic-Computer interface
+%%%% relies
+%%%% explicitly 
+%%%%  upon
+%%%% % uses  
+%%%% Enderton's  
+%%%% % underlying 
+%%%% apparatus mechanism.)
+%%%% 
+%% %uses
+%% relies upon
+%%  Enderton's 
+%% approach   $d_E$.)  
+
+%% %relies 
+%% does rely
+%% upon 
+%% Enderton's apparatus
+\item[  ii. ]
+The  $~d_M~$ 
+apparatus in
+\textsection 2.3 
+of  Mendelson's textbook
+and  the $d_H$ 
+ apparatus
+in  \textsection 0.10
+of the H\'{a}jek-Pudl\'{a}k's
+ textbook
+employ a more compressed set of logical axioms
+than $\, d_E \,$,
+but
+they use
+two rules of inference
+(formalizing
+separately
+ modus ponens and generalization).
+%% plus a smaller set of logical axioms, which Mendelson
+%% has called A1-A5.
+%% Also, the $d_H$ 
+%%  apparatus 
+%% on pages ???? 
+%% of the 
+%%  H\'{a}jek-Pudl\'{a}k textbook
+%% uses a slightly different variation of a generalization. 
+%% (In the end, 
+In the end, 
+% both
+ $~d_M~$ 
+and  $~d_H~$ 
+prove the same set of theorems
+as   $~d_E~$ with 
+only minor and unimportant changes in
+proof length.
+\item[ iii. ]
+The 
+``semantic tableaux''
+ $~d_F~$ 
+apparatus in
+Fitting's 
+and Smullyan's 
+textbooks 
+\cite{Fit,Smul}
+was
+ the main focus of our 
+investigations in \cite{ww93,ww1,wwlogos,ww5,ww6,ww14}.
+It will be rarely used
+in the current article,
+however.
+Unlike
+ $~d_E~$,  $~d_M~$  and  $~d_H~$, it
+employs no logical axioms.
+It instead
+ uses a more complicated rule of inference.
+This tableaux apparatus 
+% and also Resolution, have  been 
+ has 
+% been found to  have many 
+a wide array of
+applications
+for automated deduction,
+although it is 
+less efficient than
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+in
+% under
+% extremal
+worst-case 
+environments.
+% settings.
+%circumstances.
+\ennd
+\end{exx}
+
+
+\begin{dff}
+\label{def-2.2}
+\rm
+Each of the
+% deductive 
+methods of
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+have the property that if a theorem $\, \Psi \,$
+has a proof
+with  length $~L~$ 
+ from an arbitrary 
+axiom basis $~\alpha~$ 
+under one of these deductive systems,
+then it will have a proof from these other formalisms
+with lengths bounded by Polynomial$(L)$.
+The term 
+{\bf ``Hilbert-style''} deductive method will,
+thus, refer to any  deductive 
+% apparatus will refer to any other
+apparatus $~d~$ that
+employs
+% similarity has its 
+proof lengths
+% being 
+equivalent to within a polynomial magnitude
+to
+%of
+the comparable proof lengths from $d_E$,  $d_M$  and  $d_H$
+and which 
+also
+assures 
+that the proofs of any
+two theorems $~\Phi~$ 
+and $~\Psi~$ 
+(under $d$ from any
+axiom basis $~\alpha~$) 
+will
+always 
+have
+% by more than a constant factor 
+the sum of the lengths of the proofs
+of $~\Phi \rightarrow \Psi ~$ and $~\Phi~$ 
+% under $~d~$ from $\alpha$ always 
+formally
+bound the length of
+$~\Psi\,$'s proof. 
+\end{dff}
+
+
+\begin{exx}
+\label{ex-2.3}
+%\label{ex-basis}
+\rm
+Some added notation is
+ needed to
+explain why
+% help outline
+% an important distinction between 
+a Hilbert style
+deductive apparatus, such as $\,d_E\,$,  $\,d_H\,$
+ or $\,d_M\,$, should be distinguished from
+ $d_F$'s
+``tableaux'' apparatus.
+Let
+% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$
+once again 
+% will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us recall 
+that 
+an
+axiom basis
+ ``$\, \alpha \,$''
+is said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+%\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+
+\noindent
+%Then an 
+In this context, an
+``axiom basis''
+$\alpha$
+will be called 
+{\bf Type-M} if it contains
+\eq{totdefxs}-\eq{totdefxm}
+% \ref{totdefxs}-\ref{totdefxm}
+as theorems,  
+{\bf Type-A} if it contains 
+%only
+\eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+$\alpha$ 
+%will be 
+is
+called 
+{\bf Type-NS} if it  can prove
+none of these theorems.
+%In this context,
+%The
+%Items (a) and (b) illustrate 
+%%%Below are illustrated several
+%%%implications of this notation:
+The implications of this notation
+are formalized by Items a and b:
+
+%% 
+%% 
+%% , below, 
+%% %will 
+%% illustrate how
+%% a
+%% %% 
+%% %%  the
+%% %% prior
+%% %%  literature has
+%% %% 
+%% %% 
+%% ``Hilbert-style''
+%% deductive apparatus, such as $\,d_E\,$
+%%  or $\,d_M\,$, supports very different generalizations
+%% of the Second Incompleteness Theorem
+%% than  $\,d_F\,$'s
+%% ``tableaux-style'' apparatus:  
+%% 
+%%  the prior literature most germane
+%% to our current article is summarized as follows:
+%% 
+%% 
+%% The relationship of these constructs to 
+%% self-justification 
+%% is explained by
+%% items (a) and (b):
+\bed
+\item[   a.  ]
+The
+%%  
+%% above 
+%% evasions of the Second Incompleteness
+%% Theorem are known to be near-maximal in a mathematical sense.
+%% This is because
+%% the
+%% 
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as formalized by statement $\, ++ \,$,
+%has  implied  
+implies
+no
+natural 
+Type-S system can recognize  its
+own  consistency
+under
+any of $\, d_E \,$'s, $\, d_H \,$'s  
+or
+ $\, d_M \,$'s
+  Hilbert-style 
+versions
+of deduction:
+\begin{quote}
+{\bf ++ }
+%\footnotesize
+ \baselineskip = 1.05 \normalbaselineskip 
+{\it 
+(Solovay's  
+modification
+%Generalization 
+\cite{So94}
+%1994 Generalization \cite{So94}
+%of a 1985 theorem 
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+with
+%using 
+%some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let $ \, \alpha \, $ denote 
+%%! any consistent
+% a logic
+an axiom  
+basis  
+% axiom system 
+%basis  system 
+supporting
+% which contains 
+\eq{totdefxs}'s
+Type-S statement
+and 
+assuring
+%which assures 
+%that 
+the successor operation
+%always 
+does satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+cannot verify is
+%will be unable to recognize its
+%own  
+consistency
+under any Hilbert-style 
+%deductive 
+apparatus $d$,
+%whenever
+if  it
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+the usual % identity,
+associative, commutative 
+ distributive 
+and identity axioms.
+%   -axiom
+% properties.
+\end{quote}
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of $++$'s result. 
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+ by  Buss-Ignjatovic,
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k initial 1985 article  \cite{Pu85} 
+% implicitly
+captured 
+the majority 
+%most
+%%%  much 
+of $++$'s 
+main
+% underlying
+formalism,
+and
+Friedman did some
+related work
+% in
+\cite{Fr79a}.
+
+\item[   b.  ]
+Part of what makes
+the Pudl\'{a}k-Solovay discovery in
+ $++$ interesting is that 
+\cite{ww1,ww5,wwapal}
+%Willard
+developed two 
+% separate
+methods for 
+basis systems
+%%% $\alpha$ 
+to confirm their own consistency, whose
+natural hybridizations is precluded by $++$. These results involve
+either a Type-NS
+% basis 
+system 
+\cite{ww1,wwapal}
+ verifying its own consistency
+under 
+any  of the 
+ $d_E$ or  $d_H$
+ or $d_M$'s
+Hilbert-style methods,
+or   a Type-A 
+%basis 
+system \cite{ww93,ww1,ww5,ww6,ww14} 
+verifying
+ its
+% own 
+self-consistency
+under $d_F$'s tableaux 
+%deductive 
+apparatus.
+Also, Willard \cite{ww2,ww7} observed how one could
+refine $++$ with Adamowicz-Zbierski's
+methodology \cite{AZ1} to show 
+ Type-M  systems
+cannot recognize their semantic tableaux consistency.
+\ennd
+\end{exx}
+
+The roles of
+% Observations
+Items (a) and (b)
+in our research
+%from the current example, 
+will become more evident
+as this article progresses. 
+Essentially, our
+prior research
+% ,
+% best summarized in \cite{ww14},
+%  has 
+had focused mostly on Type-A arithmetics
+that could verify their consistency under either semantic
+tableaux deduction or some near-cousin of this concept
+(as was explained in \cite{ww14}'s short 16-page summary
+of \cite{ww93}-\cite{ww9}'s results).
+The 
+new
+$~\theta~$ operator, defined in the next section, will
+raise the question about whether a
+surprisingly powerful class of new Type-NS systems may 
+satisfy an analogous
+property in the context of
+Definition \ref{def-2.2}'s more pristine
+Hilbert-style methodology for deduction. 
+
+
+% have
+% a similar property.(The article \cite{ww14} offers a nice 16-page summary
+% of our prior results 
+% \cite{ww93}-\cite{ww9}
+% about Type-A arithmetics, but 
+% none of these results will
+% be
+%  needed
+% %  to be examined 
+% during our current article's exploration of 
+% the properties of the new % 
+% $~\theta~$ operator.)
+
+% It will be unnecessary for a reader to examine any of our
+
+ 
+%% % year-2014 
+%% Wollic-2014 paper \cite{ww14}
+%% summarized and extended our
+%% results about
+%% semantic tableaux consistency.
+%% % and this 
+%% The current 
+%% new 
+%% year-2015
+%% paper 
+%% will, now,
+%% explore whether
+%%  systems  can
+%% also corroborate
+%% their Hilbert-styled consistency
+%% under certain well-defined circumstances.
+
+%% (and seek to explore the restrictions $++$ imposes upon
+%% Hilbert-styled deduction).
+
+% 
+% (The latter topic
+% % is 
+% %very 
+% %entirely
+% %different 
+% differs
+% from the former
+% because
+% constraint  $++$ 
+% applies only 
+% to its
+% particular
+%  domain.)
+
+%in the second context.)
+
+
+%% The constraints imposed by $++$ 
+%% are challenging 
+%% because Type-NS arithmetics 
+
+This
+topic
+% subject 
+is challenging because 
+essentially all
+Type-S arithmetics
+are forbidden by $++$ from
+verifying the consistency
+of their own Hilbert-styled deductions
+(and 
+conventional forms of
+Type-NS formalisms are
+typically 
+% usually 
+quite weak).
+%% 
+%% (Thus, our efforts
+%% to design
+%% self-verifying systems
+%%  must focus on
+%% Type-NS arithmetics).
+%% 
+Our new Propositions \ref{th-3.3} and \ref{th-6.1}
+will suggest a 
+plausible partial
+solution to this 
+problem by
+% daunting challenge by
+illustrating how
+an {\it unusual class} of 
+Type-NS arithmetics will be able to construct the
+full set of integers $~0,1,2,3,...~$ by finite means 
+{\it without using}
+any of the successor, addition or multiplication
+function symbols. 
+As a result, we will suggest 
+a
+%  a {\it part-way} 
+% that an interesting
+non-trivial 
+(although diluted)
+fragment
+of what Hilbert
+and G\"{o}del
+% sought 
+% referred to
+sought
+in 
+statements 
+$*$ and $**$ 
+% will
+%  be formally  achieved 
+% become tempting 
+is likely
+% be 
+viable
+under Definition \ref{def-2.4}'s formalism.
+
+
+
+% 
+% This
+% topic
+% % subject 
+% is challenging because 
+% $++$'s 
+% Type-NS arithmetics 
+% %  obviously 
+% have sharply circumscribed powers
+% (demonstrating the 
+% broad reach
+% % ubiquitous  nature 
+% of
+% %the Second Incompleteness Theorem's reach).
+% G\"{o}del's 
+% second theorem).
+% %%  
+% %% The current article will
+% %% show, however,  that 
+% %% some Type-NS arithmetics are 
+% %% substantially
+% %%  stronger than previously 
+% %% anticipated
+% %% (and they will have useful applications 
+% %% in
+% %% computer science settings).
+% %% 
+% %% Thus in a context where 
+% %% the power of both G\"{o}del's initial
+% %% Second Incompleteness Theorem and $++$'s strengthening of it
+% %% are stunning
+% %% and
+% %% have pervasive implications,
+% %% we will show that a 
+% %% {\it partial-and-much-less-than-full}
+% %% fragment 
+% %% of what Hilbert
+% %% desired
+% %%  in statements $*$ and $**$ an be
+% %% positively  achieved.
+% %% 
+% The current article will
+% %show, however, 
+% suggest,
+% however, 
+% % that 
+% some Type-NS arithmetics are
+% % , however,  
+% % significantly 
+% %% substantially
+% more far-reaching
+% than 
+% previously 
+% anticipated. Thus,  a 
+% % well-defined 
+%   {\it
+% partial but non-trivial} fragment of what Hilbert
+% and G\"{o}del
+% % sought 
+% % referred to
+%  anticipated
+% in 
+% statements 
+% $*$ and $**$ will
+% %  be formally  achieved 
+% % become tempting 
+% look
+% % be 
+% viable
+% under Definition \ref{def-2.4}'s formalism.
+
+
+\begin{deff}
+\label{def-2.4}
+\rm
+Let
+$~\alpha~$ again 
+denote an axiom basis 
+and $~d~$ 
+designate
+ a
+deduction apparatus.
+Then the  ordered pair
+ $~(  \alpha  , d  )$
+will
+be called  {\bf Self Justifying} when:
+\begin{description}
+% \xxitch
+% \small
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+(or at least one of its axioms)
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+\end{deff}
+
+
+\begin{exx}
+\label{ex-2.5}
+\rm
+Using 
+Definition \ref{def-2.4}'s
+ notation, our research  
+in
+\cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+developed
+%\cite{ww93}-\cite{ww14}
+%has  consisted of
+% developing
+ordered pairs  $~(  \alpha  , d  )$
+that 
+were
+%are
+``Self Justifying''.
+It 
+%  has 
+also explored
+how the Second Incompleteness Theorem formalizes
+limits beyond which such formalisms cannot transgress.
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+% second
+%%% axiom 
+system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition.
+%of 
+% this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+%%%
+%%% the following 
+%%% %%% added
+%%% further
+%%% sentence,  
+%%% called
+%%% %%% that we call
+%%% {\bf SelfRef$(\alpha,d)~$}:
+\begin{quote} 
+%\xxitch
+$\oplus~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of this
+{\bf $\,$``SelfRef$(\alpha,d)$''$\,$}
+statement.
+%%% SelfRef$(\alpha,d)$'s
+%%% self-referential statement.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition.
+This is because if the 
+%  ordered
+ pair $(\alpha,d)$ is too strong
+then a classic G\"{o}del-style diagonalization argument can
+be applied to the axiom system 
+$\alpha^d~~=~~ \alpha \, + \, $ SelfRef$(\alpha,d)$,
+where the added presence of the statement 
+SelfRef$(\alpha,d)$ 
+will cause this extended version of 
+$\, \alpha\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, the machinery of the sentence
+``SelfRef$(\alpha,d)$'' is relatively easy to 
+encode,
+%make well-defined 
+via an application of the Fixed Point Theorem,
+but it
+is
+ironically 
+%%%%%{\it most often  
+{\it 
+typically
+%usually
+useless! }
+\end{exx}
+
+%\newpage
+
+
+Unlike our earlier work, which focused 
+ mostly around a 
+semantic
+tableaux apparatus for deduction,
+the current paper
+will
+explore
+%paper will explore
+\dfx{def-2.2}'s
+more pristine Hilbert-style methodologies.
+%% 
+%% analogous to 
+%% Example
+%% \ref{ex-2.1}'s 
+%% textbook
+%% methods.
+%% 
+% of 
+% $d_E$,  $d_M$
+% and  $d_H$.
+%%! 
+%%! in
+%%! the textbooks by 
+%%!  Enderton,  H\'{a}jek-Pudl\'{a}k,
+%%! Mendelson and Papadamiriyou \cite{End,HP91,Mend,Papa}.
+There are, of course, many types of generalizations
+of the Second Incompleteness Theorem known to
+arise in  Hilbert-like  settings
+\cite{BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Lo55,Kr87,Kr95,Pa71,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,wwapal}.
+Each such
+generalization
+formalizes
+a paradigm where 
+self-justification is infeasible
+under a Hilbert-style apparatus.
+
+\smallskip
+
+Our 
+main
+prior research about
+%%! 
+%%! main work about arithmetics displaying knowledge
+%%! about their 
+%%! 
+Hilbert consistency appeared in \cite{wwapal}.
+Its ISCE$(\beta)$ 
+formalism could recognize its own 
+Hilbert consistency and 
+%%could 
+prove analogs of any 
+r.e. 
+extension $~\beta$ of
+Peano Arithmetic's $\Pi_1$ theorems.
+It unfortunately required
+the use of an infinite number of constant symbols, with
+ ISCE$(\beta)$ 
+using one built-in constant
+symbol 
+$~C_i~$,
+for each power of 2. An alternative in  \cite{wwapal},
+called ISINF$(\beta)$, required use of only three constant symbols,
+but its proof lengths were impractically long. 
+
+
+%%! required
+%%! % in excess of
+%%! an impractical
+%%!  $O(N)$ length proof to construct an integer $N$.
+
+\smallskip
+
+Prior to \cite{wwapal}'s publication 
+Pavel Pudl\'{a}k
+\cite{Pupriv}
+examined this article and asked 
+us
+the crucial question about 
+whether
+we could improve 
+upon ISINF's properties
+ by using Ajtai's observations
+\cite{Aj94}
+ about
+the Pigeon Hole principle.
+%%  
+%% Sam Buss \cite{Bupriv} also asked us
+%% a 
+%% %similar 
+%% related
+%% question
+%% (during a more 
+%% informal 
+%% %abbreviated
+%% conversation).
+%% %(in a more informal manner).
+%% 
+Our prior
+partial
+answer to  Pudl\'{a}k's
+question
+was offered  
+%issue
+%appeared
+%in Sections 6 and 7 of \cite{wwapal}.
+in Sections 6 of \cite{wwapal}.
+A very different type of reply will be offered
+% 
+% We will offer 
+% % an alternate much  
+% a 
+% % much
+% more sophisticated
+% and different 
+% type of reply
+% % analysis
+% %  formalism 
+% 
+in
+the current 
+paper.
+%article.
+
+
+%%! an abbreviated version of a similar
+%%! question after we verbally summarized to him \cite{wwapal}'s
+%%! planned results.
+
+% in 1997.
+
+
+\begin{deff}
+\label{def-2.6}
+\rm
+Throughout our discussion, a
+% A
+primitive $~F~$ will be called a 
+{\bf Q-Function} 
+iff is is 
+sufficiently ambiguous
+for there to exist an UNCOUNTABLY infinite number of
+different 
+{\it 
+plausible  sequences} of
+ordered pairs in expression \eq{wow} where
+$~F(i)=a_i~$  is allowed as a
+% logically 
+permissible
+%plausible 
+%formalization 
+representation of $F$
+under some fixed axiom system $~\gamma~$. 
+\begin{equation}
+\label{wow}
+  (0,a_0) 
+ ~,~  (1,a_1)   ~,~  (2,a_2)   ~,~  (3,a_3)   ~,~  (4,a_4)~ ...
+\end{equation} 
+\end{deff}
+
+\gvs
+
+%It turns out most
+ 
+Most
+Q-Function symbols are
+unsuitable for 
+analyzing
+%producing a positive resolution to
+Hilbert's  Second Open Question or most 
+issues in
+% other prominent
+% % mathematical 
+% questions within 
+mathematics.
+ This is because the
+presence of an 
+ uncountably
+ infinite
+  number of
+different 
+plausible  sequences,
+formalized by Line
+\eq{wow} for solving
+$~F(i)=a_i~,~$  is 
+typically more of a burden than a benefit.
+An exception to this general rule
+of thumb
+ will be
+provided by
+ the next
+section's $~\theta~$ operator.
+It will be germane to 
+ $\, ++ \,$'s generalization of the Second Incompleteness
+and suggest a mechanism whereby an efficient form of
+``Type-NS''self-justifying 
+arithmetic
+can recognize its own Hilbert consistency,
+without
+viewing
+% recognizing 
+%%%%%% any of  addition, multiplication and
+even
+successor  as
+a total function.
+
+
+%% 
+%% It will enable us to develop ground terms for formulating
+%% any integer $~N~$ using
+%%  $O\{~$Log$(N)~\}~$ 
+%% logical symbols,
+%% in a context where
+%% {\it  none of the} addition, multiplication or
+%% successor  function symbols are employed
+%% by $~\theta \,$'s analog of an 
+%%  $O\{~$Log$(N)~\}~$ 
+%% lengthed 
+%% binary-like 
+%% encoding 
+%% for integers.
+%% % of an integer as a binary number. 
+%% This  alternate
+%%  $O\{~$Log$(N)~\}~$ 
+%% format
+%% for encoding an integer $~N~$ is 
+%% potneitlally useful
+%% %fascinating
+%% because  
+%% Item $\, ++ \,$'s generalization of the Second Incompleteness
+%% Theorem, due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% does not preclude evasions of its invariant when a
+%% ``Type-NS''
+%% axiom system ceases to recognize 
+%%  addition, multiplication and
+%% successor  as total functions.
+
+
+ 
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question.
+% A special class of Q-Functions
+% %  , however,
+% will
+% walk through
+%   Cantor's 
+% % the
+% world of the Uncountably Infinite
+% %% in a more 
+% in an
+% enticing manner,
+% however.
+% %% however.
+% %It 
+% They
+% will suggest a
+% %%% much
+%  {\it  diluted but non-trivial}
+% variant
+% of the aspirations
+% which 
+% %that 
+% Hilbert and G\"{o}del 
+% expressed
+%  in
+% $*$ and $**$ 
+% are 
+% %applicable to
+% feasible under
+% % plausible in the context of 
+% Hilbert Deduction.
+% (This
+% % Q-function a
+% analysis will be 
+% %%%%%%%%%%%%%% quite
+% % entirely 
+% different
+% from 
+% \cite{ww14}'s 
+% examination of
+% %formalisms
+% %the formalisms appearing in our Wollic-2014 paper
+% %%% because 
+% %it will replace 
+% semantic tableaux deduction
+% because it will
+% apply  
+% uniquely 
+% to 
+% Definition \ref{def-2.2}'s 
+% ``Hilbert-styled'' deduction methods.)
+
+ 
+%%% is replaced by the more efficient
+%%% %with the more pristine 
+%%% Hilbert-style deductive methodology.)
+
+%can be achieved.
+
+
+%%! This will suggest
+%%! a {\it limited}  
+%%! and very-much {\it down-sized} version of the formalism that
+%%!  Hilbert
+%%! advocated
+%%! is
+%%! likely
+%%! %probably
+%%! feasible
+%%! and 
+%%! germane to 
+%%! the
+%%! future
+%%! %  computational 
+%%! needs of automated
+%%! theorem provers.
+%%! Our 
+%%! exploration 
+%%!  will also provide a
+%%! % quite
+%%!  new interpretation of the
+%%! meaning of the statements $*$, $**$ and
+%%!  $***$.
+
+% by Hilbert and G\"{o}del.
+
+\vspace*{- 0.6 em} 
+
+% \section{Revisiting a World which Hilbert called
+% {\it ``Cantor's Paradise''}}
+
+\section{Main Formalism}
+\label{ss4}
+\label{seee3}
+
+%333333333333333333333333333
+\vspace*{- 0.4 em} 
+
+% OLD Title was {\it Notation and Basic Concepts}
+
+Throughout this 
+paper,
+%article,
+a function 
+ $\, H \, $ 
+will be     called
+ {\bf Non-Growth} 
+iff  
+$ H(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$.  Six  examples of  
+ non-growth functions are:
+\bee
+\parskip 0pt
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lfloor ~x^{1/y}~ \rfloor$ when $~y\geq 1~$
+%% 
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% 
+(and zero otherwise),
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+and
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\ene
+%% 
+%% 
+%% \bee
+%% \baselineskip = 0.8 \normalbaselineskip 
+%% 
+%% \item
+%% {\it Integer Subtraction} 
+%% where ``$~x-y~$'' is defined to equal zero 
+%% in {\it the special case} where
+%%  $~x \leq y,$
+%% \item
+%% {\it Integer 
+%% Division}
+%% where ``$~x \div y~$'' equals
+%% $~x~$ when $~y=0,~$ and
+%% it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+%% \item
+%% $Root(x,y)~$ which equals $~ \ulcorner ~x^{1/y}~ \urcorner$ when $~y\geq 1,~$
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% \item
+%% $Maximum(x,y),~~$
+%% \item
+%% $ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+%% and  zero otherwise. 
+%% \item
+%% $Count(x,j)~~=~~$the number of ``1'' bits
+%% among $~x$'s rightmost $~j~$ bits.
+%% \end{enumerate}
+%% 
+These operations were called
+either the 
+{\bf ``Grounding''} 
+or {\bf ``Ground-Level''} 
+ functions
+in our articles \cite{ww1,wwlogos,ww5,ww14}.
+We will
+use the latter nomenclature in the current article
+ because the notion of a  ``Ground-Level''
+function should not be confused with the very different notion
+of a ``Grounded Term'' employed by Definition
+\ref{def-3.4}. 
+
+%  TWO DEFS or ONE ???????? 
+
+
+
+Our
+starting  language $L^G$  shall also contain
+the
+two atomic 
+symbols
+%% relations 
+of ``$~=~$'' and ``$~\leq~$'' and three
+built in constants symbols, $~C_0~$, $~C_1~$ and $~C_2~$, 
+for representing
+the values of 0, 1 and 2. 
+Within this context, expressions 
+\eq{newadd} and \eq{newmult} formalize how addition and multiplication
+can be encoded as two 3-way predicates,
+%%  in  $L^G$,
+ denoted as
+Add$(x,y,z)$ and Mult$(x,y,z)$.
+
+\vspace*{- 0.6 em} 
+\beq 
+\label{newadd}
+z ~ -~x~~=~~ y~~~~ \wedge ~~~~ z~\geq~x
+\end{equation}
+
+\vspace*{- 0.6 em} 
+\begin{equation}
+\label{newmult}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+Also, 
+our constant symbols,
+ $~C_1~$ and $~C_2~,~$ for representing
+the quantities 1 and 2, 
+ allow us to
+formalize the following further
+%useful 
+%%%%%%%% function 
+operations:
+\bed
+\small
+\topsep -4pt
+\itemsep -1pt
+\it
+\item[   $~$a.  ] 
+Pred$(x)~~=~~x-1$
+(in a context where 
+%Item 1's
+%the prior paragraph's
+our prior
+definition for Subtraction implies
+Pred(0)$=0$.$~)$   
+\item[   $~$b.  ] 
+Half$(x)~~=~~x \div 2$
+(in a context 
+where 
+``$ ~x \div 2~$'' equals technically 
+$~\lfloor \, \frac{x}{2} \, \rfloor~$ 
+under 
+%our
+the 
+%preceding 
+prior
+% 
+paragraph's 
+notation).
+% of Item 2's definition for Division).  
+\item[   $~$c.  ] 
+Pred$^n(x)$ defined to be $~n~$ iterations of the
+Predecessor operation
+\item[   $~$d.  ] 
+Half$^{ \,n}(x)$ defined to be $~n~$ iterations of the
+halving operation.
+%%
+%%\item[   $~$e.  ] 
+%%Bit$(x,j)~=~$ $Count(x,j)~-~Count(x,j-1)~$
+%%(e.g. the value of $~x \,$'s rightmost
+%% $j-$th bit)
+%%\item[   $~$f.  ] 
+%%Min$(x,y)~~=~~x~-~(y-x)~~$
+%%(e.g. a quantity that represents the minimum of $x$ and $y$
+%%because our definition of ``integer subtraction'' 
+%%implies  $~y-x \,= \, 0 ~$ whenever $\, x \geq y$.$~~)$  
+%%
+\ennd
+
+
+
+Let us say
+that
+ a function symbol $H(x_1,x_2...x_j)$
+is
+ {\bf Growth Permitting}
+iff for each integer $~k \geq 2~$ there exists a 
+``growth-tuple''
+$(a_1,a_2...a_j)$ 
+satisfying
+  $~H(a_1,a_2...a_j) \, >\, k~$
+and 
+also
+% simultaneously 
+having
+each $ \, a_i \leq k$ \, 
+It will  be necessary 
+%% for  us 
+to employ
+ either an infinite number
+of constant symbols or some Growth-Permitting function 
+so that an extension of the 
+language $L^G$ can construct the 
+full
+%infinite
+collection of
+ integers of $~3,4,5,6~....~$.
+
+
+
+
+%% One  awkward aspect of this notation is that 
+%% it provides
+%%  no guarantee
+%% that    integers larger than 2 will exist         without the 
+%% presence of some 
+%% further
+%% methodology for producing larger integers.
+
+
+% 
+\smallskip
+
+
+One method for resolving this problem was presented in \cite{wwapal}.
+% 
+% It employed  an infinite number of further constant symbols. The
+% latter's
+%  ISCE$(\beta)$ 
+%  system was
+% %   shown to be 
+% compatible with                     self-justification,
+% but such an infinite number of constant symbols clearly trespassed on
+% Hilbert's goal of using a 
+% %strictly 
+% finite-sized formalism.
+% 
+Its  ISCE$(\beta)$ axiom basis
+ employed  an infinite number of further constant symbols. It
+was
+compatible with                     self-justification,
+but deviated from 
+%{\it very sharply from} 
+Hilbert's
+intended
+ goals 
+because it employed
+% by employing 
+%an  
+a {\it highly awkward}
+infinite number of 
+distinct
+ constant symbols.
+
+\smallskip
+
+The 
+% self-justifying
+``ISINF'' formalism 
+% in
+of 
+\cite{wwapal} 
+offered an alternate method for resolving this difficulty.
+%% in the context of a self-justifying logic. 
+It
+% required the use of
+used
+ {\it only
+three} constant symbols.  It   could prove analogs of all
+of Peano Arithmetic's
+$\Pi_1$ theorems, but almost all
+of
+ its proofs
+unfortunately
+ had lengths longer 
+than the number of atoms in the universe.
+Most other approaches, for resolving this dilemma, 
+% are 
+were
+also problematic
+because the
+invariant
+ $~++~$,
+which Example \ref{ex-2.3}
+attributed to the joint work of 
+  Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris,
+showed that essentially every Type-S arithmetic is unable to
+recognize its
+own consistency under a Hilbert-style deductive apparatus.
+
+\smallskip
+
+
+The challenge posed by $++$ 
+is, thus, certainly formidable.
+Our goal in 
+%the current 
+this
+article
+will be to suggest 
+how
+%that 
+a Q-function primitive $F(x)$,
+that has
+% an extraordinarily 
+a deliberately
+ambiguous
+ function definition,
+can help overcome the constraints that $++$ imposes. 
+Such an
+% ultra-ambiguous defined 
+unusual
+primitive $~F~$ will have
+an  uncountable number
+of vectors, analogous to Line \eq{wow}, that are permitted 
+solutions to
+$F$'s
+definition.
+Our basic goal
+% in this article 
+will be to outline 
+how this unusual concept is
+likely  germane to the
+self-justifying
+%% 
+%% why it is
+%% likely such Q-functions will enable one to build
+%% surprisingly efficient self-justifying logics that
+%% partially (but not fully) achieve the 
+%% %{\it  diluted portion} of the
+%% 
+aspirations  
+that
+Hilbert and G\"{o}del
+expressed in 
+%statements 
+$*$ and
+$**~$.
+
+\smallskip
+
+%\normalsize \baselineskip = 0.98 \normalbaselineskip
+%\gvs
+
+
+%% \el{wow}'s 
+%% dizzying
+%% $\aleph_1$ distinct solutions.
+%% We will traverse in the opposite direction in this article
+%% because 
+%% Definition \ref{def-2.6}
+%% %\eq{wow} 
+%% formalizes 
+%% % fascinating 
+%% %  a possible
+%% an 
+%% avenue
+%% available
+%% for evading $++$'s generalization of the 
+%% %Second 
+%% Incompleteness
+%% Theorem.
+
+% {\it in at least a partial sense.}
+
+
+%%  
+%% %They 
+%% This level of multiplicity will be
+%% %These will turn out to be 
+%% useful in the present setting
+%% because
+%% %% 
+%% %% Our hypothesis is that such
+%% %% solutions, while awkward and  disadvantageous
+%% %% in many 
+%% %% evident
+%% %% %%%%%obvious 
+%% %% respects,
+%% %% should not be discarded.
+%% %% This is 
+%% %%  because 
+%% %% 
+%% it
+%% can
+%% formalize a type of 
+%% allowed
+%% growth-permitting function,
+%% that is
+%% not prohibited by $++$'s generalization of the Second Incompleteness
+%% Theorem. 
+%% 
+%% 
+
+%\smallskip
+
+\begin{deff}
+\label{def-3.1}
+\rm
+Let us say 
+% an integer-valued
+a function symbol
+$F(x_1,x_2,...x_j)$ is {\bf `` 1-Definitive ''} iff it has only one
+solution under its definition by an axiom system $\gamma$. 
+Let us call $~F~$  {\bf ``Indeterminate''} otherwise.
+%(The remainder of this article 
+(Mathematicians
+obviously typically avoid using 
+function definitions
+%%%%%%%%%, $~F~$ 
+with 
+%that have 
+%%%%  having
+even two solutions, not to
+speak of
+what will be
+\el{wow}'s 
+dizzying
+quantity
+of possibly
+%potential
+%possible
+$\aleph_1$ distinct 
+% number of 
+solutions,
+considered in the current article.
+Our 
+main
+conjecture will be that this unconventional
+approach is 
+germane to the challenge posed
+by  $++$'s 
+broad-scale
+generalization of the 
+%Second 
+Incompleteness
+Theorem.
+This is because our conjecture will be that
+our proposed new $~\theta~$ 
+primitive
+ will represent
+an efficient Indeterminate 
+function
+that eschews  $++$'s prohibitions.) 
+\end{deff}
+
+% helpful when addressing 
+% the challenge 
+% We will traverse in 
+% an unconventional
+% %  the opposite 
+% direction in this article
+% because
+% our conjecture will be that
+% Indeterminate functions 
+% with $\aleph_1$ different 
+% allowed solutions for
+% \el{wow} 
+% formalize
+% %%  
+% %% Definition \ref{def-3.1}
+% %% % {def-2.6}
+% %% %\eq{wow} 
+% %% will formalize 
+% %% a 
+% %% possible
+% %% %feasible
+% %% %plausible
+% %% 
+% an
+% avenue
+% % available
+% for evading $++$'s 
+% broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem.)
+% 
+% (Our conjecture in this article will be that
+% %%
+% %%The next two sections
+% %%will explore
+% %%how 
+% %%
+% \el{wow}'s indeterminate ``Q-Function'' symbol $F$
+% %%%can clarify
+% is germane to
+% the
+% aspirations 
+% that 
+% Hilbert and G\"{o}del
+% % expressed
+% stated
+%  in 
+% % statements 
+% $*$ and
+% $** \, $, $\,$when
+% one employs an 
+% operator
+% $ \, F \, $ 
+% that
+% owns
+% % has
+% $\aleph_1$ distinct 
+% available
+% solutions). 
+% \end{deff}
+
+
+%%are viable,
+%%especially in an automated theorem proving setting,
+%%when Ambiguous function operatives are judiciously
+%%employed.
+%%
+
+%%!  
+%%! can
+%%! %%and germane about automated
+%%! %%deduction, will 
+%%! % theorem proving,
+%%! %computational deduction,
+%%! be satisfied by a self-justifying logic that employs
+%%! one 
+%%! %single $\aleph_1$ 
+%%!   Growth-Permitting function $F(x)$,
+%%! that is
+%%! % inherently 
+%%! ``ambiguous'',
+%%!  {\it accompanied 
+%%! by} a finite number of non-growth primitives
+%%! % {\it  non-growth}
+%%!  $G_1,~G_1,~... G_k$
+%%! that are ``unambiguous''.
+%%! 
+%%! It will also be explained how such results should have useful 
+%%! applications for automated theorem proving, even when they
+%%! employ 
+%%! only
+%%! diluted forms of  self-justifying logics.
+
+%% 
+%% \vspace*{- 1.0 em}
+%% 
+%% \subsection{Main Notation Conventions}
+%% % about Cantor's Paradise}
+%% % \large
+%% % \baselineskip = 1.8 \normalbaselineskip 
+%% 
+%% %\vspace*{- 0.7 em}
+
+{\bf More Notation:}
+$~$Let us say
+an axiom system $~\alpha~$ 
+has 
+{\bf Infinite Far Reach} iff
+it relies upon
+{\it only a  finite number} of
+distinct constant symbols
+(and/or axiom sentences) to 
+% but still can 
+prove 
+for each $n$
+the
+\el{farreach}'s invariant.
+
+%for each particular integer $n$.
+
+\vspace*{- 0.8 em} 
+
+\beq
+%% \small
+\label{farreach}
+\exists ~~x~~~ \mbox{Pred}^n(x)~\geq ~1
+\enq
+The ISINF axiomatic       framework 
+from
+   \cite{wwapal} 
+was
+    a self-justifying
+system with Infinite Far Reach.
+%% 
+%% The opening paragraph of
+%% \cite{wwapal}'s Section 6
+%% %%% quite 
+%% %was frank 
+%% warned the reader
+%% about ISINF's limitations. 
+%% These arose because
+%% 
+%% 
+%% It
+%% used the word ``unnatural'' to describe 
+%% the ISINF system.
+%% Such caution
+%% % deliberately
+%% % self-deprecating term 
+%% was appropriate because
+%% 
+Unfortunately, this result was mostly useless because
+nearly all
+   theorem-proofs
+%of trivial theorems0000000 
+from ISINF 
+were
+longer than the number of atoms in the universe.
+
+The reason
+\cite{wwapal} defined ISINF,  
+%ISINF was worthy of mention,
+despite 
+such
+%%% these
+% plainly 
+%%%  obvious 
+limitations,
+was 
+because
+ISINF 
+demonstrated some self-justifying logics,
+knowledgeable about their own Hilbert consistency,
+were
+% technically 
+able to
+prove all of Peano Arithmetic's $\Pi_1$ theorems
+together with the
+existence of
+the infinite set of integers $ \, 1,2,3,... \,  \, $.
+This result 
+% is interesting because it casts 
+did cast
+% casts 
+a
+new 
+perspective
+%light 
+on  $\,++\,$'s  
+invariant
+% $++$  (appearing on \pag2)
+by showing how some 
+Type-NS
+forms of self-justifying arithmetics
+escape $++$'s almost-ubiquitous 
+ reach
+by managing to possess infinite far reach
+ without taking
+% {\it without recognizing} 
+Successor as a total function.
+
+
+%%  of the current article.
+%% The latter result indicated that Type-S arithmetics, recognizing merely
+%% Successor as a total function, are unable to confirm their own
+%% Hilbert consistency. 
+%% Yet,
+%% %%  despite this fact, 
+%% ISINF was able to produce an 
+%% {\it  eye-squinting} caveat because it 
+%% supported the above ``Infinite Far Reach'' property
+%% without 
+%% needing
+%% %being able to prove
+%%  Line
+%% \eq{totdefxs}'s declaration that successor is a total function.
+
+%\smallskip
+
+We sent an advanced copy of \cite{wwapal}
+to
+Pudl\'{a}k.
+He
+appreciated the nature of the challenge we faced,
+concerning the delicate nature of self-justifying 
+arithmetics that are
+able to prove
+% satisfy 
+\eq{farreach}'s invariant
+{\it for each fixed $~n~$} while 
+being prohibited 
+by $++$
+from
+recognizing  successor as a total
+function.
+% (due to $++$'s restrictions).
+Pudl\'{a}k  
+%private 
+%His
+emailed communications
+\cite{Pupriv}
+suggested 
+that we look at 
+Ajtai's
+work 
+\cite{Aj94}
+about a
+%the 
+Pigeon-Hole function
+ $~ \glamb(x)~$ defined by the identities
+\eq{zm1} and \eq{zm2}.
+
+\newpage
+\vspace*{- 1.2 em} 
+\beq
+%% \small
+\label{zm1}
+\forall ~~x~~~~~ \glamb(x)~ \neq ~ 0
+\enq
+
+\vspace*{- 1.2 em} 
+
+\beq
+\label{zm2}
+%% \small
+\forall ~~x~~~ \forall ~~y~~~~ x ~ \neq~ y ~~ \Rightarrow ~~
+\glamb(x)~ \neq ~\glamb(y)
+\enq
+The relevance of 
+$~\glamb~$ 
+% Pigeon-Hole functions
+can be
+best 
+%readily 
+appreciated
+% if 
+when
+%we let
+$~\glamb^n(x)~$ 
+ denotes
+% the 
+a
+term
+ $~\glamb(~\glamb(~ ... \glamb(x)))~$
+consisting of $~n~$ iterations of the $~\glamb~$ operator.
+Then 
+% the
+\el{DUMB1}'s
+composite
+term $~S_n~$
+% , defined below, shall
+will
+% then 
+satisfy
+Pred$^n(~S_n~)~\geq ~1.~$ 
+%% 
+%% An axiom system, employing the primitive 
+%% operation
+%%  $~ \glamb~,~$ 
+%% can thus 
+%% can easily
+%% prove
+%% Line \eq{farreach}'s 
+%% assertion.
+%% %claim.
+%% %under almost all conventional logics.
+%% 
+%% 
+\beq
+\label{DUMB1}
+S_n~~~=~~~\mbox{Max}[~\glamb(0)~,~\glamb^2(0)~,~\glamb^3(0)~,~...~~\glamb^n(0)~]
+\enq
+Pudl\'{a}k
+observed
+that
+%the
+% Pigeon-Hole function 
+ $~ \glamb(x)~$  
+will
+grow too slowly (in the worst case)
+for
+one to be able to 
+deduce
+successor is a total function
+from its properties  
+% further observed that it is known 
+ \footnote{ \baselineskip = 0.94 \normalbaselineskip
+ The operation $\glamb(x)$ will grow
+at a slower rate than Successor, 
+if it equals $x+1$ for all standard
+numbers $~x~$ and if $\glamb(x)=x-1$ 
+ when $~x~$ is
+a non-standard integer. This seemingly minute detail
+implies  one cannot infer 
+Successor is a total function from
+ $\glamb$'s behavior, since the latter is contradicted by a
+ model  where
+ all  non-standard
+numbers have 
+%their 
+sizes bounded by some fixed  
+% non-standard 
+number B.
+(This 
+subtle detail, 
+raised by
+Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+it 
+%shows that
+raised the question about whether
+ a partial exception to
+Example \ref{ex-2.3}'s
+invariant $++$
+%% on \pag2,  
+might plausibly exist.)  }.
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+His insightful email \cite{Pupriv} asked
+whether 
+the inequality 
+Pred$^n(~S_n~)~\geq ~1~$
+might
+%would,
+thus, 
+% still
+enable a formalism,
+% based around
+utilizing the
+ $\, \glamb \,$ operative, 
+to 
+somehow
+improve upon \cite{wwapal}'s results ?
+
+
+% our
+% formalisms could be 
+% revised
+% %modified 
+% so that 
+% % the  Pigeon-Hole function 
+%  $~ \glamb(x)~$ 
+%  could improve upon \cite{wwapal}'s results.
+
+%%
+%%(possibly using Ajtai's methodologies \cite{Aj-focs}).
+%%Sam Buss raised, interestingly,  a 
+%%partially
+%%similar 
+%%issue during an informal conversation
+%% \cite{Bu-priv} in 1977.
+%%
+%%\smallskip
+%%
+%%These questions
+%%% by
+%%%Pudl\'{a}k and Buss 
+%%were insightful because they isolated 
+%%an
+%%important juncture where $++$'s underlying methodology does not apply.
+%%A partial answer to these questions appeared in 
+%%\cite{wwapal}'s closing section, but a more comprehensive full
+%%answer  has always eluded us.
+
+%This is  because there always seemed to appear
+%one wrinkle of details that precluded a full proof.
+
+
+\smallskip
+
+
+It  was
+initially
+ unclear 
+%%%%% to us
+whether a positive answer to 
+Pudl\'{a}k's
+ probing
+ question would resolve ISINF's main difficulties.
+This is
+because
+% Expression 
+\eq{DUMB1}'s
+term
+$~S_n~$  requires $O(~n^2~)$ logic symbols to encode
+% essentially 
+an integer quantity
+greater than
+ $~n~$
+(since its term
+$~\glamb^j(0)~$ uses $O(j)$ logic symbols).
+%an integer quantity that exceeds the quantity $~n~$ in size.
+Thus once again, the quantity $~2^{100}~,~$ whose binary encoding
+requires 100 bits,  would require in excess of 
+ $~2^{100}~$ bits to encode. 
+Such quantities, exceeding the number of atoms in the universe,
+were troubling because our 
+general
+goal has been to 
+construct self-justifying arithmetics that 
+ possessed, at least,
+some
+partial  facets of
+ pragmatic value.
+
+
+% 
+% find a partial
+% answer to Hilbert's
+% Year-1900 Second
+% Problem  
+%  that would 
+%  possess, at least,
+% some
+% partial  facets of
+%  pragmatic value.
+% 
+
+\bigskip
+\gvs
+
+The remainder of this section will outline how a different type of
+Q-Function operator will 
+be much better than
+ $~ \glamb~$ for meeting our needs.  
+During our discussion,
+Power$(x)$ will denote 
+a primitive specifying
+% that
+ $~x~$ is
+a power of
+$~2~$.  It is 
+%formally 
+encoded
+by 
+\eq{wep2} 
+because
+%under
+our Grounding language
+has
+``Logarithm$(x) \,$''$ ~ = ~ \lfloor \,$Log$_2(x) \, \rfloor \,$.
+\beq
+\vspace*{- 0.6 em}
+\label{wep2} 
+%\small
+x=1 ~~~\vee ~~~ \mbox{Logarithm}(~x~)~\neq~\mbox{Logarithm}(~x-1~)
+\enq
+In this context,
+ $\zzthe(x)$ 
+will denote the analog of
+the $\glamb(x)$ function  
+%% haphazard
+that walks among the powers of 2 in a manner 
+similar to 
+$\glamb(x)$'s
+%  haphazard
+ walk through conventional
+integers. 
+It is 
+formally
+defined by \eq{walk1}-\eq{walk4}.
+% 
+% It will thus satisfy
+% the axiomatic constraints below (which are 
+% $\zzthe(x)$'s analog of the more modest constraints given in
+% % sentences 
+% \eq{zm1} and \eq{zm2}).
+% The most important difference between these two constructs
+% is that axiom \eq{walk1} requires that 
+%  $\zzthe(x)$ maps power of 2 onto powers of 2.
+
+% {\small
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+
+\vspace*{- 1.2 em}
+\noindent
+{\it It needs to be emphasized} that 
+ \eq{walk1} -- \eq{walk4} will be the
+{\it only vehicle} our
+self-justifying axioms 
+%will 
+have available to construct
+integers $\geq \, 3$. $~$These
+axioms will be called
+%They will be henceforth called
+the {\bf $~$Up-Walking$~$} axioms. 
+(The axiom \eq{walk4} 
+is,
+% does not,
+technically, 
+unnecessary
+ to construct any
+integer  $\geq \, 1\, $, but it is helpful because
+it 
+% allows us to 
+formalizes how our methodology will treat  integers
+which are not powers of 2.)
+
+Both
+the 
+Q-functions 
+% the operators 
+$\glamb$ 
+ and $\zzthe$ are 
+challenging
+%daunting
+because there are a  
+ dizzying
+$\aleph_1$ distinct 
+vectors, analogous to 
+ Line \eq{wow}, 
+that are
+%where their definitions permits 
+representations of these functions.
+%% 
+%% Also, we may combine either operation with our 
+%% language $L^G$'s grounding function-primitives to formulate a term
+%% $~T_n~$ that defines any arbitrary integer $~n~$.
+%% 
+We will soon see that
+there is, however, a
+distinction
+%  major difference 
+between these two concepts
+from a computational complexity perspective.
+
+\begin{definition}
+\label{defx-3.2}
+\rm
+Let $~L^Q~$ 
+and $~L^{Q^*}~$ 
+denote the 
+extensions
+of $~L^G\,$'s Grounding language that contain the
+respective
+additional 
+function symbols of  
+  $\zzthe$
+ and
+$\glamb$. Then 
+$~~L^Q~$ shall be called the {\bf  Q-Grounding} language, and 
+ $~~L^{Q^*}~$ 
+will be called the  {\bf  Q* Grounding} language. 
+\end{definition}
+
+\begin{propp}
+\label{th-3.3}
+In contrast to the
+Q* Grounding language 
+that requires $O(~n^2~)$ function symbols
+for defining a term $~T^*_n~$ for representing the integer
+$~n,~$ the    Q-Grounding language
+%% will need no more than 
+needs
+% uses 
+only
+$O(~$Log$^{ \, 3\,} \,n~)$ symbols to 
+encode
+%formalize 
+a term 
+$~T_n~$ representing
+$~n$.
+\end{propp}
+
+\vspace*{- 1.0 em}
+
+\begin{center}
+% \small
+% Our proof of \phx{th-3.3} 
+\phx{th-3.3}'s 
+proof
+will rely upon the following notation  convention:
+\end{center}
+
+\vspace*{- 0.8 em}
+
+\begin{definition}
+\label{def-3.3}
+\rm
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+% Throughout this article,
+Then
+%for any  $~j \, \geq 1~,~$ 
+%the symbol 
+$~E_j~$ 
+will
+% shall
+% will 
+denote
+the quantity produced by
+\eq{ej-def}'s division operation: 
+
+\vspace*{- 0.6 em}
+
+\beq
+\small
+\label{ej-def}
+ \frac{~\mbox{Max}
+~[~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~~] }
+{~~\mbox{Half}^{\,j\,} ~ \{ ~\mbox{Max}~[
+ ~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~]~ 
+ ~ \}~~ } 
+%
+% \mbox{Max}(~\zzthe^j(1),~\zzthe^{j-1}(1),~... ~\zzthe^1(1)~ 
+\enq
+It is each to see
+$  E_j  =  2^j  $ for every
+$j   \geq 1$.
+This is
+ because \el{ej-def}'s
+twice-repeating term
+object
+ of
+$  \mbox{Max}
+~[~\zzthe^j(1),  \zzthe^{j-1}(1),...\zzthe(1) \,]$
+% is at least as large as $\, 2^j\,$.
+is a power of 2 exceeding  $\, 2^j\,$.
+%% 
+%% The definitions of the
+%% % Q-Grounding 
+%% functions of ``Half'', ``Max'' and
+%% ``$~\zzthe~$''   imply 
+%% $~E_j~=~2^j~$ for each
+%% $j \, \geq 1$. 
+%% 
+For the additional case where $~j=0~,~$
+we will 
+% formally
+define  $~E_0~=~1~$ (by 
+using the 
+%%
+%%setting it equal to 
+%%our
+%%%the
+%%
+built-in constant symbol
+of $~C_1~$).
+\end{definition}
+
+%% , which 
+%% is intended to
+%% %formally 
+%% represent the integer of ``1'').
+
+
+
+{\bf Proof of  \phx{th-3.3}:}
+%The justification of   \phx{th-3.3} is an 
+Easy consequence of
+\dfx{def-3.3}'s machinery. Thus if $~n~$ is a power of
+2 of the form $~2^j~$ then
+% the preceding
+% definition's
+expression $~E_j~$ is a term representing $~n \,$'s value
+that employs
+  $O(~$Log$^{ \, 2\,} \,n~)$ 
+logical
+symbols. On the other hand, if  
+ $~n~$ is not a power of
+2 then it can be defined 
+with   $O(~$Log$^{ \, 3\,} \,n~)$ symbols by 
+setting
+$~E_j~$ equal to the least power of 2 greater than $~n~$ and
+subtracting from $~E_j~$ those powers of 2 that are needed to
+produce $\,n\,$'s  value. 
+For example since $76~=~128~-~32~-~16~-~4~,\,$ it can
+be formalized as a term $T_{76}$ defined by
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$.
+
+% $~~~~\Box$
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\begin{definition}
+\label{def-3.4}
+\rm
+A term in mathematical logic
+is defined to be a syntactic object,  built
+out of
+solely
+ symbols for representing 
+functions,
+constants and variables.
+% 
+% The nomenclature in
+% % classical
+% logic has 
+% %formally 
+% defined
+% a {\it ``term''} to be a syntactic object,  built
+% out of symbols for representing 
+% functions,
+% constants and variables.
+% 
+% 
+Such an object is called
+% either 
+a {\bf ``Ground Term''} 
+%% (or for precision a
+%%% {\bf ``Tree-Oriented Ground Term''} )
+when it  is built {\it out of solely}
+function and
+ constant symbols.
+For example in our Q-Grounding language (which 
+uses
+%owns only
+$  C_0  $, $  C_1  $ and $  C_2  $ 
+as
+built-in
+ constants)
+% symbols), 
+the expression
+%of 
+``$\, C_2-  C_1\,$'' 
+is a 
+% such 
+a Ground term.
+Two more complex 
+examples of 
+Ground terms are
+``Max$(   C_2 ,  C_1 - C_0)$'' 
+and ``Max$( ~\zzthe(C_1)~,~C_2 ~)$''.
+Also,
+expression $~E_j~$ 
+in Line \eq{ej-def}
+should be viewed as 
+a Ground term (when one 
+views
+its
+use of the
+symbol
+ ``1'' as an %informal 
+abbreviation
+for the
+constant
+ ``$~C_1~$''). 
+\end{definition}
+
+
+\begin{remm}
+\label{rem-def-3.4}.
+\rm
+Section \textsection \ref{ss6} 
+and its   Proposition \ref{th-6.1} 
+will technically distinguish between two
+kinds of Ground terms, that it calls the
+%
+%{\bf Comment of Definition \ref{def-3.4}'s Notation:} 
+%We will  distinguish  between two
+%kinds of Ground terms in Section \textsection \ref{ss6},
+%called its 
+{\bf ``Tree-Oriented''} and 
+{\bf ``Dag-Oriented''} formats.
+The latter will differ from a more 
+conventional tree structure
+by having a 
+Directed Acyclic Graph structure replace 
+a logic's
+usual 
+ tree format for defining its quantitative values.
+Our discussion in the next two sections will be
+simplified if we use the shorter phrase of
+{\it ``Ground Term''} 
+to refer to what Section \textsection \ref{ss6} will more
+accurately call a 
+{\it ``Tree-Oriented Ground Term''}.
+(It will turn out
+% that our   
+Proposition \ref{th-6.1} 
+will later explain how Dag-Oriented Ground Terms
+differ from
+their
+ tree-oriented 
+counterparts
+% Ground Terms by allowing us
+%to reduce the 
+by reducing the
+$O(~$Log$^{ \, 3\,} \,n~)$ length of a
+tree-oriented term to a more compact
+$O(~$Log$\, \,n~)$ size.)
+ \end{remm}
+
+
+\begin{definition}
+\label{def-3.5}
+\rm
+A ground term $~T~$ will be called an 
+{\bf ``Observable''} 
+object iff there is
+%{\it only one} 
+an
+unique
+interpretation of its 
+quantified value in the
+%meaning  in our
+Q-Grounding language.
+It 
+%will be
+ is
+called an
+{\bf ``Unobservable''} iff it has multiple 
+%plausible 
+such
+interpretations
+due to $\zzthe$'s ``indeterminate'' definition
+(e.g. see Definition \ref{def-3.1}). 
+\end{definition}
+
+%%% (due to the 
+%%% %uncountably 
+%%% ambiguous nature of 
+%%% % our built-in function 
+%%% $~\zzthe~~$).
+%%% \end{definition}
+
+\begin{exx}
+\label{ex-3.6}
+\rm
+The previously mentioned ground term
+Max$( ~\zzthe(C_1)~,~C_2 ~)$ is an ``unobservable''
+because it can assume any of the plausible integer values
+of $~2 \, , \, 4 \, , \,8  \,  , \,16  \, 
+ \, ... ~$.
+On the other hand, 
+%% expression 
+\eq{xoo} 
+is an ``observable''
+that
+ represents
+ the integer value of ``3''.
+(This is because 
+its 
+twice-repeating
+term 
+``$~\mbox{Max}[ ~C_2~,  ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~$'' is bounded
+below by 4, causing the left and right sides of its subtraction
+operation to differ by 
+% an amount of 
+exactly 3.) 
+\beq
+%% \small
+\label{xoo}
+\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~] ~~-~~
+\mbox{Pred}^{\, 3 \,} \{~\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~\} 
+\enq
+Our notation 
+%thus 
+also
+implies that Line \eq{ej-def}'s
+ expression $~E_j~$ 
+is an
+ ``observable''. This implies, in turn, that 
+ \phx{th-3.3}'s term $~T_n~$ is an  ``observable''
+ employing no more than 
+ $O(~$Log$^{ \, 3\,} \,n~)$ 
+logical
+symbols.
+For example since $76~=~128~-~32~-~16~-~4~,\,$ 
+it follows that $~ T_{76}~$
+corresponds to the term
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$,
+$~$where each $~E_j~$ employs only 
+ $O(~$Log$^{ \, 2\,} \,j~)$ symbols.
+%%%%%\end{exx}
+\end{exx}
+
+
+
+Thus, \dfx{def-3.5} and Example \ref{ex-3.6} have illustrated
+%that 
+how
+the realm of ``observable'' objects is a 
+% very 
+broad and accessible world,
+of 
+non-trivial
+%% pragmatic
+ significance.
+It allows every integer $~n~$ to be represented by a
+% reasonably small 
+term $~T_n~$ with
+% an
+a tight
+ $O(~$Log$^{ \, 3\,} \,n~)$ length
+(in a context where Section \textsection \ref{ss6}'s more elaborate formalism
+will allow us to reduce this length to a yet more
+attractive  $O(~$Log$ \,n~)$ size).
+
+%% 
+%%  We will soon see
+%% how  Arithmetic's conventional $\Pi_1$ sentences can 
+%% also have
+%% % pleasingly
+%%  terse encodings.
+%% 
+
+
+%% Also, all
+%% Arithmetic's conventional logical sentences do have
+%% likewise
+%% % pleasingly
+%%  terse encodings.
+%% % under our notation convention.
+
+
+The distinction between
+``Observables'' and ``Unobservables''
+% ground terms 
+will 
+%also 
+%% 
+%% cast a
+%% delightful
+%% 
+offer a
+ new perspective on
+the aspirations
+% that 
+which
+Hilbert and G\"{o}del 
+expressed
+in 
+their
+statements $*$ and $**$
+under our proposed
+\newline
+2-part conjecture.
+It
+ will suggest
+how the Second Incompleteness Theorem
+can
+%  remain to 
+be seen as a majestic result
+from a purist perspective
+, while
+a {\it well-defined fragment} of
+%their 
+what
+Hilbert and G\"{o}del
+sought in 
+ $*$ and $**$
+%aspirations
+%% in  statements $*$ and $**$
+can 
+likely
+%almost certainly
+be 
+%part-way
+satisfied (in at least a 
+% well-defined 
+limited sense).
+
+
+\begin{remm}
+\label{rem-3.7}
+{\bf (explaining the goals of this paper):$~$}
+\rm
+Let us say 
+that
+a basis axiom system $~\alpha~$ owns
+a {\it ``Finitized Perspective''}  of the Natural Numbers
+if it requires only a 
+{\it finite number} of proper axioms
+to construct the full set of integers
+$~0,1,2,3~... ~$. All conventional arithmetics have this property.
+%It is useful to divide such 
+Such
+logics
+%arithmetics
+fall into two categories,
+called {\it Single} and {\it Double-Formatted} systems.
+%as defined below: 
+They are defined below: 
+ %These constructs are defined below:
+\bed
+\item[   a.  ]
+%An axiom basis $~\alpha~$
+%will be called
+{\bf Single-Formatted Arithmetics} consist of
+axiomatic basis systems
+% $~\alpha~$
+%all of 
+whose 
+%iff all its 
+ground terms are 
+all 
+Observables.
+(Most conventional arithmetics
+%%%% will 
+%fall into this 
+lie in this
+category
+%when the 
+because they
+employ the
+ growth
+%  function
+properties 
+ of
+the  Successor
+operation
+%function
+ in
+% a straightforward 
+%the
+%% a  conventional
+the traditional
+manner.)
+%% since the 
+%% the simple  growth function of Successor
+%% easily
+%% generates
+%% all the natural numbers). 
+%are {\it ``Single-Formatted Formatted''} logics. 
+\item[   b.  ]
+{\bf Double-Formatted Arithmetics} 
+% representing 
+represent
+systems
+%%%consisting of
+%%%%axiomatic 
+%%%logics
+%%%%%basis systems
+whose ground terms 
+may be either
+ Observables
+or Unobservables.
+(Axiomizations
+for Q-Grounded logics 
+%%  of
+%% the
+%% % our
+%% Q-Grounding language
+are
+%%% will 
+%obviously 
+%%% be
+``Double-Formatted''
+because they
+allow $\theta$'s analog of 
+\el{wow}'s function symbol $F$ 
+to have
+an uncountable number of 
+different allowed
+representations).
+% 
+% (Our
+% Q-Grounding language 
+% gives support to such a system.
+% This is because it
+% can  have its function primitives
+% defined by a finite number of 
+% proper axioms,)
+% %axiom-sentences.) 
+\ennd
+The distinction between 
+categories
+%Items 
+(a) and (b) is
+significant
+% important
+because 
+Example \ref{ex-2.3}
+%%%  \pag2 
+%%% had 
+already explained how
+ statement $++$'s generalization
+of the Second Incompleteness Theorem applies to 
+any formalism recognizing Successor as a total function.
+Thus, Item (b)'s Double-Formatted logics 
+are useful, if one wishes to consider alternatives
+to
+%formalism that do not recognize
+successor as a total function.
+%More precisely, 
+In this context,
+Hilbert's 
+%  famous 
+%Year-1900 
+Second 
+Open
+Problem
+can be viewed
+ as a  {\it  2-part question}, 
+composed of sub-queries Q-1 and Q-2:
+%%%%%
+%%%%% {\it  2-part question}.
+%%%%%The separation of Hilbert's question into two parts,
+%%%%%called Q-1 and Q-2, will allow 
+%%%%%%% 
+%%%%%%% This 
+%%%%%%% bipartite
+%%%%%%% distinction 
+%%%%%%% is useful because it
+%%%%%%% can enable 
+%%%%%%% 
+%%%%%the academic community to better
+%%%%%  with
+%%%%% what Hilbert and G\"{o}del were
+%%%%%seeking to accomplish
+%%%%%in 
+%%%%%their
+%%%%%statements
+%%%%%of $*$, $**$ and $***$.
+\bed
+\small
+\item[ {\bf Question Q-1$~~$}] {\it Are any axiom systems
+able to
+ prove 
+theorems 
+verifying
+ their own consistency in a robust sense?$~~$}
+The answer to Q-1 is clearly ``No'' because the combination
+ G\"{o}del's initial 1931 result \cite{Go31} with 
+%the 
+%further
+Hilbert-Bernays's result 
+\cite{HB39} 
+and the Pudl\'{a}k-Solovay invariant $++$
+(from Example \ref{ex-2.3})
+%% \pag2) 
+imply 
+arithmetics of ordinary strength cannot prove
+their own consistency in a robust sense.
+\item[ {\bf Question Q-2$~~$}]
+ {\it Can 
+logic systems
+%arithmetic logics
+%axiomatizations of Arithmetic
+%  , at least, 
+%somehow 
+``appreciate'' 
+% (not formally ``prove'')
+ their
+own consistency in some 
+{\bf REDUCED} sense, that is diluted
+but not fully immaterial?}
+$~~\,$The answer to 
+%question 
+Q-2 is 
+complex
+%%% more complex than Q-1
+%less clear-cut
+because 
+%several types of 
+some
+arithmetics,
+such as \cite{ww93,ww1,ww5,wwapal,ww9,ww14}'s paradigms,
+ can
+formalize
+% ``recognize''
+their 
+own consistency 
+using Example \ref{ex-2.5}'s
+% a 
+Fixed-Point {\it ``I am consistent''}
+axiom.
+\ennd
+
+
+%% %  sentence $\,\oplus\,$.
+%% %%Using
+%% %%%%%%%%%Under
+%% % Using the notation from 
+%% Under
+%% Lines
+%% \eq{totdefxs}--\eq{totdefxm}'s notation,
+%% these paradigms include:
+%% % both:
+%% \bee
+%% \small
+%% \baselineskip = .86 \normalbaselineskip 
+%% \item
+%%  Type-A arithmetics
+%% \cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+%% %capable of 
+%% recognizing their self-consistencies under
+%% either the deductive mechanics of semantic tableaux or one
+%% of its cousins. (See especially \cite{ww14}'s
+%% recent Wollic-2014 paper.)
+%% \item
+%%  Type-NS arithmetics recognizing their Hilbert consistency,
+%% such as the formalisms of \cite{ww1,wwapal}
+%% %further 
+%% improved, possibly,
+%% with the added techniques introduced in
+%% this  article.
+%% \ene
+%% 
+%% \ennd
+
+
+A theme of this article will be that
+% distinction 
+the distinguishing
+between questions Q-1 and Q-2 and between
+Single and Double-Formatted Logics 
+is 
+related
+% likely central 
+to the mystery
+% that has enshrouded 
+enshrouding
+the Second Incompleteness Theorem.
+This is
+%is germane to the aspirations of automated theorem proving
+%will be germane to this article 
+because there 
+%is no doubt
+can be no doubt that
+% can be no question 
+%%%%%%%% that 
+the Second
+Incompleteness Theorem is  fully
+robust
+% result 
+from a purist 
+%pristine 
+mathematical perspective.
+Yet,
+it is still problematic to fully
+% 
+%  simultaneously
+% % at the same time,
+% it is 
+% hard to 
+% entirely
+% 
+dismiss
+ Hilbert's 1926
+suggestion that 
+ some 
+specialized forms of logics should
+%declaration 
+%% 
+%% concerns
+%% in $\,*\,$ 
+%% that 
+%% {\it ``the honor of human understanding''}
+%% requires
+%% examining
+%% % explaining 
+%% % considering
+%% how  logic systems can
+%% 
+possess
+a type of well-defined 
+ knowledge about their
+own 
+internal
+consistency.
+(This is because it is
+highly
+ awkward to explain how and why
+human beings 
+are able to
+%can 
+%manage to 
+motivate 
+their 
+%cogitations,
+cognitive process,
+% themselves to think,
+ if they do not own
+some type of 
+% instinctive
+internal
+knowledge about their own
+ consistency.)
+
+% sufficient
+% % enough 
+%  knowledge about their
+% % own 
+% internal
+% consistency
+% to motivate 
+% cognition.
+
+% Bad change above
+%cogitation.
+% themselves to cogitate. 
+
+%%It is also
+%%especially 
+%%% very 
+%%tempting 
+%%to divide Hilbert's Year-1900
+%%Open Question into its Q-1 and Q-2 separate parts
+%% during the 21st century,
+%%as computers share with humans cogitative abilities.
+%%
+%%Maybe DELETE above sentence ??? 
+\end{remm}
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+The next 
+two
+sections will 
+describe our 
+2-part
+conjecture about how 
+%a
+Double-Formatted Logics are 
+likely to 
+%produce some 
+cast
+new perspectives
+on 
+this topic.
+%the nature of the Second Incompleteness Theorem.
+Before starting this subject, it should be mentioned
+that other unusual interpretations of the Second Incompleteness
+Theorem have followed
+from Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and also
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a preferable Hilbert-style deductive apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspective  
+should
+% certainly,
+ be examined by researchers
+interested in the
+Second 
+Incompleteness Theorem,
+although 
+%but
+it is
+%% 
+% they are 
+unrelated to 
+our particular
+% the next section's 
+%specific analysis of
+type of
+Hilbert-styled self-justifying effects,
+studied in the current article.
+
+
+%% systems 
+%% formalizing
+%% %verifying 
+%% their
+%% own consistency
+%% %%%%%Definition \ref{def-2.2}'s
+%% %%% approximate 
+%% under
+%% Hilbert-styled 
+%% deduction.
+
+
+%deduction.
+%  Hilbert deduction.
+
+%methods.
+%formalism.
+
+
+%% It is,
+%% % They
+%% %are, 
+%% however,  not germane to the next section's
+%% perspective.
+
+%methodology.
+%main formalisms.
+%methods.
+%results.
+
+ % \baselineskip = 1.8 \normalbaselineskip 
+
+%\section{
+%\small 
+%Improving \cite{wwapal}'s Results with a 
+%``Double-Formatted'' Logic  }
+
+\section{First Half of our 2-Part Conjecture}
+
+% \label{ss32}
+\label{ss5}
+
+
+The only aspect of our prior research that will be directly
+related
+to our 2-part conjecture is the ISCE axiom system,
+defined in \cite{wwapal}'s Sections 3 \& 4.
+The next several paragraphs will review 
+\cite{wwapal}'s results, for the reader's convenience.
+%% 
+%% This section will
+%% review \cite{wwapal}'s results in sufficient detail
+%% so that a reader need not examine \cite{wwapal}'s formal 
+%% text,
+%% 
+%% %%%%%definition of the ISCE axiom system. 
+%% 
+%% During our discussion, 
+%% 
+
+
+During our 
+discussion,
+%review of \cite{wwapal}'s results,
+$~L^G~$ will once again denote
+our Grounding-level language built out of our six
+non-growth functions,
+consisting
+ of 
+the
+Subtraction, Division,
+Maximum, Logarithm, Root and Count operations.
+Also, $\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ will denote
+three constant symbols designating the
+integers values of 
+``0'', ``1'' and  ``2''.
+In a context where Pred$(x)$ is an abbreviation for
+``$\,x \,- \, 1\,$''
+(or more precisely  
+``$\,x \,- \, C_1\,$'' ), 
+the ISCE axiom system from \cite{wwapal}
+used
+\eq{start}'s axiom
+ statement 
+to define
+ $\,C_0\,$, $\,C_1\,$ and $\,C_2~$:
+% these three constants:  
+\begin{equation}
+\label{start}
+\mbox{Pred}(    C_0    )  =  C_0~ \, \wedge ~ \,
+C_1 \neq C_0~ \, \wedge ~ \,
+\mbox{Pred}(    C_1    )  =  C_0 ~ \, \wedge ~ \,
+\mbox{Pred}(    C_2    )  =  C_1 
+\end{equation}
+%Also,
+The  challenge 
+\cite{wwapal} 
+faced was its formalism could
+not use any of the
+%  conventional 
+function-operations of
+successor, addition or multiplication to infer the existence
+of larger integers from the initial constants of 
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$. This was because
+the Pudl\'{a}k-Solovay result $++$ 
+indicated
+  the 
+presumption  successor is a total function 
+precludes   
+most
+%axiom 
+systems
+from recognizing their
+own Hilbert
+consistency.
+
+Our article
+\cite{wwapal} 
+considered two alternatives
+%to a conventional Successor 
+%function symbol 
+for overcoming these difficulties,
+ called
+the  {\bf Additive} and  {\bf Multiplicative Naming}
+conventions.
+They defined 
+some
+further constant symbols $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+where 
+%respectively
+$~C_j~=~2^{j-1}~$ and $~C^*_j~=~2^{\,  2^{ \,j-2}}~$.
+
+The definition of these
+% new 
+constants
+%  symbols
+is 
+easy
+%straightforward
+under $L^G\,$'s
+% Grounding-level 
+language.
+% called $L^G~,~$
+%all of whose function objects are non-growth primitives. 
+This is because
+Lines
+\eq{newadd}  and \eq{newmult}
+%had 
+specify how
+% that
+two 3-way predicates, called
+Add$(,x,y,z)$  and  Mult$(,x,y,z)\,$, 
+%can 
+%do
+encode the identities of
+% can be encoded to specify,  respectively,
+$x=y+z$ and $x*y=z$.
+Our additive and multiplicative 
+% naming 
+conventions
+can,
+%will,
+ then, define 
+ $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+%by  using
+via
+an infinite number of instances  of
+%utilizing respectively
+ \eq{addcov} and
+\eq{multcov}'s
+%{\it infinitely long}
+axiom schemas:
+% two infinite schemas of  axiom-sentences:
+%% 
+%% that belong to our 
+%%  ``Additive'' and  ``Multiplicative
+%%  Naming Conventions'',
+%% then the values for 
+%% $~C_j~$ and 
+%% $~C^*_j~$ can be easily derived from $j-2$ instances of
+%% %respectively \eq{addcov} and \eq{multcov}'s 
+%% these
+%% schema:
+%% 
+
+
+{
+%\small
+\beq
+\label{addcov}
+\mbox{Add}(~C_{j-1}~,~C_{j-1}~,~C_{j}~)
+\enq
+\beq
+\label{multcov}
+\mbox{Mult}(~C^*_{j-1}~,~C^*_{j-1}~,~C^*_{j}~)
+\enq}
+The methodology in
+ \cite{wwapal} 
+%% employed \eq{addcov}  and  \eq{multcov}'s schema in a context where it 
+presumed
+% assured
+%the Y of 
+the ``names'' for its constants $ C_j $ 
+and $ C^*_j $ 
+had nice compact encodings using $O(~Log(j)~)$ bits.  
+Its formalism calculated
+%, thereby, 
+the values of ``unnamed'' integers from 
+named entities via the {\it non-growth} Subtraction and
+Division primitives. For instance since $~20~=~32-8-4~,~$
+the quantity 20 
+can be encoded as $~C_6-C_4-C_3$.
+%%%%%%%%%%%%%%%% under \eq{addcov}'s naming convention.
+
+
+%% required
+%% $O(~Log(j)~)$ bits.  
+%% Thus, the length of these encodings was 
+%% much 
+%% smaller
+%% than the respective
+%% numbers
+%% % magnitudes of 
+%%  $2^{j-1}~$ and $2^{2^{j-2}}$ 
+%% %that 
+%% these constants represent.
+
+The challenge \cite{wwapal} 
+faced was to determine whether 
+%it was possible to formulate 
+self-justification
+was possible
+under 
+either
+\eq{addcov}'s
+% ``Additive'' 
+or  \eq{multcov}'s 
+% ``Multiplicative'' 
+%% naming
+schema.
+It found 
+%that 
+ \eq{multcov}'s 
+multiplicative
+% naming 
+convention was  incompatible 
+with self-justification (due to its 
+%%very 
+speedy growth rate),
+but
+%In contrast,
+\eq{addcov}'s additive 
+% naming 
+schema did,
+conveniently,
+ permit self-justification. 
+
+\medskip
+
+Our new proposed Double-Formatted form of a self-justifying
+axiom system is easiest to describe, if we first
+review \cite{wwapal}'s Single-Formatted formalism
+and then incrementally refine it.
+
+The extension of our base-language $~L^G~$
+that includes the Additive Naming Convention (ANC)'s
+additional constants 
+ $~C_3,~C_4,~ C_5,~ ...~$ will be called
+an {\bf ANC-Based Language} and be denoted
+as  $~L^{ANC}~$. 
+Also if
+ $\, t \,$ denotes   any term in $\, L^{ANC} \,$'s   
+language, then 
+the quantifiers in 
+the two wffs of
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and
+$\exists ~ v \leq t~~ \Psi (v)$
+will be  called $\, L^{ANC} \,$'s   
+{\bf ``Bounded 
+Quantifiers''}.
+
+
+\begin{deff}
+\label{def-3.8}
+\rm
+The analogs of a conventional arithmetic's
+$\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+formulae
+in the
+language $L^{ANC}$ will be denoted as
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$
+ and $\Sigma^{ANC}_n$.
+Thus,
+a formula will be defined to be
+$\Delta^{ANC}_0$  iff all its quantifiers are bounded.  
+The
+%%%%%%%%%  below 
+definitions 
+of  $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$
+formulae are also  quite conventional:
+\bee
+%\small
+\parskip -2 pt
+\baselineskip = 0.8 \normalbaselineskip 
+\item
+Every  
+$\Delta_0^{ANC}$ formula is considered to
+be 
+also 
+a
+$\Pi_0^{ANC}$  and 
+an
+$\Sigma_0^{ANC}  $ expression.
+%% 
+%% ``$~\Pi_0^{ANC}~ \,$''  and 
+%% %  also 
+%%  ``$~\Sigma_0^{ANC}~ \, $''. 
+%% 
+\item
+A
+formula
+is  called
+ $ \,\Pi_n^{ANC} \,$
+when it
+% is 
+can be
+encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  
+where
+%with
+$\Phi$ is  $\Sigma_{n-1}^{ANC}$
+\item
+A formula
+is  called
+ $\Sigma_n^{ANC}$
+when it can be encoded as 
+$\exists v_1~ ...~ \exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{ANC}$.
+\ene
+\end{deff}
+
+
+
+%%\begin{deff}
+%%\label{def3.9}
+%%\rm
+
+ \parskip 1pt
+
+
+Given an initial axiom system $\beta,$
+the Theorem 3 of \cite{wwapal} defined a 
+self-justifying logic, called
+ISCE$(\beta)$ 
+that could prove all 
+$~\beta\,$'s $\Pi_1^{ANC}$ theorems and 
+verify its own consistency under a Hilbert-style deductive
+apparatus. It consisted of the following four
+groups of axioms:
+%
+%  \newpage
+\begin{description}
+ \parskip 0pt
+\item
+{\bf GROUP-ZERO:} 
+This 
+schema
+%  axiom group 
+will
+use \el{start}'s axiom to define the constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ 
+and 
+%employed
+%an infinite number of instances of
+\el{addcov}'s Additive Naming 
+schema
+%convention
+to define 
+ the further constants
+ $    C_3,  C_4,  C_5,  ...   $  
+\item
+{\bf GROUP-1:}
+It is convenient  to
+define
+ ISCE's Group-1 and Group-2
+axioms using a notation that 
+will support \cite{wwapal}'s Theorem 3
+in a 
+% slightly
+ more 
+general sense than
+appeared in \cite{wwapal},
+%%% under a slightly different notation convention,
+% is transparently equivalent
+% (but slightly different) from \cite{wwapal}'s counterpart,
+so that
+% a 
+the
+new
+% proposed
+%%%% second
+ ``IQFS''
+formalism
+(appearing later in this section)
+% shall
+will
+%proposal shall
+% framework will 
+be easier to describe.
+Let us
+therefore
+ say  a $\Pi_1^{ANC} $  sentence is {\bf Simple}
+iff the only built-in constants it employs are
+$\,C_0\,$, $\,C_1\,$ and $\,C_2$.
+Then ISCE's Group-1 scheme will
+allowed to
+ be any finite set of 
+simple $\Pi_1^{ANC} $  axioms, called $~S~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following properties:
+\bee
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+axioms
+will be capable of proving all $\Delta^{ANC}_0 $  
+statements which
+are true.
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+scheme
+will also be capable of proving 
+that
+the ``=" and ``$\leq$" predicates
+% own 
+support
+their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+Any finite set 
+$\Pi_1^{ANC} $  axioms
+with the above properties can be used to define $~S~$
+and 
+support
+%prove 
+an analog of
+\cite{wwapal}'s Theorem 3,
+by a trivial generalization
+\footnote{A formal proof of this generalization of
+\cite{wwapal}'s results is 
+%absolutely 
+entirely
+routine.
+%  and omitted here for the sake of brevity.}
+For the sake of brevity, it  is
+omitted.} 
+%
+%here,}
+ of
+the methodologies from Sections 3 and 4 of
+\cite{wwapal}. (Thus,
+any such finite set $~S~$ supporting Conditions (1) and (2)
+can  be employed
+by
+ISCE's
+Group-1 part.)
+%%% 
+%%% and it is unimportant which
+%%% particular defining
+%%%  set is used.
+% 
+% BBB111
+%  
+% This
+% % schema
+% axiom group 
+% consisted of a finite
+% set of 
+% $\Pi_1^{ANC} $ axioms
+% %, CALLED $F$,
+%  defining ISCE's
+% Grounding  function primitives. 
+% %This means that
+% For each  such function $G$ and set of numbers 
+% $    {k},   {k_1},    {k_2}, ...    {k_m}$,
+% %the combination of 
+% the Group-Zero and Group-1 axioms 
+% %must
+% will
+%  imply 
+% $ G(    {k_1},    {k_2}, ...    {k_m}) \,=\,    {k}   $ when
+% this sentence is true
+% \footnote{ \f55 
+% Our
+%  $\Pi^{ANC}_1$
+% encoding for the
+% Group-1 scheme  needs,
+% technically, 
+% % employ
+%  only
+% employ
+%  the three constant symbol $C_0$, $C_1$ and $C_2$ for the
+% union of all 
+% the
+%  Group-Zero and Group-1 axioms
+% to satisfy 
+% their
+% %its
+% %the 
+% above requirements.} .
+% The Group-1 schema
+% of \cite{wwapal} 
+% will also
+% assign the ``=" and ``$<$" predicates
+% their conventional
+% %  logical
+% properties.
+% %footnoted property.}
+% %%
+% %%(Any finite 
+% %%set of  $\Pi_1^{ANC} $
+% %%sentences  meeting these conditions is
+% %%suitable.)
+% %%
+\item
+{\bf GROUP-2:}
+Let
+$\ulcorner \, \Phi \, \urcorner$ denote $\Phi$'s G\"{o}del number, and
+$\mbox{HilbPrf}_{ \beta  }(x,y)$
+denote a 
+%%%%%%%%%%%  $\Delta _0^{ANC+}$ 
+$\Delta _0^{ANC}$ 
+formula indicating  $y$ is a
+Hilbert-styled
+proof
+from axiom system $\beta  $ of the theorem 
+$x$.
+%
+% Suppose that
+%$~\beta~$ uses the same Grounding function symbols as
+%ISCE$^{ANC}(\beta)$,
+%and it therefore generates
+%a set of 
+%% $\Pi_1^{ANC+} $ theorems.
+% $\Pi_1^{ANC} $ 
+%theorems.
+%
+For each 
+%$\Pi_1^{ANC+} $
+$\Pi_1^{ANC} $
+ sentence  $\Phi$,
+the Group-2 schema
+for ISCE$(\beta)$ 
+%
+%was defined in  \cite{wwapal} 
+%did 
+will 
+contain
+% an 
+one
+axiom of the form:
+%% 
+%% \begin{equation}
+%% \small
+%% \label{group2nold}
+%% \forall ~x~\forall ~y~
+%% ~~\{~~[~~ \sigma_{~ \ulcorner \, \Phi \, \urcorner
+%%  ~}(x)~\wedge~
+%% \{~ \mbox{HilbPrf}~_\beta
+%% ~(~ x ~,~y~)~~]~~
+%% \Rightarrow ~~ \Phi~~ \}
+%% \end{equation}
+%% % {\bf IMPORTANT CLARIFICATION:} 
+%% %{\small 
+%% %%{{\bf DECIPHERING LINE \eq{group2nold}:$~$}
+%% {{\bf Clarification:$~$ }
+%%  \el{group2nold} is {\it helpful}
+%% because   ISCE(\beta)$  can             infer
+%% \eq{group2old}'s {\it simpler statement}
+%% directly
+%% from the combination of
+%% \eq{group2nold},
+%% % it,
+%% the Group-1 schema and \el{deltf}'s definition of
+%% ``$~\sigma~$''.}
+%% 
+\begin{equation}
+% \small
+\label{group2old}
+\forall ~y~~~\{~ \mbox{HilbPrf}~_\beta
+~(~ \ulcorner \Phi \urcorner ~,~y~)~~
+\Rightarrow ~~ \Phi~~\}
+\end{equation}
+\item
+{\bf GROUP-3:}
+This last part of
+%%%%%%%%%%%%%%%  \cite{wwapal}'s
+ISCE$(\beta)$
+formalism was
+ a single 
+self-referencing
+$\Pi_1^{ANC}$
+sentence  
+stating:
+ %% essentially declaring:
+\begin{quote}
+% \small
+%%%%%%%%%%%%% $ \oplus ~ \oplus ~~~$
+$ \oplus  \oplus ~~~$
+ ``There 
+%is 
+exists
+no
+Hilbert-style proof of 0=1 from the union of the Group-0, 1 and 2
+axioms  with {\it THIS  SENTENCE} (referring to itself)''.
+\end{quote}
+\end{description}
+%{\bf CLARIFICATION:}
+{\bf Clarifying $ \oplus  \oplus$'s Meaning:}
+ $~$Several of our articles
+\cite{ww1,ww5,wwapal,ww9}
+employed
+self-referential
+  $\Pi_1^{ANC}$ constructions,
+similar 
+to 
+%%%%%%%%%the sentence
+ $ \oplus  \oplus \,$
+as Example \ref{ex-2.5} had mentioned.
+%% 
+%% whose 
+%% % precise implications were outlined in
+%% significance was explained by
+%% %formalized by 
+%% Example \ref{ex-2.5}.
+%% 
+A reader can find
+several
+%detailed
+slightly different
+ illustrations about how 
+$~ \oplus  \oplus ~ $
+%  $\, \oplus  \oplus $'s 
+%   self-referential statement 
+is encoded in  these articles.
+
+
+% 
+% Each of these articles provide examples of
+% how analogs for
+% $\, \oplus  \oplus $'s 
+% self-referential
+% statement
+% are encoded.
+
+
+ 
+% If the reader wishes to see
+% a formal encoding  for
+% $\, \oplus  \oplus $'s 
+% %self-referential
+% % Fixed-Point 
+% statement,
+% %it 
+% one such example
+% is provided by  
+% \cite{wwapal}'s 
+%  Lemma 1.
+% 
+
+
+\begin{deff}
+\label{def-3.9x10}
+\rm
+Let $~I(~\bullet~)~$ denote
+an operation that maps
+an initial axiom basis $\, \beta \,$ onto an alternate
+system  $\,I(\beta)\, $.
+(One example of
+such an operation is the
+  ISCE$( \, \bullet \, )$ 
+framework,
+that maps 
+an initial axiom basis of 
+  $~\beta~$ onto 
+the alternate formalism of
+ ISCE$(\beta).~)~$ 
+Such an operation  $~I(~\bullet~)~$
+is  called {\bf Consistency Preserving}
+iff  $\,I(\beta)\, $ is consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemas is
+consistent.
+\end{deff}
+
+
+%Most of our research in 
+% \cite{ww93}-\cite{ww14}
+% has 
+
+Several of our research projects 
+%centered around
+had employed
+ \dfx{def-3.9x10}'s
+framework.
+For instance, 
+%% 
+%% the 
+%% 
+%% 
+%% Its
+%% %%%  main
+%% % central 
+%% focus in
+\cite{wwapal} 
+demonstrated
+%consisted of showing
+ the   ISCE$( \, \bullet \, )$ 
+mapping was consistency preserving.
+Thus if PA+ denotes the extension of
+Peano Arithmetic that 
+includes
+PA's traditional  Addition and Multiplication
+functions
+%% 
+%% 1n addition to the conventional
+%% functions of addition and multiplication
+%% contains 
+%% 
+%% 
+plus $L^G\,$'s six
+added
+%previously mentions
+ Grounding-level function 
+primitives,
+%functions,
+then
+ ISCE$( \, $PA+$ \, )$
+will 
+be automatically
+%be
+ consistent
+(because  PA+ was consistent).
+% consistent whenever PA+ is consistent.
+Hence while Peano Arithmetic is unable to
+verify its own consistency (on account of G\"{o}del's
+seminal 1931 discovery), it is sufficiently agile to
+prove the following relative-consistency statement:
+\begin{center}
+%% \small
+$\#~~~$ If PA is consistent then 
+ ISCE$( \, $PA+$ \, )$ is  
+ self-justifying.
+ \end{center}
+This
+%The above
+% statement
+ relative-consistency statement
+%does offer 
+provides
+a partial 
+positive
+answer to
+the
+Q-2 version of Hilbert's Second  Question.
+%This is because it formalizes one respect 
+It 
+captures
+% Brad change encapsulizes 
+one
+% positive 
+respect
+in which
+%such as  
+ISCE$( \, $PA+$ \, )$
+can {\it appreciate} its own consistency.
+This respect is, obviously,
+only
+of a limited nature
+because $++$'s generalization of the Second
+Incompleteness Theorem indicates 
+that
+no Type-S arithmetic
+can
+% simultaneously 
+recognize 
+% {\it both} 
+its Hilbert consistency and
+take
+successor 
+to be
+ a total function.
+%The consistency-preservation property of 
+% ISCE$( \, \bullet \, )$
+%dies, however,
+It does,  however,
+ raise the following
+enticing
+ question:
+\begin{quote}
+$\# \, \#~ $
+Can the infinite number of
+distinct
+ constant symbols, employed by
+ISCE's Group-Zero schema, be reduced to a finite size
+by a Type-NS Self-Justifying Logic,
+without resorting to \cite{wwapal}'s inefficient
+``ISINF'' 
+methodology (which requires 
+a proof 
+having an expensive
+ $\Omega(N)$ length for constructing integers $N$
+whose binary encoding uses $O(~$Log$(N)~)$ bits) ?
+\end{quote}
+The remainder of this section will outline how an encouraging
+answer to 
+$\, \# \, \# \, $'s query
+is likely to
+%%%should,
+% conveniently
+arrive,
+%be plausible
+when one 
+% carefully
+%delicately 
+modifies ISCE's formalism
+with  the Q-function operative of $~\zzthe~$.
+
+\begin{deff}
+\label{def-3.10}
+\rm
+Let $L^Q$ 
+% once 
+again denote the extension of
+$~L^G\,$'s Grounding language that includes
+the
+% further
+ Q-function symbol of $\, \theta $.
+Then
+$\Delta^Q_0$,
+ $\Pi^Q_n$ and $\Sigma^Q_n$
+will,
+intuitively,
+%similarly
+ denote the
+%  1-to-1 
+analogs of
+\dfx{def-3.8}'s
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$'s
+formulae
+in $~L^G\,$'s  language.
+In particular, if $~\Phi~$ 
+is one of an
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+formula,
+then
+% the formula 
+$~\Phi^Q~$
+will be called
+% respectively
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+when
+%if 
+it
+differs from $~\Phi~$ 
+only
+by
+ replacing each constant $~C_J~$
+from the set $~C_3,C_4,C_5...~$ 
+with Line \eq{ej-def}'s 
+% mathematically equivalent term of
+term  $~E_{J-1}~$. 
+\end{deff}
+
+\parskip 2pt
+
+%% 444444444444444
+
+\begin{example}
+\label{ex-3.11}
+\rm
+Suppose $~\Phi$ 
+is one of a 
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+sentence that employs the three constant symbols
+of $C_4$,   $C_6$ and  $C_{10}\,$
+for
+ representing the
+three numbers
+of 8, 32 and 512. 
+Let us recall 
+that
+ $E_3$,   $E_5$ and  $E_9\,$
+%  do
+formulate these three quantities
+under  Line \eq{ej-def}'s notation.
+Then $~\Phi^Q$  will have an 
+identical definition as
+ $~\Phi$ 
+except each $C_j$ is replaced by
+$E_{j-1}$.
+
+
+A formula is,
+moreover,
+ defined to lie in one
+of the 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+classes 
+{\it only if} it is constructed in such a manner.
+This fact
+% brad assures 
+ensures
+that all the terms employed in these
+three classes of sentences are 
+{\it ``Observable''} terms.
+Hence ``Unobservable'' ground terms are allowed in
+$~L^Q\,$'s language, but {\it they are excluded}
+from occurring in the 
+{\it ``end-product''} 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+theorems 
+that 
+%%will now be discussed.
+%it proves !
+do
+encapsulate
+%formalize
+ the {\it intended use of 
+%its
+this 
+formalism.}
+\end{example}
+
+\begin{deff}
+\label{def-3.12}
+\rm
+The term {\bf IQFS($ ~\bullet~$)$~$}  refers
+to the self-justifying analog of
+  ISCE($ ~\bullet~$)$~$
+%that will be employed 
+under  $L^Q\,$'s
+language.
+(The acronym ``IQFS'' stands for 
+``Introspective Q-Function Semantics''.)
+In a context where $~\beta~$ is 
+%some i
+an initial axiom
+system that proves theorems 
+%under
+in 
+the 
+language  $L^Q$, the 
+system
+%formalism 
+ IQFS($ \, \beta\,$)
+%$~$
+will 
+be defined as a
+ 4-part 
+formalism,
+analogous to  ISCE($\beta$),
+except for the following 
+%relatively modest 
+changes: 
+\bed
+\parskip 0pt
+\item[  a.  ]
+The Group-Zero schema of
+ IQFS will
+differ from ISCE's analog
+by replacing 
+\el{addcov}'s ``Additive Naming'' schema with  
+the
+Up-Walking axioms,
+given in Lines  \eq{walk1}--\eq{walk4}.
+(This is because
+the language  $L^Q$ differs from
+ $L^{ANC}$ by 
+having the 
+ Q-function operator of $~\zzthe~$
+define the formal quantities that are represented by
+the constant symbols
+of $~C_3,C_4,C_5~~....~$ 
+under  $L^{ANC}.~~)$ 
+Otherwise both
+these
+Group-Zero 
+schemes will be 
+identical.
+Thus,
+ they 
+will 
+both 
+use \el{start}'s axiom to define the 
+three initial constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,~$.  
+\item[  b.  ]
+The Group-1 scheme of IQFS will be identical to ISCE's counterpart
+except it will reflect Item a's modification of the Group-Zero
+scheme for $\, L^Q\,$'s revised language
+(e.g. 
+the footnote \footnote{\f55
+In a context where a $\Pi_1^Q $  sentence is
+called  ``Simple''
+when it contains no $~E_j~$ term with $~j \geq 2~,~$ 
+the Group-1 scheme of IQFS will be analogous to
+ISCE's counterpart by consisting of
+ any finite set of 
+simple $\Pi_1^Q $  axioms, called $~S^*~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following properties:
+\bee
+\item
+  The union of $~S^*~$ with IQFS's Group-Zero
+axioms
+will be capable of proving all $\Delta^Q_0 $ 
+statements which  are true.
+\item
+  The union of $~S^*~$ with IQFS's Group-Zero
+scheme
+will also be capable of proving 
+that
+the ``=" and
+ ``$\leq$" predicates
+support their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+The above two properties are the 1-to-1 analogs of
+their counterparts used by ISCE's Group-1 scheme.
+As was the case with ISCE's formalism,
+any finite set
+of simple 
+$\Pi_1^Q $  axioms
+with the above properties can be used to define $~S^*~.~$
+Once again,
+it is unimportant which 
+particular
+finite-sized
+definition for $S^*$
+% such set 
+is used.} describes 
+such a
+%the
+resulting
+%% quite
+ straightforward revision of
+the Group-1 scheme.)
+\item[  c.  ]
+All the $\Pi_1^Q$ axioms lying in IQFS's
+%Group-1 and 
+Group-2 scheme will be identical to their counterparts
+under ISCE, except  they
+will
+ employ 
+\dfx{def-3.10}'s machinery for translating 
+ $ \,\Pi_1^{ANC} \,$ 
+sentences into their equivalent
+ $ \,\Pi_1^Q \,$ counterparts. 
+\item[  d.  ]
+The Group-3 axiom of  IQFS
+will be similar to ISCE's Group-3 
+{\it ``I am consistent''} 
+axiom-statement, except 
+the latter's notion of ``I'' will reflect the above
+changes in the Groups 0, 1 and 2 schemes. 
+It
+%Thus, the new
+%Group-3 axiom 
+will,
+thus,
+be a $\Pi_1^Q$ sentence declaring that
+{\it ``There is no 
+Hilbert-style
+proof of 0=1 from the union of the preceding axioms
+with THIS SENTENCE (looking at itself)''.}
+\ennd
+\end{deff}
+
+The properties of IQFS
+ are
+interesting
+% especially intriguing 
+from a 
+% Computer Science 
+Complexity perspective
+because
+\phx{th-3.3} showed 
+% that 
+every integer $\,n\,$ 
+can
+%could 
+be encoded
+%under it by
+by
+%via
+a
+ term $T_n$ that has an $O\{~[~$Log$(n)~]^3~\}$ length.
+This is
+unlike the 
+%much 
+%%%%%%%%%% far worse 
+asymptote  $\Omega(n^2)$ that results
+when the 
+$~\zzthe$ primitive 
+(from Lines  \eq{walk1}-\eq{walk4})
+is replaced by 
+Lines \ref{zm1} and  \ref{zm2}'s
+less efficient
+primitive of 
+$~\glamb~$.  
+
+
+% function
+
+
+%% It
+%% is
+%% also
+%%  almost
+%% as good as the 
+%% %encoded
+%%  $O\{~$Log$(n)~\cdot ~ $LogLog$(n)~\}$ length
+%% that ISCE produces.
+%% % while 
+%% Moreover
+%% IQFS,$~$ {\it unlike ISCE,$~$}
+%% requires only a finite number of constant symbols
+%% to define any 
+%% %starting 
+%% integer $N$. 
+
+%In other words, the 
+
+
+An open question of fundamental interest is whether or not
+IQFS framework is
+% fundamentally 
+analogous to ISCE by also
+satisfying 
+Definition \ref{def-3.9x10}'s Consistency Preservation property.
+The
+% close 
+similarity between the definitions of these two
+axiomatic frameworks
+ strongly 
+suggests that
+\cite{wwapal}'s proof of ISCE's consistency preservation
+property 
+%should 
+ought to
+be extensible to IQFS.
+
+The first half of our 2-part conjecture is,
+%thus,
+the presumption
+ that IQFS
+does have this 
+desirable
+characteristic. (Unfortunately, we suspect a
+formally
+rigorous proof of 
+this consistency preservation property
+ will be 
+% much
+%significantly
+ more 
+complex
+%complicated 
+than 
+it was for
+ISCE's
+analog in Theorem 3 of \cite{wwapal}
+because a more complicated mathematical infrastructure will
+be needed to overcome certain bottle necks that would
+otherwise arise during the
+proof.) 
+
+
+%% More will be said about this conjecture
+%% in
+%% 
+
+If 
+our conjecture
+does hold then IQFS will be 
+% much 
+substantially
+more germane to the
+goals that Hilbert  sought in statement $*$ than
+was ISCE.
+This is
+because the former framework
+differs from the latter by
+ needing only three starting constants,
+representing the integers of 0, 1 and 2, in order to
+construct the 
+full
+infinite span of integers $~3,4,5,~...~$.
+
+%\bigskip
+
+\smallskip
+
+Moreover,
+our conjecture that  IQFS will 
+duplicate ISCE's consistency-preservation property is based on
+more than the analogous definitions between these two frameworks.
+It is also because  Item  $++$'s
+ Pudl\'{a}k-Solovay
+generalization of the Second Incompleteness Theorem
+% 
+% (from the Item  $++$ given 
+% in Example
+% \ref {ex-2.3} )
+% is
+%   proven 
+% by Appendix A 
+% 
+is demonstrated by the Proposition A.1
+(in Appendix A)
+to be inapplicable to 
+ IQFS's formalism.
+It is within this combined context that IQFS formalism
+seems to
+become especially enticing, when it 
+differs from ISCE by needing only three starting constant
+symbols to construct the full infinite range of integers.
+
+%from a purely finite basis.
+
+
+\parskip 2 pt
+
+\section{Second Half of our 2-Part Conjecture}
+
+\label{ss6}
+
+Both the virtues and drawbacks of \textsection \ref{ss5}'s
+conjecture
+are highlighted by Proposition \ref{th-3.3}'s 
+characterization of
+the  $O\{~[~$Log$(n)~]^3~\}$ 
+%invariant for 
+quantity of logical symbols 
+used for encoding
+% needed to to encode 
+an integer $~n~$ as a
+ grounded term
+ $T_n$. 
+Thus,
+this
+ amount is clearly
+%much
+significantly
+ better than the alternate
+$O(~n^2~)$ 
+length
+that arises when the $~\theta~$ function symbol
+is replaced by the less efficient primitive 
+$~\glamb~,~$ 
+%%%%% operator 
+defined by Lines 
+\eq{zm1} and \eq{zm2}.
+On the other hand, one would ideally 
+prefer our ground terms to resemble the conventional encoding
+of a binary number that uses 
+ $O\{~$Log$(n)~\}$ 
+logical symbols to encode an
+arbitrary  number $~n~$.
+
+It turns out  it is possible to improve 
+Proposition \ref{th-3.3}'s encodings 
+to such a compressed  $O\{~$Log$(n)~\}$ size,
+if one adds only a minor
+wiggle to the logic's notation
+%%%%%%%%%%%%%%%%%% conventions 
+convention.
+This distinction arises because most traditional
+logic
+languages
+% , as typically formalized in textbooks,
+will
+formalize
+% both conventional terms and 
+Definition \ref{def-3.4}'s 
+``Ground terms''
+as tree-like structures.
+An 
+easy
+alternative modification of this 
+%perspective 
+construct
+will
+% would 
+allow these terms to
+own the more general structure of
+a Directed Acyclic Graph (Dag).
+ 
+% Kozen has noted ????.
+%some computer scientists have noted.
+
+This distinction has been
+traditionally viewed
+ as an unimportant 
+wrinkle
+%issue 
+because it can be 
+readily
+%easily
+proven that every Dag-oriented  term can be 
+converted into its Tree-oriented counterpart with 
+merely a
+% only an usually unimportant  
+Polynomial increase in space. 
+The reason for our special interest in an
+alternate Dag-formulated
+%distinction
+%between a Tree-oriented and Dag-oriented
+base-language for logic 
+is that the latter 
+ushers in
+ more efficiently encoded Ground terms. 
+Thus,   Proposition \ref{th-6.1} 
+will
+indicate
+that
+Proposition \ref{th-3.3}'s earlier
+ ground terms can 
+have their lengths
+ compressed from an
+$O\{~[~$Log$(n)~]^3~\}$ 
+to an   $O\{~$Log$(n)~\}$ 
+magnitude in a Dag context.
+
+%in the latter context.
+
+
+\begin{propp}
+\label{th-6.1}
+% \rm
+Let us consider a Dag-analog of 
+\textsection \ref{ss4}'s formalism where one again:
+\bee
+\item
+$\theta$ is the only available growth permitting function symbol,
+\item
+the only built-in constant symbols are the 
+entities 
+ $~C_0~$, $~C_1~$ and $~C_2~$ 
+for representing
+the values of 0, 1 and 2, and
+\item
+three function symbols for
+representing
+ integer-subtraction, integer-division
+and the maximum operation 
+are, once 
+again, available. 
+\ene
+%Within this notation, 
+In this context,
+any integer $~n~$ can be encoded
+by a Dag-oriented Ground term 
+$~T_n^*~$
+using only
+ $O\{~$Log$(n)~\}$ logical 
+symbols. (As the pointers needed to
+separate these  $O\{~$Log$(n)~\}$ logical objects 
+will have encodings using 
+ $O\{~$LogLog$(n)~\}$ bits, the total amount of 
+memory to encode a
+ Dag-oriented Ground term will require
+ $O\{~$Log$(n)~\cdot ~~$LogLog$(n)~\}$ bits.)
+\end{propp}
+
+The proof for Proposition \ref{th-6.1}
+rests essentially on a more elaborate version of the
+argument that   \textsection \ref{ss4}
+used to justify Proposition  \ref{th-3.3}.
+It
+% essentially rests on 
+uses
+the fact that 
+Proposition  \ref{th-3.3}'s ground term $T_n$
+will have
+% has
+many repeating subterms that can be compressed into
+% one 
+single objects under a Dag-style notation.
+%% 
+%% Our
+%% %We supply in the attached 
+%% $\, 1 ~\frac{1}{2} \,$ page
+%% proof in 
+%% 
+Appendix B  provides
+a  $\, 1 ~\frac{1}{2} \,$ page
+formal proof
+about how 
+%the 
+a more
+nicely
+compressed object $T_n^*$ 
+can be encoded
+using only $O\{~$Log$(n)~\}$ logical symbols.
+
+Let  IQFS$^*$ denote the analog of our  
+IQFS framework that has   
+Proposition \ref{th-6.1}'s
+ Dag-oriented Ground Terms replace
+  \textsection \ref{ss4}'s earlier 
+Tree-oriented Ground Terms
+in an accordingly revised language.
+The second half of our 2-part conjecture is that there is no
+difference between 
+ IQFS$^*$  and
+IQFS
+ from the perspective of
+Definition \ref{def-3.9x10}'s
+Consistency Preservation property.
+Both these formalism are thus anticipated to 
+be analogous to \cite{wwapal}'s ISCE 
+framework, insofar as
+  IQFS$(\beta)$  and  IQFS$^*(\beta)$ 
+will be consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemes is
+consistent.
+
+Although
+ IQFS$^*$ and 
+IQFS will 
+possess
+ similar properties under the second half of
+our 2-part conjecture, the distinction between these
+two formalisms should not be viewed as inconsequential.
+This is because the Ground terms in the  
+ IQFS$^*$ use  $O\{~$Log$(n)~\}$ logical symbols,
+similar to the classic encodings of an integer $~n~$ as
+a binary number. Thus, our conjecture about the self-justifying
+properties of  IQFS$^*$ 
+will be more meaningful than its analog
+for IQFS
+(because  IQFS$^*$  possesses 
+%% owns 
+much greater levels of efficiency).
+
+
+%is much more efficient.
+
+%\end{document}
+
+
+
+\section{On the Epistemological Significance of  Our 2-Part Conjecture}
+\label{ss7}
+
+\parskip 2pt
+
+
+All  our published articles
+(since 1993) 
+ about self-justifying arithmetics
+have emphasized that our evasions of the
+Second Incompleteness Effect  rested upon using arithmetics
+that were much weaker than traditional arithmetics in, at least,
+some 
+particular
+well-defined respects. The attached Appendix A 
+has emphasized
+that there are good reasons for conjecturing that IQFS satisfies
+a consistency-preservation property similar to ISCE, but this 
+does not reply to a second question that many 
+researcher
+% will
+may
+ want
+to have
+ addressed.
+
+% skeptics may initially raise.
+
+% will 
+% understandably
+% not, by itself, change many reader's  skepticism
+% about employing  forms of arithmetic that are weaker than its
+% traditional counterparts.
+
+It concerns the fact that our proposed   IQFS formalisms 
+%even more controversial is that they are what
+employs what
+\textsection \ref{ss4} calls a
+``Double-Formatted'' logic, where a 
+ground-term can correspond to either 
+what Definition \ref{def-3.5}
+calls an 
+ ``Observable''
+or an
+ ``Unobservable''
+object.
+ The latter will possess no
+fully
+ evident meaning
+because
+% our mathematical notation allows an 
+Unobservable ground-terms 
+can 
+%possess 
+represent any of
+multiple different 
+allowed values.
+%meanings.
+Many  
+researchers 
+will thus wish
+to
+% have 
+receive
+a reply to
+% thus raise 
+the following 
+%skeptical 
+resulting
+question:
+%
+% to be
+% addressed:
+% answered:
+\begin{quote}
+\it
+ $  \bullet ~~~$ 
+Does 
+% not 
+the presence of   
+``Unobservable'' ground-terms in 
+the language of  IQFS cause it
+to lie fundamentally outside the domain
+of modern mathematical logic
+(thus making its partial evasion of the Second Incompleteness
+Theorem
+mostly
+ irrelevant) ? 
+\end{quote}
+Our 
+% 
+% positive
+reply to  $ \, \bullet \,$'s skeptical inquiry
+has two quite
+positive
+% distinct 
+facets.
+%  
+% will have
+%  two 
+% % equally important 
+% %parts. 
+% very positive
+% aspects.
+% % facets.
+% 
+It is given below: 
+\bee
+\parskip 0pt
+\item
+% Our
+Propositions 
+\ref{th-3.3} and \ref{th-6.1} have 
+illustrated how
+%emphasized that each
+every 
+integer $~n~$ can be represented by a ground term $T_n$
+with two 
+%%%%% 
+distinctly 
+different forms of efficiency.
+% different permissible types of efficiency.
+%Moreover our 
+In this context,
+Definition \ref{def-3.10}
+and Example \ref{ex-3.11}
+have noted that our language $L^Q$ is sufficiently
+agile so that each of $L^{ANC}\,$'s
+$\Pi_n^{ANC}$ 
+sentences can be directly translated into
+their equivalent $\Pi_n^Q$ counterparts. The presence of
+``Unobservable'' objects
+ in $L^Q$ 
+% 
+is 
+thus 
+% has
+no major
+drawback
+% disadvantage
+because
+% when 
+all the conventional arithmetic objects from   
+ $L^{ANC}$ have efficient translations in 
+ $L^Q\,$'s language.
+\item
+Moreover, there is an added sense where it is
+propitious
+% nice
+that  $L^Q$
+utilize
+% possesses
+ {\it both} 
+``Observable'' and ``Unobservable'' ground-terms.
+This is because human
+beings,
+ in their common colloquial verbal languages,
+use
+references to
+ ambiguous linguistic objects in their everyday
+mundane
+ speech habits
+(similar to ``Unobservable'' ground-terms).
+% and logical sentences that possess such ambiguities). 
+Thus assuming our 2-part conjecture is
+correct, our proposed IQFS
+formalism
+ will own the 
+pleasing
+simultaneous abilities to:
+\bed
+% \baselineskip = 1.05  \normalbaselineskip 
+ \baselineskip = 1.1  \normalbaselineskip 
+% \small
+\parskip 0 pt
+\item[  a.  ]
+verify its
+% (his) 
+own consistency, 
+\item[  b.  ]
+prove analogs of all of Peano
+Arithmetic's $\Pi_1$ theorems
+(despite the fact it starts with only
+$C_0~$, $C_1$ 
+and $C_2\,$ as
+its 
+% three
+ initial constant symbols), and
+\item[  c.  ]
+ resemble the ambiguity 
+that humans display when attempting to decipher the meaning
+of human-type 
+language
+objects (in that Unobservable ground-term
+contain a level of built-in ambiguity
+that is
+ analogous to
+many mundane
+colloquial objects, that typically lack full specification
+% inherent levels of ambiguity 
+under an
+ordinary
+conventional
+human language).
+%
+% sentences, that lie outside the domain of formal mathematics
+\ennd
+\ene
+The combination of Items 1, 2a, 2b and 2c thus suggests
+that
+some perhaps
+ 5-10 \%
+fragment of 
+the aspirations that 
+Hilbert and G\"{o}del 
+had
+expressed
+in statements $*$ and $**$ 
+% thus 
+can be positively acheived
+(within the obvious context where
+the initial objectives of Hilbert's Consistency Program
+were definitively shown to 
+ have been over-ambitious by the Incompleteness
+Theorem).
+
+ 
+%% The current paper will be possibly the last paper I publish about
+%% self-justifying logics in a context where it is being submitted
+%% for publication when I am over 66 years old. I would therefore
+%% like to answer two other further questions that a reader may have
+%% about self-justifying logics in the Remarks 7.1 and 7.2 below:
+
+\medskip
+
+It  
+also
+should
+ be mentioned that the infinite
+number of axiom
+sentences,
+ appearing in the Group-2 schemas
+for ISCE$(\beta)$, IQFS$(\beta)$  and IQFS$^*(\beta)$,
+can be
+% should be able to be
+nicely
+ reduced to a 
+purely
+finite size, with almost no loss 
+in
+%of useful
+information. This was done in \cite{ww14} 
+for the Group-2
+scheme
+of its IS$_D(\beta)$ formalism, 
+with the latter
+%%%%%%%%%%%%%%%% still 
+%where the
+% 
+% germane Group-2 scheme was
+% reduced to one
+% single  axiom sentence 
+% while the resulting 
+% 
+% latter
+%formalism still
+% produced
+producing
+isomorphic counterparts
+of all of $~\beta \,$'s
+full set of
+ $\Pi_1$ theorems
+(e.g. see Sections 5 and 6 of    \cite{ww14}).
+The same methods will
+% trivially 
+routinely
+generalize for the
+% 
+% Analogs of the techniques from Sections 5 and 6 of 
+% \cite{ww14} 
+% % will easily 
+% apply to each of the 
+% 
+ISCE,
+ IQFS and IQFS$^*$ frameworks, if our 2-part conjecture
+does hold true.
+
+
+% 777777
+
+
+\section{Summary of Results}
+\label{ss8}
+
+\baselineskip = 1.21 \normalbaselineskip 
+% \parskip 2pt
+
+\parskip 4pt
+
+The  
+% importance 
+% % significance
+% of the 
+Second Incompleteness Theorem's
+%important 
+%underlying
+vital significance 
+% implications
+in establishing a
+90-95 \%
+refutation of the objectives of
+Hilbert's Consistency Program
+% are, 
+is, 
+of course, undeniable. It would, nevertheless, be of interest if
+some 5-10 \%
+fragment of 
+%its
+Hilbert's
+ initially intended 
+ objectives 
+% 
+% that Hilbert and G\"{o}del set forth in
+% their
+% statements $*$ and $**$ 
+% 
+was
+%were
+%could be
+partially
+ achieved.
+
+This is because it is difficult to fathom how human beings
+can
+maintain
+their psychological motive to engage in cognition without owning some
+type of qualified instinctive faith in their own consistency.
+Moreover, the close similarity between the defining structures of the
+ISCE and IQFS frameworks strongly suggests
+\cite{wwapal}'s proof of ISCE's consistency preservation property
+should generalize for both 
+IQFS  and 
+ IQFS$^*$ under a more elaborate
+% and sophisticated
+inductive machinery.
+Such a self-justifying arithmetic
+would be of interest
+%is very satisfying 
+% be very tempting 
+because it would be
+supportive of  each of
+the invariants of
+1, 2a, 2b and 2c 
+listed in Section
+% from
+% \textsection 
+\ref{ss7}. 
+
+
+%More precisely,
+
+This is because
+ if IQFS satisfies an analog of
+ISCE's consistency preservation property (as we conjecture),
+then IQFS(PA+) would be 
+% a
+philosophically 
+%interesting
+curious.
+% 
+%  This is 
+% % formalism
+% % 
+% % response
+% % to the open questions raised by Hilbert and G\"{o}del
+% % in $*$ and $**$
+% % 
+% because IQFS(PA+) would
+% % then 
+% 
+It would then
+be a self-justifying
+% arithmetic
+logic
+which
+% that
+proves
+% 
+all 
+Peano Arithmetic $\Pi_1^Q$ theorems, that
+% which
+is
+% {\bf  NOT MUCH HARMED} 
+ {\it NOT MUCH DILUTED}
+by the presence of unobservable 
+ground terms in $L^Q\,$'s language (because such
+unobservables
+%%%%% do {\it never physically appear} in 
+% do  
+{\it never formally appear} in 
+IQFS(PA+)'s end-product $\Pi_1^Q$ theorems).
+
+
+%. and it would thus be very tempting.
+
+
+The importance of $++$'s generalization of the
+Second Incompleteness Effect, due
+ to the joint work of 
+  Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+% \cite{Ne86,Pu85,So94,WP87},
+has been repeatedly 
+emphasized.
+%mentioned in this article. 
+%
+%has been repeatedly emphasized and
+%cannot
+%  can never
+% be understated.
+It  demonstrated conventional Type-S arithmetics
+cannot verify their own 
+Hilbert
+consistency.
+It is for this reason that
+our examination of self-justifying logics has focused 
+on arithmetics where either:
+%our subsequent research explored 
+%arithmetics with:
+%% 
+%% For instance, 
+%% this vital invariance
+%%  was
+%% %has been 
+%% cited, in detail,
+%% % mentioned 
+%% in all our published papers
+%% except  \cite{ww93},
+%% (The latter article could not cite 
+%% this topic because our 
+%% 1994
+%% telephone conversations with
+%% Robert  Solovay 
+%% stimulated Solovay's observation about
+%% hybridizing \cite{Pu85,Ne86,WP87}'s
+%% formalisms to establish \cite{So94}
+%% that no
+%% %%  
+%% %%  \cite{ww93} was published.)
+%% %%  
+%% %%  had taken
+%% %%   place
+%% %%  that 
+%% %%  %he 
+%% %%  Solovay
+%% %%  \cite{So94}
+%% %%  %began
+%% %%  %in 1994, that Solovay 
+%% %%  %%%%%%%%%noted
+%% %%  %%%%%%%%%1
+%% %%   observed
+%% %%  % in 1994 
+%% %%  % he know 
+%% %%  how
+%% %%  % to extend 
+%% %%  the work of
+%% %%    Pudl\'{a}k  Nelson and Wilkie-Paris,
+%% %%  \cite{Pu85,Ne86,WP87} 
+%% %%  implied
+%% %%  %show  
+%% %%  all
+%% %%  natural
+%% %%  
+%% natural
+%% Type-S axiom systems 
+%% %are 
+%% can corroborate its
+%% % were
+%% % unable to verify their 
+%% own Hilbert consistency.)
+%% It is within this context that our research has explored two
+%% methods for constructing self-justifying logics where
+%% either:
+%% % 
+%% % types of formalisms where an formalism
+%% % $~\alpha ~$
+%% % can partially
+%% % corroborate its own consistency where either:
+%% % 
+%% 
+\bed
+\parskip 1pt
+\baselineskip = 1.2  \normalbaselineskip 
+\item[   I.  ]  
+a system $~\alpha~$ 
+% Can
+does
+corroborate its own consistency
+under
+both
+ semantic tableaux deduction and its
+ Tab(1)
+generalization
+%in a context where
+ when 
+it 
+% formally
+views
+%under
+%\cite{ww14}'s notion of 
+% ``Type-A'' formalisms that view
+%
+%%% addition as a total function and 
+multiplication
+formally as a 3-way relation
+(as
+was
+ encapsulated by  \cite{ww14}'s
+summary of
+ \cite{ww93,ww1,ww5,ww6}'s
+% \cite{ww93,ww1,ww5,ww6,ww9}'s
+results).
+\item[  II.  ]   
+or alternatively
+a system
+ $~\alpha~$  recognizes its
+own Hilbert consistency
+in a context where it 
+is a ``Type-NS'' logic
+that views
+{\it BOTH}
+  addition and  multiplication as 
+being 3-way relations.
+% (instead of being
+% total functions).
+\ennd
+It is within the context of Topic II where our
+IQFS(PA)  and 
+ IQFS$^*$(PA) frameworks are conjectured to be
+self-justifying 
+arithmetics
+% that are
+ capable of proving
+%  isomorphic counterparts of all of
+all Peano Arithmetic's $\Pi_1$ theorems.
+
+\smallskip
+
+% arithmetics, capable of proving analogs of
+% Peano Arithmetics $\Pi_1$ theorems translated into the
+% language of $L^Q$.
+
+Such an approach will be no full remedy
+%%%%  rrremm
+% 
+% Such a perspective will 
+% have only limited advantages 
+% %be no panacea 
+% 
+when the
+traditional growth properties of the  addition,
+multiplication and successor function operations are replaced
+by an
+alternative  $~\theta~$ function symbol,
+with its
+% surprising
+use of
+a surprisingly modified
+% an
+ ``indeterminate'' function definition.
+It does, however, suggest that 
+{\it a well-defined fragment} of what Hilbert and 
+G\"{o}del sought in statements $*$ and $**$ 
+% is, alas,  feasible in some
+should be 
+likely
+ feasible under
+some special  theoretical 
+circumstances, 
+that are 
+%%%% of 
+%%  , at least,
+% quite
+%limited but  
+partially
+tempting
+%  significance 
+on account of
+% our
+ Propositions 
+% \ref{th-3.3} and \ref{th-6.1}.
+\ref{th-3.3}, \ref{th-6.1} and A.1.
+
+
+
+Thus, our 2-part conjecture 
+raises the prospect of
+a 
+logic
+%formalism 
+simultaneously recognizing the
+% 
+% results can simultaneously
+% recognize the
+%  
+% % undeniable
+central 
+and crucial
+role of the Second Incompleteness Theorem
+in 
+modern
+mathematics,
+%mathematical logic, 
+while also postulating how
+%a 
+% thinking 
+human
+beings can
+formalize
+% formally  own 
+enough of
+a {\it fragmented knowledge}
+about 
+their
+%his/her 
+own 
+internal
+consistency
+% for it 
+to gain 
+% for gaining
+%acquiring
+the psychological stamina 
+% to allow it
+% needed
+for
+motivating
+% engage in
+cognition.
+
+% In other words, \textsection \ref{ss4}'s crucial
+
+\smallskip
+
+This is
+%ultimately
+ because \textsection \ref{ss4}'s 
+% notion of 
+discussion about
+ Double-Formatted arithmetics
+illustrates 
+%suggests 
+how
+to separate the
+crucial
+ notions of ``Observable'' and 
+ ``Unobservable'' ground terms, so
+that 
+% one can recognize 
+%postulate how to
+% {\it simultaneously} 
+the historic
+% importance 
+significance
+of the
+Second Incompleteness Theorem
+is 
+potentially
+compatible with
+% and that
+% while also
+%%while
+% also 
+%% appreciating that
+ a {\it partial fragment} of 
+the aspirations
+% 
+% that
+% %the Y that
+% Hilbert and G\"{o}del
+% %% raised
+% % 
+% expressed 
+% in $*$ and $**$.
+% 
+$*$ and $**$ of
+%that
+%the Y that
+Hilbert and G\"{o}del
+being realized.
+
+\medskip
+
+{\bf Acknowledgments:}
+%%%%As  several Sections 1-4, 
+%\textsection \ref{ss2}, 
+I am
+% much
+%very 
+grateful to
+%was influenced by an emailed letter from 
+Pavel Pudl\'{a}k
+for suggesting 
+\cite {Pupriv}
+I investigate how to apply
+% an analog of 
+Ajtai's study
+\cite{Aj94} of Pigeon-Hole effects 
+for
+refining my prior results about self-justifying logics.
+(The combination of
+ Pudl\'{a}k's
+insightful             suggestion
+% \cite {Pupriv}
+and our
+subsequent
+% further 
+        distinguishing
+between the
+$~\glamb   ~$ and $~\theta~$ operators
+has led to the 
+conjectured
+ improvement of 
+\cite{wwapal}'s ISCE formalism.)
+% I am very grateful to Pudl\'{a}k for making this
+% suggestion. 
+I also thank Bradley Armour-Garb for
+%  many
+several
+comments about how to improve 
+% the 
+this paper's
+presentation.
+
+% in this article.
+
+\small
+\parskip 2 pt
+
+\baselineskip =  0.92 \normalbaselineskip 
+\baselineskip =  0.80 \normalbaselineskip 
+\bibliographystyle{abbrv}
+\bibliography{aa}
+
+%\setlength{\textwidth}{5.0 in}
+
+\gvs
+
+\parskip 0 pt
+
+\section*{Appendix A: An Added Justification for Our Conjecture}
+
+\parskip 3 pt
+
+Section \ref{ss5} indicated that
+our conjecture that  IQFS will 
+duplicate ISCE's consistency-preservation property was based on
+more than the analogous definitions between the
+IQFS and ISCE frameworks.
+It was also because the 
+ Pudl\'{a}k-Solovay
+generalization of the Second Incompleteness Effect
+(from the Item  $++$ given 
+in Example
+\ref {ex-2.3} )
+can be
+  proven 
+ to be inapplicable to 
+ IQFS's formalism.
+This is because our Proposition A.1 below
+demonstrates that the union of the Groups 0, 1 and 2
+axioms of IQFS(PA+) are unable to prove that successor is
+a total function.
+
+{\bf Proposition A.1}
+{\it 
+Let $~\beta~$ denote any
+consistent
+ extension of 
+Page 16's axiom system PA+
+% (employing the $L^G$ language)
+that serves as the input argument for the ISCE($\beta$)
+and IQFS($\beta$)  formalisms.
+Then the union of the Groups 0, 1 and 2 axiom schemes for
+IQFS($\beta$)  are unable to prove that any of the operations
+of successor, addition and/or multiplication are total functions.
+Hence, extensions of these three groups of axioms are not precluded
+from verifying their own consistency by $++$'s generalization of
+the Second Incompleteness Effect.}
+
+
+{
+{\it Proof:}  
+Our article
+ \cite{wwapal}
+showed
+ ISCE($\beta$) 
+was a consistent self-justifying axiom
+system whenever  $~\beta~$ satisfies Proposition A.1's hypothesis.
+% Hence, the invariant $++$ implies that 
+% ISCE($\beta$) is unable to confirm that successor is a total function.
+It then easily follows that there exists a 
+non-standard
+model $M_b$ 
+for  ISCE($\beta$) 
+that contains some non-standard
+integer $b$ which represents some power 
+\newline
+of 2.
+
+Let  $M_b^b$ 
+denote the subset of  the model  $M_b$ 
+which consists of the set of integers that are
+no larger than $b$.  
+Then  $M_b^b$  is also a model of 
+ ISCE($\beta$) because the latter axiom system contains no growth functions.
+
+Now let us define a function operator $~\theta~$ so
+that 
+\bee
+\item $~~\theta(x)~=~2x~~$ when $x$ is a standard number which is power of 2. 
+\item $~~\theta(x)~=~\frac{x}{2}~~$ when $x$ is a non-standard number which is power of 2. 
+\item $~~\theta(x)~=~0~~$ whenever $x$ is not a power of 2. 
+\ene
+ Also, let $N_b^b$ 
+denote a model identical to   $M_b^b$ 
+except that $N_b^b$
+ contains the added function symbol of $\theta$.
+It is easy to show that 
+ $N_b^b$
+ is 
+a
+model for the Groups 0, 1 and 2 axioms for 
+ IQFS($\beta$)
+  formalisms because Items 1-3 imply that
+$~\theta~$ will satisfy Lines
+ \eq{walk1}-\eq{walk4}'s four ``Up-Walking'' requirements.
+Moreover since the element $~b~$ owns no successor under the model
+ $N_b^b$, our proof has shown that 
+ Groups 0, 1 and 2 axioms for 
+ IQFS($\beta$) do not imply that successor is a total function.
+Hence $++$'s generalization of the Second Incompleteness Theorem
+does not apply to this set of axioms
+(because 
+these three groups of axioms
+ can be formalized by 
+a model that fails to recognize
+successor as a total function).
+ $~~~\Box$
+
+\smallskip
+
+It is useful to close this appendix with the reminder that 
+Proposition A.1 does not formally prove the consistency of
+ IQFS($\beta$). 
+It merely makes this prospect look likely
+because it demonstrates no
+%  analog
+counterpart
+ of $++$'s machinery applies
+to  IQFS($\beta$). 
+It is for this reason that we conjecture that
+ IQFS will 
+duplicate ISCE's 
+analogous
+consistency-preservation property.
+
+
+
+% \newpage
+
+\section*{Appendix B: Providing Proposition \ref{th-6.1}'s Proof}
+
+ 
+Our proof for Proposition \ref{th-6.1}
+will rest upon
+an  essentially   more elaborate version of the
+\textsection \ref{ss4}'s proof for 
+Proposition  \ref{th-3.3}.
+It will use the fact that 
+Proposition  \ref{th-3.3}'s ground term $T_n$ has
+many repeating subterms that can be compressed into
+single objects under a Dag-style notation.
+
+Throughout our proof of Proposition \ref{th-6.1},
+$~G~$ will denote our Directed Acyclic Graph (Dag),
+and 
+the symbol $~M~$ will be an abbreviation for
+$~\lceil~1\, + \, $Log$_2(n)~\rceil~$.
+This directed graph will consist of approximately
+$~5 \, \cdot \, $Log$(n)~$ nodes.
+The first five of its six groups of nodes
+in $G$'s graph
+are defined below 
+in roughly
+% bottom-to-top order:
+bottom-up order:
+\bee
+\item The bottom nodes in $G$'s
+graph
+will be the three  
+built-in constant symbols of
+ $~C_0~$, $~C_1~$ and $~C_2~$ 
+that represent
+the values of 0, 1 and 2.
+%\ene
+%\end{document}
+\item
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+For each $~j \leq M\,$, the next $j$ levels of
+$G$'s directed graph will define 
+nodes $A_j$ that formalize the quantity
+ $~\zzthe^j(1)~$. In a context where 
+$~A_0~$ 
+is an abbreviation for $~C_1~$,
+the remaining $A_j$ are defined by:
+\beq
+A_j ~~ =~~~ \zzthe(~A_{j-1}~)
+\enq
+\item
+For each $~j \leq M\,$, let $B_j$
+denote the value of Max$(A_0,A_1,A_2,...A_j)$.
+In a context where 
+$~B_0~$ 
+is an abbreviation for the entity $~C_1~$,
+the remaining $B_j$ are defined
+chronologically in $G$'s directed graph by:
+\beq
+B_j ~~ =~~~\mbox{Max}(A_j,B_{j-1})
+\enq
+\item
+For each $~j \leq M\,$, let $D_j$
+denote the value of $~2^{-j} \, \cdot \, B_M~$.
+In a context where  
+$~D_0~$ 
+was defined by the prior entry $B_M$ in
+our directed graph, 
+the remaining $D_j$ nodes in our graph
+will be defined via \eq{usediv}'s 
+Division operation:
+\beq
+\label{usediv}
+D_j ~~ =~~\frac{D_{j-1}}{2} ~~~~~~\mbox{e.g.}~~~~~~ 
+D_j ~~ =~~\frac{D_{j-1}}{C_2}
+\enq
+\item
+For each $~j \leq M\,$, 
+the node
+ $E_j$
+will represent the quantity
+$~2^j~$.
+These nodes in $G$'s graph will 
+be defined by 
+\eq{usediv2}'s 
+Division operation:
+\beq
+\label{usediv2}
+E_j ~~ =~~\frac{D_M}{D_{M-j}}
+\enq
+\ene
+
+
+Some added notation is needed to describe the last part of
+$G$'s graph for formalizing $n$'s representation as a
+Dag-oriented ground term
+employing $O \{ ~$Log$(n)~\}$
+logic symbols.
+ Let $T_n$
+ denote 
+Proposition \ref{th-3.3}'s formulation of $~n~$ 
+as a Tree-oriented ground term, and 
+$T_n^*$
+denote its Dag counterpart.
+Our proof of 
+Proposition \ref{th-3.3} noted 
+$T_n$ could be constructed by setting $E_M$ equal to
+the least power of 2 greater than $~n~$ and
+then  subtracting from it
+those powers of 2 which are needed to produce the quantity $n$.
+
+The exact same methodology will now
+be used to construct
+our $T_n^*$ representation of $~n~,~$ except 
+ we
+will now
+obviously
+use the methodologies from Items 1-5 to
+assure
+%  that 
+no more than $O\{~$Log$(n)~\}$
+graph nodes are used to construct
+all 
+\el{usediv2}'s 
+ $E_j$ terms.
+(For example since $86~=~128-32-8-2$  
+%% $118~=~128-8-2$
+which in turn equals ``$~E_7-E_5-E_3-E_1~$'',
+ the final stage of 
+$G$'s construction of $T^*_{86}$ will
+first
+ set node
+$F_1$ equal to ``$~E_7-E_5~$'',
+then
+ set node
+$F_2$ equal to 
+ ``$~F_1-E_3~$'' 
+and lastly have the desired
+output node
+$F_3$ represent the final answer as the
+quantity of  ``$~F_2-E_1~$''.)
+
+It is easy to see that this methodology will never use more
+than  
+  $O\{~$Log$(n)~\}$ logical symbols to encode
+ $T_n^*$ as a Dag-oriented ground term.
+Moreover, the needed pointers in the Dag graph $G$
+will require no more than LogLog$(n)$ bits to
+%separate its
+distinguish between these
+   $O\{~$Log$(n)~\}$ 
+separate objects.
+%logical symbols.
+Hence
+if one selects to use a pointer methodology to formulate
+$G$'s graph,
+then
+%   our full graph $G$ will need 
+no more than
+ $O\{~$Log$(n)~\cdot~$LogLog$(n)~\}$ bits
+will be needed
+to encode
+all these pointers.  $~~\Box$
+
+
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/2011-2019/2015-lfcs.bak b/nachlass/collected_dew_materials/2011-2019/2015-lfcs.bak
new file mode 100644
index 0000000..68a4b23
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2015-lfcs.bak
@@ -0,0 +1,7915 @@
+%%  2015 sept 9 after submission CHANGES ONE SENTENCE IN EXAMPLE 2.5
+
+%% 2015 sept 7  2.20 pm (after finding addrees)
+%% after reding conclusion to BOB
+% chipped off end
+
+
+%% www.cs.albany.edu/~dew/algor
+
+
+%% 2015 home august 24 11.2  am 
+
+%% 2015 home august 22 1.1  pm (single space)
+
+
+ % 2015 july 4 3.4 am after spell 10.1 am after sinatra
+
+ % 2015 july 2 3.15 pm 
+
+%% 2015 july 2 2.50 pm upstairs 
+
+%% 2015 july 1  10.30  am downstairs
+
+
+
+%% notarized notes 2015 april 2 6.3  am april 4 notarize again 
+
+%% home 2014 feb 8 1.15am (new email address)
+
+%% home 2015 feb6 4.3 am   suny 2.40 pm home 6.15 pm
+
+
+%% gmail dan.willard.albany and Prof.DanEdwardWillard
+%% gmail password cpZ9ar48s
+
+
+%%% SUNY JAN 11 Brad Copy 8.4  pm
+
+%%  SUNY   jan11 5/30pm spell check
+
+%% 2015 HOME jan 10  9.4  pm  pm New Abstratct
+
+%% 512 6932
+
+%% Towards a Restructuring of Hilbert's Consistency Program
+
+% www.cs.albany.edu/~dew/algor/
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[12pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+
+
+
+% \addtolength{\oddsidemargin}{-0.95in}
+ \addtolength{\oddsidemargin}{-0.9in}
+%\addtolength{\oddsidemargin}{-1.0in}
+% \addtolength{\oddsidemargin}{-0.95in}
+
+\setlength{\textheight}{9.6 in}
+\setlength{\textheight}{8.8 in}
+\setlength{\textheight}{9.3 in}
+\setlength{\textheight}{9.55 in}
+%above too short
+
+
+\setlength{\textwidth}{6.3 in}
+%% PRINT
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{5.4 in}
+
+
+\setlength{\textwidth}{6.9 in}
+%% \setlength{\textwidth}{6.7 in}
+
+% \setlength{\textwidth}{6.3 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+% Above IDeall
+
+
+%% \setlength{\textwidth}{6.4 in}
+%%%% above brad with 11 point
+
+%\setlength{\textwidth}{6.0 in}
+%\setlength{\textwidth}{5.7 in}
+
+%\setlength{\textwidth}{6.4 in}
+
+%\setlength{\textwidth}{5.5 in}
+
+%\addtolength{\topmargin}{-1.0in}
+\addtolength{\topmargin}{-0.95in}
+%\addtolength{\topmargin}{-1.0in}
+%\addtolength{\topmargin}{1.2in}
+
+%\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pag5{Page 5}
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+
+% \title{ A 2-Part Conjecture about How  
+
+
+
+
+\title{Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After
+A New $~\theta~$ Function Primitive
+%AFTER A NEW ``$~\theta~$'' Function Primitive
+Is Added to
+A New $~\theta~$ Function Primitive
+Arithmetic's Formalism}
+
+
+\title{Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After A New $~\theta~$ Function Primitive
+%AFTER A NEW ``$~\theta~$'' Function Primitive
+Is Added to Arithmetic's Formalism}
+
+
+\title{On  How the Introducing of a 
+ New $~\theta~$ Function Symbol
+Into Arithmetic's Formalism Is Germane
+to Devising Axiom Systems that Can 
+Appreciate Fragments of Their Own
+Hilbert Consistency}
+
+
+
+%% 
+%% \title{On the 
+%% Likelihood
+%% That a 
+%% Curtailed but 
+%% Well-Defined
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Should be
+%% Feasible
+%% for the
+%% Case of
+%% Hilbert Deduction}
+
+
+% \title{On the Almost-Certain Likelihood
+% That a Sharply Curtailed but 
+% Well-Defined
+% %Significant
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Ought to be
+% %Plausible 
+% Feasible
+% for the
+% % Even the Challenging
+% Case of
+% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+% \title{\large \bf On the Revival of a Modified and Diluted Version of
+% Hilbert's Consistency Program (Extended Abstract)} 
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E. Willard \thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+%Email = dew@cs.albany.edu.}}
+%\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{University at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+%\begin{abstract
+\baselineskip = 1.17 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+
+%  PERTECT TITLE ABOVE ALTHOUHG PERHAPS OLD MANUSCRIPT BETTEDR
+
+\begin{abstract}
+
+%\Large
+
+% \baselineskip = 1.8 \normalbaselineskip 
+%aaaaaaaaaaa
+
+\large
+\LARGE
+ \normalsize
+
+
+ It is known that the combined work of Pudl\'{a}k and Solovay
+ \cite{Pu85,So94}, enhanced by some added techniques of Nelson and
+ Wilkie-Paris \cite{Ne86,WP87}, implies no reasonable axiom system can verify
+ its own Hilbert consistency, when it recognizes Successor as a total function
+ and treats addition and multiplication as 3-way relations (as Example
+ \ref{ex-2.3} will explain).  These considerations will lead us to examine
+ unconventional axiomatizations for arithmetic that continue to view addition
+ and multiplication as 3-way relations, but which replace the successor
+ function symbol with an entirely new operator, called the ``$~\theta~$''
+ primitive.
+
+\medskip
+
+%% This $~\theta~$ operator
+%% will
+%% allow us 
+%% to encode any integer $~n~$ by a term $~T_n~$
+%% whose length will exceed the $O(~$Log$~n~)$ length of a
+%% binary encoding 
+%% by
+%% only  the
+%% relatively
+%%  small magnitudes formalized by
+%% Proposition \ref{th-3.3}  and Remark \ref{rem-def-3.4}.
+% Proposition 3.3  and Remark 3.6
+
+It is likely that this paradigm can be combined 
+with our prior results from \cite{wwapal} 
+%%% 
+%% with the prior results 
+%% in our APAL 2006 paper
+%% REMOVE NEXT lINE
+to construct  axiom systems that are
+seriously
+diluted but
+able to verify their Hilbert-style 
+consistency
+in some interesting fragmentary respects.
+
+\end{abstract}
+
+\bigskip
+\bigskip
+\bigskip
+\LARGE
+
+% ttttt THIS PAPER SHOULD BE MASTER DRAFDT for future articles.
+
+
+\bigskip
+\bigskip
+\bigskip
+
+
+\normalsize
+
+{\bf Keywords:}
+Bounded Arithmetic, 
+G\"{o}del's Second Incompleteness Theorem, Hilbert's Second
+Open Question,
+Semantic Tableaux Deduction, 
+and Hilbert
+Deduction.
+
+
+
+\bigskip
+
+
+% {\bf Keywords:}
+% G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+% Open Question,
+% Hilbert-styled Deduction (and its Frege-like analogs).
+
+
+
+
+% \bigskip
+% 
+% 
+% 
+% {\bf Mathematics Subject Classification:}
+% 03B52; 03F25; 03F45; 03H13 
+% 
+% 
+% 
+% \bigskip
+% \bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+%\begin{abstract}
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+\newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+\def\gvxs{ }
+
+
+\def\gvxs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+\def\gvxs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 0pt}
+
+\def\gvxs{ \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 7pt}
+\def\gvxs{ \large \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+
+
+\def\gvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvxs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvxs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+
+\noindent
+
+
+\newpage
+
+\def\gvxs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvxs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvxs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvxs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvxs{ }
+
+
+\gvxs
+
+\footnotesize
+
+
+\def\gvxs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvxs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvxs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvxs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvxs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvxs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvxs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+ \def\gvxs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+
+
+ \def\gvxs{ \Large \baselineskip = 2.15 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \Large \baselineskip = 2.18 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \Large \baselineskip = 2.11 \normalbaselineskip  \parskip - 2pt}
+\def\hvxs{ \Large \baselineskip = 2.18 \normalbaselineskip  \parskip 3pt}
+
+
+
+
+
+
+
+
+
+
+ \def\gvxs{ \large \baselineskip = 2.75 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \large \baselineskip = 2.78 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \large \baselineskip = 2.71 \normalbaselineskip  \parskip - 2pt}
+\def\hvxs{ \large \baselineskip = 2.78 \normalbaselineskip  \parskip 3pt}
+
+
+ \def\fvxs{ \normalsize \baselineskip = 1,46 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 1,47 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 1,47 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \normalsize \baselineskip = 1,48 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \normalsize \baselineskip = 1,41 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\hvxs{ \normalsize \baselineskip = 1,48 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 1pt}
+
+
+
+ \def\fvxs{ \normalsize \baselineskip = 1,66 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 1,67 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 1,67 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \normalsize \baselineskip = 1,68 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \normalsize \baselineskip = 1,61 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\hvxs{ \normalsize \baselineskip = 1,68 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 1pt}
+
+
+
+ \def\fvxs{ \normalsize \baselineskip = 0.96 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 0.97 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 0.97 \normalbaselineskip  \parskip 2pt}
+  \def\svxs{ \normalsize \baselineskip = 0.91 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+
+\def\hvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 2pt}
+
+
+ % \def\svxs{ }
+
+%moving
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\lvxs{ \normalsize \baselineskip = 0.99 \normalbaselineskip  \parskip 2pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+
+\def\rvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\lvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\hvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 2.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 2.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 2.01 \normalbaselineskip  \parskip 2pt}
+
+%moving
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip
+  3pt}
+\def\lvxs{ \normalsize \baselineskip = 0.99 \normalbaselineskip  \parskip 2pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+
+\def\tempvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+
+%\def\tempvxs{ }
+
+\lvxs
+\nvxs
+
+%%% IGNORE BELOW:
+
+%% NOTE TO ME: Page 5 is cited in paper. needs to  BE UPDATED
+%%% AND prehaps GO BACK OLD FORM.
+
+
+%%% iiiii
+\section{Introduction}
+
+\label{ss1}
+\label{ss2}
+
+\vspace*{- 0.5 em}
+
+We have published a series of articles about generalizations
+and boundary-case exceptions for the Second Incompleteness Theorem
+in \cite{ww93}-\cite{ww14}. One theme of this literature was that
+such boundary-case exceptions will arise when multiplication
+is treated as a 3-way relation by a system which verifies its own consistency
+%in a 
+under
+semantic tableaux deduction.
+% instead of Hilbert deduction 
+%%%%%%% context.
+A 15-page summary of
+this research  appeared 
+in
+\cite{ww14}, but the latter is 
+not
+% unnecessary to examine  as 
+a prerequisite for reading this paper.
+
+
+%%  our prior research 
+%% about this topic
+%% was provided in our
+%% Wollic-2014 paper \cite{ww14}, but it is unnecessary for a reader
+%% to examine \cite{ww14} as a prerequisite for this paper.
+
+\smallskip
+
+The main shortcoming in our prior research was that our formalisms
+were mostly unable to recognize their own consistency under 
+Hilbert-style deductive methods. This was because a version
+of the Second Incompleteness Theorem, 
+due to the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87}, indicated the mere assumption that the successor
+operation was a total function was sufficient to trigger off the power
+of the Second Incompleteness Theorem
+for logics that verified their own consistency relative to Hilbert
+deduction. In particular, only our paper \cite{wwapal}
+% (which was 
+%by coincidence invited by editor Sergey Artemov to appear in APAL)
+was 
+% exclusively 
+devoted to addressing this problem.
+Its ISCE formalism could verify its own Hilbert consistency, but it
+unfortunately required deploying an infinite number of separate constant
+symbols because the presence of even a successor function symbol
+would trigger off the Second Incompleteness Effect for systems corroborating
+their own Hilbert consistency.
+
+The current article will introduce a modification of \cite{wwapal}'s
+ISCE formalism, called IQFS, that 
+addresses this
+% should help partially resolve this
+challenge. It will introduce a 
+% new
+function symbol, called the ``$~\theta~$''
+primitive, that will enable an arithmetic to construct the infinite 
+collection of 
+% positive 
+integers, 
+{\it without using} counterparts of any of the
+%  conventional growth 
+% function 
+% primitives of 
+successor, addition or multiplication
+function-mappings.
+
+\medskip
+
+It is almost certain that \cite{wwapal}'s proof of ISCE's
+consistency preservation property will generalize for the
+current article's IQFS formalism. 
+We will provide no proof supporting our  conjecture in this
+extended abstract, but the intuition 
+behind it
+% supporting our conjecture 
+will become quite evident. 
+Its gist will be that our proposed
+new $~\theta~$
+primitive  
+%will be 
+should likely be
+germane to 
+%{\it some special} 
+{\it some unusual} 
+axiom systems owning a
+{\it small 
+% quite
+fragmentized}  knowledge about their own consistency.
+
+
+%% 
+%% The proof of 
+%% IQFS's self-justifying properties will, 
+%% however, be much longer
+%% than
+%% %
+%%  \cite{wwapal}'s proof of its analogous Theorem 3
+%% (if the latter result is proven with the same level of detail
+%% and
+%% %  100 \% 
+%% rigor that Theorem 3's proof did receive).
+%% We will sketch in this short conference abstract the intuitive
+%% reason why IQFS should satisfy 
+%% an analog of ISCE's consistency preservation property, but
+%% % provide 
+%% no formal proof
+%% will be provided.
+
+% a formal consistency preservation property, exactly analogous to ISCE.
+
+%%  I do
+%% not want to write up such a proof in a context where I am currently
+%% suffering from Diabetes and Hypertension. It is essentially
+%% 99 \%
+%% certain that this article's proposed new 
+%% $~\theta~$ operator should support our conjecture.
+%% The purpose of this article will be to invite the
+%% research community to investigate how one can 
+%% expand the mini-formalism from 
+%% \cite{wwapal}'s Theorem 3 to rigorously prove our
+%% stated conjecture.
+
+% 
+% \section{More Detailed Description of Goals}
+
+%\label{ss2}
+
+
+% This article will define a new
+%  ``$~\theta~$'' 
+% function symbol
+% that should enable unusual logics
+% %% 
+% %% % 5-10 \% 
+% %% boundary-case
+% %% effect where a 
+% %% system can own
+% %% 
+% to own a
+% {\it diluted but tangible} knowledge about 
+% their own
+% % its 
+% %own 
+% consistency.
+
+\bigskip
+
+
+%% {\bf More Detailed Description of Goals}
+%% %
+%% % It is  known the
+%% %  
+%% G\"{o}del's
+
+%More precisely, 
+
+During this article, we will often note
+the 
+Second Incompleteness Theorem 
+was
+%known to be
+ published in two 
+% quite 
+different 
+ forms during
+1931-1939.
+Its initial 1931 variant, formalized by Theorem XI
+in  G\"{o}del's
+% millineal 
+paper \cite{Go31},
+% 
+% 
+% Its Theorem XI,
+% later known as the ``Second  Incompleteness
+% Theorem'',
+% %,
+% %appearing in G\"{o}del's millennial paper \cite{Go31}. 
+% 
+demonstrated
+% that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+%  Russell-Whitehead 
+Principia Mathematicae formalism
+% $\, P \,$
+could
+% could
+ verify
+its own consistency.
+The widely quoted more general
+result, that 
+every consistent r.e. 
+extension
+ of Peano Arithmetic must
+be unable to prove a theorem affirming its
+own consistency, 
+was 
+first
+published 
+%% 
+%% (see \footnote{ Boolos states in \cite{Bool}
+%% that it has been open to scholarly debate
+%% whether or not the 1939
+%% Hilbert-Bernays generalization of the Second Incompleteness Theorem
+%% is or (is not) a straightforward generalization of
+%% G\"{o}del's initial result} ) 
+%% 
+in the 1939 edition of
+the Hilbert-Bernays
+textbook \cite{HB39}.
+
+% It has been considered
+% to be the definitive demonstration of the broad reach of
+% the Second Incompleteness Effect.
+
+%%  
+%% It also established, beyond any reasonable doubt, that any type
+%% of formalism possessing a conventional knowledge of its own consistency,
+%% must rely upon a 
+%% foundational structure
+%%  fundamentally different from Peano Arithmetic. 
+%% (This is because the 
+%% Hilbert-Bernays 
+%% textbook formalized the forerunner of
+%% what has now been known as the 
+%% Hilbert-Bernays Derivability Conditions \cite{HB39,HP91,Lo55,Mend},
+%% as a mechanism for
+%% % foreseeing 
+%% envisioning
+%% the 
+%% astonishing
+%% broad generality of the
+%% Second Incompleteness Effect.) 
+ 
+
+It is, thus, fascinating that Hilbert,
+as the co-author of 
+% an important 
+a
+%very 
+% historic
+generalization of the Second Incompleteness Theorem,
+never withdrew the
+% chose to  never fully withdraw his 
+1926 justification
+ \cite{Hil26}
+for his consistency program:
+\begin{quote}
+\small
+\baselineskip = 0.9 \normalbaselineskip 
+$*~$~ 
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding''}
+% itself.''}
+\end{quote}
+
+
+%% Indeed, 
+%% %Instead,
+%% Hilbert 
+%% always insisted  some
+%% new formalism would revive his consistency program
+%% and had its
+%% motto
+%% ({\it ``Wir m\"{u}ssen wissen,  $~$Wir werden wissen''} )  
+%% engraved on his tombstone.
+
+
+% Moreover, it 
+% 
+It 
+is also 
+known
+\cite{Da97,Go5,Yo5}
+that G\"{o}del
+was
+% also
+doubtful about the generality of the Second Incompleteness
+Theorem for at least two years after its publication.
+He thus inserted the following
+cautious caveat into
+his famous
+1931 
+% millennial
+paper  \cite{Go31}:
+% whose closing section 
+%%% 
+%%% One of the closing paragraphs of
+%%%  \cite{Go31} 
+%%% thus
+%%% included
+%
+%
+%%% contained the following cautious disclaimer:
+%caveat:
+% \newpage
+\begin{quote}
+\small
+\baselineskip = 0.9 \normalbaselineskip 
+\it
+$~**~~$ 
+``It must be
+% expressly 
+noted that
+Theorem XI
+%'s incompleteness result
+(e.g. the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and  there might
+conceivably be ... ''
+%  finite proofs} which cannot
+% be stated in P or in ...''
+\end{quote}
+The 
+% above 1931
+statement $**$ has 
+had
+%been subject to 
+numerous
+%many
+different
+interpretations
+ \footnote{
+ Some 
+ scholars
+ have interpreted
+ $\,**\,$
+ as
+ %as, possibly,'
+ anticipating
+ % 
+ %  attempts
+ % to confirm Peano Arithmetic's consistency,
+ % via 
+ % either
+ % 
+ Gentzen's formalism or 
+  G\"{o}del's Dialetica interpretation.}.
+All
+ G\"{o}del's 
+biographers
+\cite{Da97,Go5,Yo5}
+%%%have
+have noted
+% 
+% 
+% 
+% (with
+% some 
+% scholars
+% viewing it as germane to
+% Gentzen's formalism or 
+%  G\"{o}del's Dialetica interpretation).
+% 
+% \gvxs
+% \nvxs
+% \bvxs
+% %\parskip 1pt
+% 
+% \noindent
+% The 
+% % above 1931
+% statement $**$ has 
+% had
+% %been subject to 
+% numerous
+% %many
+% different
+% interpretations
+% (with
+% some 
+% scholars
+% viewing it as germane to
+% Gentzen's formalism or 
+%  G\"{o}del's Dialetica interpretation).
+% % 
+% % \footnote{
+% % Some 
+% % scholars
+% % have interpreted
+% % $\,**\,$
+% % as
+% % %as, possibly,'
+% % anticipating
+% % % 
+% % %  attempts
+% % % to confirm Peano Arithmetic's consistency,
+% % % via 
+% % % either
+% % % 
+% % Gentzen's formalism or 
+% %  G\"{o}del's Dialetica interpretation.}.
+% % 
+% All
+%  G\"{o}del's 
+% biographers
+% \cite{Da97,Go5,Yo5}
+% %%%have
+% noted
+% % 
+% % 
+% 
+% 
+%
+his
+% % that G\"{o}del's
+initial intention
+was
+to
+establish
+%achieve
+Hilbert's proposed objectives, before 
+%he proved
+proving
+%proving
+% G\"{o}del proved
+a result 
+% 
+% however,
+% %%%%%his
+% G\"{o}del
+% did  originally 
+% seek 
+% % goal was 
+% to
+% establish
+% %achieve
+% Hilbert's proposed objectives before 
+% proving
+% % G\"{o}del proved
+% a result 
+% 
+leading
+%that led 
+in the opposite direction.
+Yourgrau \cite{Yo5}
+records
+%furthermore,
+ how
+von Neumann 
+% surprisingly
+%did
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive 
+%%%  achievement of a'
+ termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+It is known 
+ G\"{o}del 
+began to 
+more fully
+endorse 
+the Second Incompleteness
+Theorem
+during a 1933
+%% Vienna
+lecture  \cite{Go33},
+and he 
+% told biographers he
+strongly
+%completely 
+% fully
+embraced it
+after learning about Turing's work
+\cite{Tur36}.
+
+\smallskip
+
+Our research
+in \cite{ww93}-\cite{ww14}
+%has been 
+has been
+% is
+related to issues 
+%analogous
+ similar 
+to those
+%that were 
+raised by Hilbert and 
+G\"{o}del
+in
+ statements  $*$ and $**$.
+This is because it is counter-intuitive and awkward to
+%presume that 
+explain how
+human beings can maintain the
+psychological drive and 
+needed energy-desire to cogitate, without
+being stimulated by 
+{\it some type (?)} of instinctive faith in their own
+consistency. The new ``$~\theta~$'' primitive, introduced in the 
+current article, will
+further
+ reinforce this perspective.
+
+%  (under a definition of 
+% % formal  consistency
+% such
+% % this concept 
+% that is suitably 
+% gentle and
+% % delicate
+% soft
+%  to 
+% be consistent with the 
+% Incompleteness Theorem's requirements).
+
+% preclude a violation of the 
+% restrictions imposed by the Incompleteness Theorem).
+
+
+\smallskip
+
+We emphasize 
+ that 
+%our current 
+the present
+paper will differ from
+all our prior research (except for \cite{wwapal}'s
+trial-balloon result) 
+{\it by changing the focus from
+semantic tableaux deduction to a Hilbert-style
+deductive methodology.}
+
+\smallskip
+
+%\parskip 0pt
+
+
+%% 
+%% Accordingly, our research in
+%% \cite{ww93}-\cite{ww14} 
+%% has explored both generalizations and
+%% boundary-case exceptions of the Incompleteness Effect, so as
+%% to determine what type of boundary-case evasions are permitted.
+%% Our prior research in \cite{ww93}-\cite{ww14} 
+%% had used mostly cut-free forms of deduction to
+%% evade the 
+%% restrictions imposed by the
+%% Second Incompleteness Effect. The current article will instead
+%% focus on more pristine Hilbert-Frege methods of deduction.
+%% They are likely to support an evasion of the Second Incompleteness
+%% Effect when our axiom systems replace the traditional  
+%% growth properties of the addition, multiplication and successor 
+%% function symbols with our new $~\theta~$ primitive.
+%% 
+%% \smallskip
+%% 
+%% The motivation for this replacement will be
+%% explained during the next section of this article.
+%% It is needed
+%% essentially
+%% because
+%% a
+%%  version of the Second Incompleteness Theorem,
+%% due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% %will show 
+%% demonstrates
+%% that 
+%% if an axiom system $~\alpha~$
+%% proves any 
+%% of \eq{totxtefs} - \eq{totxtefm}'s totality statements
+%% then it is incapable of confirming its own consistency
+%% under a Hilbert-style deductive method.
+%% 
+%% 
+%% 
+%% 
+%% 
+%% \smallskip
+
+Our results will suggest  it is possible to obtain a
+{\it part-way 5-10 \%
+positive} interpretation
+for what
+Hilbert and G\"{o}del 
+% advanced
+ were seeking 
+% a Consistency Program
+to establish
+%%  seeking to accomplish
+% contemplating 
+in their 
+statements
+$*$ and $**$, within a context where 
+it is  known that the 
+Second Incompleteness Effect precludes a full achievement of
+%these 
+Hilbert's
+objectives
+from ever transpiring.
+The last five minutes of
+a 
+recent
+ 60-minute YouTube 
+presentation
+ by Harvey Friedman
+\cite{Fr14},
+entitled 
+% the 
+{\it ``The Blessing and Curse of Kurt  G\"{o}del''},
+suggested that it is
+%the themes of Hilbert and G\"{o}del's
+%remarks $*$ and $**$ by indicating that 
+%it is 
+interesting to explore futuristic partial 
+boundary-like evasions of the Second Incompleteness Theorem,
+despite the stunning strength of 
+G\"{o}del's result.
+It is within this context where our proposed use of a new
+$~\theta~$ primitive 
+symbol to replace the growth properties
+of the traditional addition, multiplication and successor function
+symbols may be of potential interest.
+
+\smallskip
+
+The development of our $~\theta~$ primitive 
+was 
+% partially 
+influenced by a private email
+communication 
+we
+% had 
+received
+from  
+Pavel Pudl\'{a}k \cite{Pupriv},
+as \textsection \ref{ss4}
+% \ref{ss3}  \& \ref{ss4} 
+%shall 
+will
+explain.
+We also emphasize that the 
+%conventional 
+usual
+interpretation of
+the
+% Second 
+Incompleteness Theorem, as precluding
+Hilbert's Consistency Program from
+% ever 
+achieving its initially
+specified objectives, is certainly correct.
+{\it Our only caveat} is that 
+some  
+{\it very  tiny} 5-10 \%
+% perhaps
+{\it fragmentized part} 
+of Hilbert's and G\"{o}del's aspirations in
+$*$ and $**$ ought to be viable.
+
+%fragment of its objectives ought to be viable.
+
+
+%% the latter should not lead  one
+%% to
+%% ignoring the role that a
+%% human's instinctive faith in his/her's internal
+%% consistency 
+%% %crucially stimulates and motivates
+%% plays in stimulating and motivating
+%% human cognition.
+
+%% 
+%% It is
+%% from this 
+%% special
+%% perspective where our prior research and
+%% new 
+%% results 
+%% %2-part conjecture 
+%% will
+%% % does 
+%% suggest that 
+%% an approximate
+%% %at least a
+%% 5-10 \%
+%% fragment 
+%% % fraction
+%% of what Hilbert and G\"{o}del 
+%% %suggested in 
+%% had
+%% sought
+%% in $*$ and $**$ 
+%% %could
+%% should be
+%%  plausibly 
+%% %is
+%% %% be formally 
+%%  feasible.
+%% 
+
+
+%\gvxs
+
+\vspace*{- 0.6 em}
+
+\section{Starting Perspective}
+%  222222}
+\label{ss3}
+
+\vspace*{- 0.6 em}
+
+%%! 
+%%! This article will be written in a style so that its
+%%! overall theme (if not full details)
+%%! should become 
+%%! {\it  quickly} comprehensible to a reader who has
+%%! examined 
+%%! only
+%%!  one of the
+%%! % introductory 
+%%! logic textbooks by say Enderton,
+%%! Fitting, H\'{a}jek-Pudl\'{a}k,
+%%!  Mendelson
+%%! or Papadimitriou \cite{End,Fi96,HP91,Mend,Papa}.
+%%! %% 
+%%! %% We will rely mostly upon the
+%%! %% precise
+%%! %% deductive calculi notation employed
+%%! %% in Section 2.4 of Enderton's textbook,
+%%! %% but any of 
+%%! %% the 
+%%! %% similar
+%%! %% Hilbert-style deductive calculi of 
+%%! %%  H\'{a}jek-Pudl\'{a}k,
+%%! %%  Mendelson
+%%! %% or Papadimitriou \cite{HP91,Mend,Papa}.
+%%! %% will also be suitable for achieving our results.
+%%! 
+%%! %% 
+%%! %%  
+%%! %% ( Papadimitriyou's textbook
+%%! %% generously states it employs a deductive notation
+%%! %% that 
+%%! %% has
+%%! %% stemmed from its predecessor in 
+%%! %% Enderton's textbook.)
+%%! 
+%%! 
+
+
+
+%% In order to make our 
+%% results
+%% %research 
+%% apply to the 
+%% formalism
+
+It is helpful to employ a flexible vocabulary so
+% that our
+our 
+results
+will apply 
+%research 
+%to the 
+%formalisms
+to any of the
+%% 
+%% accessible to
+%% some 
+%% readers who are acquainted with only one of the 
+%% 
+textbook 
+formalisms of
+% settings outlined by 
+%  
+say 
+Enderton,
+Fitting, H\'{a}jek-Pudl\'{a}k,
+or Mendelson
+\cite{End,Fit,HP91,Mend}.
+% 
+% ,
+% %% or
+% %%  Papadimitriou 
+% %% \cite{End,Fit,HP91,Mend,Papa},
+% %% 
+% %% 
+% %the widest
+% % possible  
+% %audience, 
+% it is 
+% helpful
+% % useful to 
+% use
+% %employ 
+% a
+% % very 
+% flexible
+%  vocabulary.
+% %%
+% %%that 
+% %%allows a reader to 
+% %%%quickly 
+% %%%translate results 
+% %%traverse
+% %%from
+% %%one textbook to another.
+% % Therefore, let us define a
+Let us 
+% thereby
+call an
+%\newpage
+%
+%\noindent
+ordered pair $(\alpha,d)$ a
+    {\bf ``Generalized Arithmetic''}
+% therefore
+iff its 
+% first and second 
+two components 
+%% 
+%%  Each of the 
+%% textbooks \cite{End,Fi96,Mend,Papa,HP96} have
+%% employed
+%% substantially
+%%  different variants of Basis and 
+%%  Deductive-Apparatus structures. 
+%% 
+are 
+% described
+formalized
+% defined 
+as below:
+%% 
+%% Their
+%% definitions in
+%%  Items (1) and (2)
+%% %simple 
+%% %%%definitions of these two notions
+%% %given below,
+%%     allow one to easily translate 
+%% %theorems 
+%% formalisms
+%% % methodologies
+%% from
+%% one textbook 
+%% % source
+%% to another:
+%% 
+%% %\njp
+%% % \newpage
+%% \parskip 2pt
+\bee
+\item
+The {\bf ``Axiom Basis''} $~\alpha~$ 
+for an arbitrary arithmetic
+shall be defined as
+the set of 
+{\it proper axioms} employed by the 
+formalism  $(  \alpha  , d  )$.
+\item
+An arithmetic's {\bf ``Deductive Apparatus''} $~d~$ 
+is defined as
+the 
+{\it combination} of its formal rules for inference
+and 
+%its
+the
+ built-in
+ logical axioms ``$~L_d~$''
+%  (that are 
+% implicitly 
+employed by these rules.
+\ene
+
+%%%\item
+%%%The term {\bf ``Deductive Apparatus''} $~d~$ will
+%%%refer to the 
+%%%{\it combination} of the rules of inference
+%%%used by an arithmetic and its
+%%%the logical axioms ``$~L_d~$'' that 
+%%%render meaning to
+%%%%are an automatic part of
+%%%$~d\,$'s machinery.
+
+
+\begin{exx}
+\label{ex-2.1}
+%\label{ex-basis}
+\rm
+This notation 
+allows one to
+ conveniently separate  the logical axioms
+$~L_d~,~$ associated
+with  $(  \alpha  , d  )~$, from 
+ $\, \alpha \,$'s
+ ``basis axioms''.
+%basis axioms
+It also allows one to  isolate
+and compare
+%  , conveniently, 
+various
+apparatus techniques,
+%technique,
+% employed in the exact formalisms 
+including the
+ $~d_E~$,  
+ $~d_M~$, 
+ $~d_H~$,  
+and  $~d_F~$
+methods
+%that we will now define:
+defined below:
+%% 
+%%  Three
+%% examples of this are illustrated below,
+%% in a context where
+%% are the deductive apparatus machineries defined
+%% in  Enderton's, Mendelson's and Fitting's textbooks
+%% \cite{End,Fi96,Mend}.
+%% 
+\bed
+\item[   i. ]
+The  $~d_E~$ apparatus,
+formalized in 
+\textsection
+ 2.4 of  Enderton's textbook, 
+% will
+uses only  modus ponens
+as a rule of inference.
+The latter will be accompanied
+by 
+a
+4-part 
+system of
+ logical axioms,
+called $~L_{d_E}~$, $\,$ to endow
+ $~d_E~$ 
+with an
+ability to  support
+% apparatus
+% agility so that it  supports
+%can satisfy
+%the analog of 
+G\"{o}del's Completeness Theorem.
+%% '
+%% (similar to  other'
+%% % full-scale '
+%% deductive methodologies).'
+%% 
+%%%% 
+%%%% (Papadimitriyou's
+%%%% % in-depth  exploration
+%%%% textbook \cite{Papa}  about
+%%%% % examination  of 
+%%%% the Logic-Computer interface
+%%%% relies
+%%%% explicitly 
+%%%%  upon
+%%%% % uses  
+%%%% Enderton's  
+%%%% % underlying 
+%%%% apparatus mechanism.)
+%%%% 
+%% %uses
+%% relies upon
+%%  Enderton's 
+%% approach   $d_E$.)  
+
+%% %relies 
+%% does rely
+%% upon 
+%% Enderton's apparatus
+\item[  ii. ]
+The  $~d_M~$ 
+apparatus in
+\textsection 2.3 
+of  Mendelson's textbook
+and  the $d_H$ 
+ apparatus
+in  \textsection 0.10
+of the H\'{a}jek-Pudl\'{a}k's
+ textbook
+employ a more compressed set of logical axioms
+than $\, d_E \,$,
+but
+they 
+instead
+use
+two rules of inference
+% (formalizing separately
+( modus ponens and generalization).
+%% plus a smaller set of logical axioms, which Mendelson
+%% has called A1-A5.
+%% Also, the $d_H$ 
+%%  apparatus 
+%% on pages ???? 
+%% of the 
+%%  H\'{a}jek-Pudl\'{a}k textbook
+%% uses a slightly different variation of a generalization. 
+%% (In the end, 
+In the end, 
+% both
+ $~d_M~$ 
+and  $~d_H~$ 
+prove the same
+% set of 
+theorems
+as   $~d_E~$ with 
+only
+%  minor and 
+unimportant changes in
+proof length.
+\item[ iii. ]
+The 
+``semantic tableaux''
+ $\,d_F \,$ 
+apparatus in
+Fitting's 
+and Smullyan's 
+textbooks 
+\cite{Fit,Smul}
+was
+ the 
+% main 
+focus of our 
+investigations in \cite{ww93,ww1,ww5,ww6,ww14}.
+It will be rarely used
+in the current article,
+however.
+Unlike
+ $~d_E~$,  $~d_M~$  and  $~d_H~$, it
+employs no logical axioms.
+It instead
+ uses a more complicated rule of inference.
+This tableaux apparatus 
+% and also Resolution, have  been 
+ has 
+% been found to  have many 
+many
+%a wide array of
+applications
+% underfor 
+in
+automated deduction,
+although it is 
+less efficient than
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+in
+% under
+% extremal
+worst-case 
+environments.
+% settings.
+%circumstances.
+\ennd
+\end{exx}
+
+\tvxs
+
+\begin{dff}
+\label{def-2.2}
+\rm
+Each of the
+% deductive 
+methods of
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+have the property that if a theorem $\, \Psi \,$
+has a proof
+with  length $~L~$ 
+ from an arbitrary 
+axiom basis $~\alpha~$ 
+under one of these deductive systems,
+then it will have a proof from these other formalisms
+with lengths bounded by Polynomial$(L)$.
+The term 
+{\bf ``Hilbert-style''} deductive method will,
+thus, refer to any  deductive 
+% apparatus will refer to any other
+apparatus $~d~$ that
+has a modus ponens rule and
+employs
+% similarity has its 
+proof lengths
+% being 
+equivalent to within a polynomial magnitude
+to
+%of
+the comparable proof lengths from $d_E$,  $d_M$  and  $d_H$.
+\end{dff}
+
+
+%% and which 
+%% also
+%% assures 
+%% that the proofs of any
+%% two theorems $~\Phi~$ 
+%% and $~\Psi~$ 
+%% (under $d$ from any
+%% axiom basis $~\alpha~$) 
+%% will
+%% always 
+%% have
+%% % by more than a constant factor 
+%% the sum of the lengths of the proofs
+%% of $~\Phi \rightarrow \Psi ~$ and $~\Phi~$ 
+%% % under $~d~$ from $\alpha$ always 
+%% formally
+%% bound the length of
+%% $~\Psi\,$'s proof. 
+%% \end{dff}
+
+
+\begin{exx}
+\label{ex-2.3}
+%\label{ex-basis}
+\rm
+Some added notation is
+ needed to
+explain why
+% help outline
+% an important distinction between 
+a Hilbert style
+deductive apparatus, such as $\,d_E\,$,  $\,d_H\,$
+ or $\,d_M\,$, should be distinguished from
+ $d_F$'s
+``tableaux'' apparatus.
+Let
+% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$
+once again 
+% will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us recall 
+that 
+an
+axiom basis
+ ``$\, \alpha \,$''
+is said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+
+\noindent
+%Then an 
+In this context, an
+``axiom basis''
+$\alpha$
+will be called 
+{\bf Type-M} if it contains
+\eq{totdefxs}-\eq{totdefxm}
+% \ref{totdefxs}-\ref{totdefxm}
+as theorems,  
+{\bf Type-A} if it contains 
+%only
+\eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+$\alpha$ 
+%will be 
+is
+called 
+{\bf Type-NS} if it  can prove
+none of these theorems.
+%In this context,
+%The
+%Items (a) and (b) illustrate 
+%%%Below are illustrated several
+%%%implications of this notation:
+The implications of this notation
+are formalized by Items a and b:
+
+%% 
+%% 
+%% , below, 
+%% %will 
+%% illustrate how
+%% a
+%% %% 
+%% %%  the
+%% %% prior
+%% %%  literature has
+%% %% 
+%% %% 
+%% ``Hilbert-style''
+%% deductive apparatus, such as $\,d_E\,$
+%%  or $\,d_M\,$, supports very different generalizations
+%% of the Second Incompleteness Theorem
+%% than  $\,d_F\,$'s
+%% ``tableaux-style'' apparatus:  
+%% 
+%%  the prior literature most germane
+%% to our current article is summarized as follows:
+%% 
+%% 
+%% The relationship of these constructs to 
+%% self-justification 
+%% is explained by
+%% items (a) and (b):
+\bed
+\item[   a.  ]
+The
+%%  
+%% above 
+%% evasions of the Second Incompleteness
+%% Theorem are known to be near-maximal in a mathematical sense.
+%% This is because
+%% the
+%% 
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as formalized by statement $\, ++ \,$,
+%has  implied  
+implies
+no
+natural 
+Type-S system can recognize  its
+own  consistency
+under
+any of $\, d_E \,$'s, $\, d_H \,$'s  
+or
+ $\, d_M \,$'s
+  Hilbert-style 
+versions
+of deduction:
+\begin{quote}
+{\bf ++ }
+\small
+% \footnotesize
+ \baselineskip = 0.9 \normalbaselineskip 
+{\it 
+(Solovay's  
+modification
+%Generalization 
+\cite{So94}
+%1994 Generalization \cite{So94}
+%of a 1985 theorem 
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+with
+%using 
+%some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let $ \, \alpha \, $ denote 
+%%! any consistent
+% a logic
+an axiom  
+basis  
+% axiom system 
+%basis  system 
+%supporting
+% which contains 
+able to prove
+Line
+\eq{totdefxs}'s
+Type-S statement
+and 
+assuring
+%which assures 
+%that 
+the successor operation
+%always 
+does satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+the identity
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+cannot verify its
+%will be unable to recognize its
+%own  
+consistency
+under any Hilbert-style 
+%deductive 
+apparatus $d$,
+%whenever
+if  it
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+the usual % identity,
+associative, commutative 
+ distributive 
+and identity axioms.
+%   -axiom
+% properties.
+\end{quote}
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of $++$'s result. 
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+by
+Buss-Ignjatovic,
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k's initial 1985 article  \cite{Pu85} 
+% implicitly
+did capture 
+essentially
+the 
+{\it great majority} 
+%most
+%%%  much 
+of $++$'s 
+% main
+% underlying
+formalism,
+and
+Friedman did 
+related work
+in
+\cite{Fr79a}.
+
+\item[   b.  ]
+Part of what makes
+% the Pudl\'{a}k-Solovay discovery in
+ $++$ interesting is that 
+\cite{ww93,ww1,ww5,wwapal}
+%Willard
+developed various 
+% separate
+methods for 
+basis systems
+%%% $\alpha$ 
+to confirm their own consistency, whose
+main
+further  improvements are 
+prohibited by either the invariant $++$ 
+or by
+\cite{ww2,ww7}'s hybridization of 
+$++$'s formalism with some further methods of 
+ Adamowicz-Zbierski
+\cite{AZ1}. (As a consequence of these facts, it is known
+that 
+some 
+Type-A 
+and Type-NS
+arithmetics can verify their
+respective
+ semantic tableaux and 
+ Hilbert-styled consistencies,
+but 
+Type-S arithmetics cannot verify their Hilbert consistency and
+most Type-M systems cannot verify their semantic tableaux
+consistency.)
+ \ennd
+\end{exx}
+
+% natural hybridizations is precluded by $++$. These results involve
+% either a Type-NS
+% % basis 
+% system 
+% 
+%  verifying its own consistency
+% under 
+% any  of the 
+%  $d_E$ or  $d_H$
+%  or $d_M$'s
+% Hilbert-style methods,
+% or   a Type-A 
+% %basis 
+% system \cite{ww93,ww1,ww5,ww6,ww14} 
+% verifying
+%  its
+% % own 
+% self-consistency
+% under $d_F$'s tableaux 
+% %deductive 
+% apparatus.
+% Also, Willard \cite{ww2,ww7} observed how one could
+% refine $++$ with Adamowicz-Zbierski's
+% methodology \cite{AZ1} to show 
+%  Type-M  systems
+% cannot recognize their semantic tableaux consistency.
+% \ennd
+% \end{exx}
+
+\lvxs
+\nvxs
+
+
+% \tempvxs
+
+
+%% A more detailed 15-page summary,
+%% % of our prior research, 
+%% germane to
+%% % the
+%%  Item (b),
+%% % , above, 
+%% %can be found in our article
+%% appears in
+%%   \cite{ww14}.
+
+% NNN NEED TO REWRITE NEXT TWO PARAGRAPHS
+
+
+A full 15-page summary
+% of our prior research, 
+of
+ Item (b)'s results
+% , above, 
+%can be found in our article
+can be found in
+  \cite{ww14}.
+It does not need
+ to be read,
+however,
+ as a prerequisite for understanding
+the current paper. This is because our goal
+%%%  in the current paper
+will be to explore axiom basis systems that can recognize their
+own 
+Hilbert-styled
+consistency, and the invariant $++$ indicates that each of the
+classes of Type-S, Type-A and Type-M 
+arithmetics
+are irrelevant to 
+this objective.
+%our goals.
+
+
+%% %% formalisms 
+%% own
+%% % contain 
+%% excessive
+%% growth properties that 
+%% lie outside 
+%% our goals.
+
+%this goal.
+
+%are incompatible with our goals.
+ 
+Instead, the \textsection \ref{seee3} will introduce a new
+growth function symbol, called the ``$~\theta~$'' primitive,
+that allows us to reside within the domain of a Type-NS arithmetic
+because {\it none of the  identities in
+Lines 
+ \eq{totdefxs}-\eq{totdefxm}} 
+will be provable consequences
+of $\theta$'s speedy but unconventional growth properties.
+
+This $\theta$ primitive will be attractive because
+Proposition \ref{th-3.3}
+and Remark \ref{rem-def-3.4} will imply  it supports
+respective $O(~$Log$^3 \, n \,)$ and $O(~$Log$ \, n \,)$
+growth speeds for constructing arbitrary integers $~n$
+(depending on what linguistic notation one uses for encoding integers).
+As a result,
+%%% of this fact,  
+\textsection \ref{ss32}
+ will conjecture that
+a seemingly minor ``IQFS'' modification of \cite{wwapal}'s
+ISCE formalism is an arithmetic that possesses some interesting abilities
+to confirm its own 
+% Hilbert-style 
+Hilbert 
+consistency.
+
+We might add that the discussion in this article will be
+{\it entirely self-contained}
+because 
+ \textsection \ref{ss32}
+ will summarize  \cite{wwapal}'s
+definition of the ISCE formalism.
+
+\begin{deff}
+\label{def-2.4}
+\rm
+Let
+$~\alpha~$ again 
+denote an axiom basis 
+and $~d~$ 
+designate
+ a
+deduction apparatus.
+%  
+%  During our discussion about the open questions
+%  raised by Hilbert's and 
+%   G\"{o}del's
+%  statements 
+%  $*$ and $**\,$,
+%  an 
+%  %
+%  % Then the  
+%  
+In this context, an
+ordered pair
+ $~(  \alpha  , d  )$
+will
+be called  {\bf Self Justifying} when:
+\begin{description}
+% \xxitch
+% \small
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+(or at least one of its axioms)
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+\end{deff}
+
+
+\begin{exx}
+\label{ex-2.5}
+ \baselineskip = 1.04 \normalbaselineskip 
+\rm
+% Using 
+% Definition \ref{def-2.4}'s
+%  notation, our research  
+% 
+Our research  
+in
+\cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+developed
+%\cite{ww93}-\cite{ww14}
+%has  consisted of
+% developing
+ordered pairs  $~(  \alpha  , d  )$
+that 
+were
+%are
+``Self Justifying''.
+It 
+%  has 
+also explored
+how the Second Incompleteness Theorem formalizes
+limits beyond which such formalisms cannot transgress.
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+% second
+%%% axiom 
+system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition.
+%of 
+% this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+%%%
+%%% the following 
+%%% %%% added
+%%% further
+%%% sentence,  
+%%% called
+%%% %%% that we call
+%%% {\bf SelfRef$(\alpha,d)~$}:
+\begin{quote}
+\small
+% \baselineskip = 0.95 \normalbaselineskip  
+%\xxitch
+$\oplus~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of this
+{\bf $\,$``SelfRef$(\alpha,d)$''$\,$}
+statement.
+%%% SelfRef$(\alpha,d)$'s
+%%% self-referential statement.
+Both Kleene and 
+Rogers  \cite{Kl38,Ro67}
+% 
+% Each of
+% Kleene, 
+% Rogers and Jeroslow 
+%  \cite{Kl38,Ro67,Je71}
+% 
+ noted
+$\alpha ^d$ 
+may
+% , however,  
+be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition.
+This is because if the 
+%  ordered
+ pair $(\alpha,d)$ is too strong
+then a classic G\"{o}del-style diagonalization argument can
+be applied to the axiom system 
+$\alpha^d~~=~~ \alpha \, + \, $ SelfRef$(\alpha,d)$,
+where the added presence of 
+% statement 
+SelfRef$(\alpha,d)$'s axiomatic   statement 
+will cause 
+%% this extended version of 
+$\, \alpha^d\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, the machinery of the sentence
+``SelfRef$(\alpha,d)$'' is relatively easy to 
+encode,
+%make well-defined 
+via an application of the Fixed Point Theorem,
+but it
+is
+ironically 
+%%%%%{\it most often  
+{\it 
+typically
+%usually
+useless! }
+\end{exx}
+
+%\newpage
+
+
+Unlike our earlier work, which focused 
+ mostly around a 
+semantic
+tableaux apparatus for deduction,
+the current paper
+will
+explore
+%paper will explore
+\dfx{def-2.2}'s
+more pristine Hilbert-style methodologies.
+%% 
+%% analogous to 
+%% Example
+%% \ref{ex-2.1}'s 
+%% textbook
+%% methods.
+%% 
+% of 
+% $d_E$,  $d_M$
+% and  $d_H$.
+%%! 
+%%! in
+%%! the textbooks by 
+%%!  Enderton,  H\'{a}jek-Pudl\'{a}k,
+%%! Mendelson and Papadamiriyou \cite{End,HP91,Mend,Papa}.
+There are, of course, many types of generalizations
+of the Second Incompleteness Theorem known to
+arise in  Hilbert-like  settings
+\cite{BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Lo55,Kr87,Kr95,Pa71,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,wwapal}.
+Each such
+generalization
+formalizes
+a paradigm where 
+self-justification is infeasible
+under a Hilbert-style apparatus.
+
+\smallskip
+
+Our 
+main
+prior research about
+%%! 
+%%! main work about arithmetics displaying knowledge
+%%! about their 
+%%! 
+Hilbert consistency appeared in \cite{wwapal}.
+Its ISCE$(\beta)$ 
+system could recognize its
+own 
+Hilbert consistency and 
+%%could 
+prove analogs of 
+the  $\Pi_1$ theorems of
+any 
+% r.e. 
+extension $~\beta$ of
+Peano Arithmetic.
+%  's $\Pi_1$ theorems.
+It unfortunately required
+% the 
+use of  infinitely many
+% number of 
+constant symbols, with
+ ISCE$(\beta)$ 
+using one built-in constant
+% symbol 
+$C_i$
+for each power of 2.
+(Such a series of constants seriously
+ deviated from 
+Hilbert's
+intended
+ goals
+in statement $*\,$, as
+we will explain during
+\textsection \ref{ss32}'s detailed review of
+\cite{wwapal}'s results.)
+
+
+%  An alternative in  \cite{wwapal},
+% called ISINF$(\beta)$, required use of only three constant symbols,
+% but its proof lengths were impractically long. 
+
+
+%%! required
+%%! % in excess of
+%%! an impractical
+%%!  $O(N)$ length proof to construct an integer $N$.
+
+\smallskip
+
+Prior to \cite{wwapal}'s publication, 
+Pavel Pudl\'{a}k
+\cite{Pupriv}
+examined this article and asked 
+%us
+% the crucial question about 
+whether
+we could improve 
+upon ISCE
+% 's properties
+ by using Ajtai's observations
+\cite{Aj94}
+ about
+the Pigeon Hole principle.
+%%  
+%% Sam Buss \cite{Bupriv} also asked us
+%% a 
+%% %similar 
+%% related
+%% question
+%% (during a more 
+%% informal 
+%% %abbreviated
+%% conversation).
+%% %(in a more informal manner).
+%% 
+Our prior
+partial
+answer to  Pudl\'{a}k's
+question
+was offered  
+%issue
+%appeared
+%in Sections 6 and 7 of \cite{wwapal}.
+in Sections 6 of \cite{wwapal}.
+A 
+% very 
+different type of reply will be offered
+% 
+% We will offer 
+% % an alternate much  
+% a 
+% % much
+% more sophisticated
+% and different 
+% type of reply
+% % analysis
+% %  formalism 
+% 
+in
+the current 
+paper.
+%article.
+
+
+%%! an abbreviated version of a similar
+%%! question after we verbally summarized to him \cite{wwapal}'s
+%%! planned results.
+
+% in 1997.
+
+
+\begin{deff}
+\label{def-2.6}
+\rm
+Throughout our discussion, a
+% A
+primitive $~F~$ will be called a 
+{\bf Q-Function} 
+iff is is 
+sufficiently ambiguous
+for there to exist an uncountably infinite set of
+% different 
+distinct
+{\it 
+plausible  sequences} of
+ordered pairs in expression \eq{wow} where
+$~F(i)=a_i~$  is allowed as a
+% logically 
+permissible
+%plausible 
+%formalization 
+representation of $F$
+under some fixed axiom system $~\gamma~$. 
+\begin{equation}
+\label{wow}
+  (0,a_0) 
+ ~,~  (1,a_1)   ~,~  (2,a_2)   ~,~  (3,a_3)   ~,~  (4,a_4)~ ...
+\end{equation} 
+\end{deff}
+
+% \gvxs2
+
+\vspace*{- 0.6 em} 
+%It turns out most
+
+%  
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question or most 
+% issues in
+% % other prominent
+% % % mathematical 
+% % questions within 
+% mathematics.
+
+Most
+Q-Function symbols are
+awkward to employ.
+ This is because the
+presence of an 
+ uncountably
+ infinite
+  number of
+different 
+plausible  sequences,
+formalized by Line
+\eq{wow} for solving
+$~F(i)=a_i~,~$  is 
+typically more of a burden than a benefit.
+A
+possible
+%potential
+ exception to this general rule
+of thumb
+ will be
+provided by
+ the next
+section's $~\theta~$ operator: It
+% because it
+ will,
+conveniently,
+ lie outside the scope of
+ $\, ++ \,$'s generalization of the Second Incompleteness
+Theorem.
+This fact will ultimately lead to our
+main  conjecture
+about stronger variants of Type-NS logics
+recognizing their own Hilbert consistency.
+
+
+
+% an enticing manner.
+
+
+%  It 
+% will
+% % should 
+% provide an
+% % enticing
+%  avenue for
+% Type-NS axiom systems to recognize their own
+% Hilbert consistency (if
+% \textsection \ref{ss5}'s 
+% ``IQFS''
+% %anticipated
+% conjecture is
+% correct). 
+
+
+%% 
+%% and suggest a mechanism whereby an efficient form of
+%% ``Type-NS''self-justifying 
+%% arithmetic
+%% can recognize its own Hilbert consistency,
+%% without
+%% viewing
+%% % recognizing 
+%% %%%%%% any of  addition, multiplication and
+%% even
+%% successor  as
+%% a total function.
+
+%In other words,
+% \smallskip
+
+
+%% 
+%% 
+%% \medskip
+%% Thus in a context where the partial drawbacks of
+%% our 
+%% new $~\theta~$ primitive
+%% will be beyond doubt, it will
+%% % simultaneously
+%% renew the
+%% serious
+%%  question about whether a  
+%% {\it 
+%% part-way 
+%% 5-10 \% fragment} of
+%% %positive} interpretation
+%% %can be assigned to
+%% Hilbert's and G\"{o}del's
+%% goals in $*$ and $**$
+%% can be acheived.
+
+
+%% 
+%% Our
+%% suggestion
+%% % conjecture
+%%  will be that
+%% Q-functions 
+%% might
+%% allow one to assign a
+%% {\it 
+%% % part-way 
+%% 5-10 \%
+%% positive} interpretation
+%% for what
+%% Hilbert and G\"{o}del 
+%% were
+%% seeking
+%% %referring to
+%% % a Consistency Program
+%% % to establish
+%% %%  seeking to accomplish
+%% % contemplating 
+%% in their 
+%% famous
+%% %often quoted
+%% statements
+%% $*$ and $**$.
+%% 
+
+%% 
+%% It will enable us to develop ground terms for formulating
+%% any integer $~N~$ using
+%%  $O\{~$Log$(N)~\}~$ 
+%% logical symbols,
+%% in a context where
+%% {\it  none of the} addition, multiplication or
+%% successor  function symbols are employed
+%% by $~\theta \,$'s analog of an 
+%%  $O\{~$Log$(N)~\}~$ 
+%% lengthed 
+%% binary-like 
+%% encoding 
+%% for integers.
+%% % of an integer as a binary number. 
+%% This  alternate
+%%  $O\{~$Log$(N)~\}~$ 
+%% format
+%% for encoding an integer $~N~$ is 
+%% potneitlally useful
+%% %fascinating
+%% because  
+%% Item $\, ++ \,$'s generalization of the Second Incompleteness
+%% Theorem, due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% does not preclude evasions of its invariant when a
+%% ``Type-NS''
+%% axiom system ceases to recognize 
+%%  addition, multiplication and
+%% successor  as total functions.
+
+
+ 
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question.
+% A special class of Q-Functions
+% %  , however,
+% will
+% walk through
+%   Cantor's 
+% % the
+% world of the Uncountably Infinite
+% %% in a more 
+% in an
+% enticing manner,
+% however.
+% %% however.
+% %It 
+% They
+% will suggest a
+% %%% much
+%  {\it  diluted but non-trivial}
+% variant
+% of the aspirations
+% which 
+% %that 
+% Hilbert and G\"{o}del 
+% expressed
+%  in
+% $*$ and $**$ 
+% are 
+% %applicable to
+% feasible under
+% % plausible in the context of 
+% Hilbert Deduction.
+% (This
+% % Q-function a
+% analysis will be 
+% %%%%%%%%%%%%%% quite
+% % entirely 
+% different
+% from 
+% \cite{ww14}'s 
+% examination of
+% %formalisms
+% %the formalisms appearing in our Wollic-2014 paper
+% %%% because 
+% %it will replace 
+% semantic tableaux deduction
+% because it will
+% apply  
+% uniquely 
+% to 
+% Definition \ref{def-2.2}'s 
+% ``Hilbert-styled'' deduction methods.)
+
+ 
+%%% is replaced by the more efficient
+%%% %with the more pristine 
+%%% Hilbert-style deductive methodology.)
+
+%can be achieved.
+
+
+%%! This will suggest
+%%! a {\it limited}  
+%%! and very-much {\it down-sized} version of the formalism that
+%%!  Hilbert
+%%! advocated
+%%! is
+%%! likely
+%%! %probably
+%%! feasible
+%%! and 
+%%! germane to 
+%%! the
+%%! future
+%%! %  computational 
+%%! needs of automated
+%%! theorem provers.
+%%! Our 
+%%! exploration 
+%%!  will also provide a
+%%! % quite
+%%!  new interpretation of the
+%%! meaning of the statements $*$, $**$ and
+%%!  $***$.
+
+% by Hilbert and G\"{o}del.
+
+
+
+% \section{Revisiting a World which Hilbert called
+% {\it ``Cantor's Paradise''}}
+
+%\section{Main Formalism}
+
+% \section{Deploying a New ``$~\theta~$'' Primitive}
+
+% \vspace*{- 0.9 em} 
+
+
+
+%\section{Need for a New ``$~\theta~$'' Primitive}
+
+
+%\section{Mysterious New ``$~\theta~$'' Primitive}
+
+
+% \section{The Surprisingly Useful ``$~\theta~$'' Primitive}
+
+%\section{The  ``$~\theta~$'' Primitive and Its Potential Uses}
+
+\vspace*{- 0.5 em} 
+\section{Arithmetic Under The  ``$~\theta~$'' Primitive}
+\label{ss4}
+\label{seee3}
+
+%333333333333333333333333333
+
+\vspace*{- 0.5 em} 
+
+%\vspace*{- 0.9 em} 
+
+
+% OLD Title was {\it Notation and Basic Concepts}
+
+% Throughout this 
+% paper,
+% %article,
+% % a 
+
+A function 
+ $\, H \, $ 
+will be     called
+a
+ {\bf Non-Growth} operation 
+iff  
+$ H(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$.  Six  examples of  
+ non-growth functions are:
+\bee
+%\small
+\footnotesize
+\parskip - 3pt
+ \baselineskip = 0.6 \normalbaselineskip 
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lfloor ~x^{1/y}~ \rfloor$ when $~y\geq 1~$
+%% 
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% 
+(and zero otherwise),
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+and
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\ene
+%% 
+%% 
+%% \bee
+%% \baselineskip = 0.8 \normalbaselineskip 
+%% 
+%% \item
+%% {\it Integer Subtraction} 
+%% where ``$~x-y~$'' is defined to equal zero 
+%% in {\it the special case} where
+%%  $~x \leq y,$
+%% \item
+%% {\it Integer 
+%% Division}
+%% where ``$~x \div y~$'' equals
+%% $~x~$ when $~y=0,~$ and
+%% it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+%% \item
+%% $Root(x,y)~$ which equals $~ \ulcorner ~x^{1/y}~ \urcorner$ when $~y\geq 1,~$
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% \item
+%% $Maximum(x,y),~~$
+%% \item
+%% $ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+%% and  zero otherwise. 
+%% \item
+%% $Count(x,j)~~=~~$the number of ``1'' bits
+%% among $~x$'s rightmost $~j~$ bits.
+%% \end{enumerate}
+%% 
+These operations were called
+either the 
+{\bf ``Grounding''} 
+or {\bf ``Ground-Level''} 
+ functions
+in our articles \cite{ww1,ww5,ww14}.
+% We will use the
+We will rely upon the
+% The 
+latter nomenclature in the current article
+ because the notion of a  ``Ground-Level''
+function should not be confused with the
+% very 
+quite
+different notion
+of a ``Grounded Term'' (employed by Definition
+\ref{def-3.4}). 
+
+%  TWO DEFS or ONE ???????? 
+
+
+
+Our
+starting  language $L^G$ 
+will
+% shall 
+also contain
+the
+two atomic 
+symbols
+%% relations 
+of ``$~=~$'' and ``$~\leq~$'' and three
+built in constants symbols, $~C_0~$, $~C_1~$ and $~C_2~$, 
+for representing
+the values of 0, 1 and 2. 
+Within this context, 
+ Expressions 
+% Lines
+\eq{newadd} and \eq{newmult} formalize how addition and multiplication
+can be encoded as two 3-way predicates,
+%%  in  $L^G$,
+ denoted as
+Add$(x,y,z)$ and Mult$(x,y,z)$.
+% 
+% (Their
+% %  unusual
+% particular
+% definitions
+% are 
+% % quite
+% %highly
+% useful because they
+% allow our ``Type-NS'' arithmetic to evade
+% satisfying
+% %  Lines 
+%  \eq{totdefxs}-\eq{totdefxm}'s
+% forbidden
+% function-existence
+%  conditions.)
+% 
+%%%%undesirable constraints.)
+% 
+% they do not imply addition
+% and multiplication are total functions (e.g.
+% they permit our arithmetic to be a 
+% ``Type NS system''.) 
+% 
+%  further conditions.)
+% 
+% (These 
+% definitions
+% % for Add$(x,y,z)$ and Mult$(x,y,z)$
+% {\bf notably  allow} a
+% %two 3-way predicates are consistent with a 
+% ``Type NS system'' to
+% {\it evade satisfying} Lines  \eq{totdefxs}-\eq{totdefxm}
+% {\it forbidden}
+% constraints.) 
+% %  further conditions.)
+
+\newpage
+%bbbbbb
+{ \small
+\vspace*{- 1.2 em} 
+\beq 
+\label{newadd}
+z ~ -~x~~=~~ y~~~~ \wedge ~~~~ z~\geq~x
+\end{equation}
+
+\vspace*{- 1.2 em} 
+\begin{equation}
+\label{newmult}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation} }
+$~~~~$Also, 
+our constant symbols,
+ $~C_1~$ and $~C_2~,~$ for representing
+the quantities 1 and 2, 
+will  allow us to
+formalize the following further
+%useful 
+%%%%%%%% function 
+operations:
+\bed
+\small
+\topsep -4pt
+\itemsep -1pt
+\it
+\item[   $~$a.  ] 
+Pred$(x)~~=~~x-1$
+(in a context where 
+%Item 1's
+%the prior paragraph's
+our prior
+definition for Subtraction implies
+Pred(0)$=0$.$~)$   
+\item[   $~$b.  ] 
+Half$(x)~~=~~x \div 2$
+(in a context 
+where 
+``$ ~x \div 2~$'' equals technically 
+$~\lfloor \, \frac{x}{2} \, \rfloor~$ 
+under 
+%our
+% the prior paragraph's 
+our
+notation).
+% of Item 2's definition for Division).  
+\item[   $~$c.  ] 
+Pred$^n(x)$ defined to be $~n~$ iterations of the
+Predecessor operation
+\item[   $~$d.  ] 
+Half$^{ \,n}(x)$ defined to be $~n~$ iterations of the
+halving operation.
+%%
+%%\item[   $~$e.  ] 
+%%Bit$(x,j)~=~$ $Count(x,j)~-~Count(x,j-1)~$
+%%(e.g. the value of $~x \,$'s rightmost
+%% $j-$th bit)
+%%\item[   $~$f.  ] 
+%%Min$(x,y)~~=~~x~-~(y-x)~~$
+%%(e.g. a quantity that represents the minimum of $x$ and $y$
+%%because our definition of ``integer subtraction'' 
+%%implies  $~y-x \,= \, 0 ~$ whenever $\, x \geq y$.$~~)$  
+%%
+\ennd
+
+\nvxs
+
+Let us say
+that
+ a function symbol $H(x_1,x_2...x_j)$
+is
+ {\bf Growth Permitting}
+iff for each integer $~k \geq 2~$ there exists a 
+``growth-tuple''
+$(a_1,a_2...a_j)$ 
+satisfying
+  $~H(a_1,a_2...a_j) \, >\, k~$
+and 
+also
+% simultaneously 
+having
+each $ \, a_i \leq k$ \, 
+It will  be necessary 
+%% for  us 
+to employ
+ either an infinite number
+of constant symbols or some Growth-Permitting function 
+so that an extension of the 
+language $L^G$ can construct the 
+full
+%infinite
+collection of
+ integers of $~3,4,5,6~....~$.
+
+
+
+
+%% One  awkward aspect of this notation is that 
+%% it provides
+%%  no guarantee
+%% that    integers larger than 2 will exist         without the 
+%% presence of some 
+%% further
+%% methodology for producing larger integers.
+
+
+% 
+\smallskip
+
+
+One method for resolving this problem was presented in \cite{wwapal}.
+% 
+% It employed  an infinite number of further constant symbols. The
+% latter's
+%  ISCE$(\beta)$ 
+%  system was
+% %   shown to be 
+% compatible with                     self-justification,
+% but such an infinite number of constant symbols clearly trespassed on
+% Hilbert's goal of using a 
+% %strictly 
+% finite-sized formalism.
+% 
+Its  ISCE$(\beta)$ axiom basis
+deployed  an infinite number of 
+% further 
+distinct
+constant symbols. It
+was
+compatible with   self-justification,
+but deviated from 
+%{\it very sharply from} 
+Hilbert's
+intended
+ goals 
+because it employed
+% by employing 
+%an  
+a {\it highly awkward}
+infinite number of 
+distinct
+ constant symbols.
+(The reader will 
+better
+appreciate this point when
+\textsection \ref{ss32}
+reviews 
+% the
+% properties of 
+ISCE's defining formalism.
+This  difficulty is fundamental 
+% to avoid 
+because
+the Invariant $++$'s generalization of the
+Second Incompleteness Theorem indicates that
+Type-S arithmetics are unable to confirm their
+own Hilbert consistency.)
+
+
+
+\medskip
+
+
+
+%% The 
+%% % self-justifying
+%% ``ISINF'' formalism 
+%% % in
+%% of 
+%% \cite{wwapal} 
+%% offered an alternate method for resolving this difficulty.
+%% %% in the context of a self-justifying logic. 
+%% It
+%% % required the use of
+%% used
+%%  {\it only
+%% three} constant symbols.  It   could prove analogs of all
+%% of Peano Arithmetic's
+%% $\Pi_1$ theorems, but almost all
+%% of
+%%  its proofs
+%% unfortunately
+%%  had lengths longer 
+%% than the number of atoms in the universe.
+%% Most other approaches, for resolving this dilemma, 
+%% % are 
+%% were
+%% also problematic
+%% because
+%% Example \ref{ex-2.3}'s
+%% invariant
+%%  $~++~$
+%% % 
+%% % which Example \ref{ex-2.3}
+%% % attributed to the joint work of 
+%% %   Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris,
+%% % 
+%% showed that essentially every Type-S arithmetic is unable to
+%% recognize its
+%% own consistency under a Hilbert-style deductive apparatus.
+%% 
+%\smallskip
+
+
+% The challenge posed by $++$ 
+% is, thus,
+% substantial.
+% % certainly formidable.
+
+Our goal in 
+%the current 
+this
+article
+will be to suggest 
+how
+%that 
+a Q-function primitive $F(x)$,
+that has
+% an extraordinarily 
+a deliberately
+ambiguous
+ function definition,
+can help overcome the constraints that $++$ imposes. 
+Such an
+% ultra-ambiguous defined 
+unusual
+primitive $~F~$ will have
+an  uncountable number
+of vectors, analogous to Line \eq{wow}, that are permitted 
+solutions to
+$F$'s
+definition.
+Our basic goal
+% in this article 
+will be to outline 
+how this unusual concept is
+likely  germane to the
+self-justifying
+axiom system satisfying
+{\it  a very diluted} but not
+immaterial subset of
+%% 
+%% why it is
+%% likely such Q-functions will enable one to build
+%% surprisingly efficient self-justifying logics that
+%% partially (but not fully) achieve the 
+%% %{\it  diluted portion} of the
+%% 
+%aspirations  
+%that
+Hilbert's and G\"{o}del's goals in 
+%expressed in 
+statements 
+$*$ and
+$**~$.
+
+\smallskip
+
+%\normalsize \baselineskip = 0.98 \normalbaselineskip
+%\gvxs
+
+
+%% \el{wow}'s 
+%% dizzying
+%% $\aleph_1$ distinct solutions.
+%% We will traverse in the opposite direction in this article
+%% because 
+%% Definition \ref{def-2.6}
+%% %\eq{wow} 
+%% formalizes 
+%% % fascinating 
+%% %  a possible
+%% an 
+%% avenue
+%% available
+%% for evading $++$'s generalization of the 
+%% %Second 
+%% Incompleteness
+%% Theorem.
+
+% {\it in at least a partial sense.}
+
+
+%%  
+%% %They 
+%% This level of multiplicity will be
+%% %These will turn out to be 
+%% useful in the present setting
+%% because
+%% %% 
+%% %% Our hypothesis is that such
+%% %% solutions, while awkward and  disadvantageous
+%% %% in many 
+%% %% evident
+%% %% %%%%%obvious 
+%% %% respects,
+%% %% should not be discarded.
+%% %% This is 
+%% %%  because 
+%% %% 
+%% it
+%% can
+%% formalize a type of 
+%% allowed
+%% growth-permitting function,
+%% that is
+%% not prohibited by $++$'s generalization of the Second Incompleteness
+%% Theorem. 
+%% 
+%% 
+
+%\smallskip
+
+\begin{deff}
+\label{def-3.1}
+\rm
+Let us say 
+% an integer-valued
+a function symbol
+$F(x_1,x_2,...x_j)$ is {\bf `` 1-Definitive ''} iff it has only one
+solution under its definition by an axiom system $\gamma$. 
+Let us call $~F~$  {\bf ``Indeterminate''} otherwise.
+%(The remainder of this article 
+(Mathematicians
+obviously typically avoid using 
+function definitions
+%%%%%%%%%, $~F~$ 
+with 
+%that have 
+%%%%  having
+even two solutions, not to
+speak of
+what will be
+\el{wow}'s 
+surprising
+%dizzying
+quantity
+of
+%potential
+%possible
+%% possibly $\aleph_1$ distinct 
+ potentially infinitely many distinct 
+% number of 
+solutions.
+This is
+because such objects are typically more of a burden than a benefit.
+Our conjecture
+% , in this paper, 
+will be that a special new
+Indeterminate function, called ``$~\theta~$'', will 
+be different and
+offer
+% an
+%%%a surprisingly
+%%% efficient means to eschew $++$'s prohibitions.)
+a 
+%surprisingly efficient 
+means to eschew $++$'s 
+%prohibitions.)
+prohibitions with pleasing levels of
+% quantitative
+efficency.)
+% efficency.)
+
+
+\end{deff}
+
+
+% considered in the current article.
+% Our 
+% % main
+% conjecture will be that this unconventional
+% approach is 
+% germane to the challenge posed
+% by  $++$'s 
+% %
+% % broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem
+% %
+% %This is because our conjecture will be that
+% %
+% because the  new
+% proposed $~\theta~$ 
+% primitive
+%  will represent
+% an efficient Indeterminate 
+% function
+% that eschews  $++$'s prohibitions.) 
+% \end{deff}
+
+% helpful when addressing 
+% the challenge 
+% We will traverse in 
+% an unconventional
+% %  the opposite 
+% direction in this article
+% because
+% our conjecture will be that
+% Indeterminate functions 
+% with $\aleph_1$ different 
+% allowed solutions for
+% \el{wow} 
+% formalize
+% %%  
+% %% Definition \ref{def-3.1}
+% %% % {def-2.6}
+% %% %\eq{wow} 
+% %% will formalize 
+% %% a 
+% %% possible
+% %% %feasible
+% %% %plausible
+% %% 
+% an
+% avenue
+% % available
+% for evading $++$'s 
+% broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem.)
+% 
+% (Our conjecture in this article will be that
+% %%
+% %%The next two sections
+% %%will explore
+% %%how 
+% %%
+% \el{wow}'s indeterminate ``Q-Function'' symbol $F$
+% %%%can clarify
+% is germane to
+% the
+% aspirations 
+% that 
+% Hilbert and G\"{o}del
+% % expressed
+% stated
+%  in 
+% % statements 
+% $*$ and
+% $** \, $, $\,$when
+% one employs an 
+% operator
+% $ \, F \, $ 
+% that
+% owns
+% % has
+% $\aleph_1$ distinct 
+% available
+% solutions). 
+% \end{deff}
+
+
+%%are viable,
+%%especially in an automated theorem proving setting,
+%%when Ambiguous function operatives are judiciously
+%%employed.
+%%
+
+%%!  
+%%! can
+%%! %%and germane about automated
+%%! %%deduction, will 
+%%! % theorem proving,
+%%! %computational deduction,
+%%! be satisfied by a self-justifying logic that employs
+%%! one 
+%%! %single $\aleph_1$ 
+%%!   Growth-Permitting function $F(x)$,
+%%! that is
+%%! % inherently 
+%%! ``ambiguous'',
+%%!  {\it accompanied 
+%%! by} a finite number of non-growth primitives
+%%! % {\it  non-growth}
+%%!  $G_1,~G_1,~... G_k$
+%%! that are ``unambiguous''.
+%%! 
+%%! It will also be explained how such results should have useful 
+%%! applications for automated theorem proving, even when they
+%%! employ 
+%%! only
+%%! diluted forms of  self-justifying logics.
+
+%% 
+%% \vspace*{- 1.0 em}
+%% 
+%% \subsection{Main Notation Conventions}
+%% % about Cantor's Paradise}
+%% % \large
+%% % \baselineskip = 1.8 \normalbaselineskip 
+%% 
+%% %\vspace*{- 0.7 em}
+
+%\gvxs
+
+{\bf More Notation:}
+$~$Let us say
+an axiom system $~\alpha~$ 
+has 
+{\bf Infinite Far Reach} iff
+it relies upon
+{\it only a  finite number} of
+axioms to
+% distinct constant symbols
+% (and/or axiom sentences) to 
+prove 
+for each $n$
+the
+\el{farreach}'s invariant.
+
+%for each particular integer $n$.
+
+\vspace*{- 0.8 em} 
+
+\beq
+%% \small
+\label{farreach}
+\exists ~~x~~~ \mbox{Pred}^n(x)~\geq ~1
+\enq
+
+%\newpage
+
+
+\nvxs
+
+\noindent
+The ``ISINF'' axiomatic       framework 
+from
+   \cite{wwapal} 
+was
+    a self-justifying
+system with Infinite Far Reach.
+%% 
+%% The opening paragraph of
+%% \cite{wwapal}'s Section 6
+%% %%% quite 
+%% %was frank 
+%% warned the reader
+%% about ISINF's limitations. 
+%% These arose because
+%% 
+%% 
+%% It
+%% used the word ``unnatural'' to describe 
+%% the ISINF system.
+%% Such caution
+%% % deliberately
+%% % self-deprecating term 
+%% was appropriate because
+%% 
+Unfortunately, this result was mostly useless because
+nearly all
+   theorem-proofs
+%of trivial theorems0000000 
+from ISINF 
+were
+longer than the number of atoms in the universe.
+
+\newpage
+\parskip 0pt
+
+The reason
+\cite{wwapal} defined ISINF,  
+%ISINF was worthy of mention,
+despite 
+its evident impractical  characteristics,
+% 
+% such
+% %%% these
+% % plainly 
+% %%%  obvious 
+% limitations,
+% 
+was 
+because
+ISINF 
+demonstrated some
+unusual
+ self-justifying logics,
+knowledgeable about their own Hilbert consistency,
+were
+{\it technically} 
+able to
+prove all of Peano Arithmetic's $\Pi_1$ theorems
+together with the
+existence of
+the infinite set of integers $ \, 1,2,3,... \,  \, $.
+This result 
+% is interesting because it casts 
+did cast
+% casts 
+a
+new 
+perspective
+%light 
+on  $\,++\,$'s  
+invariant
+% $++$  (appearing on \pag2)
+by showing how
+{\it  some  unusual}
+Type-NS
+forms of self-justifying arithmetics
+did
+escape $++$'s almost-ubiquitous 
+ reach
+by managing to possess infinite far reach
+ without taking
+% {\it without recognizing} 
+Successor as a total function.
+
+
+%%  of the current article.
+%% The latter result indicated that Type-S arithmetics, recognizing merely
+%% Successor as a total function, are unable to confirm their own
+%% Hilbert consistency. 
+%% Yet,
+%% %%  despite this fact, 
+%% ISINF was able to produce an 
+%% {\it  eye-squinting} caveat because it 
+%% supported the above ``Infinite Far Reach'' property
+%% without 
+%% needing
+%% %being able to prove
+%%  Line
+%% \eq{totdefxs}'s declaration that successor is a total function.
+
+%\smallskip
+
+We sent an advanced copy of \cite{wwapal}
+to
+Pudl\'{a}k.
+He
+appreciated the nature of the challenge we faced.
+% 
+% ,
+% concerning the delicate nature of self-justifying 
+% arithmetics that are
+% able to prove
+% % satisfy 
+% \eq{farreach}'s invariant
+% {\it for each fixed $~n~$} while 
+% being prohibited 
+% by $++$
+% from
+% recognizing  successor as a total
+% function.
+% % (due to $++$'s restrictions).
+% 
+% 
+Pudl\'{a}k's
+subsequent  
+%private 
+%His
+emailed communications
+\cite{Pupriv}
+suggested 
+that we look at 
+Ajtai's
+work 
+\cite{Aj94}
+about a
+%the 
+Pigeon-Hole function
+ $~ \glamb(x)~$ defined by the identities
+\eq{zm1} and \eq{zm2}.
+
+% \newpage
+\vspace*{- 1.2 em} 
+\beq
+%% \small
+\label{zm1}
+\forall ~~x~~~~~ \glamb(x)~ \neq ~ 0
+\enq
+
+\vspace*{- 1.2 em} 
+
+\beq
+\label{zm2}
+%% \small
+\forall ~~x~~~ \forall ~~y~~~~ x ~ \neq~ y ~~ \Rightarrow ~~
+\glamb(x)~ \neq ~\glamb(y)
+\enq
+The relevance of 
+$~\glamb~$ 
+% Pigeon-Hole functions
+can be
+best 
+%readily 
+appreciated
+% if 
+when
+%we let
+$~\glamb^n(x)~$ 
+ denotes
+% the 
+a
+term
+ $~\glamb(~\glamb(~ ... \glamb(x)))~$
+consisting of $~n~$ iterations of the $~\glamb~$ operator.
+Then the below
+% the
+%%% \el{DUMB1}'s composite
+term $~S_n~$
+% , defined below, shall
+will
+% then 
+satisfy
+Pred$^n(~S_n~)~\geq ~1.~$ 
+%% 
+%% An axiom system, employing the primitive 
+%% operation
+%%  $~ \glamb~,~$ 
+%% can thus 
+%% can easily
+%% prove
+%% Line \eq{farreach}'s 
+%% assertion.
+%% %claim.
+%% %under almost all conventional logics.
+%% 
+%% 
+\beq
+% \vspace*{- 0.5 em}
+\label{DUMB1}
+S_n~~~=~~~\mbox{Max}[~\glamb(0)~,~\glamb^2(0)~,~\glamb^3(0)~,~...~~\glamb^n(0)~]
+\enq
+Pudl\'{a}k
+observed
+that
+%the
+% Pigeon-Hole function 
+ $~ \glamb(x)~$  
+will
+grow too slowly
+(under well-defined non-standard models)
+% (in the worst case)
+for
+one to be able to 
+deduce
+successor is a total function
+from its properties.
+%%   
+%% % further observed that it is known 
+%%  \footnote{
+%% \tiny
+%%  \baselineskip = 0.94 \normalbaselineskip
+%%  The operation $\glamb(x)$ will grow
+%% at a slower rate than Successor, 
+%% if it equals $x+1$ for all standard
+%% numbers $~x~$ and if $\glamb(x)=x-1$ 
+%%  when $~x~$ is
+%% a non-standard integer. This seemingly minute detail
+%% implies  one cannot infer 
+%% Successor is a total function from
+%%  $\glamb$'s behavior.}.
+%% 
+%% 
+%% 
+%%  since the latter is contradicted by a
+%%  model  where
+%%  all  non-standard
+%% numbers have 
+%% %their 
+%% sizes bounded by some fixed  
+%% % non-standard 
+%% number B.
+%% (This 
+%% subtle 
+%% %detail, 
+%% raised by
+%% Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+%% it 
+%% %shows that
+%% raised the question about whether
+%%  a partial exception to
+%% Example \ref{ex-2.3}'s
+%% invariant $++$
+%% %% on \pag2,  
+%% might plausibly exist.)  }.
+%% 
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+%%% 
+%%% 
+%%% 
+%%%  (in the worst case)
+%%% for
+%%% one to be able to 
+%%% deduce
+%%% successor is a total function
+%%% from its properties  
+%%% % further observed that it is known 
+%%%  \footnote{
+%%% \tiny
+%%%  \baselineskip = 0.94 \normalbaselineskip
+%%%  The operation $\glamb(x)$ will grow
+%%% at a slower rate than Successor, 
+%%% if it equals $x+1$ for all standard
+%%% numbers $~x~$ and if $\glamb(x)=x-1$ 
+%%%  when $~x~$ is
+%%% a non-standard integer. This seemingly minute detail
+%%% implies  one cannot infer 
+%%% Successor is a total function from
+%%%  $\glamb$'s behavior.}.
+%% 
+%% 
+%%  since the latter is contradicted by a
+%%  model  where
+%%  all  non-standard
+%% numbers have 
+%% %their 
+%% sizes bounded by some fixed  
+%% % non-standard 
+%% number B.
+%% (This 
+%% subtle 
+%% %detail, 
+%% raised by
+%% Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+%% it 
+%% %shows that
+%% raised the question about whether
+%%  a partial exception to
+%% Example \ref{ex-2.3}'s
+%% invariant $++$
+%% %% on \pag2,  
+%% might plausibly exist.)  }.
+%% 
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+His insightful email \cite{Pupriv} asked
+whether 
+the inequality 
+Pred$^n(~S_n~)~\geq ~1~$
+might
+%would,
+thus, 
+% still
+enable a formalism,
+% based around
+utilizing the
+ $\, \glamb \,$ operative, 
+to 
+somehow
+improve upon \cite{wwapal}'s results ?
+
+
+% our
+% formalisms could be 
+% revised
+% %modified 
+% so that 
+% % the  Pigeon-Hole function 
+%  $~ \glamb(x)~$ 
+%  could improve upon \cite{wwapal}'s results.
+
+%%
+%%(possibly using Ajtai's methodologies \cite{Aj-focs}).
+%%Sam Buss raised, interestingly,  a 
+%%partially
+%%similar 
+%%issue during an informal conversation
+%% \cite{Bu-priv} in 1977.
+%%
+%%\smallskip
+%%
+%%These questions
+%%% by
+%%%Pudl\'{a}k and Buss 
+%%were insightful because they isolated 
+%%an
+%%important juncture where $++$'s underlying methodology does not apply.
+%%A partial answer to these questions appeared in 
+%%\cite{wwapal}'s closing section, but a more comprehensive full
+%%answer  has always eluded us.
+
+%This is  because there always seemed to appear
+%one wrinkle of details that precluded a full proof.
+
+
+\smallskip
+
+
+It  was
+initially
+ unclear 
+%%%%% to us
+whether a positive answer to 
+Pudl\'{a}k's
+ probing
+ question would resolve ISINF's main difficulties.
+This is
+because
+% Expression 
+\eq{DUMB1}'s
+term
+$~S_n~$  requires $O(~n^2~)$ logic symbols to encode
+% essentially 
+an integer quantity
+greater than
+ $~n~$
+(since its term
+$~\glamb^j(0)~$ uses $O(j)$ logic symbols).
+%an integer quantity that exceeds the quantity $~n~$ in size.
+Thus once again, the quantity $~2^{100}~,~$ whose binary encoding
+requires 100 bits,  would require in excess of 
+ $~2^{100}~$ bits to encode. 
+Such large  quantities are obviously undesirable.
+
+
+% Such impractical quantities are obviously distant
+% from what is desired.
+
+
+% Such quantities, exceeding the number of atoms in the universe,
+% were troubling because our 
+% general
+% goal has been to 
+% construct self-justifying arithmetics
+% with  pragmatic features.
+
+
+%%  that 
+%%  possessed, at least,
+%% some
+%% partial  facets of
+%%  pragmatic value.
+
+
+% 
+% find a partial
+% answer to Hilbert's
+% Year-1900 Second
+% Problem  
+%  that would 
+%  possess, at least,
+% some
+% partial  facets of
+%  pragmatic value.
+% 
+
+\medskip
+\nvxs
+
+The remainder of this section will outline how a different type of
+Q-Function operator will 
+be
+%  much 
+better than
+ $~ \glamb~$ for meeting our needs.  
+During our discussion,
+Power$(x)$ will denote 
+a primitive specifying
+% that
+ $~x~$ is
+a power of
+$~2~$.  
+Its formal encoding
+in $L^G\,$'s language
+ is illustrated by \eq{wep2}. 
+%% 
+%% It is 
+%% %formally 
+%% encoded
+%% by 
+%% \eq{wep2} 
+%% because
+%% %under
+%% our Grounding language
+%% has
+%% ``Logarithm$(x) \,$''$ ~ = ~ \lfloor \,$Log$_2(x) \, \rfloor \,$.
+\beq
+\vspace*{- 0.6 em}
+\label{wep2} 
+%\small
+x=1 ~~~\vee ~~~ \mbox{Logarithm}(~x~)~\neq~\mbox{Logarithm}(~x-1~)
+\enq
+In this context,
+ $\zzthe(x)$ 
+will denote the analog of
+the $\glamb(x)$ function  
+%% haphazard
+that walks among the powers of 2 in a manner 
+similar to 
+$\glamb(x)$'s
+%  haphazard
+ walk through conventional
+integers. 
+It is 
+% formally
+defined by \eq{walk1}-\eq{walk4}.
+% 
+% It will thus satisfy
+% the axiomatic constraints below (which are 
+% $\zzthe(x)$'s analog of the more modest constraints given in
+% % sentences 
+% \eq{zm1} and \eq{zm2}).
+% The most important difference between these two constructs
+% is that axiom \eq{walk1} requires that 
+%  $\zzthe(x)$ maps power of 2 onto powers of 2.
+
+{
+%\small
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq} }
+
+\vspace*{- 1.2 em}
+\noindent
+{\it It needs to be emphasized} that 
+ \eq{walk1} -- \eq{walk4} will be the
+{\it only vehicle} our
+proposed formalisms will
+%self-justifying axioms 
+%will 
+have available to construct
+integers $\geq \, 3$. $~$These
+axioms will be called
+%They will be henceforth called
+the {\bf $~$Up-Walking$~$} axioms. 
+(The axiom \eq{walk4} 
+is
+% does not,
+%% , technically, 
+unnecessary
+ to construct any
+integer  $\geq \, 1\, $, but it is helpful 
+for 
+% because
+% it 
+% % allows us to 
+% formalizes 
+formalizing
+how our methodology will treat  integers
+which are not powers of 2.)
+
+% \gvxs
+% \svxs
+
+\smallskip
+
+
+Both
+$\glamb$ 
+ and $\zzthe$ are
+% unusual
+% complicated
+% entities 
+Q-functions
+%because they 
+that
+own
+% potentially
+ infinitely many distinct 
+% $\aleph_1$
+%  distinct 
+vectors, 
+%analogous 
+similar
+to 
+ Line \eq{wow},
+for representing them.
+We will soon see
+their 
+underlying
+% computational
+ complexities 
+% properties 
+are 
+%sharply 
+%quite
+surprisingly
+different.
+
+
+%% 
+%% they have sharply
+%% contrasting
+%% %%very
+%% %sharply 
+%% %% different
+%% computational
+%% % complexity 
+%% properties.
+%% 
+
+%% Both
+%% the 
+%% Q-functions 
+%% % the operators 
+%% $\glamb$ 
+%%  and $\zzthe$ are 
+%% %awkward 
+%% challenging
+%% to analyze
+%% %challenging
+%% %daunting
+%% because there are
+%% % a  
+%% % dizzying
+%% $\aleph_1$ distinct 
+%% vectors, analogous to 
+%%  Line \eq{wow}, 
+%% that are
+%% %where their definitions permits 
+%% representations of these functions.
+%% %% 
+%% %% Also, we may combine either operation with our 
+%% %% language $L^G$'s grounding function-primitives to formulate a term
+%% %% $~T_n~$ that defines any arbitrary integer $~n~$.
+%% %% 
+%% We will soon see that
+%% there is, however, a
+%% distinction
+%% %  major difference 
+%% between these
+%%  two concepts
+%% from a
+%% % computational 
+%% complexity perspective.
+
+\begin{definition}
+\label{defx-3.2}
+\rm
+Let $~L^Q~$ 
+and $~L^{Q^*}~$ 
+denote the 
+extensions
+of $~L^G\,$'s Grounding language that contain the
+respective
+additional 
+function symbols of  
+  $\zzthe$
+ and
+$\glamb$. Then 
+$~~L^Q~$ shall be called the {\bf  Q-Grounding} language, and 
+ $~~L^{Q^*}~$ 
+will be called the  {\bf  Q* Grounding} language. 
+\end{definition}
+
+\begin{propp}
+\label{th-3.3}
+In contrast to the
+Q* Grounding language 
+that requires $O(~n^2~)$ function symbols
+for defining a term $~T^*_n~$ for representing the integer
+$~n,~$ the    Q-Grounding language
+%% will need no more than 
+needs
+% uses 
+only
+$O(~$Log$^{ \, 3\,} \,n~)$ symbols to 
+encode
+%formalize 
+a term 
+$~T_n~$ representing
+$~n$.
+\end{propp}
+
+\vspace*{- 1.0 em}
+
+\begin{center}
+% \small
+% Our proof of \phx{th-3.3} 
+\phx{th-3.3}'s 
+proof
+will rely upon the following notation  convention:
+\end{center}
+
+\vspace*{- 0.8 em}
+
+\begin{definition}
+\label{def-3.3}
+\rm
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+% Throughout this article,
+Then
+%for any  $~j \, \geq 1~,~$ 
+%the symbol 
+$~E_j~$ 
+will
+% shall
+% will 
+denote
+the quantity produced by
+\eq{ej-def}'s division operation: 
+
+\vspace*{- 0.6 em}
+
+\beq
+\small
+\label{ej-def}
+ \frac{~\mbox{Max}
+~[~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~~] }
+{~~\mbox{Half}^{\,j\,} ~ \{ ~\mbox{Max}~[
+ ~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~]~ 
+ ~ \}~~ } 
+%
+% \mbox{Max}(~\zzthe^j(1),~\zzthe^{j-1}(1),~... ~\zzthe^1(1)~ 
+\enq
+It is each to see
+$  E_j  =  2^j  $ for every
+$j   \geq 1$.
+This is
+ because \el{ej-def}'s
+twice-repeating
+ term
+% object  of
+``$\,  \mbox{Max}
+~[~\zzthe^j(1),  \zzthe^{j-1}(1),...\zzthe(1) \,]\,$''
+% is at least as large as $\, 2^j\,$.
+is a power of 2 exceeding  $\, 2^j\,$.
+%% 
+%% The definitions of the
+%% % Q-Grounding 
+%% functions of ``Half'', ``Max'' and
+%% ``$~\zzthe~$''   imply 
+%% $~E_j~=~2^j~$ for each
+%% $j \, \geq 1$. 
+%% 
+For the additional case where $~j=0~,~$
+we will 
+% formally
+define  $~E_0~=~1~$ (by 
+using the 
+%%
+%%setting it equal to 
+%%our
+%%%the
+%%
+built-in constant symbol
+of $~C_1~$).
+\end{definition}
+
+%% , which 
+%% is intended to
+%% %formally 
+%% represent the integer of ``1'').
+
+
+
+{\bf Proof of  \phx{th-3.3}:}
+%The justification of   \phx{th-3.3} is an 
+Easy consequence of
+\dfx{def-3.3}'s machinery. Thus if $~n~$ is a power of
+2 of the form $~2^j~$ then
+% the preceding
+% definition's
+expression $~E_j~$ is a term representing $~n \,$'s value
+that employs
+  $O(~$Log$^{ \, 2\,} \,n~)$ 
+logical
+symbols. On the other hand, if  
+ $~n~$ is not a power of
+2 then it can be defined 
+with   $O(~$Log$^{ \, 3\,} \,n~)$ symbols by 
+setting
+$~E_j~$ equal to the least power of 2 greater than $~n~$ and
+subtracting from $~E_j~$ those powers of 2 that are needed to
+produce $\,n\,$'s  value. 
+For example since $76~=~128~-~32~-~16~-~4~,\,$ it can
+be formalized as a term $T_{76}$ defined by
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$.
+
+% $~~~~\Box$
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\begin{definition}
+\label{def-3.4}
+\rm
+A term in mathematical logic
+is defined to be a syntactic object,  built
+out of
+solely
+ symbols for representing 
+functions,
+constants and variables.
+% 
+% The nomenclature in
+% % classical
+% logic has 
+% %formally 
+% defined
+% a {\it ``term''} to be a syntactic object,  built
+% out of symbols for representing 
+% functions,
+% constants and variables.
+% 
+% 
+Such an object is called
+% either 
+a {\bf ``Ground Term''} 
+%% (or for precision a
+%%% {\bf ``Tree-Oriented Ground Term''} )
+when it  is built {\it out of solely}
+function and
+ constant symbols.
+For example in our Q-Grounding language (which 
+uses
+%owns only
+$  C_0  $, $  C_1  $ and $  C_2  $ 
+as its
+built-in
+ constants),
+% symbols), 
+the expression
+%of 
+``$\, C_2-  C_1\,$'' 
+is 
+% such 
+a Ground term.
+Two more complex 
+examples of 
+Ground terms are
+``Max$(   C_2 ,  C_1 - C_0)$'' 
+and ``Max$( ~\zzthe(C_1)~,~C_2 ~)$''.
+Also,
+expression $~E_j~$ 
+in Line \eq{ej-def}
+should be viewed as 
+a Ground term (when one 
+views
+its
+use of the
+symbol
+ ``1'' as an %informal 
+abbreviation
+for the
+constant
+ ``$~C_1~$''). 
+\end{definition}
+
+
+\begin{remm}
+\label{rem-def-3.4}
+{\bf (sharply improving upon 
+ \phx{th-3.3}'s result) : }
+\rm
+A longer version of this article
+will 
+%technically 
+distinguish between two
+kinds of Ground terms, which it calls the
+%
+%{\bf Comment of Definition \ref{def-3.4}'s Notation:} 
+%We will  distinguish  between two
+%kinds of Ground terms in Section \textsection \ref{ss6},
+%called its 
+{\bf ``Tree-Oriented''} and 
+{\bf ``Dag-Oriented''} formats.
+The latter will differ from a more 
+conventional tree structure
+by having a 
+Directed Acyclic Graph structure replace 
+a logic's
+usual 
+ tree format for defining its quantitative values.
+It will turn out that
+ Dag-Oriented Ground Terms
+will allow one to compress multiple repeating
+terms into single objects.
+This will
+%and thus 
+reduce the number of logical symbols in 
+ \phx{th-3.3}'s 
+$O(~$Log$^{ \, 3\,} \,n~)$ sized 
+%ground 
+terms to a 
+ more compact
+$O(~$Log$\, \,n~)$ quantity.
+(This is almost analogous to the 
+% (in a context where the pointers to our
+$O(~$Log$\, \,n~)$ 
+size of an integer's binary encoding,
+except that we will need  $O(~$LogLog$\, \,n~)$
+further bits to encode the pointers to 
+each 
+specified
+object.)
+ \end{remm}
+
+
+%% quantity
+%% $O(~$Log$\, \,n~)$ 
+%% objects will require  $O(~$LogLog$\, \,n~)$
+%% bits per pointer). 
+%% %(analogous to the classical binary encoding of an integer).
+
+% 
+%  {\it This is the same length
+% as would occur in a conventional
+% $O(~$Log$\, \,n~)$ sized binary encoding of an integer.}
+% We will refer to this improvement later in the current
+% article because it will suggest that 
+%  \phx{th-3.3}'s 
+% $O(~$Log$^{ \, 3\,} \,n~)$ sized ground terms attain a length
+% not too different from the binary encoding of an integer,
+% after further refinements are undertaken.
+% 
+
+
+
+\begin{definition}
+\label{def-3.5}
+\rm
+A ground term
+% $~T~$ will be 
+is
+called an 
+{\bf ``Observable''} 
+object iff it has a
+%{\it only one} 
+% an
+unique
+interpretation of its 
+quantified value in the
+%meaning  in our
+Q-Grounding language.
+It 
+%will be
+ is
+called an
+{\bf ``Unobservable''} iff it has multiple 
+%plausible 
+such
+interpretations
+due to $\zzthe$'s ``indeterminate'' definition
+(e.g. see Definition \ref{def-3.1}). 
+\end{definition}
+
+%%% (due to the 
+%%% %uncountably 
+%%% ambiguous nature of 
+%%% % our built-in function 
+%%% $~\zzthe~~$).
+%%% \end{definition}
+
+\begin{exx}
+\label{ex-3.6}
+\rm
+The previously mentioned ground term
+Max$( ~\zzthe(C_1)~,~C_2 ~)$ is an ``unobservable''
+because it can assume any of the plausible integer values
+of $~2 \, , \, 4 \, , \,8  \,  , \,16  \, 
+ \, ... ~$.
+On the other hand,
+
+\newpage
+ 
+\gvxs
+\nvxs
+\parskip 0pt
+
+\noindent
+\el{xoo} 
+%is 
+provides
+an 
+example of an
+``observable''
+that
+ represents
+ the integer value of ``3''.
+(This is because 
+its 
+twice-repeating
+term 
+``$~\mbox{Max}[ ~C_2~,  ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~$'' is bounded
+below by 4, causing the left and right sides of its subtraction
+operation to differ by 
+% an amount of 
+exactly 3.) 
+\beq
+%% \small
+\label{xoo}
+\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~] ~~-~~
+\mbox{Pred}^{\, 3 \,} \{~\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~\} 
+\enq
+Our notation 
+%thus 
+also
+implies that Line \eq{ej-def}'s
+ expression $~E_j~$ 
+is an
+ ``observable''. This implies, in turn, that 
+ \phx{th-3.3}'s term $~T_n~$ is an  ``observable''
+ (employing 
+conveniently
+no more than 
+ $O(~$Log$^{ \, 3\,} \,n~)$ 
+logical
+symbols).
+\end{exx}
+% 
+% 
+% For example since $76~=~128~-~32~-~16~-~4~,\,$ 
+% it follows that $~ T_{76}~$
+% corresponds to the term
+% $~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$,
+% $~$where each $~E_j~$ employs only 
+%  $O(~$Log$^{ \, 2\,} \,j~)$ symbols.
+% %%%%%\end{exx}
+
+
+\nvxs
+\hvxs
+
+
+Thus, \dfx{def-3.5} and Example \ref{ex-3.6} have illustrated
+%that 
+how
+the realm of ``observable'' objects is a 
+% very 
+broad and accessible world,
+of 
+potential usefulness.
+% 
+% non-trivial
+% %% pragmatic
+%  significance.
+% 
+It allows every integer $~n~$ to be represented by a
+% reasonably small 
+term $~T_n~$ with
+% an
+a tight
+ $O(~$Log$^{ \, 3\,} \,n~)$ length
+(in a context where
+Remark \ref{rem-def-3.4}'s refinement
+% Section \textsection \ref{ss6}'s more elaborate formalism
+%will allow us to 
+will 
+reduce this length
+almost to a more compact
+% 
+% nearly
+%  to a 
+% % far 
+% more
+% attractive  
+% 
+$O(~$Log$ \,n~)$ magnitude).
+
+
+The distinction between
+``Observables'' and ``Unobservables''
+% ground terms 
+will 
+%also 
+%% 
+%% cast a
+%% delightful
+%% 
+offer a
+ new perspective on
+the aspirations
+% that 
+which
+Hilbert and G\"{o}del 
+expressed
+in 
+their
+statements $*$ and $**$.
+%
+% under our proposed formalism.
+It
+ will suggest
+how the Second Incompleteness Theorem
+can
+%  remain to 
+be seen as a majestic result
+from a purist perspective, while
+a {\it well-defined fragment} of
+%their 
+what
+Hilbert and G\"{o}del
+sought in 
+ $*$ and $**$
+%aspirations
+%% in  statements $*$ and $**$
+can 
+likely
+%almost certainly
+be 
+%part-way
+satisfied (in at least a 
+% well-defined 
+limited sense).
+% 
+% 
+% 
+% 
+% \begin{remm}
+% \label{rem-3.7}
+% {\bf (explaining the goals of this paper):$~$}
+% \rm
+% Let us say 
+% that
+% a basis axiom system $~\alpha~$ owns
+% a {\it ``Finitized Perspective''}  of the Natural Numbers
+% if it requires only a 
+% {\it finite number} of proper axioms
+% to construct the full set of integers
+% $~0,1,2,3~... ~$. All conventional arithmetics have this property.
+% %It is useful to divide such 
+% Such
+% logics
+% %arithmetics
+% fall into two categories,
+% called {\it Single} and {\it Double-Formatted} systems.
+% %as defined below: 
+% They are defined below: 
+%  %These constructs are defined below:
+% \bed
+% \item[   a.  ]
+% %An axiom basis $~\alpha~$
+% %will be called
+% {\bf Single-Formatted Arithmetics} consist of
+% axiomatic basis systems
+% % $~\alpha~$
+% %all of 
+% whose 
+% %iff all its 
+% ground terms are 
+% all 
+% Observables.
+% (Most conventional arithmetics
+% %%%% will 
+% %fall into this 
+% lie in this
+% category
+% %when the 
+% because they
+% employ the
+%  growth
+% %  function
+% properties 
+%  of
+% the  Successor
+% operation
+% %function
+%  in
+% % a straightforward 
+% %the
+% %% a  conventional
+% the traditional
+% manner.)
+% %% since the 
+% %% the simple  growth function of Successor
+% %% easily
+% %% generates
+% %% all the natural numbers). 
+% %are {\it ``Single-Formatted Formatted''} logics. 
+% \item[   b.  ]
+% {\bf Double-Formatted Arithmetics} 
+% % representing 
+% represent
+% systems
+% %%%consisting of
+% %%%%axiomatic 
+% %%%logics
+% %%%%%basis systems
+% whose ground terms 
+% may be either
+%  Observables
+% or Unobservables.
+% (Axiomizations
+% for Q-Grounded logics 
+% %%  of
+% %% the
+% %% % our
+% %% Q-Grounding language
+% are
+% %%% will 
+% %obviously 
+% %%% be
+% ``Double-Formatted''
+% because they
+% allow $\theta$'s analog of 
+% \el{wow}'s function symbol $F$ 
+% to have
+% an uncountable number of 
+% different allowed
+% representations).
+% % 
+% % (Our
+% % Q-Grounding language 
+% % gives support to such a system.
+% % This is because it
+% % can  have its function primitives
+% % defined by a finite number of 
+% % proper axioms,)
+% % %axiom-sentences.) 
+% % \ennd
+% The distinction between 
+% categories
+% %Items 
+% (a) and (b) is
+% significant
+% % important
+% because 
+% Example \ref{ex-2.3}
+% %%%  \pag2 
+% %%% had 
+% already explained how
+%  statement $++$'s generalization
+% of the Second Incompleteness Theorem applies to 
+% any formalism recognizing Successor as a total function.
+% Thus, Item (b)'s Double-Formatted logics 
+% are useful, if one wishes to consider alternatives
+% to
+% %formalism that do not recognize
+% successor as a total function.
+% %More precisely, 
+% In this context,
+% 
+This will be because
+Hilbert's 
+%  famous 
+%Year-1900 
+Second 
+Open
+Problem
+can be viewed
+ as a  {\it  2-part question}, 
+composed of sub-queries Q-1 and Q-2:
+%%%%%
+%%%%% {\it  2-part question}.
+%%%%%The separation of Hilbert's question into two parts,
+%%%%%called Q-1 and Q-2, will allow 
+%%%%%%% 
+%%%%%%% This 
+%%%%%%% bipartite
+%%%%%%% distinction 
+%%%%%%% is useful because it
+%%%%%%% can enable 
+%%%%%%% 
+%%%%%the academic community to better
+%%%%%  with
+%%%%% what Hilbert and G\"{o}del were
+%%%%%seeking to accomplish
+%%%%%in 
+%%%%%their
+%%%%%statements
+%%%%%of $*$, $**$ and $***$.
+\bed
+\small
+\item[ {\bf Question Q-1$~~$}] {\it Are any axiom systems
+able to
+ prove 
+theorems 
+verifying
+ their own consistency in a robust sense?$~~$}
+The answer to Q-1 is clearly ``No'' because the combination
+ G\"{o}del's initial 1931 result \cite{Go31} with 
+%the 
+%further
+Hilbert-Bernays's result 
+\cite{HB39} 
+and the Pudl\'{a}k-Solovay invariant $++$
+(from Example \ref{ex-2.3})
+%% \pag2) 
+imply 
+arithmetics of ordinary strength cannot prove
+their own consistency in a robust sense.
+\item[ {\bf Question Q-2$~~$}]
+ {\it Can 
+logic systems
+%arithmetic logics
+%axiomatizations of Arithmetic
+%  , at least, 
+%somehow 
+``appreciate'' 
+% (not formally ``prove'')
+ their
+own consistency in some 
+{\bf REDUCED} sense, that is diluted
+but not fully immaterial?}
+$~~\,$The answer to 
+%question 
+Q-2 is 
+complex
+%%% more complex than Q-1
+%less clear-cut
+because 
+%several types of 
+some
+arithmetics,
+such as \cite{ww93,ww1,ww5,wwapal,ww9,ww14}'s paradigms,
+ can
+formalize
+% ``recognize''
+their 
+own consistency 
+using Example \ref{ex-2.5}'s
+% a 
+Fixed-Point {\it ``I am consistent''}
+axiom.
+Moreover,
+ Definition \ref{def-3.5}'s 
+% further 
+separation of
+the concepts  of ``Observables'' from ``Unobservables''
+%
+%the notions of Observable from Unobservable objects 
+%
+raises
+some
+% further
+% very
+subtle issues beyond these distinctions.
+\ennd
+
+
+%% %  sentence $\,\oplus\,$.
+%% %%Using
+%% %%%%%%%%%Under
+%% % Using the notation from 
+%% Under
+%% Lines
+%% \eq{totdefxs}--\eq{totdefxm}'s notation,
+%% these paradigms include:
+%% % both:
+%% \bee
+%% \small
+%% \baselineskip = .86 \normalbaselineskip 
+%% \item
+%%  Type-A arithmetics
+%% \cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+%% %capable of 
+%% recognizing their self-consistencies under
+%% either the deductive mechanics of semantic tableaux or one
+%% of its cousins. (See especially \cite{ww14}'s
+%% recent Wollic-2014 paper.)
+%% \item
+%%  Type-NS arithmetics recognizing their Hilbert consistency,
+%% such as the formalisms of \cite{ww1,wwapal}
+%% %further 
+%% improved, possibly,
+%% with the added techniques introduced in
+%% this  article.
+%% \ene
+%% 
+%% \ennd
+
+One theme
+in 
+% the remainder  of 
+this article will be that
+the
+Second Incompleteness Theorem represents a
+$100 \, \% $ 
+full
+% comprehensive
+reply to question Q-1 
+but only a
+% and a
+ 90 \% 
+% adequate
+ reply to question Q-2.   
+Our
+tiny
+% only 
+%tiny
+ caveat to Q-2 will be related to Hilbert's 
+insistence that {\it some type of ``new formalism''} 
+% was needed
+will be needed
+to explain how
+% it is 
+humans
+motivate themselves to engage
+in cognition. 
+
+%The next section of this article will
+%  note 
+
+Our discussion will
+observe
+% that
+mathematicians 
+% had
+made no distinction between 
+Unobservable and  Observable ground terms during the
+% early 
+1930's.
+% We 
+It
+will suggest 
+a 
+% tiny new 
+revised
+interpretation can be assigned to the 
+% historic 
+%often-quoted
+statements $*$ and $** \,$
+of
+%by 
+Hilbert and G\"{o}del,
+when one 
+views
+them
+ from the perspective of 
+arithmetics that 
+rely upon indeterminate growth functions,
+similar to $\theta$.
+
+%employ $\theta-$like
+% growth functions.
+
+% logics that allow deploying unobservable  ground terms.
+
+ 
+% owns two types of ground terms.
+
+% looks more closely at these two types of
+% ground terms.
+
+
+%% where the remaining  5-10 \% fraction of
+%% unresolved issues is connected to the 
+%% fundamental
+%% distinction
+%%  separating
+%% % separation of
+%% Unobservable from  Observable ground terms.
+
+
+% that the final
+% tiny remaining
+%  5-10 \%
+% gap, pointed to
+% in
+% % by
+% Hilbert's and G\"{o}del's 
+% statements $*$ and $** \,$,  can be viewed as
+% being related to this 
+% fundamental
+% distinction.
+
+
+%% Some 
+%% %other insightful 
+%% different
+%% approaches to these dilemmas
+
+%Some other 
+
+Other insightful
+approaches
+to
+% the 
+Incompleteness paradigms
+are related to 
+ Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+% Kleene-like 
+{\it ``I am consistent''} axiom-sentences,
+similar to Example \ref{ex-2.5}'s
+``SelfRef'' statement.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a 
+% preferable 
+Hilbert-style 
+deductive 
+apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspectives  
+should
+% certainly,
+ be examined by researchers
+interested in the
+% Second 
+Incompleteness Theorem,
+but they are unrelated to 
+our 
+objectives
+%IQFS formalism.
+
+\smallskip
+
+
+%% During our 
+%% % description 
+%% investigation
+%% of IQFS, 
+
+Also during the next chapter,
+the reader should keep
+in mind that 
+Proposition \ref{th-3.3}'s
+% characterization of the 
+$O(~$Log$^3 \, n \,)$
+lengths for encoding $T_n$'s ground terms can be reduced to
+% an essentially 
+% a more compact
+%nearly an
+ essentially an
+  $O(~$Log$ \, n \,)$ complexity,
+when these terms are encoded 
+using
+%under
+ Remark \ref{rem-def-3.4}'s 
+more compressed 
+Directed Acyclic Graph
+% formalism.
+methodology.
+This fact will make IQFS's formalism look
+% much 
+% significantly more tempting.
+%%%% very
+quite
+  tempting.
+
+% cccc ddddddd 
+
+%% the next chapter's discussion.
+%% 
+%% 
+%% Their insights are important but 
+%% unrelated to 
+%% our particular
+%% % the next section's 
+%% %specific analysis of
+%% %%% type of
+%% Hilbert-styled self-justifying effects,
+%% explored in the next chapter.
+%% 
+
+
+
+\gvxs
+
+\vspace*{- 0.6 em}
+
+\section{Proposed New IQFS Formalism}
+
+ \label{ss32}
+\label{ss5}
+
+\vspace*{- 0.6 em}
+
+
+\nvxs
+
+% \rvxs
+
+
+\parskip 1 pt
+
+The only aspect of our prior research that will be 
+related to our proposed new IQFS formalism
+is the ISCE framework,
+defined in \cite{wwapal}'s Sections 3 \& 4.
+The next several paragraphs will review 
+\cite{wwapal}'s results, 
+so that a reader 
+can omit examining \cite{wwapal}. 
+
+
+% will not need to examine 
+% \cite{wwapal}'s 
+% formal treatment.
+%results.
+%%%%%%%%%%%%%%%for the reader's convenience.
+%% 
+%% This section will
+%% review \cite{wwapal}'s results in sufficient detail
+%% so that a reader need not examine \cite{wwapal}'s formal 
+%% text,
+%% 
+%% %%%%%definition of the ISCE axiom system. 
+%% 
+%% During our discussion, 
+%% 
+
+% \lvxs
+% \parskip 1 pt
+
+\smallskip
+
+During our 
+discussion,
+%review of \cite{wwapal}'s results,
+$~L^G~$ will  again denote
+our
+ ``Grounding-level'' 
+language 
+that 
+formalizes
+%employs
+\textsection \ref{seee3}'s
+six non-growth
+%  functions of
+operations of
+ Subtraction, Division,
+Maximum, Logarithm, Root and Count determination.
+%%% 
+%%%  the six ``Grounding-level'' 
+%%% % non-growth 
+%%% functions defined on Page 5.
+%  
+%  consisting
+%   of 
+%  the
+%  Subtraction, Division,
+%  Maximum, Logarithm, Root and Count operations.
+%  
+Also, $\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ will denote
+three constant symbols designating the
+integers 
+%values 
+of 
+``0'', ``1'' and  ``2''.
+In a context where Pred$(x)$ is an abbreviation for
+``$\,x \,- \, 1\,$'',
+%% (or more precisely ``$\,x \,- \, C_1\,$'' ), 
+the ISCE axiom system
+% from \cite{wwapal}
+used
+\eq{start}'s axiom
+ statement 
+to define
+ $\,C_0\,$, $\,C_1\,$ and $\,C_2~$:
+% these three constants:  
+\begin{equation}
+\label{start}
+\small
+\mbox{Pred}(    C_0    )  =  C_0~ \, \wedge ~ \,
+C_1 \neq C_0~ \, \wedge ~ \,
+\mbox{Pred}(    C_1    )  =  C_0 ~ \, \wedge ~ \,
+\mbox{Pred}(    C_2    )  =  C_1 
+\end{equation}
+%Also,
+The  challenge 
+\cite{wwapal} 
+faced was its formalism could
+not use any of the
+%  conventional 
+operations 
+%function-operations 
+of
+successor, addition or multiplication to infer the existence
+of larger integers from the initial constants of 
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$
+(without violating $++$'s generalization of the Second Incompleteness
+Theorem).
+% 
+%  This was because
+% the Pudl\'{a}k-Solovay result $++$ 
+% indicated
+%   the 
+% presumption  successor is a total function 
+% precludes   
+% most
+% %axiom 
+% systems
+% from recognizing their
+% own Hilbert
+% consistency.
+
+\smallskip
+
+
+Our article
+\cite{wwapal} 
+considered two 
+methods for achieving these tasks,
+%alternatives
+%to a conventional Successor 
+%function symbol 
+% for overcoming these difficulties,
+ called
+the  {\bf Additive} and  {\bf Multiplicative Naming}
+conventions.
+They defined 
+some
+further constant symbols $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+where 
+%respectively
+$~C_j~=~2^{j-1}~$ and $~C^*_j~=~2^{\,  2^{ \,j-2}}~$.
+
+\smallskip
+
+
+The definition of these
+% new 
+constants
+%  symbols
+is 
+easy
+%straightforward
+under $L^G\,$'s
+% Grounding-level 
+language.
+% called $L^G~,~$
+%all of whose function objects are non-growth primitives. 
+This is because
+Lines
+\eq{newadd}  and \eq{newmult}
+%had 
+specify how
+% that
+two 3-way predicates, called
+Add$(,x,y,z)$  and  Mult$(,x,y,z)\,$, 
+%can 
+%do
+encode the identities of
+% can be encoded to specify,  respectively,
+$x=y+z$ and $x*y=z$.
+Our additive and multiplicative 
+% naming 
+conventions
+can,
+%will,
+ then, define 
+ $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+%by  using
+via
+an infinite number of instances  of
+%utilizing respectively
+ \eq{addcov} and
+\eq{multcov}'s
+%{\it infinitely long}
+axioms:
+%%%%%%%%%% schemas:
+% two infinite schemas of  axiom-sentences:
+%% 
+%% that belong to our 
+%%  ``Additive'' and  ``Multiplicative
+%%  Naming Conventions'',
+%% then the values for 
+%% $~C_j~$ and 
+%% $~C^*_j~$ can be easily derived from $j-2$ instances of
+%% %respectively \eq{addcov} and \eq{multcov}'s 
+%% these
+%% schema:
+%% 
+
+
+{
+\small
+\beq
+\label{addcov}
+\mbox{Add}(~C_{j-1}~,~C_{j-1}~,~C_{j}~)
+\enq
+\beq
+\label{multcov}
+\mbox{Mult}(~C^*_{j-1}~,~C^*_{j-1}~,~C^*_{j}~)
+\enq}
+The methodology in
+ \cite{wwapal} 
+%% employed \eq{addcov}  and  \eq{multcov}'s schema in a context where it 
+presumed
+% assured
+%the Y of 
+the ``names'' for its constants $ C_j $ 
+and $ C^*_j $ 
+had nice compact encodings using $O(~Log(j)~)$ bits.  
+Its formalism calculated
+%, thereby, 
+the values of ``unnamed'' integers from 
+named entities via the {\it non-growth} Subtraction and
+Division primitives. For instance since $~20~=~32-8-4~,~$
+the quantity 20 
+can be encoded as $~C_6-C_4-C_3$.
+%%%%%%%%%%%%%%%% under \eq{addcov}'s naming convention.
+
+
+%% required
+%% $O(~Log(j)~)$ bits.  
+%% Thus, the length of these encodings was 
+%% much 
+%% smaller
+%% than the respective
+%% numbers
+%% % magnitudes of 
+%%  $2^{j-1}~$ and $2^{2^{j-2}}$ 
+%% %that 
+%% these constants represent.
+
+\smallskip
+
+
+The challenge \cite{wwapal} 
+faced was to determine whether 
+%it was possible to formulate 
+self-justification
+was possible
+under 
+%% either
+\eq{addcov}'s
+% ``Additive'' 
+or  \eq{multcov}'s 
+% ``Multiplicative'' 
+%% naming
+schema.
+It found 
+%that 
+ \eq{multcov}'s 
+multiplicative
+% naming 
+convention was  incompatible 
+with self-justification (due to its 
+%%very 
+speedy growth rate),
+but
+%In contrast,
+\eq{addcov}'s additive 
+% naming 
+schema did
+% conveniently,
+ permit self-justification. 
+
+\medskip
+
+Our new proposed IQFS
+axiom system is easiest to describe, if we first
+review \cite{wwapal}'s definition of ISCE
+and then
+explain how our IQFS framework 
+%% 
+%% will improve upon
+%% it (by not requiring
+%% the definition of an infinite number of separate
+%% constant symbols).
+%% 
+can  incrementally refine it.
+The extension of our base-language $~L^G~$
+that includes the Additive Naming Convention (ANC)'s
+additional constants 
+ $~C_3,~C_4,~ C_5,~ ...~$
+will be called
+an {\bf ANC-Based Language}.
+It will be denoted
+as  $~L^{ANC}~$. 
+Also if
+ $\, t \,$ denotes   any term in $\, L^{ANC} \,$'s   
+language, then 
+the quantifiers in 
+the two wffs of
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and
+$\exists ~ v \leq t~~ \Psi (v)$
+will be  called $\, L^{ANC} \,$'s   
+{\bf ``Bounded 
+Quantifiers''}.
+
+
+\begin{deff}
+\label{def-3.8}
+\rm
+The analogs of
+% a
+ conventional
+% arithmetic's
+$\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+formulae
+in the
+language $L^{ANC}$ will be denoted as
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$
+ and $\Sigma^{ANC}_n$.
+Thus,
+a formula will be defined to be
+$\Delta^{ANC}_0$  iff all its quantifiers are bounded.  
+The
+%%%%%%%%%  below 
+definitions 
+of  $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$
+formulae are
+%  also  
+quite conventional:
+\bee
+%\small
+\parskip -2 pt
+\baselineskip = 0.8 \normalbaselineskip 
+\item
+\small
+Every  
+$\Delta_0^{ANC}$ formula is considered to
+be 
+also 
+a
+$\Pi_0^{ANC}$  and 
+an
+$\Sigma_0^{ANC}  $ expression.
+%% 
+%% ``$~\Pi_0^{ANC}~ \,$''  and 
+%% %  also 
+%%  ``$~\Sigma_0^{ANC}~ \, $''. 
+%% 
+\item
+A
+formula
+is  called
+ $ \,\Pi_n^{ANC} \,$
+when it
+% is 
+can be
+encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  
+where
+%with
+$\Phi$ is  $\Sigma_{n-1}^{ANC}$
+\item
+A formula
+is  called
+ $\Sigma_n^{ANC}$
+when it can be encoded as 
+$\exists v_1~ ...~ \exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{ANC}$.
+\ene
+\end{deff}
+
+
+
+%%\begin{deff}
+%%\label{def3.9}
+%%\rm
+
+ \parskip 0pt
+
+
+Given an initial axiom system $\beta,$
+the Theorem 3 of \cite{wwapal} defined a 
+self-justifying logic, called
+ISCE$(\beta)$ 
+that could prove all 
+$~\beta\,$'s $\Pi_1^{ANC}$ theorems and 
+verify its own consistency under a Hilbert-style deductive
+apparatus. It consisted of the following four
+groups of axioms:
+%
+%  \newpage
+\begin{description}
+\small
+ \parskip 2pt
+\item
+{\bf GROUP-ZERO:} 
+This 
+schema
+%  axiom group 
+will
+use \el{start}'s axiom to define the constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ 
+and 
+%employed
+%an infinite number of instances of
+\el{addcov}'s Additive Naming 
+schema
+%convention
+to define 
+ the further constants
+ $    C_3,  C_4,  C_5,  ...   $  
+\item
+{\bf GROUP-1:}
+It is convenient  to
+define
+ ISCE's Group-1 and Group-2
+axioms using a notation that 
+will support \cite{wwapal}'s Theorem 3
+in a 
+% slightly
+ more 
+general sense than
+appeared in \cite{wwapal},
+%%% under a slightly different notation convention,
+% is transparently equivalent
+% (but slightly different) from \cite{wwapal}'s counterpart,
+so our
+% a 
+new
+% proposed
+%%%% second
+ ``IQFS''
+formalism
+(appearing later in this section)
+% shall
+will
+%proposal shall
+% framework will 
+be easier to define.
+Let us
+therefore
+ say  a $\Pi_1^{ANC} $  sentence is {\bf Simple}
+iff the only built-in constants it employs are
+$\,C_0\,$, $\,C_1\,$ and $\,C_2$.
+Then ISCE's Group-1 scheme will
+allowed to
+ be any finite set of 
+simple $\Pi_1^{ANC} $  axioms, called $~S~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following 
+two
+properties:
+\bee
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+axioms
+%will be capable of proving 
+%do 
+will
+prove
+all true $\Delta^{ANC}_0 $
+sentences.  
+%
+% statements which
+% are true.
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+scheme
+%will be capable of proving 
+will
+% do
+ prove
+that
+the ``=" and ``$\leq$" predicates
+% own 
+support
+their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+Any finite set 
+$\Pi_1^{ANC} $  axioms
+with the above properties can be used to define $~S~$
+and 
+support
+%prove 
+an analog of
+\cite{wwapal}'s Theorem 3,
+by a trivial generalization
+of
+\cite{wwapal}'s results.
+% 
+% \footnote{A formal proof of this generalization of
+% \cite{wwapal}'s results is 
+% %absolutely 
+% entirely
+% routine.
+% %  and omitted here for the sake of brevity.}
+% For the sake of brevity, it  is
+% omitted.} 
+% 
+%  of
+% the methodologies from Sections 3 and 4 of
+% \cite{wwapal}. (Thus,
+% any such finite set $~S~$ supporting Conditions (1) and (2)
+% can  be employed
+% by
+% ISCE's
+% Group-1 part.)
+%%% 
+%%% and it is unimportant which
+%%% particular defining
+%%%  set is used.
+% 
+% BBB111
+%  
+% This
+% % schema
+% axiom group 
+% consisted of a finite
+% set of 
+% $\Pi_1^{ANC} $ axioms
+% %, CALLED $F$,
+%  defining ISCE's
+% Grounding  function primitives. 
+% %This means that
+% For each  such function $G$ and set of numbers 
+% $    {k},   {k_1},    {k_2}, ...    {k_m}$,
+% %the combination of 
+% the Group-Zero and Group-1 axioms 
+% %must
+% will
+%  imply 
+% $ G(    {k_1},    {k_2}, ...    {k_m}) \,=\,    {k}   $ when
+% this sentence is true
+% \footnote{ \f55 
+% Our
+%  $\Pi^{ANC}_1$
+% encoding for the
+% Group-1 scheme  needs,
+% technically, 
+% % employ
+%  only
+% employ
+%  the three constant symbol $C_0$, $C_1$ and $C_2$ for the
+% union of all 
+% the
+%  Group-Zero and Group-1 axioms
+% to satisfy 
+% their
+% %its
+% %the 
+% above requirements.} .
+% The Group-1 schema
+% of \cite{wwapal} 
+% will also
+% assign the ``=" and ``$<$" predicates
+% their conventional
+% %  logical
+% properties.
+% %footnoted property.}
+% %%
+% %%(Any finite 
+% %%set of  $\Pi_1^{ANC} $
+% %%sentences  meeting these conditions is
+% %%suitable.)
+% %%
+\item
+{\bf GROUP-2:}
+Let
+$\ulcorner \, \Phi \, \urcorner$ denote $\Phi$'s G\"{o}del number, and
+$\mbox{HilbPrf}_{ \beta  }(x,y)$
+denote a 
+%%%%%%%%%%%  $\Delta _0^{ANC+}$ 
+$\Delta _0^{ANC}$ 
+formula indicating  $y$ is a
+Hilbert-styled
+proof
+from axiom system $\beta  $ of the theorem 
+$x$.
+%
+% Suppose that
+%$~\beta~$ uses the same Grounding function symbols as
+%ISCE$^{ANC}(\beta)$,
+%and it therefore generates
+%a set of 
+%% $\Pi_1^{ANC+} $ theorems.
+% $\Pi_1^{ANC} $ 
+%theorems.
+%
+For each 
+%$\Pi_1^{ANC+} $
+$\Pi_1^{ANC} $
+ sentence  $\Phi$,
+the Group-2 schema
+for ISCE$(\beta)$ 
+%
+%was defined in  \cite{wwapal} 
+%did 
+will 
+contain
+% an 
+one
+axiom of the form:
+%% 
+%% \begin{equation}
+%% \small
+%% \label{group2nold}
+%% \forall ~x~\forall ~y~
+%% ~~\{~~[~~ \sigma_{~ \ulcorner \, \Phi \, \urcorner
+%%  ~}(x)~\wedge~
+%% \{~ \mbox{HilbPrf}~_\beta
+%% ~(~ x ~,~y~)~~]~~
+%% \Rightarrow ~~ \Phi~~ \}
+%% \end{equation}
+%% % {\bf IMPORTANT CLARIFICATION:} 
+%% %{\small 
+%% %%{{\bf DECIPHERING LINE \eq{group2nold}:$~$}
+%% {{\bf Clarification:$~$ }
+%%  \el{group2nold} is {\it helpful}
+%% because   ISCE(\beta)$  can             infer
+%% \eq{group2old}'s {\it simpler statement}
+%% directly
+%% from the combination of
+%% \eq{group2nold},
+%% % it,
+%% the Group-1 schema and \el{deltf}'s definition of
+%% ``$~\sigma~$''.}
+%% 
+\begin{equation}
+% \small
+\label{group2old}
+\forall ~y~~~\{~ \mbox{HilbPrf}~_\beta
+~(~ \ulcorner \Phi \urcorner ~,~y~)~~
+\Rightarrow ~~ \Phi~~\}
+\end{equation}
+\item
+{\bf GROUP-3:}
+This last part of
+%%%%%%%%%%%%%%%  \cite{wwapal}'s
+ISCE$(\beta)$
+% formalism 
+was
+ a single 
+self-referencing
+$\Pi_1^{ANC}$
+sentence  
+stating:
+ %% essentially declaring:
+\begin{quote}
+% \small
+%%%%%%%%%%%%% $ \oplus ~ \oplus ~~~$
+$ \oplus  \oplus ~~~$
+ ``There 
+%is 
+exists
+no
+Hilbert-style proof of 0=1 from the union of the Group-0, 1 and 2
+axioms  with {\it THIS  SENTENCE} (referring to itself)''.
+\end{quote}
+\end{description}
+%{\bf CLARIFICATION:}
+{\bf Clarifying $ \oplus  \oplus$'s Meaning:}
+ $~$Several of our articles
+\cite{ww1,ww5,wwapal,ww9}
+employed
+self-referential
+  $\Pi_1^{ANC}$ constructions,
+similar 
+to 
+%%%%%%%%%the sentence
+ $ \oplus  \oplus \,$,
+as Example \ref{ex-2.5} had mentioned.
+%% 
+%% whose 
+%% % precise implications were outlined in
+%% significance was explained by
+%% %formalized by 
+%% Example \ref{ex-2.5}.
+%% 
+A reader can find
+several
+%detailed
+slightly different
+ illustrations about how 
+$~ \oplus  \oplus ~ $
+%  $\, \oplus  \oplus $'s 
+%   self-referential statement 
+is encoded in  these articles.
+
+
+% 
+% Each of these articles provide examples of
+% how analogs for
+% $\, \oplus  \oplus $'s 
+% self-referential
+% statement
+% are encoded.
+
+
+ 
+% If the reader wishes to see
+% a formal encoding  for
+% $\, \oplus  \oplus $'s 
+% %self-referential
+% % Fixed-Point 
+% statement,
+% %it 
+% one such example
+% is provided by  
+% \cite{wwapal}'s 
+%  Lemma 1.
+% 
+
+
+\begin{deff}
+\label{def-3.9x10}
+\rm
+Let $~I(~\bullet~)~$ denote
+an operation that maps
+an initial axiom basis $\, \beta \,$ onto an alternate
+system  $\,I(\beta)\, $.
+(One example of
+such an operation is the
+  ISCE$( \, \bullet \, )$ 
+framework,
+that maps 
+an initial axiom basis of 
+  $~\beta~$ onto 
+the alternate formalism of
+ ISCE$(\beta).~)~$ 
+Such an operation  $~I(~\bullet~)~$
+is  called {\bf Consistency Preserving}
+iff  $\,I(\beta)\, $ is consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemas is
+consistent.
+\end{deff}
+
+
+%Most of our research in 
+% \cite{ww93}-\cite{ww14}
+% has 
+
+Several of our research projects 
+%centered around
+%had 
+employed
+ \dfx{def-3.9x10}'s
+framework.
+For instance, 
+%% 
+%% the 
+%% 
+%% 
+%% Its
+%% %%%  main
+%% % central 
+%% focus in
+\cite{wwapal} 
+demonstrated
+%consisted of showing
+ the   ISCE$( \, \bullet \, )$ 
+mapping was consistency preserving.
+Thus if PA+ denotes the extension of
+Peano Arithmetic that 
+includes
+PA's traditional  Addition and Multiplication
+functions
+%% 
+%% 1n addition to the conventional
+%% functions of addition and multiplication
+%% contains 
+%% 
+%% 
+plus $L^G\,$'s six
+added
+%previously mentions
+ Grounding-level function 
+primitives,
+%functions,
+then
+ ISCE$( \, $PA+$ \, )$
+will 
+be automatically
+%be
+ consistent
+(because  PA+ was consistent).
+% consistent whenever PA+ is consistent.
+Hence while Peano Arithmetic is unable to
+verify its own consistency,
+%  (on account of G\"{o}del's
+% seminal 1931 discovery), 
+it is sufficiently agile to
+prove the following relative-consistency statement:
+\begin{center}
+%% \small
+$\#~~~$ If PA is consistent then 
+ ISCE$( \, $PA+$ \, )$ is  
+ self-justifying.
+ \end{center}
+This
+%The above
+% statement
+ relative-consistency statement
+%does offer 
+provides
+a partial 
+positive
+answer to
+the
+Q-2 version of Hilbert's Second  Question.
+It 
+captures
+% Brad change encapsulizes 
+one
+% positive 
+respect
+in which
+%such as  
+ISCE$( \, $PA+$ \, )$
+can {\it appreciate} its own consistency.
+% 
+% \newpage
+% 
+% \svxs
+% 
+% \noindent
+%This is because it formalizes one respect 
+This respect is, obviously,
+only
+of a limited nature
+because $++$'s generalization of the Second
+Incompleteness Theorem indicates 
+that
+no Type-S arithmetic
+can
+% simultaneously 
+recognize 
+% {\it both} 
+its Hilbert consistency and
+take
+successor 
+to be
+ a total function.
+%The consistency-preservation property of 
+% ISCE$( \, \bullet \, )$
+%dies, however,
+It does,  however,
+ raise the following
+enticing
+ question:
+\newpage
+
+\lvxs
+\parskip 0pt
+
+\begin{quote}
+$\# \, \#~ $
+\small
+Can the infinite number of
+distinct
+ constant symbols, employed by
+ISCE's Group-Zero schema, be reduced to a finite size
+by a Type-NS Self-Justifying Logic,
+without resorting to \cite{wwapal}'s inefficient
+``ISINF'' 
+methodology (which requires 
+a proof 
+having an expensive
+ $\Omega(N)$ length for constructing integers $N$
+whose binary encoding uses $O(~$Log$(N)~)$ bits) ?
+\end{quote}
+The remainder of this section will outline how an encouraging
+answer to 
+$\, \# \, \# \, $'s query
+is likely to
+%%%should,
+% conveniently
+arrive,
+%be plausible
+when one 
+% carefully
+%delicately 
+modifies ISCE's formalism
+with  the Q-function operative of $~\zzthe~$.
+
+\begin{deff}
+\label{def-3.10}
+\rm
+Let $L^Q$ 
+% once 
+again denote the extension of
+$~L^G\,$'s Grounding language that includes
+the
+% further
+ Q-function symbol of $\, \theta $.
+Then
+$\Delta^Q_0$,
+ $\Pi^Q_n$ and $\Sigma^Q_n$
+will,
+intuitively,
+%similarly
+ denote the
+%  1-to-1 
+analogs of
+\dfx{def-3.8}'s
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$'s
+formulae
+in $~L^G\,$'s  language.
+In particular, if $~\Phi~$ 
+is one of an
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+formula,
+then
+% the formula 
+$~\Phi^Q~$
+will be called
+% respectively
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+when
+%if 
+it
+differs from $~\Phi~$ 
+only
+by
+ replacing each constant $~C_J~$
+from the set $~C_3,C_4,C_5...~$ 
+with Line \eq{ej-def}'s 
+% mathematically equivalent term of
+term  $~E_{J-1}~$. 
+\end{deff}
+
+\parskip 2pt
+
+%% 444444444444444
+
+\begin{example}
+\label{ex-3.11}
+\rm
+Suppose $~\Phi$ 
+is one of a 
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+sentence that employs the three constant symbols
+of $C_4$,   $C_6$ and  $C_{10}\,$
+for
+ representing the
+three numbers
+of 8, 32 and 512. 
+Let us recall 
+that
+ $E_3$,   $E_5$ and  $E_9\,$
+%  do
+formulate these three quantities
+under  Line \eq{ej-def}'s notation.
+Then $~\Phi^Q$  will have an 
+identical definition as
+ $~\Phi$ 
+except each $C_j$ is replaced by
+$E_{j-1}$.
+
+
+A formula is,
+moreover,
+ defined to lie in one
+of the 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+classes 
+{\it only if} it is constructed in such a manner.
+This fact
+% brad assures 
+ensures
+that all the terms employed in these
+three classes of sentences are 
+{\it ``Observable''} terms.
+Hence ``Unobservable'' ground terms are allowed in
+$~L^Q\,$'s language, but {\it they are excluded}
+from occurring in the 
+{\it ``end-product''} 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+theorems 
+that 
+%%will now be discussed.
+%it proves !
+do
+encapsulate
+%formalize
+ the {\it intended use of 
+%its
+this 
+formalism.}
+\end{example}
+
+
+\begin{deff}
+  \label{def-3.12}
+\rm
+The term {\bf IQFS($ ~\bullet~$)$~$}
+will  refer
+to the self-justifying analog of
+  ISCE($ ~\bullet~$)$~$
+%that will be employed 
+under  $L^Q\,$'s
+language.
+(The acronym ``IQFS'' stands for 
+``Introspective Q-Function Semantics''.)
+In a context where $~\beta~$ is 
+%some i
+an initial axiom
+system that proves theorems 
+%under
+in 
+the 
+language  $L^Q$, the 
+system
+%formalism 
+ IQFS($ \, \beta\,$)
+%$~$
+will 
+be defined as a
+ 4-part 
+formalism,
+analogous to  ISCE($\beta$),
+except for the following 
+%relatively modest 
+three 
+relatively
+modest
+changes: 
+\bed
+\small
+\parskip 0pt
+\item[  a.  ]
+The Group-Zero schema of
+ IQFS will
+differ from ISCE's analog
+by replacing 
+\el{addcov}'s ``Additive Naming'' schema with  
+the
+Up-Walking axioms,
+given in Lines  \eq{walk1}--\eq{walk4}.
+(This is because
+the language  $L^Q$ differs from
+ $L^{ANC}$ by 
+having the 
+ Q-function operator of $~\zzthe~$
+define the formal quantities that are represented by
+the constant symbols
+of $~C_3,C_4,C_5~~....~$ 
+under  $L^{ANC}.~~)$ 
+%% 
+%% Otherwise both
+%% these
+%% Group-Zero 
+%% schemes will be 
+%% identical.
+%% Thus,
+%%  they 
+%% will 
+%% both 
+%% use \el{start}'s axiom to define the 
+%% three initial constants of
+%% $\,C_0\,$, $\,C_1\,$ and $\,C_2\,~$.
+%% 
+\item[  b.  ]
+All the $\Pi_1^Q$ axioms lying in IQFS's
+Group-1 and Group-2 schemes will be
+% identical
+analogous to their counterparts
+under ISCE, except  they
+will
+ employ 
+\dfx{def-3.10}'s machinery for translating 
+ $ \,\Pi_1^{ANC} \,$ 
+sentences into
+essentially their
+% equivalent
+ $ \,\Pi_1^Q \,$ counterparts. 
+\item[  c.  ]
+The Group-3 axiom of  IQFS
+will be similar to ISCE's Group-3 
+{\it ``I am consistent''} 
+axiom-statement, except 
+the latter's notion of ``I'' will reflect the above
+changes in the Groups 0, 1 and 2 schemes. 
+It
+%Thus, the new
+%Group-3 axiom 
+will,
+thus,
+be a $\Pi_1^Q$ sentence declaring that
+{\it ``There is no 
+Hilbert-style
+proof of 0=1 from the union of the preceding axioms
+with THIS SENTENCE (looking at itself)''.}
+\ennd
+\end{deff}
+
+%\noindent
+
+\bvxs
+\parskip 2pt
+
+{\bf REVISITING THE Q-2 VERSION OF
+  HILBERT'S SECOND OPEN
+QUESTION FROM THE PERSPECTIVE OF  ``IQFS'' .}
+$~$
+Let us recall
+% that 
+\textsection \ref{ss4} indicated 
+that   Hilbert's Second Open Problem could be
+divided into two sub-queries, 
+that were
+called Q-1 and Q-2.
+The former query asked whether axiom systems could
+verify their own consistency in a robust sense, and the latter
+inquired whether some 
+{\it weaker but non-trivial} forms of
+self-justification might exist.
+The 
+Q-1
+paradigm
+% 
+% former query
+% addressed the larger part of Hilbert's
+% open question. It
+% % We already noted that the Q-1 version of this query
+% 
+was definitively resolved in a negative direction
+by the combination of G\"{o}del's initial Second Incompleteness
+Effect,
+its
+generalization
+appearing
+% documented 
+in the 
+Hilbert-Bernays textbook \cite{HB39} and  the
+Result $++$ due to the combined work of
+ combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87}.
+In contrast,
+we noted in
+\textsection \ref{ss4} 
+ that there
+were other
+issues raised by the Q-2 version of Hilbert's
+question that were 
+not yet
+fully
+resolved.
+
+% \gvxs
+ 
+\smallskip
+
+It is within 
+this
+context where 
+Definition \ref{def-3.12}'s IQFS framework is helpful.
+The strong similarity between the definitions of ISCE and IQFS,
+{\it by itself,} 
+suggests that IQFS is likely to satisfy a
+consistency-preservation property analogous to ISCE.
+Moreover, all the techniques that were used to prove
+either $++$'s generalization of the Second Incompleteness
+Theorem
+% 
+%  (due to the combined 
+% work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+% \cite{Ne86,Pu85,So94,WP87}) 
+% 
+or the related subsequent 
+results of
+% 
+% generalizations
+% %% of the Second Incompleteness Effect 
+% that
+% Example \ref{ex-2.3} attributed
+% to 
+% 
+Buss-Ignjatovic,
+H\'{a}jek, 
+\v{S}vejdar 
+and Willard 
+ \cite{BI95,Ha7,Sv7,ww1}
+% \cite{BI95,Ha7,Sv7,ww1,wwlogos}
+lose their relevance in IQFS's context.
+
+\medskip
+
+This is because a longer version of the current article
+demonstrates
+the Groups 0, 1 and 2 axioms of IQFS
+{\it  are unable}
+to prove 
+% an invariant implying
+successor is a total function, 
+while the 
+incompleteness results
+of   \cite{BI95,Ha7,Ne86,Pu85,So94,Sv7,WP87,ww1}
+require taking
+successor as a total function.
+
+% is
+% precisely what is needed for these formalisms to 
+% become applicable
+%  to IQFS.
+
+% (because they require a counterpart of a successor
+% function operation).
+
+
+
+\smallskip
+
+The properties of IQFS
+ are
+interesting
+% especially intriguing 
+from a 
+% Computer Science 
+Complexity perspective
+because
+\phx{th-3.3} showed 
+% that 
+every integer $\,n\,$ 
+can
+%could 
+be encoded
+%under it by
+by
+%via
+a
+ term $T_n$ that has an $O\{~[~$Log$(n)~]^3~\}$ length.
+This is
+unlike the 
+%much 
+%%%%%%%%%% far worse 
+asymptote  $\Omega(n^2)$ that results
+when the 
+$~\zzthe$ primitive 
+(from Lines  \eq{walk1}-\eq{walk4})
+is replaced by 
+Lines \ref{zm1} and  \ref{zm2}'s
+less efficient
+primitive of 
+$~\glamb~$.  
+Moreover, Remark \ref{rem-def-3.4}
+indicated
+% our
+that these
+ $O\{~[~$Log$(n)~]^3~\}$ lengths
+could be reduced
+to almost an  $O\{~$Log$(n)~\}$ size if 
+$L^Q\,$'s Ground terms were encoded as 
+Directed Acyclic Graphs (rather than as tree-like objects).
+
+We consider it 99 \% likely that
+{\it BOTH} 
+Definition \ref{def-3.12}'s
+%precise
+specified
+formulation of 
+% the
+ IQFS($ ~\bullet~)$
+%construct 
+and its 
+% more compressed Dag 
+implied Dag
+refinement
+(using  Remark \ref{rem-def-3.4}'s additional machinery)
+%modification
+will satisfy 
+a consistency preservation property analogous to ISCE.
+If this 
+2-part
+conjecture is correct, it will support our
+hypothesis that the Q-2 version of Hilbert's Second Open
+Question 
+% would 
+does
+support some 
+{\it fragmentary}
+positive results,
+in the context of
+% with regards to
+Hilbert-styled deductive methods using the
+$~\zzthe$ primitive for formulating growth among integers.
+
+Moreover, a reader should not be
+especially
+ concerned that the Group-2
+axiom schemas for ISCE and IQFS involve employing
+an infinite number of separate incarnations
+of \el{group2old}'s axiom schema.
+This is because these
+Group-2 schemas
+can be
+% should be able to be
+nicely
+ reduced to a 
+purely
+finite size, with almost no loss 
+in
+%of useful
+information. This was done in \cite{ww14} 
+for the Group-2
+scheme
+of its IS$_D(\beta)$ formalism, 
+with the latter
+%%%%%%%%%%%%%%%% still 
+%where the
+% 
+% germane Group-2 scheme was
+% reduced to one
+% single  axiom sentence 
+% while the resulting 
+% 
+% latter
+%formalism still
+% produced
+producing
+isomorphic counterparts
+of all of $~\beta \,$'s
+full set of
+ $\Pi_1$ theorems
+(e.g. see 
+%Sections 5 and 6 
+\textsection $\,$5
+of    \cite{ww14}).
+The same methods will
+% trivially 
+%routinely
+easily
+generalize for
+%%%% the
+% 
+% Analogs of the techniques from Sections 5 and 6 of 
+% \cite{ww14} 
+% % will easily 
+% apply to each of the 
+% 
+ IQFS, 
+% axiom framework, 
+if it does satisfy
+Definition \ref{def-3.9x10}'s
+Consistency Preservation property (as we
+conjecture it does).
+
+% it does).
+
+\vspace*{- 1.0 em}
+
+%\section{Broader Perspectives Produced by These Results}
+
+
+\section{Concluding Remarks}
+
+\vspace*{- 0.8 em}
+
+\lvxs
+
+\label{nnnew}
+
+%\Large
+% \baselineskip = 1.8 \normalbaselineskip 
+
+There is no question that the 
+% Second 
+Incompleteness Theorem
+%does imply
+% demonstrates
+illustrates 
+that 
+90-95 \% of the initial objectives of
+Hilbert's Consistency Program were overly ambitious.
+It would, nevertheless, be of interest if
+some 5-10 \%
+fragment of 
+Hilbert's
+% initially 
+intended 
+goals
+% objectives 
+were
+partially
+ achieved.
+
+%% bbbbbb
+
+\smallskip
+
+
+This is because it is difficult to fathom how humans
+can
+maintain
+their psychological motive to engage in cognition without owning some
+type of
+% qualified 
+instinctive faith in their own consistency.
+%% 
+%% Moreover, the close similarity between the defining structures of the
+%% ISCE and IQFS frameworks strongly suggests
+%% \cite{wwapal}'s proof of ISCE's consistency preservation property
+%% should generalize for both 
+%% IQFS  and 
+%%  IQFS$^*$ under a more elaborate
+%% % and sophisticated
+%% inductive machinery.
+%% 
+Moreover, it is fascinating that 
+the distinction between
+Unobservable and Observable ground terms, using 
+Proposition \ref{th-3.3}'s  and Remark \ref{rem-def-3.4}'s
+ $\, \theta \, $ operator,
+% 
+% whose $O(~$Log$^3\,n )$ and $O(~$Log$~n )$ complexities
+% are characterized
+% by Proposition \ref{th-3.3}  and Remark \ref{rem-def-3.4},
+% 
+%%%%%%
+$\,$does 
+seem to 
+lend credibility to a
+% fraction 
+{\it  partial  subset}
+% {\it fragment}
+of the goals
+that
+Hilbert and G\"{o}del 
+advanced
+%% were seeking 
+in
+% aspiring to in
+their
+statements $*$ and $**~$.
+
+\smallskip
+
+\nvxs
+
+Also, the last five minutes of a YouTube lecture
+by Harvey Friedman,
+entitled
+% the 
+{\it ``The Blessing and Curse of Kurt  G\"{o}del''},
+raised the question 
+\cite{Fr14}
+of whether 
+some type of 
+{\it sharply  circumscribed} boundary-case exception
+to 
+the Second Incompleteness
+Theorem 
+might be possible.
+
+
+% 
+% could be evaded with some
+% % new
+% non-recursive  function symbol
+% (which under
+% \cite{Fr14}'s
+% % Friedman's
+% hypothetical
+%  example
+% involved deploying the laws of Physics
+% instead of
+% %  rather than 
+% Lines  \eq{walk1}-\eq{walk4}'s
+% indeterminate definition).
+
+
+
+% (in a context where the broader ambitions of these
+% two statements 
+% are clearly untenable).
+
+%% infeasible
+
+
+%Thus,
+
+% \medskip
+
+
+Our proposed IQFS 
+% axiom system 
+framework
+is intended to
+be no full remedy,
+when the
+traditional growth properties of the  addition,
+multiplication and successor function operations are replaced
+by an
+alternative  $~\theta~$ function symbol.
+It is only a partial solution, similar to our
+alternative class of 
+partially
+positive
+ results in
+\cite{ww93,ww1,ww5,ww14},
+involving 
+%axiom systems 
+arithmetics
+that
+ sacrifice
+%      sacrificed 
+their
+understanding that multiplication is a total function
+for the sake of gaining an appreciation of their 
+semantic
+tableaux consistency.
+
+\smallskip
+
+Neither of these
+% results 
+formalisms
+are perfect, and 
+imperfections 
+will 
+% be ever-present
+always be 
+ present
+%result
+%be inevitable
+when one
+%  explores
+considers
+ the
+%  tight
+ dilemma posed by the
+% 
+% must
+%  always
+% % have to 
+% be tolerated
+% 
+% when examining the  dilemma posed by the
+Second Incompleteness 
+Effect. 
+It is within such a  context that
+{\it a well-defined fragment} of 
+%what 
+the goals
+% which
+that
+Hilbert and 
+G\"{o}del sought in
+$*$ and $**$ 
+should be
+%% 
+%% looks 
+%% %part-way 
+%% like it is probably
+%% 
+%         realistically feasible 
+possible to 
+reach
+%realize
+under
+%  some
+certain
+% % might be plausibly 
+% %possible 
+% should be 
+% possible 
+% to obtain
+{\it meticulously defined weak-logic settings$\,$,}
+if IQFS satisfies an
+analog of ISCE's
+ consistency-preservation
+property (as we conjecture it 
+will almost certainly
+ do). 
+
+%\textsection \ref{ss5}'s conjectures about IQFS do
+%old to be ture.
+
+
+%  special
+% % broad
+% % potential 
+% interest.
+% 
+
+
+   \medskip
+
+{\bf Acknowledgments:}
+%%%%As  several Sections 1-4, 
+%\textsection \ref{ss2}, 
+I am
+% much
+%very 
+grateful to
+%was influenced by an emailed letter from 
+Pavel Pudl\'{a}k
+for suggesting 
+\cite {Pupriv}
+I investigate how to apply
+% an analog of 
+Ajtai's study
+\cite{Aj94} of Pigeon-Hole effects 
+for
+refining my prior results about self-justifying logics.
+(The combination of
+ Pudl\'{a}k's
+insightful             suggestion
+% \cite {Pupriv}
+and our
+subsequent
+% further 
+        distinguishing
+between the
+$~\glamb   ~$ and $~\theta~$ operators
+has led to the 
+conjectured
+ improvement of 
+\cite{wwapal}'s ISCE formalism.)
+% I am very grateful to Pudl\'{a}k for making this
+% suggestion. 
+I also thank Bradley Armour-Garb
+and Seth Chaiken
+ for
+%  many
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%several
+comments
+%  about how to improve 
+that improved 
+ the 
+%this paper's
+presentation.
+
+
+\footnotesize
+% \tiny
+\parskip -3 pt
+
+%\baselineskip =  0.92 \normalbaselineskip 
+\baselineskip =  0.5 \normalbaselineskip 
+%\baselineskip =  0.65 \normalbaselineskip 
+\bibliographystyle{abbrv}
+\bibliography{aa}
+
+ \end{document}
+\newpage
+
+rrrrrrrrrrrrrr
+
+
+\Large
+
+
+On How the Introducing of a New $~\theta~$ Function Symbol Into Arithmetic's
+Formalism Is Germane to Devising Axiom Systems that Can Appreciate Fragments
+of Their Own Hilbert Consistency
+
+
+Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After A New $~\theta~$ Function Primitive
+Is Added to Arithmetic's Formalism
+
+AFTER A NEW ``$~\theta~$'' Function Primitive
+
+
+ \baselineskip = 1.5 \normalbaselineskip 
+
+It is known that the combined work of Pudlak and Solovay, enhanced by some
+added techniques of Nelson and Wilkie-Paris, implies no reasonable axiom
+system can verify its own Hilbert consistency, when it recognizes Successor as
+a total function and treats addition and multiplication as 3-way relations.
+These considerations will lead us to examine unconventional axiomatizations
+for arithmetic that continue to view addition and multiplication as 3-way
+relations, but which replace the successor function symbol with an entirely
+new operator, called the $\theta$ primitive.
+
+
+
+It is likely that this paradigm can be combined with the prior results in our
+APAL 2006 paper to construct axiom systems that are seriously diluted but able
+to verify their Hilbert-style consistency in some interesting fragmentary
+respects.
+
+
+
+%% This operator will allow us to encode any integer  n  by a term $T_n$ whose
+%% length will exceed the O(Log n) length of a binary encoding by only the
+%% relatively small magnitudes formalized by our Proposition 3.3 and Remark 3.6.
+%% It is likely that this paradigm can be combined with the prior results in our
+%% APAL-2006 paper to construct axiom systems that are seriously diluted but able
+%% to verify their Hilbert-style consistency in certain interesting respects.
+
+
+
+{\bf Keywords:}
+
+G\"{o}del's Second Incompleteness Theorem, 
+Hilbert's Second Open Question,
+Bounded Arithmetic, 
+Distinction Between Semantic Tableaux and Hilbert Deduction, 
+Weak Arithmetics.
+
+\end{document}
+
+% \parskip 2pt
+
+The 
+significance
+% roles 
+of
+% Observations
+(a) and (b)
+%in our research
+%from the current example, 
+will become
+% more 
+evident
+as this article progresses. 
+Essentially, our
+prior research
+% ,
+% best summarized in \cite{ww14},
+%  has 
+%
+% had 
+focused
+% mostly 
+on Type-A arithmetics
+that could verify their consistency under
+% either 
+semantic tableaux deduction 
+and/or its near cousins.
+%% 
+%% or some near-cousin of this concept
+%% (e.g. see \cite{ww14}'s  summary
+%% of \cite{ww93}-\cite{ww9}'s results).
+%% The 
+%% 
+(A 15-page summary of this research appears in
+\cite{ww14},
+ but 
+it
+%the latter
+does need to be examined.)
+%%% 
+%%%  is 
+%%% not
+%%% % unnecessary to examine  as 
+%%% a prerequisite for reading this paper.
+%%% 
+%%% 
+%%%  but this technical
+%%% material does not need to be examined.) 
+%%% does hot need to examine
+%%% this material for the 
+%%% 
+%%% 
+Our
+% new
+$~\theta~$ operator, defined in the next section, will
+raise the question about whether a
+% surprising
+% powerful 
+new
+class of 
+%new 
+Type-NS systems 
+will
+%may 
+satisfy an analogous
+property in the context of
+Definition \ref{def-2.2}'s more pristine
+Hilbert-style methodology for deduction. 
+
+\medskip
+
+% have
+% a similar property.(The article \cite{ww14} offers a nice 16-page summary
+% of our prior results 
+% \cite{ww93}-\cite{ww9}
+% about Type-A arithmetics, but 
+% none of these results will
+% be
+%  needed
+% %  to be examined 
+% during our current article's exploration of 
+% the properties of the new % 
+% $~\theta~$ operator.)
+
+% It will be unnecessary for a reader to examine any of our
+
+ 
+%% % year-2014 
+%% Wollic-2014 paper \cite{ww14}
+%% summarized and extended our
+%% results about
+%% semantic tableaux consistency.
+%% % and this 
+%% The current 
+%% new 
+%% year-2015
+%% paper 
+%% will, now,
+%% explore whether
+%%  systems  can
+%% also corroborate
+%% their Hilbert-styled consistency
+%% under certain well-defined circumstances.
+
+%% (and seek to explore the restrictions $++$ imposes upon
+%% Hilbert-styled deduction).
+
+% 
+% (The latter topic
+% % is 
+% %very 
+% %entirely
+% %different 
+% differs
+% from the former
+% because
+% constraint  $++$ 
+% applies only 
+% to its
+% particular
+%  domain.)
+
+%in the second context.)
+
+
+%% The constraints imposed by $++$ 
+%% are challenging 
+%% because Type-NS arithmetics 
+
+This
+topic
+% subject 
+is 
+interesting
+%challenging 
+because 
+%% essentially all
+Type-S arithmetics
+are forbidden by $++$ from
+verifying the consistency
+of their own Hilbert-styled deductions
+%%%%%%%%%%%%%%%%%%% (and conventional 
+%forms of
+(while
+Type-NS formalisms are
+%typically 
+ usually 
+quite weak).
+%% 
+%% (Thus, our efforts
+%% to design
+%% self-verifying systems
+%%  must focus on
+%% Type-NS arithmetics).
+%% 
+Our new 
+$~\theta~$ operator,
+together with
+Proposition \ref{th-3.3}
+and Remark \ref{rem-def-3.4}, 
+% and \ref{th-6.1}
+will suggest a 
+%possible
+% plausible 
+partial
+solution to this 
+problem by
+% daunting challenge by
+illustrating how
+an {\it unusual class} of 
+Type-NS arithmetics can efficiently construct the
+full set of integers $~0,1,2,3,...~$
+% by finite means 
+{\it without using}
+any of the successor, addition or multiplication
+% functional 
+operations.
+
+% function symbols. 
+
+As a result, we will suggest 
+a
+%  a {\it part-way} 
+% that an interesting
+%%% non-trivial (although diluted)
+{\it small fragment}
+of what Hilbert
+and G\"{o}del
+% sought 
+% referred to
+did seek
+%sought
+in 
+statements 
+$*$ and $**$ 
+% will
+%  be formally  achieved 
+% become tempting 
+is likely
+% be 
+viable
+under Definition \ref{def-2.4}'s formalism.
+
+
+
+% 
+% This
+% topic
+% % subject 
+% is challenging because 
+% $++$'s 
+% Type-NS arithmetics 
+% %  obviously 
+% have sharply circumscribed powers
+% (demonstrating the 
+% broad reach
+% % ubiquitous  nature 
+% of
+% %the Second Incompleteness Theorem's reach).
+% G\"{o}del's 
+% second theorem).
+% %%  
+% %% The current article will
+% %% show, however,  that 
+% %% some Type-NS arithmetics are 
+% %% substantially
+% %%  stronger than previously 
+% %% anticipated
+% %% (and they will have useful applications 
+% %% in
+% %% computer science settings).
+% %% 
+% %% Thus in a context where 
+% %% the power of both G\"{o}del's initial
+% %% Second Incompleteness Theorem and $++$'s strengthening of it
+% %% are stunning
+% %% and
+% %% have pervasive implications,
+% %% we will show that a 
+% %% {\it partial-and-much-less-than-full}
+% %% fragment 
+% %% of what Hilbert
+% %% desired
+% %%  in statements $*$ and $**$ an be
+% %% positively  achieved.
+% %% 
+% The current article will
+% %show, however, 
+% suggest,
+% however, 
+% % that 
+% some Type-NS arithmetics are
+% % , however,  
+% % significantly 
+% %% substantially
+% more far-reaching
+% than 
+% previously 
+% anticipated. Thus,  a 
+% % well-defined 
+%   {\it
+% partial but non-trivial} fragment of what Hilbert
+% and G\"{o}del
+% % sought 
+% % referred to
+%  anticipated
+% in 
+% statements 
+% $*$ and $**$ will
+% %  be formally  achieved 
+% % become tempting 
+% look
+% % be 
+% viable
+% under Definition \ref{def-2.4}'s formalism.
+
+
+
+\end{document}
+
+% \textsextion
+
+%\setlength{\textwidth}{5.0 in}
+
+\gvxs
+
+Line 1
+
+
+Line 1
+
+
+Line 1
+
+
+Line 1
+
+
+%% eeeeee
+
+\newpage
+
+A theme of this article will be that
+% distinction 
+the distinguishing
+between questions Q-1 and Q-2 and 
+the separation of Observables from
+Unobservables
+is 
+related
+% likely central 
+to the mystery
+% that has enshrouded 
+enshrouding
+the Second Incompleteness Theorem.
+This is
+%is germane to the aspirations of automated theorem proving
+%will be germane to this article 
+because there 
+%is no doubt
+can be no doubt that
+% can be no question 
+%%%%%%%% that 
+the Second
+Incompleteness Theorem is  fully
+robust
+% result 
+from a purist 
+%pristine 
+mathematical perspective.
+Yet,
+it is still problematic to fully
+% 
+%  simultaneously
+% % at the same time,
+% it is 
+% hard to 
+% entirely
+% 
+dismiss
+ Hilbert's 1926
+suggestion that 
+ some 
+specialized forms of logics should
+%declaration 
+%% 
+%% concerns
+%% in $\,*\,$ 
+%% that 
+%% {\it ``the honor of human understanding''}
+%% requires
+%% examining
+%% % explaining 
+%% % considering
+%% how  logic systems can
+%% 
+possess
+a type of well-defined 
+ knowledge about their
+own 
+internal
+consistency.
+(This is because it is
+highly
+ awkward to explain how and why
+human beings 
+are able to
+%can 
+%manage to 
+motivate 
+their 
+%cogitations,
+cognitive process,
+% themselves to think,
+ if they do not own
+some type of 
+% instinctive
+internal
+knowledge about their own
+ consistency.)
+
+% sufficient
+% % enough 
+%  knowledge about their
+% % own 
+% internal
+% consistency
+% to motivate 
+% cognition.
+
+% Bad change above
+%cogitation.
+% themselves to cogitate. 
+
+%%It is also
+%%especially 
+%%% very 
+%%tempting 
+%%to divide Hilbert's Year-1900
+%%Open Question into its Q-1 and Q-2 separate parts
+%% during the 21st century,
+%%as computers share with humans cogitative abilities.
+%%
+%%Maybe DELETE above sentence ??? 
+%\end{remm}
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\smallskip
+
+The next 
+section will 
+formalize our
+% new
+ proposed IQFS formalism.
+% 
+% describe our 
+% 2-part
+% conjecture about how 
+% %a
+% Double-Formatted Logics are 
+% likely to 
+% %produce some 
+% cast
+% new perspectives
+% on 
+% this topic.
+% %the nature of the Second Incompleteness Theorem.
+% 
+Before starting this subject, it should be mentioned
+that other unusual interpretations of the Second Incompleteness
+Theorem have followed
+from Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and also
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a 
+% preferable 
+Hilbert-style 
+deductive 
+apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspective  
+should
+% certainly,
+ be examined by researchers
+interested in the
+Second 
+Incompleteness Theorem,
+although 
+%but
+it is
+%% 
+% they are 
+unrelated to 
+our particular
+% the next section's 
+%specific analysis of
+%%% type of
+Hilbert-styled self-justifying effects,
+studied in the current article.
+
+
+%% systems 
+%% formalizing
+%% %verifying 
+%% their
+%% own consistency
+%% %%%%%Definition \ref{def-2.2}'s
+%% %%% approximate 
+%% under
+%% Hilbert-styled 
+%% deduction.
+
+
+%deduction.
+%  Hilbert deduction.
+
+%methods.
+%formalism.
+
+
+%% It is,
+%% % They
+%% %are, 
+%% however,  not germane to the next section's
+%% perspective.
+
+%methodology.
+%main formalisms.
+%methods.
+%results.
+
+ % \baselineskip = 1.8 \normalbaselineskip 
+
+%\section{
+%\small 
+%Improving \cite{wwapal}'s Results with a 
+%``Double-Formatted'' Logic  }
+
+\newpage
+xxxxxxxxxxx
+
diff --git a/nachlass/collected_dew_materials/2011-2019/2015-lfcs.tex b/nachlass/collected_dew_materials/2011-2019/2015-lfcs.tex
new file mode 100644
index 0000000..68a4b23
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/2015-lfcs.tex
@@ -0,0 +1,7915 @@
+%%  2015 sept 9 after submission CHANGES ONE SENTENCE IN EXAMPLE 2.5
+
+%% 2015 sept 7  2.20 pm (after finding addrees)
+%% after reding conclusion to BOB
+% chipped off end
+
+
+%% www.cs.albany.edu/~dew/algor
+
+
+%% 2015 home august 24 11.2  am 
+
+%% 2015 home august 22 1.1  pm (single space)
+
+
+ % 2015 july 4 3.4 am after spell 10.1 am after sinatra
+
+ % 2015 july 2 3.15 pm 
+
+%% 2015 july 2 2.50 pm upstairs 
+
+%% 2015 july 1  10.30  am downstairs
+
+
+
+%% notarized notes 2015 april 2 6.3  am april 4 notarize again 
+
+%% home 2014 feb 8 1.15am (new email address)
+
+%% home 2015 feb6 4.3 am   suny 2.40 pm home 6.15 pm
+
+
+%% gmail dan.willard.albany and Prof.DanEdwardWillard
+%% gmail password cpZ9ar48s
+
+
+%%% SUNY JAN 11 Brad Copy 8.4  pm
+
+%%  SUNY   jan11 5/30pm spell check
+
+%% 2015 HOME jan 10  9.4  pm  pm New Abstratct
+
+%% 512 6932
+
+%% Towards a Restructuring of Hilbert's Consistency Program
+
+% www.cs.albany.edu/~dew/algor/
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[12pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+
+
+
+% \addtolength{\oddsidemargin}{-0.95in}
+ \addtolength{\oddsidemargin}{-0.9in}
+%\addtolength{\oddsidemargin}{-1.0in}
+% \addtolength{\oddsidemargin}{-0.95in}
+
+\setlength{\textheight}{9.6 in}
+\setlength{\textheight}{8.8 in}
+\setlength{\textheight}{9.3 in}
+\setlength{\textheight}{9.55 in}
+%above too short
+
+
+\setlength{\textwidth}{6.3 in}
+%% PRINT
+
+\setlength{\textwidth}{6.0 in}
+\setlength{\textwidth}{5.4 in}
+
+
+\setlength{\textwidth}{6.9 in}
+%% \setlength{\textwidth}{6.7 in}
+
+% \setlength{\textwidth}{6.3 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+
+
+% \setlength{\textwidth}{7.0 in}
+% Above IDeall
+
+
+%% \setlength{\textwidth}{6.4 in}
+%%%% above brad with 11 point
+
+%\setlength{\textwidth}{6.0 in}
+%\setlength{\textwidth}{5.7 in}
+
+%\setlength{\textwidth}{6.4 in}
+
+%\setlength{\textwidth}{5.5 in}
+
+%\addtolength{\topmargin}{-1.0in}
+\addtolength{\topmargin}{-0.95in}
+%\addtolength{\topmargin}{-1.0in}
+%\addtolength{\topmargin}{1.2in}
+
+%\addtolength{\topmargin}{-.95in}
+%\addtolength{\topmargin}{+.7in}
+%%% delete above for pdf
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pag5{Page 5}
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+
+% \title{ A 2-Part Conjecture about How  
+
+
+
+
+\title{Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After
+A New $~\theta~$ Function Primitive
+%AFTER A NEW ``$~\theta~$'' Function Primitive
+Is Added to
+A New $~\theta~$ Function Primitive
+Arithmetic's Formalism}
+
+
+\title{Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After A New $~\theta~$ Function Primitive
+%AFTER A NEW ``$~\theta~$'' Function Primitive
+Is Added to Arithmetic's Formalism}
+
+
+\title{On  How the Introducing of a 
+ New $~\theta~$ Function Symbol
+Into Arithmetic's Formalism Is Germane
+to Devising Axiom Systems that Can 
+Appreciate Fragments of Their Own
+Hilbert Consistency}
+
+
+
+%% 
+%% \title{On the 
+%% Likelihood
+%% That a 
+%% Curtailed but 
+%% Well-Defined
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Should be
+%% Feasible
+%% for the
+%% Case of
+%% Hilbert Deduction}
+
+
+% \title{On the Almost-Certain Likelihood
+% That a Sharply Curtailed but 
+% Well-Defined
+% %Significant
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Ought to be
+% %Plausible 
+% Feasible
+% for the
+% % Even the Challenging
+% Case of
+% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+% \title{\large \bf On the Revival of a Modified and Diluted Version of
+% Hilbert's Consistency Program (Extended Abstract)} 
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E. Willard \thanks{This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+%Email = dew@cs.albany.edu.}}
+%\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{University at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+%\begin{abstract
+\baselineskip = 1.17 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+
+%  PERTECT TITLE ABOVE ALTHOUHG PERHAPS OLD MANUSCRIPT BETTEDR
+
+\begin{abstract}
+
+%\Large
+
+% \baselineskip = 1.8 \normalbaselineskip 
+%aaaaaaaaaaa
+
+\large
+\LARGE
+ \normalsize
+
+
+ It is known that the combined work of Pudl\'{a}k and Solovay
+ \cite{Pu85,So94}, enhanced by some added techniques of Nelson and
+ Wilkie-Paris \cite{Ne86,WP87}, implies no reasonable axiom system can verify
+ its own Hilbert consistency, when it recognizes Successor as a total function
+ and treats addition and multiplication as 3-way relations (as Example
+ \ref{ex-2.3} will explain).  These considerations will lead us to examine
+ unconventional axiomatizations for arithmetic that continue to view addition
+ and multiplication as 3-way relations, but which replace the successor
+ function symbol with an entirely new operator, called the ``$~\theta~$''
+ primitive.
+
+\medskip
+
+%% This $~\theta~$ operator
+%% will
+%% allow us 
+%% to encode any integer $~n~$ by a term $~T_n~$
+%% whose length will exceed the $O(~$Log$~n~)$ length of a
+%% binary encoding 
+%% by
+%% only  the
+%% relatively
+%%  small magnitudes formalized by
+%% Proposition \ref{th-3.3}  and Remark \ref{rem-def-3.4}.
+% Proposition 3.3  and Remark 3.6
+
+It is likely that this paradigm can be combined 
+with our prior results from \cite{wwapal} 
+%%% 
+%% with the prior results 
+%% in our APAL 2006 paper
+%% REMOVE NEXT lINE
+to construct  axiom systems that are
+seriously
+diluted but
+able to verify their Hilbert-style 
+consistency
+in some interesting fragmentary respects.
+
+\end{abstract}
+
+\bigskip
+\bigskip
+\bigskip
+\LARGE
+
+% ttttt THIS PAPER SHOULD BE MASTER DRAFDT for future articles.
+
+
+\bigskip
+\bigskip
+\bigskip
+
+
+\normalsize
+
+{\bf Keywords:}
+Bounded Arithmetic, 
+G\"{o}del's Second Incompleteness Theorem, Hilbert's Second
+Open Question,
+Semantic Tableaux Deduction, 
+and Hilbert
+Deduction.
+
+
+
+\bigskip
+
+
+% {\bf Keywords:}
+% G\"{o}del's Second Incompleteness Theorem, Consistency, Hilbert's Second
+% Open Question,
+% Hilbert-styled Deduction (and its Frege-like analogs).
+
+
+
+
+% \bigskip
+% 
+% 
+% 
+% {\bf Mathematics Subject Classification:}
+% 03B52; 03F25; 03F45; 03H13 
+% 
+% 
+% 
+% \bigskip
+% \bigskip
+
+
+ 
+% {\bf Please Cite this Paper as:}
+% {\rm http://arxiv.org/abs/1108.6330}, 
+%  appearing in Cornell Archives 
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+%\begin{abstract}
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+\newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+\def\gvxs{ }
+
+
+\def\gvxs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+\def\gvxs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \baselineskip = 1.0 \normalbaselineskip  \parskip 0pt}
+
+\def\gvxs{ \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 7pt}
+\def\gvxs{ \large \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 6pt}
+
+
+\def\gvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvxs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvxs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+
+\noindent
+
+
+\newpage
+
+\def\gvxs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvxs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvxs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvxs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvxs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvxs{ }
+
+
+\gvxs
+
+\footnotesize
+
+
+\def\gvxs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvxs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvxs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvxs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvxs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvxs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvxs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvxs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvxs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+ \def\gvxs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvxs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+
+
+ \def\gvxs{ \Large \baselineskip = 2.15 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \Large \baselineskip = 2.18 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \Large \baselineskip = 2.11 \normalbaselineskip  \parskip - 2pt}
+\def\hvxs{ \Large \baselineskip = 2.18 \normalbaselineskip  \parskip 3pt}
+
+
+
+
+
+
+
+
+
+
+ \def\gvxs{ \large \baselineskip = 2.75 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \large \baselineskip = 2.78 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \large \baselineskip = 2.71 \normalbaselineskip  \parskip - 2pt}
+\def\hvxs{ \large \baselineskip = 2.78 \normalbaselineskip  \parskip 3pt}
+
+
+ \def\fvxs{ \normalsize \baselineskip = 1,46 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 1,47 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 1,47 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \normalsize \baselineskip = 1,48 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \normalsize \baselineskip = 1,41 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\hvxs{ \normalsize \baselineskip = 1,48 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 1pt}
+
+
+
+ \def\fvxs{ \normalsize \baselineskip = 1,66 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 1,67 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 1,67 \normalbaselineskip  \parskip 2pt}
+ \def\lvxs{ \normalsize \baselineskip = 1,68 \normalbaselineskip  \parskip 2pt}
+ \def\svxs{ \normalsize \baselineskip = 1,61 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\hvxs{ \normalsize \baselineskip = 1,68 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.6 \normalbaselineskip  \parskip 1pt}
+
+
+
+ \def\fvxs{ \normalsize \baselineskip = 0.96 \normalbaselineskip  \parskip 2pt}
+ \def\fvxs{ \normalsize \baselineskip = 0.97 \normalbaselineskip  \parskip 2pt}
+ \def\gvxs{ \normalsize \baselineskip = 0.97 \normalbaselineskip  \parskip 2pt}
+  \def\svxs{ \normalsize \baselineskip = 0.91 \normalbaselineskip  \parskip - 2pt}
+ % \def\svxs{ }
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 2pt}
+
+\def\hvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.35 \normalbaselineskip  \parskip 2pt}
+
+
+ % \def\svxs{ }
+
+%moving
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\lvxs{ \normalsize \baselineskip = 0.99 \normalbaselineskip  \parskip 2pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+
+\def\rvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\lvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\hvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 2.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 2.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 2.01 \normalbaselineskip  \parskip 2pt}
+
+%moving
+\def\rvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip
+  3pt}
+\def\lvxs{ \normalsize \baselineskip = 0.99 \normalbaselineskip  \parskip 2pt}
+\def\hvxs{ \normalsize \baselineskip = 0.98 \normalbaselineskip  \parskip 3pt}
+\def\bvxs{ \normalsize \baselineskip = 1.0 \normalbaselineskip  \parskip 3pt}
+\def\nvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+\def\tvxs{ \normalsize \baselineskip = 1.01 \normalbaselineskip  \parskip 2pt}
+
+\def\tempvxs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 3pt}
+
+%\def\tempvxs{ }
+
+\lvxs
+\nvxs
+
+%%% IGNORE BELOW:
+
+%% NOTE TO ME: Page 5 is cited in paper. needs to  BE UPDATED
+%%% AND prehaps GO BACK OLD FORM.
+
+
+%%% iiiii
+\section{Introduction}
+
+\label{ss1}
+\label{ss2}
+
+\vspace*{- 0.5 em}
+
+We have published a series of articles about generalizations
+and boundary-case exceptions for the Second Incompleteness Theorem
+in \cite{ww93}-\cite{ww14}. One theme of this literature was that
+such boundary-case exceptions will arise when multiplication
+is treated as a 3-way relation by a system which verifies its own consistency
+%in a 
+under
+semantic tableaux deduction.
+% instead of Hilbert deduction 
+%%%%%%% context.
+A 15-page summary of
+this research  appeared 
+in
+\cite{ww14}, but the latter is 
+not
+% unnecessary to examine  as 
+a prerequisite for reading this paper.
+
+
+%%  our prior research 
+%% about this topic
+%% was provided in our
+%% Wollic-2014 paper \cite{ww14}, but it is unnecessary for a reader
+%% to examine \cite{ww14} as a prerequisite for this paper.
+
+\smallskip
+
+The main shortcoming in our prior research was that our formalisms
+were mostly unable to recognize their own consistency under 
+Hilbert-style deductive methods. This was because a version
+of the Second Incompleteness Theorem, 
+due to the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87}, indicated the mere assumption that the successor
+operation was a total function was sufficient to trigger off the power
+of the Second Incompleteness Theorem
+for logics that verified their own consistency relative to Hilbert
+deduction. In particular, only our paper \cite{wwapal}
+% (which was 
+%by coincidence invited by editor Sergey Artemov to appear in APAL)
+was 
+% exclusively 
+devoted to addressing this problem.
+Its ISCE formalism could verify its own Hilbert consistency, but it
+unfortunately required deploying an infinite number of separate constant
+symbols because the presence of even a successor function symbol
+would trigger off the Second Incompleteness Effect for systems corroborating
+their own Hilbert consistency.
+
+The current article will introduce a modification of \cite{wwapal}'s
+ISCE formalism, called IQFS, that 
+addresses this
+% should help partially resolve this
+challenge. It will introduce a 
+% new
+function symbol, called the ``$~\theta~$''
+primitive, that will enable an arithmetic to construct the infinite 
+collection of 
+% positive 
+integers, 
+{\it without using} counterparts of any of the
+%  conventional growth 
+% function 
+% primitives of 
+successor, addition or multiplication
+function-mappings.
+
+\medskip
+
+It is almost certain that \cite{wwapal}'s proof of ISCE's
+consistency preservation property will generalize for the
+current article's IQFS formalism. 
+We will provide no proof supporting our  conjecture in this
+extended abstract, but the intuition 
+behind it
+% supporting our conjecture 
+will become quite evident. 
+Its gist will be that our proposed
+new $~\theta~$
+primitive  
+%will be 
+should likely be
+germane to 
+%{\it some special} 
+{\it some unusual} 
+axiom systems owning a
+{\it small 
+% quite
+fragmentized}  knowledge about their own consistency.
+
+
+%% 
+%% The proof of 
+%% IQFS's self-justifying properties will, 
+%% however, be much longer
+%% than
+%% %
+%%  \cite{wwapal}'s proof of its analogous Theorem 3
+%% (if the latter result is proven with the same level of detail
+%% and
+%% %  100 \% 
+%% rigor that Theorem 3's proof did receive).
+%% We will sketch in this short conference abstract the intuitive
+%% reason why IQFS should satisfy 
+%% an analog of ISCE's consistency preservation property, but
+%% % provide 
+%% no formal proof
+%% will be provided.
+
+% a formal consistency preservation property, exactly analogous to ISCE.
+
+%%  I do
+%% not want to write up such a proof in a context where I am currently
+%% suffering from Diabetes and Hypertension. It is essentially
+%% 99 \%
+%% certain that this article's proposed new 
+%% $~\theta~$ operator should support our conjecture.
+%% The purpose of this article will be to invite the
+%% research community to investigate how one can 
+%% expand the mini-formalism from 
+%% \cite{wwapal}'s Theorem 3 to rigorously prove our
+%% stated conjecture.
+
+% 
+% \section{More Detailed Description of Goals}
+
+%\label{ss2}
+
+
+% This article will define a new
+%  ``$~\theta~$'' 
+% function symbol
+% that should enable unusual logics
+% %% 
+% %% % 5-10 \% 
+% %% boundary-case
+% %% effect where a 
+% %% system can own
+% %% 
+% to own a
+% {\it diluted but tangible} knowledge about 
+% their own
+% % its 
+% %own 
+% consistency.
+
+\bigskip
+
+
+%% {\bf More Detailed Description of Goals}
+%% %
+%% % It is  known the
+%% %  
+%% G\"{o}del's
+
+%More precisely, 
+
+During this article, we will often note
+the 
+Second Incompleteness Theorem 
+was
+%known to be
+ published in two 
+% quite 
+different 
+ forms during
+1931-1939.
+Its initial 1931 variant, formalized by Theorem XI
+in  G\"{o}del's
+% millineal 
+paper \cite{Go31},
+% 
+% 
+% Its Theorem XI,
+% later known as the ``Second  Incompleteness
+% Theorem'',
+% %,
+% %appearing in G\"{o}del's millennial paper \cite{Go31}. 
+% 
+demonstrated
+% that
+no extension
+of
+% axiom systems,
+% roughly corresponding to
+the
+%  Russell-Whitehead 
+Principia Mathematicae formalism
+% $\, P \,$
+could
+% could
+ verify
+its own consistency.
+The widely quoted more general
+result, that 
+every consistent r.e. 
+extension
+ of Peano Arithmetic must
+be unable to prove a theorem affirming its
+own consistency, 
+was 
+first
+published 
+%% 
+%% (see \footnote{ Boolos states in \cite{Bool}
+%% that it has been open to scholarly debate
+%% whether or not the 1939
+%% Hilbert-Bernays generalization of the Second Incompleteness Theorem
+%% is or (is not) a straightforward generalization of
+%% G\"{o}del's initial result} ) 
+%% 
+in the 1939 edition of
+the Hilbert-Bernays
+textbook \cite{HB39}.
+
+% It has been considered
+% to be the definitive demonstration of the broad reach of
+% the Second Incompleteness Effect.
+
+%%  
+%% It also established, beyond any reasonable doubt, that any type
+%% of formalism possessing a conventional knowledge of its own consistency,
+%% must rely upon a 
+%% foundational structure
+%%  fundamentally different from Peano Arithmetic. 
+%% (This is because the 
+%% Hilbert-Bernays 
+%% textbook formalized the forerunner of
+%% what has now been known as the 
+%% Hilbert-Bernays Derivability Conditions \cite{HB39,HP91,Lo55,Mend},
+%% as a mechanism for
+%% % foreseeing 
+%% envisioning
+%% the 
+%% astonishing
+%% broad generality of the
+%% Second Incompleteness Effect.) 
+ 
+
+It is, thus, fascinating that Hilbert,
+as the co-author of 
+% an important 
+a
+%very 
+% historic
+generalization of the Second Incompleteness Theorem,
+never withdrew the
+% chose to  never fully withdraw his 
+1926 justification
+ \cite{Hil26}
+for his consistency program:
+\begin{quote}
+\small
+\baselineskip = 0.9 \normalbaselineskip 
+$*~$~ 
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding''}
+% itself.''}
+\end{quote}
+
+
+%% Indeed, 
+%% %Instead,
+%% Hilbert 
+%% always insisted  some
+%% new formalism would revive his consistency program
+%% and had its
+%% motto
+%% ({\it ``Wir m\"{u}ssen wissen,  $~$Wir werden wissen''} )  
+%% engraved on his tombstone.
+
+
+% Moreover, it 
+% 
+It 
+is also 
+known
+\cite{Da97,Go5,Yo5}
+that G\"{o}del
+was
+% also
+doubtful about the generality of the Second Incompleteness
+Theorem for at least two years after its publication.
+He thus inserted the following
+cautious caveat into
+his famous
+1931 
+% millennial
+paper  \cite{Go31}:
+% whose closing section 
+%%% 
+%%% One of the closing paragraphs of
+%%%  \cite{Go31} 
+%%% thus
+%%% included
+%
+%
+%%% contained the following cautious disclaimer:
+%caveat:
+% \newpage
+\begin{quote}
+\small
+\baselineskip = 0.9 \normalbaselineskip 
+\it
+$~**~~$ 
+``It must be
+% expressly 
+noted that
+Theorem XI
+%'s incompleteness result
+(e.g. the Second Incompleteness Theorem) 
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and  there might
+conceivably be ... ''
+%  finite proofs} which cannot
+% be stated in P or in ...''
+\end{quote}
+The 
+% above 1931
+statement $**$ has 
+had
+%been subject to 
+numerous
+%many
+different
+interpretations
+ \footnote{
+ Some 
+ scholars
+ have interpreted
+ $\,**\,$
+ as
+ %as, possibly,'
+ anticipating
+ % 
+ %  attempts
+ % to confirm Peano Arithmetic's consistency,
+ % via 
+ % either
+ % 
+ Gentzen's formalism or 
+  G\"{o}del's Dialetica interpretation.}.
+All
+ G\"{o}del's 
+biographers
+\cite{Da97,Go5,Yo5}
+%%%have
+have noted
+% 
+% 
+% 
+% (with
+% some 
+% scholars
+% viewing it as germane to
+% Gentzen's formalism or 
+%  G\"{o}del's Dialetica interpretation).
+% 
+% \gvxs
+% \nvxs
+% \bvxs
+% %\parskip 1pt
+% 
+% \noindent
+% The 
+% % above 1931
+% statement $**$ has 
+% had
+% %been subject to 
+% numerous
+% %many
+% different
+% interpretations
+% (with
+% some 
+% scholars
+% viewing it as germane to
+% Gentzen's formalism or 
+%  G\"{o}del's Dialetica interpretation).
+% % 
+% % \footnote{
+% % Some 
+% % scholars
+% % have interpreted
+% % $\,**\,$
+% % as
+% % %as, possibly,'
+% % anticipating
+% % % 
+% % %  attempts
+% % % to confirm Peano Arithmetic's consistency,
+% % % via 
+% % % either
+% % % 
+% % Gentzen's formalism or 
+% %  G\"{o}del's Dialetica interpretation.}.
+% % 
+% All
+%  G\"{o}del's 
+% biographers
+% \cite{Da97,Go5,Yo5}
+% %%%have
+% noted
+% % 
+% % 
+% 
+% 
+%
+his
+% % that G\"{o}del's
+initial intention
+was
+to
+establish
+%achieve
+Hilbert's proposed objectives, before 
+%he proved
+proving
+%proving
+% G\"{o}del proved
+a result 
+% 
+% however,
+% %%%%%his
+% G\"{o}del
+% did  originally 
+% seek 
+% % goal was 
+% to
+% establish
+% %achieve
+% Hilbert's proposed objectives before 
+% proving
+% % G\"{o}del proved
+% a result 
+% 
+leading
+%that led 
+in the opposite direction.
+Yourgrau \cite{Yo5}
+records
+%furthermore,
+ how
+von Neumann 
+% surprisingly
+%did
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive 
+%%%  achievement of a'
+ termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+It is known 
+ G\"{o}del 
+began to 
+more fully
+endorse 
+the Second Incompleteness
+Theorem
+during a 1933
+%% Vienna
+lecture  \cite{Go33},
+and he 
+% told biographers he
+strongly
+%completely 
+% fully
+embraced it
+after learning about Turing's work
+\cite{Tur36}.
+
+\smallskip
+
+Our research
+in \cite{ww93}-\cite{ww14}
+%has been 
+has been
+% is
+related to issues 
+%analogous
+ similar 
+to those
+%that were 
+raised by Hilbert and 
+G\"{o}del
+in
+ statements  $*$ and $**$.
+This is because it is counter-intuitive and awkward to
+%presume that 
+explain how
+human beings can maintain the
+psychological drive and 
+needed energy-desire to cogitate, without
+being stimulated by 
+{\it some type (?)} of instinctive faith in their own
+consistency. The new ``$~\theta~$'' primitive, introduced in the 
+current article, will
+further
+ reinforce this perspective.
+
+%  (under a definition of 
+% % formal  consistency
+% such
+% % this concept 
+% that is suitably 
+% gentle and
+% % delicate
+% soft
+%  to 
+% be consistent with the 
+% Incompleteness Theorem's requirements).
+
+% preclude a violation of the 
+% restrictions imposed by the Incompleteness Theorem).
+
+
+\smallskip
+
+We emphasize 
+ that 
+%our current 
+the present
+paper will differ from
+all our prior research (except for \cite{wwapal}'s
+trial-balloon result) 
+{\it by changing the focus from
+semantic tableaux deduction to a Hilbert-style
+deductive methodology.}
+
+\smallskip
+
+%\parskip 0pt
+
+
+%% 
+%% Accordingly, our research in
+%% \cite{ww93}-\cite{ww14} 
+%% has explored both generalizations and
+%% boundary-case exceptions of the Incompleteness Effect, so as
+%% to determine what type of boundary-case evasions are permitted.
+%% Our prior research in \cite{ww93}-\cite{ww14} 
+%% had used mostly cut-free forms of deduction to
+%% evade the 
+%% restrictions imposed by the
+%% Second Incompleteness Effect. The current article will instead
+%% focus on more pristine Hilbert-Frege methods of deduction.
+%% They are likely to support an evasion of the Second Incompleteness
+%% Effect when our axiom systems replace the traditional  
+%% growth properties of the addition, multiplication and successor 
+%% function symbols with our new $~\theta~$ primitive.
+%% 
+%% \smallskip
+%% 
+%% The motivation for this replacement will be
+%% explained during the next section of this article.
+%% It is needed
+%% essentially
+%% because
+%% a
+%%  version of the Second Incompleteness Theorem,
+%% due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% %will show 
+%% demonstrates
+%% that 
+%% if an axiom system $~\alpha~$
+%% proves any 
+%% of \eq{totxtefs} - \eq{totxtefm}'s totality statements
+%% then it is incapable of confirming its own consistency
+%% under a Hilbert-style deductive method.
+%% 
+%% 
+%% 
+%% 
+%% 
+%% \smallskip
+
+Our results will suggest  it is possible to obtain a
+{\it part-way 5-10 \%
+positive} interpretation
+for what
+Hilbert and G\"{o}del 
+% advanced
+ were seeking 
+% a Consistency Program
+to establish
+%%  seeking to accomplish
+% contemplating 
+in their 
+statements
+$*$ and $**$, within a context where 
+it is  known that the 
+Second Incompleteness Effect precludes a full achievement of
+%these 
+Hilbert's
+objectives
+from ever transpiring.
+The last five minutes of
+a 
+recent
+ 60-minute YouTube 
+presentation
+ by Harvey Friedman
+\cite{Fr14},
+entitled 
+% the 
+{\it ``The Blessing and Curse of Kurt  G\"{o}del''},
+suggested that it is
+%the themes of Hilbert and G\"{o}del's
+%remarks $*$ and $**$ by indicating that 
+%it is 
+interesting to explore futuristic partial 
+boundary-like evasions of the Second Incompleteness Theorem,
+despite the stunning strength of 
+G\"{o}del's result.
+It is within this context where our proposed use of a new
+$~\theta~$ primitive 
+symbol to replace the growth properties
+of the traditional addition, multiplication and successor function
+symbols may be of potential interest.
+
+\smallskip
+
+The development of our $~\theta~$ primitive 
+was 
+% partially 
+influenced by a private email
+communication 
+we
+% had 
+received
+from  
+Pavel Pudl\'{a}k \cite{Pupriv},
+as \textsection \ref{ss4}
+% \ref{ss3}  \& \ref{ss4} 
+%shall 
+will
+explain.
+We also emphasize that the 
+%conventional 
+usual
+interpretation of
+the
+% Second 
+Incompleteness Theorem, as precluding
+Hilbert's Consistency Program from
+% ever 
+achieving its initially
+specified objectives, is certainly correct.
+{\it Our only caveat} is that 
+some  
+{\it very  tiny} 5-10 \%
+% perhaps
+{\it fragmentized part} 
+of Hilbert's and G\"{o}del's aspirations in
+$*$ and $**$ ought to be viable.
+
+%fragment of its objectives ought to be viable.
+
+
+%% the latter should not lead  one
+%% to
+%% ignoring the role that a
+%% human's instinctive faith in his/her's internal
+%% consistency 
+%% %crucially stimulates and motivates
+%% plays in stimulating and motivating
+%% human cognition.
+
+%% 
+%% It is
+%% from this 
+%% special
+%% perspective where our prior research and
+%% new 
+%% results 
+%% %2-part conjecture 
+%% will
+%% % does 
+%% suggest that 
+%% an approximate
+%% %at least a
+%% 5-10 \%
+%% fragment 
+%% % fraction
+%% of what Hilbert and G\"{o}del 
+%% %suggested in 
+%% had
+%% sought
+%% in $*$ and $**$ 
+%% %could
+%% should be
+%%  plausibly 
+%% %is
+%% %% be formally 
+%%  feasible.
+%% 
+
+
+%\gvxs
+
+\vspace*{- 0.6 em}
+
+\section{Starting Perspective}
+%  222222}
+\label{ss3}
+
+\vspace*{- 0.6 em}
+
+%%! 
+%%! This article will be written in a style so that its
+%%! overall theme (if not full details)
+%%! should become 
+%%! {\it  quickly} comprehensible to a reader who has
+%%! examined 
+%%! only
+%%!  one of the
+%%! % introductory 
+%%! logic textbooks by say Enderton,
+%%! Fitting, H\'{a}jek-Pudl\'{a}k,
+%%!  Mendelson
+%%! or Papadimitriou \cite{End,Fi96,HP91,Mend,Papa}.
+%%! %% 
+%%! %% We will rely mostly upon the
+%%! %% precise
+%%! %% deductive calculi notation employed
+%%! %% in Section 2.4 of Enderton's textbook,
+%%! %% but any of 
+%%! %% the 
+%%! %% similar
+%%! %% Hilbert-style deductive calculi of 
+%%! %%  H\'{a}jek-Pudl\'{a}k,
+%%! %%  Mendelson
+%%! %% or Papadimitriou \cite{HP91,Mend,Papa}.
+%%! %% will also be suitable for achieving our results.
+%%! 
+%%! %% 
+%%! %%  
+%%! %% ( Papadimitriyou's textbook
+%%! %% generously states it employs a deductive notation
+%%! %% that 
+%%! %% has
+%%! %% stemmed from its predecessor in 
+%%! %% Enderton's textbook.)
+%%! 
+%%! 
+
+
+
+%% In order to make our 
+%% results
+%% %research 
+%% apply to the 
+%% formalism
+
+It is helpful to employ a flexible vocabulary so
+% that our
+our 
+results
+will apply 
+%research 
+%to the 
+%formalisms
+to any of the
+%% 
+%% accessible to
+%% some 
+%% readers who are acquainted with only one of the 
+%% 
+textbook 
+formalisms of
+% settings outlined by 
+%  
+say 
+Enderton,
+Fitting, H\'{a}jek-Pudl\'{a}k,
+or Mendelson
+\cite{End,Fit,HP91,Mend}.
+% 
+% ,
+% %% or
+% %%  Papadimitriou 
+% %% \cite{End,Fit,HP91,Mend,Papa},
+% %% 
+% %% 
+% %the widest
+% % possible  
+% %audience, 
+% it is 
+% helpful
+% % useful to 
+% use
+% %employ 
+% a
+% % very 
+% flexible
+%  vocabulary.
+% %%
+% %%that 
+% %%allows a reader to 
+% %%%quickly 
+% %%%translate results 
+% %%traverse
+% %%from
+% %%one textbook to another.
+% % Therefore, let us define a
+Let us 
+% thereby
+call an
+%\newpage
+%
+%\noindent
+ordered pair $(\alpha,d)$ a
+    {\bf ``Generalized Arithmetic''}
+% therefore
+iff its 
+% first and second 
+two components 
+%% 
+%%  Each of the 
+%% textbooks \cite{End,Fi96,Mend,Papa,HP96} have
+%% employed
+%% substantially
+%%  different variants of Basis and 
+%%  Deductive-Apparatus structures. 
+%% 
+are 
+% described
+formalized
+% defined 
+as below:
+%% 
+%% Their
+%% definitions in
+%%  Items (1) and (2)
+%% %simple 
+%% %%%definitions of these two notions
+%% %given below,
+%%     allow one to easily translate 
+%% %theorems 
+%% formalisms
+%% % methodologies
+%% from
+%% one textbook 
+%% % source
+%% to another:
+%% 
+%% %\njp
+%% % \newpage
+%% \parskip 2pt
+\bee
+\item
+The {\bf ``Axiom Basis''} $~\alpha~$ 
+for an arbitrary arithmetic
+shall be defined as
+the set of 
+{\it proper axioms} employed by the 
+formalism  $(  \alpha  , d  )$.
+\item
+An arithmetic's {\bf ``Deductive Apparatus''} $~d~$ 
+is defined as
+the 
+{\it combination} of its formal rules for inference
+and 
+%its
+the
+ built-in
+ logical axioms ``$~L_d~$''
+%  (that are 
+% implicitly 
+employed by these rules.
+\ene
+
+%%%\item
+%%%The term {\bf ``Deductive Apparatus''} $~d~$ will
+%%%refer to the 
+%%%{\it combination} of the rules of inference
+%%%used by an arithmetic and its
+%%%the logical axioms ``$~L_d~$'' that 
+%%%render meaning to
+%%%%are an automatic part of
+%%%$~d\,$'s machinery.
+
+
+\begin{exx}
+\label{ex-2.1}
+%\label{ex-basis}
+\rm
+This notation 
+allows one to
+ conveniently separate  the logical axioms
+$~L_d~,~$ associated
+with  $(  \alpha  , d  )~$, from 
+ $\, \alpha \,$'s
+ ``basis axioms''.
+%basis axioms
+It also allows one to  isolate
+and compare
+%  , conveniently, 
+various
+apparatus techniques,
+%technique,
+% employed in the exact formalisms 
+including the
+ $~d_E~$,  
+ $~d_M~$, 
+ $~d_H~$,  
+and  $~d_F~$
+methods
+%that we will now define:
+defined below:
+%% 
+%%  Three
+%% examples of this are illustrated below,
+%% in a context where
+%% are the deductive apparatus machineries defined
+%% in  Enderton's, Mendelson's and Fitting's textbooks
+%% \cite{End,Fi96,Mend}.
+%% 
+\bed
+\item[   i. ]
+The  $~d_E~$ apparatus,
+formalized in 
+\textsection
+ 2.4 of  Enderton's textbook, 
+% will
+uses only  modus ponens
+as a rule of inference.
+The latter will be accompanied
+by 
+a
+4-part 
+system of
+ logical axioms,
+called $~L_{d_E}~$, $\,$ to endow
+ $~d_E~$ 
+with an
+ability to  support
+% apparatus
+% agility so that it  supports
+%can satisfy
+%the analog of 
+G\"{o}del's Completeness Theorem.
+%% '
+%% (similar to  other'
+%% % full-scale '
+%% deductive methodologies).'
+%% 
+%%%% 
+%%%% (Papadimitriyou's
+%%%% % in-depth  exploration
+%%%% textbook \cite{Papa}  about
+%%%% % examination  of 
+%%%% the Logic-Computer interface
+%%%% relies
+%%%% explicitly 
+%%%%  upon
+%%%% % uses  
+%%%% Enderton's  
+%%%% % underlying 
+%%%% apparatus mechanism.)
+%%%% 
+%% %uses
+%% relies upon
+%%  Enderton's 
+%% approach   $d_E$.)  
+
+%% %relies 
+%% does rely
+%% upon 
+%% Enderton's apparatus
+\item[  ii. ]
+The  $~d_M~$ 
+apparatus in
+\textsection 2.3 
+of  Mendelson's textbook
+and  the $d_H$ 
+ apparatus
+in  \textsection 0.10
+of the H\'{a}jek-Pudl\'{a}k's
+ textbook
+employ a more compressed set of logical axioms
+than $\, d_E \,$,
+but
+they 
+instead
+use
+two rules of inference
+% (formalizing separately
+( modus ponens and generalization).
+%% plus a smaller set of logical axioms, which Mendelson
+%% has called A1-A5.
+%% Also, the $d_H$ 
+%%  apparatus 
+%% on pages ???? 
+%% of the 
+%%  H\'{a}jek-Pudl\'{a}k textbook
+%% uses a slightly different variation of a generalization. 
+%% (In the end, 
+In the end, 
+% both
+ $~d_M~$ 
+and  $~d_H~$ 
+prove the same
+% set of 
+theorems
+as   $~d_E~$ with 
+only
+%  minor and 
+unimportant changes in
+proof length.
+\item[ iii. ]
+The 
+``semantic tableaux''
+ $\,d_F \,$ 
+apparatus in
+Fitting's 
+and Smullyan's 
+textbooks 
+\cite{Fit,Smul}
+was
+ the 
+% main 
+focus of our 
+investigations in \cite{ww93,ww1,ww5,ww6,ww14}.
+It will be rarely used
+in the current article,
+however.
+Unlike
+ $~d_E~$,  $~d_M~$  and  $~d_H~$, it
+employs no logical axioms.
+It instead
+ uses a more complicated rule of inference.
+This tableaux apparatus 
+% and also Resolution, have  been 
+ has 
+% been found to  have many 
+many
+%a wide array of
+applications
+% underfor 
+in
+automated deduction,
+although it is 
+less efficient than
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+in
+% under
+% extremal
+worst-case 
+environments.
+% settings.
+%circumstances.
+\ennd
+\end{exx}
+
+\tvxs
+
+\begin{dff}
+\label{def-2.2}
+\rm
+Each of the
+% deductive 
+methods of
+ $  d_E  $,  $  d_M  $  and  $  d_H  $   
+have the property that if a theorem $\, \Psi \,$
+has a proof
+with  length $~L~$ 
+ from an arbitrary 
+axiom basis $~\alpha~$ 
+under one of these deductive systems,
+then it will have a proof from these other formalisms
+with lengths bounded by Polynomial$(L)$.
+The term 
+{\bf ``Hilbert-style''} deductive method will,
+thus, refer to any  deductive 
+% apparatus will refer to any other
+apparatus $~d~$ that
+has a modus ponens rule and
+employs
+% similarity has its 
+proof lengths
+% being 
+equivalent to within a polynomial magnitude
+to
+%of
+the comparable proof lengths from $d_E$,  $d_M$  and  $d_H$.
+\end{dff}
+
+
+%% and which 
+%% also
+%% assures 
+%% that the proofs of any
+%% two theorems $~\Phi~$ 
+%% and $~\Psi~$ 
+%% (under $d$ from any
+%% axiom basis $~\alpha~$) 
+%% will
+%% always 
+%% have
+%% % by more than a constant factor 
+%% the sum of the lengths of the proofs
+%% of $~\Phi \rightarrow \Psi ~$ and $~\Phi~$ 
+%% % under $~d~$ from $\alpha$ always 
+%% formally
+%% bound the length of
+%% $~\Psi\,$'s proof. 
+%% \end{dff}
+
+
+\begin{exx}
+\label{ex-2.3}
+%\label{ex-basis}
+\rm
+Some added notation is
+ needed to
+explain why
+% help outline
+% an important distinction between 
+a Hilbert style
+deductive apparatus, such as $\,d_E\,$,  $\,d_H\,$
+ or $\,d_M\,$, should be distinguished from
+ $d_F$'s
+``tableaux'' apparatus.
+Let
+% the symbols
+ $Add(x,y,z)$ and    $Mult(x,y,z)$
+once again 
+% will
+denote 
+two 
+3-way predicate symbols
+specifying
+that
+$x+y=z$ and
+$x*y=z$.
+Also, let us recall 
+that 
+an
+axiom basis
+ ``$\, \alpha \,$''
+is said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} if 
+%%%%%%%%% $\, \alpha \,$ 
+it
+includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems.
+
+% {\small
+{\vspace*{- 0.6 em}
+{
+\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+}
+
+\vspace*{- 1.4 em}
+
+\noindent
+%Then an 
+In this context, an
+``axiom basis''
+$\alpha$
+will be called 
+{\bf Type-M} if it contains
+\eq{totdefxs}-\eq{totdefxm}
+% \ref{totdefxs}-\ref{totdefxm}
+as theorems,  
+{\bf Type-A} if it contains 
+%only
+\eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+$\alpha$ 
+%will be 
+is
+called 
+{\bf Type-NS} if it  can prove
+none of these theorems.
+%In this context,
+%The
+%Items (a) and (b) illustrate 
+%%%Below are illustrated several
+%%%implications of this notation:
+The implications of this notation
+are formalized by Items a and b:
+
+%% 
+%% 
+%% , below, 
+%% %will 
+%% illustrate how
+%% a
+%% %% 
+%% %%  the
+%% %% prior
+%% %%  literature has
+%% %% 
+%% %% 
+%% ``Hilbert-style''
+%% deductive apparatus, such as $\,d_E\,$
+%%  or $\,d_M\,$, supports very different generalizations
+%% of the Second Incompleteness Theorem
+%% than  $\,d_F\,$'s
+%% ``tableaux-style'' apparatus:  
+%% 
+%%  the prior literature most germane
+%% to our current article is summarized as follows:
+%% 
+%% 
+%% The relationship of these constructs to 
+%% self-justification 
+%% is explained by
+%% items (a) and (b):
+\bed
+\item[   a.  ]
+The
+%%  
+%% above 
+%% evasions of the Second Incompleteness
+%% Theorem are known to be near-maximal in a mathematical sense.
+%% This is because
+%% the
+%% 
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as formalized by statement $\, ++ \,$,
+%has  implied  
+implies
+no
+natural 
+Type-S system can recognize  its
+own  consistency
+under
+any of $\, d_E \,$'s, $\, d_H \,$'s  
+or
+ $\, d_M \,$'s
+  Hilbert-style 
+versions
+of deduction:
+\begin{quote}
+{\bf ++ }
+\small
+% \footnotesize
+ \baselineskip = 0.9 \normalbaselineskip 
+{\it 
+(Solovay's  
+modification
+%Generalization 
+\cite{So94}
+%1994 Generalization \cite{So94}
+%of a 1985 theorem 
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+with
+%using 
+%some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let $ \, \alpha \, $ denote 
+%%! any consistent
+% a logic
+an axiom  
+basis  
+% axiom system 
+%basis  system 
+%supporting
+% which contains 
+able to prove
+Line
+\eq{totdefxs}'s
+Type-S statement
+and 
+assuring
+%which assures 
+%that 
+the successor operation
+%always 
+does satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+the identity
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then $~\alpha~$
+cannot verify its
+%will be unable to recognize its
+%own  
+consistency
+under any Hilbert-style 
+%deductive 
+apparatus $d$,
+%whenever
+if  it
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+the usual % identity,
+associative, commutative 
+ distributive 
+and identity axioms.
+%   -axiom
+% properties.
+\end{quote}
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of $++$'s result. 
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+by
+Buss-Ignjatovic,
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k's initial 1985 article  \cite{Pu85} 
+% implicitly
+did capture 
+essentially
+the 
+{\it great majority} 
+%most
+%%%  much 
+of $++$'s 
+% main
+% underlying
+formalism,
+and
+Friedman did 
+related work
+in
+\cite{Fr79a}.
+
+\item[   b.  ]
+Part of what makes
+% the Pudl\'{a}k-Solovay discovery in
+ $++$ interesting is that 
+\cite{ww93,ww1,ww5,wwapal}
+%Willard
+developed various 
+% separate
+methods for 
+basis systems
+%%% $\alpha$ 
+to confirm their own consistency, whose
+main
+further  improvements are 
+prohibited by either the invariant $++$ 
+or by
+\cite{ww2,ww7}'s hybridization of 
+$++$'s formalism with some further methods of 
+ Adamowicz-Zbierski
+\cite{AZ1}. (As a consequence of these facts, it is known
+that 
+some 
+Type-A 
+and Type-NS
+arithmetics can verify their
+respective
+ semantic tableaux and 
+ Hilbert-styled consistencies,
+but 
+Type-S arithmetics cannot verify their Hilbert consistency and
+most Type-M systems cannot verify their semantic tableaux
+consistency.)
+ \ennd
+\end{exx}
+
+% natural hybridizations is precluded by $++$. These results involve
+% either a Type-NS
+% % basis 
+% system 
+% 
+%  verifying its own consistency
+% under 
+% any  of the 
+%  $d_E$ or  $d_H$
+%  or $d_M$'s
+% Hilbert-style methods,
+% or   a Type-A 
+% %basis 
+% system \cite{ww93,ww1,ww5,ww6,ww14} 
+% verifying
+%  its
+% % own 
+% self-consistency
+% under $d_F$'s tableaux 
+% %deductive 
+% apparatus.
+% Also, Willard \cite{ww2,ww7} observed how one could
+% refine $++$ with Adamowicz-Zbierski's
+% methodology \cite{AZ1} to show 
+%  Type-M  systems
+% cannot recognize their semantic tableaux consistency.
+% \ennd
+% \end{exx}
+
+\lvxs
+\nvxs
+
+
+% \tempvxs
+
+
+%% A more detailed 15-page summary,
+%% % of our prior research, 
+%% germane to
+%% % the
+%%  Item (b),
+%% % , above, 
+%% %can be found in our article
+%% appears in
+%%   \cite{ww14}.
+
+% NNN NEED TO REWRITE NEXT TWO PARAGRAPHS
+
+
+A full 15-page summary
+% of our prior research, 
+of
+ Item (b)'s results
+% , above, 
+%can be found in our article
+can be found in
+  \cite{ww14}.
+It does not need
+ to be read,
+however,
+ as a prerequisite for understanding
+the current paper. This is because our goal
+%%%  in the current paper
+will be to explore axiom basis systems that can recognize their
+own 
+Hilbert-styled
+consistency, and the invariant $++$ indicates that each of the
+classes of Type-S, Type-A and Type-M 
+arithmetics
+are irrelevant to 
+this objective.
+%our goals.
+
+
+%% %% formalisms 
+%% own
+%% % contain 
+%% excessive
+%% growth properties that 
+%% lie outside 
+%% our goals.
+
+%this goal.
+
+%are incompatible with our goals.
+ 
+Instead, the \textsection \ref{seee3} will introduce a new
+growth function symbol, called the ``$~\theta~$'' primitive,
+that allows us to reside within the domain of a Type-NS arithmetic
+because {\it none of the  identities in
+Lines 
+ \eq{totdefxs}-\eq{totdefxm}} 
+will be provable consequences
+of $\theta$'s speedy but unconventional growth properties.
+
+This $\theta$ primitive will be attractive because
+Proposition \ref{th-3.3}
+and Remark \ref{rem-def-3.4} will imply  it supports
+respective $O(~$Log$^3 \, n \,)$ and $O(~$Log$ \, n \,)$
+growth speeds for constructing arbitrary integers $~n$
+(depending on what linguistic notation one uses for encoding integers).
+As a result,
+%%% of this fact,  
+\textsection \ref{ss32}
+ will conjecture that
+a seemingly minor ``IQFS'' modification of \cite{wwapal}'s
+ISCE formalism is an arithmetic that possesses some interesting abilities
+to confirm its own 
+% Hilbert-style 
+Hilbert 
+consistency.
+
+We might add that the discussion in this article will be
+{\it entirely self-contained}
+because 
+ \textsection \ref{ss32}
+ will summarize  \cite{wwapal}'s
+definition of the ISCE formalism.
+
+\begin{deff}
+\label{def-2.4}
+\rm
+Let
+$~\alpha~$ again 
+denote an axiom basis 
+and $~d~$ 
+designate
+ a
+deduction apparatus.
+%  
+%  During our discussion about the open questions
+%  raised by Hilbert's and 
+%   G\"{o}del's
+%  statements 
+%  $*$ and $**\,$,
+%  an 
+%  %
+%  % Then the  
+%  
+In this context, an
+ordered pair
+ $~(  \alpha  , d  )$
+will
+be called  {\bf Self Justifying} when:
+\begin{description}
+% \xxitch
+% \small
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+(or at least one of its axioms)
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+\end{deff}
+
+
+\begin{exx}
+\label{ex-2.5}
+ \baselineskip = 1.04 \normalbaselineskip 
+\rm
+% Using 
+% Definition \ref{def-2.4}'s
+%  notation, our research  
+% 
+Our research  
+in
+\cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+developed
+%\cite{ww93}-\cite{ww14}
+%has  consisted of
+% developing
+ordered pairs  $~(  \alpha  , d  )$
+that 
+were
+%are
+``Self Justifying''.
+It 
+%  has 
+also explored
+how the Second Incompleteness Theorem formalizes
+limits beyond which such formalisms cannot transgress.
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+% second
+%%% axiom 
+system $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition.
+%of 
+% this definition.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+%%%
+%%% the following 
+%%% %%% added
+%%% further
+%%% sentence,  
+%%% called
+%%% %%% that we call
+%%% {\bf SelfRef$(\alpha,d)~$}:
+\begin{quote}
+\small
+% \baselineskip = 0.95 \normalbaselineskip  
+%\xxitch
+$\oplus~~~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+discussed
+how
+to
+encode
+approximate
+ analogs of this
+{\bf $\,$``SelfRef$(\alpha,d)$''$\,$}
+statement.
+%%% SelfRef$(\alpha,d)$'s
+%%% self-referential statement.
+Both Kleene and 
+Rogers  \cite{Kl38,Ro67}
+% 
+% Each of
+% Kleene, 
+% Rogers and Jeroslow 
+%  \cite{Kl38,Ro67,Je71}
+% 
+ noted
+$\alpha ^d$ 
+may
+% , however,  
+be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii of   self-justification's
+definition.
+This is because if the 
+%  ordered
+ pair $(\alpha,d)$ is too strong
+then a classic G\"{o}del-style diagonalization argument can
+be applied to the axiom system 
+$\alpha^d~~=~~ \alpha \, + \, $ SelfRef$(\alpha,d)$,
+where the added presence of 
+% statement 
+SelfRef$(\alpha,d)$'s axiomatic   statement 
+will cause 
+%% this extended version of 
+$\, \alpha^d\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, the machinery of the sentence
+``SelfRef$(\alpha,d)$'' is relatively easy to 
+encode,
+%make well-defined 
+via an application of the Fixed Point Theorem,
+but it
+is
+ironically 
+%%%%%{\it most often  
+{\it 
+typically
+%usually
+useless! }
+\end{exx}
+
+%\newpage
+
+
+Unlike our earlier work, which focused 
+ mostly around a 
+semantic
+tableaux apparatus for deduction,
+the current paper
+will
+explore
+%paper will explore
+\dfx{def-2.2}'s
+more pristine Hilbert-style methodologies.
+%% 
+%% analogous to 
+%% Example
+%% \ref{ex-2.1}'s 
+%% textbook
+%% methods.
+%% 
+% of 
+% $d_E$,  $d_M$
+% and  $d_H$.
+%%! 
+%%! in
+%%! the textbooks by 
+%%!  Enderton,  H\'{a}jek-Pudl\'{a}k,
+%%! Mendelson and Papadamiriyou \cite{End,HP91,Mend,Papa}.
+There are, of course, many types of generalizations
+of the Second Incompleteness Theorem known to
+arise in  Hilbert-like  settings
+\cite{BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Lo55,Kr87,Kr95,Pa71,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,wwapal}.
+Each such
+generalization
+formalizes
+a paradigm where 
+self-justification is infeasible
+under a Hilbert-style apparatus.
+
+\smallskip
+
+Our 
+main
+prior research about
+%%! 
+%%! main work about arithmetics displaying knowledge
+%%! about their 
+%%! 
+Hilbert consistency appeared in \cite{wwapal}.
+Its ISCE$(\beta)$ 
+system could recognize its
+own 
+Hilbert consistency and 
+%%could 
+prove analogs of 
+the  $\Pi_1$ theorems of
+any 
+% r.e. 
+extension $~\beta$ of
+Peano Arithmetic.
+%  's $\Pi_1$ theorems.
+It unfortunately required
+% the 
+use of  infinitely many
+% number of 
+constant symbols, with
+ ISCE$(\beta)$ 
+using one built-in constant
+% symbol 
+$C_i$
+for each power of 2.
+(Such a series of constants seriously
+ deviated from 
+Hilbert's
+intended
+ goals
+in statement $*\,$, as
+we will explain during
+\textsection \ref{ss32}'s detailed review of
+\cite{wwapal}'s results.)
+
+
+%  An alternative in  \cite{wwapal},
+% called ISINF$(\beta)$, required use of only three constant symbols,
+% but its proof lengths were impractically long. 
+
+
+%%! required
+%%! % in excess of
+%%! an impractical
+%%!  $O(N)$ length proof to construct an integer $N$.
+
+\smallskip
+
+Prior to \cite{wwapal}'s publication, 
+Pavel Pudl\'{a}k
+\cite{Pupriv}
+examined this article and asked 
+%us
+% the crucial question about 
+whether
+we could improve 
+upon ISCE
+% 's properties
+ by using Ajtai's observations
+\cite{Aj94}
+ about
+the Pigeon Hole principle.
+%%  
+%% Sam Buss \cite{Bupriv} also asked us
+%% a 
+%% %similar 
+%% related
+%% question
+%% (during a more 
+%% informal 
+%% %abbreviated
+%% conversation).
+%% %(in a more informal manner).
+%% 
+Our prior
+partial
+answer to  Pudl\'{a}k's
+question
+was offered  
+%issue
+%appeared
+%in Sections 6 and 7 of \cite{wwapal}.
+in Sections 6 of \cite{wwapal}.
+A 
+% very 
+different type of reply will be offered
+% 
+% We will offer 
+% % an alternate much  
+% a 
+% % much
+% more sophisticated
+% and different 
+% type of reply
+% % analysis
+% %  formalism 
+% 
+in
+the current 
+paper.
+%article.
+
+
+%%! an abbreviated version of a similar
+%%! question after we verbally summarized to him \cite{wwapal}'s
+%%! planned results.
+
+% in 1997.
+
+
+\begin{deff}
+\label{def-2.6}
+\rm
+Throughout our discussion, a
+% A
+primitive $~F~$ will be called a 
+{\bf Q-Function} 
+iff is is 
+sufficiently ambiguous
+for there to exist an uncountably infinite set of
+% different 
+distinct
+{\it 
+plausible  sequences} of
+ordered pairs in expression \eq{wow} where
+$~F(i)=a_i~$  is allowed as a
+% logically 
+permissible
+%plausible 
+%formalization 
+representation of $F$
+under some fixed axiom system $~\gamma~$. 
+\begin{equation}
+\label{wow}
+  (0,a_0) 
+ ~,~  (1,a_1)   ~,~  (2,a_2)   ~,~  (3,a_3)   ~,~  (4,a_4)~ ...
+\end{equation} 
+\end{deff}
+
+% \gvxs2
+
+\vspace*{- 0.6 em} 
+%It turns out most
+
+%  
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question or most 
+% issues in
+% % other prominent
+% % % mathematical 
+% % questions within 
+% mathematics.
+
+Most
+Q-Function symbols are
+awkward to employ.
+ This is because the
+presence of an 
+ uncountably
+ infinite
+  number of
+different 
+plausible  sequences,
+formalized by Line
+\eq{wow} for solving
+$~F(i)=a_i~,~$  is 
+typically more of a burden than a benefit.
+A
+possible
+%potential
+ exception to this general rule
+of thumb
+ will be
+provided by
+ the next
+section's $~\theta~$ operator: It
+% because it
+ will,
+conveniently,
+ lie outside the scope of
+ $\, ++ \,$'s generalization of the Second Incompleteness
+Theorem.
+This fact will ultimately lead to our
+main  conjecture
+about stronger variants of Type-NS logics
+recognizing their own Hilbert consistency.
+
+
+
+% an enticing manner.
+
+
+%  It 
+% will
+% % should 
+% provide an
+% % enticing
+%  avenue for
+% Type-NS axiom systems to recognize their own
+% Hilbert consistency (if
+% \textsection \ref{ss5}'s 
+% ``IQFS''
+% %anticipated
+% conjecture is
+% correct). 
+
+
+%% 
+%% and suggest a mechanism whereby an efficient form of
+%% ``Type-NS''self-justifying 
+%% arithmetic
+%% can recognize its own Hilbert consistency,
+%% without
+%% viewing
+%% % recognizing 
+%% %%%%%% any of  addition, multiplication and
+%% even
+%% successor  as
+%% a total function.
+
+%In other words,
+% \smallskip
+
+
+%% 
+%% 
+%% \medskip
+%% Thus in a context where the partial drawbacks of
+%% our 
+%% new $~\theta~$ primitive
+%% will be beyond doubt, it will
+%% % simultaneously
+%% renew the
+%% serious
+%%  question about whether a  
+%% {\it 
+%% part-way 
+%% 5-10 \% fragment} of
+%% %positive} interpretation
+%% %can be assigned to
+%% Hilbert's and G\"{o}del's
+%% goals in $*$ and $**$
+%% can be acheived.
+
+
+%% 
+%% Our
+%% suggestion
+%% % conjecture
+%%  will be that
+%% Q-functions 
+%% might
+%% allow one to assign a
+%% {\it 
+%% % part-way 
+%% 5-10 \%
+%% positive} interpretation
+%% for what
+%% Hilbert and G\"{o}del 
+%% were
+%% seeking
+%% %referring to
+%% % a Consistency Program
+%% % to establish
+%% %%  seeking to accomplish
+%% % contemplating 
+%% in their 
+%% famous
+%% %often quoted
+%% statements
+%% $*$ and $**$.
+%% 
+
+%% 
+%% It will enable us to develop ground terms for formulating
+%% any integer $~N~$ using
+%%  $O\{~$Log$(N)~\}~$ 
+%% logical symbols,
+%% in a context where
+%% {\it  none of the} addition, multiplication or
+%% successor  function symbols are employed
+%% by $~\theta \,$'s analog of an 
+%%  $O\{~$Log$(N)~\}~$ 
+%% lengthed 
+%% binary-like 
+%% encoding 
+%% for integers.
+%% % of an integer as a binary number. 
+%% This  alternate
+%%  $O\{~$Log$(N)~\}~$ 
+%% format
+%% for encoding an integer $~N~$ is 
+%% potneitlally useful
+%% %fascinating
+%% because  
+%% Item $\, ++ \,$'s generalization of the Second Incompleteness
+%% Theorem, due to the
+%% combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+%% \cite{Ne86,Pu85,So94,WP87},
+%% does not preclude evasions of its invariant when a
+%% ``Type-NS''
+%% axiom system ceases to recognize 
+%%  addition, multiplication and
+%% successor  as total functions.
+
+
+ 
+% Most
+% Q-Function symbols are
+% unsuitable for 
+% analyzing
+% %producing a positive resolution to
+% Hilbert's  Second Open Question.
+% A special class of Q-Functions
+% %  , however,
+% will
+% walk through
+%   Cantor's 
+% % the
+% world of the Uncountably Infinite
+% %% in a more 
+% in an
+% enticing manner,
+% however.
+% %% however.
+% %It 
+% They
+% will suggest a
+% %%% much
+%  {\it  diluted but non-trivial}
+% variant
+% of the aspirations
+% which 
+% %that 
+% Hilbert and G\"{o}del 
+% expressed
+%  in
+% $*$ and $**$ 
+% are 
+% %applicable to
+% feasible under
+% % plausible in the context of 
+% Hilbert Deduction.
+% (This
+% % Q-function a
+% analysis will be 
+% %%%%%%%%%%%%%% quite
+% % entirely 
+% different
+% from 
+% \cite{ww14}'s 
+% examination of
+% %formalisms
+% %the formalisms appearing in our Wollic-2014 paper
+% %%% because 
+% %it will replace 
+% semantic tableaux deduction
+% because it will
+% apply  
+% uniquely 
+% to 
+% Definition \ref{def-2.2}'s 
+% ``Hilbert-styled'' deduction methods.)
+
+ 
+%%% is replaced by the more efficient
+%%% %with the more pristine 
+%%% Hilbert-style deductive methodology.)
+
+%can be achieved.
+
+
+%%! This will suggest
+%%! a {\it limited}  
+%%! and very-much {\it down-sized} version of the formalism that
+%%!  Hilbert
+%%! advocated
+%%! is
+%%! likely
+%%! %probably
+%%! feasible
+%%! and 
+%%! germane to 
+%%! the
+%%! future
+%%! %  computational 
+%%! needs of automated
+%%! theorem provers.
+%%! Our 
+%%! exploration 
+%%!  will also provide a
+%%! % quite
+%%!  new interpretation of the
+%%! meaning of the statements $*$, $**$ and
+%%!  $***$.
+
+% by Hilbert and G\"{o}del.
+
+
+
+% \section{Revisiting a World which Hilbert called
+% {\it ``Cantor's Paradise''}}
+
+%\section{Main Formalism}
+
+% \section{Deploying a New ``$~\theta~$'' Primitive}
+
+% \vspace*{- 0.9 em} 
+
+
+
+%\section{Need for a New ``$~\theta~$'' Primitive}
+
+
+%\section{Mysterious New ``$~\theta~$'' Primitive}
+
+
+% \section{The Surprisingly Useful ``$~\theta~$'' Primitive}
+
+%\section{The  ``$~\theta~$'' Primitive and Its Potential Uses}
+
+\vspace*{- 0.5 em} 
+\section{Arithmetic Under The  ``$~\theta~$'' Primitive}
+\label{ss4}
+\label{seee3}
+
+%333333333333333333333333333
+
+\vspace*{- 0.5 em} 
+
+%\vspace*{- 0.9 em} 
+
+
+% OLD Title was {\it Notation and Basic Concepts}
+
+% Throughout this 
+% paper,
+% %article,
+% % a 
+
+A function 
+ $\, H \, $ 
+will be     called
+a
+ {\bf Non-Growth} operation 
+iff  
+$ H(a_1,a_2,...a_j) 
+\leq   Maximum(a_1,a_2,...a_j)$
+for all  $a_1,a_2,...a_j$.  Six  examples of  
+ non-growth functions are:
+\bee
+%\small
+\footnotesize
+\parskip - 3pt
+ \baselineskip = 0.6 \normalbaselineskip 
+\item
+{\it Integer Subtraction} 
+where ``$~x-y~$'' is defined to equal zero 
+in {the special case} where
+ $~x \leq y,$
+\item
+{\it Integer 
+Division}
+where ``$~x \div y~$'' equals
+$~x~$ when $~y=0~$ and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+\item
+$Root(x,y)~$ which equals $~ \lfloor ~x^{1/y}~ \rfloor$ when $~y\geq 1~$
+%% 
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% 
+(and zero otherwise),
+\item
+$Maximum(x,y),~~$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+and
+\item
+$Count(x,j)~~=~~$the number of ``1'' bits
+among $~x$'s rightmost $~j~$ bits.
+\ene
+%% 
+%% 
+%% \bee
+%% \baselineskip = 0.8 \normalbaselineskip 
+%% 
+%% \item
+%% {\it Integer Subtraction} 
+%% where ``$~x-y~$'' is defined to equal zero 
+%% in {\it the special case} where
+%%  $~x \leq y,$
+%% \item
+%% {\it Integer 
+%% Division}
+%% where ``$~x \div y~$'' equals
+%% $~x~$ when $~y=0,~$ and
+%% it equals $~\lfloor ~x/y ~\rfloor~$ otherwise,
+%% \item
+%% $Root(x,y)~$ which equals $~ \ulcorner ~x^{1/y}~ \urcorner$ when $~y\geq 1,~$
+%% and
+%% it equals $~x~$ when $~y=0.$
+%% \item
+%% $Maximum(x,y),~~$
+%% \item
+%% $ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$ when $~x \geq 2,~$
+%% and  zero otherwise. 
+%% \item
+%% $Count(x,j)~~=~~$the number of ``1'' bits
+%% among $~x$'s rightmost $~j~$ bits.
+%% \end{enumerate}
+%% 
+These operations were called
+either the 
+{\bf ``Grounding''} 
+or {\bf ``Ground-Level''} 
+ functions
+in our articles \cite{ww1,ww5,ww14}.
+% We will use the
+We will rely upon the
+% The 
+latter nomenclature in the current article
+ because the notion of a  ``Ground-Level''
+function should not be confused with the
+% very 
+quite
+different notion
+of a ``Grounded Term'' (employed by Definition
+\ref{def-3.4}). 
+
+%  TWO DEFS or ONE ???????? 
+
+
+
+Our
+starting  language $L^G$ 
+will
+% shall 
+also contain
+the
+two atomic 
+symbols
+%% relations 
+of ``$~=~$'' and ``$~\leq~$'' and three
+built in constants symbols, $~C_0~$, $~C_1~$ and $~C_2~$, 
+for representing
+the values of 0, 1 and 2. 
+Within this context, 
+ Expressions 
+% Lines
+\eq{newadd} and \eq{newmult} formalize how addition and multiplication
+can be encoded as two 3-way predicates,
+%%  in  $L^G$,
+ denoted as
+Add$(x,y,z)$ and Mult$(x,y,z)$.
+% 
+% (Their
+% %  unusual
+% particular
+% definitions
+% are 
+% % quite
+% %highly
+% useful because they
+% allow our ``Type-NS'' arithmetic to evade
+% satisfying
+% %  Lines 
+%  \eq{totdefxs}-\eq{totdefxm}'s
+% forbidden
+% function-existence
+%  conditions.)
+% 
+%%%%undesirable constraints.)
+% 
+% they do not imply addition
+% and multiplication are total functions (e.g.
+% they permit our arithmetic to be a 
+% ``Type NS system''.) 
+% 
+%  further conditions.)
+% 
+% (These 
+% definitions
+% % for Add$(x,y,z)$ and Mult$(x,y,z)$
+% {\bf notably  allow} a
+% %two 3-way predicates are consistent with a 
+% ``Type NS system'' to
+% {\it evade satisfying} Lines  \eq{totdefxs}-\eq{totdefxm}
+% {\it forbidden}
+% constraints.) 
+% %  further conditions.)
+
+\newpage
+%bbbbbb
+{ \small
+\vspace*{- 1.2 em} 
+\beq 
+\label{newadd}
+z ~ -~x~~=~~ y~~~~ \wedge ~~~~ z~\geq~x
+\end{equation}
+
+\vspace*{- 1.2 em} 
+\begin{equation}
+\label{newmult}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation} }
+$~~~~$Also, 
+our constant symbols,
+ $~C_1~$ and $~C_2~,~$ for representing
+the quantities 1 and 2, 
+will  allow us to
+formalize the following further
+%useful 
+%%%%%%%% function 
+operations:
+\bed
+\small
+\topsep -4pt
+\itemsep -1pt
+\it
+\item[   $~$a.  ] 
+Pred$(x)~~=~~x-1$
+(in a context where 
+%Item 1's
+%the prior paragraph's
+our prior
+definition for Subtraction implies
+Pred(0)$=0$.$~)$   
+\item[   $~$b.  ] 
+Half$(x)~~=~~x \div 2$
+(in a context 
+where 
+``$ ~x \div 2~$'' equals technically 
+$~\lfloor \, \frac{x}{2} \, \rfloor~$ 
+under 
+%our
+% the prior paragraph's 
+our
+notation).
+% of Item 2's definition for Division).  
+\item[   $~$c.  ] 
+Pred$^n(x)$ defined to be $~n~$ iterations of the
+Predecessor operation
+\item[   $~$d.  ] 
+Half$^{ \,n}(x)$ defined to be $~n~$ iterations of the
+halving operation.
+%%
+%%\item[   $~$e.  ] 
+%%Bit$(x,j)~=~$ $Count(x,j)~-~Count(x,j-1)~$
+%%(e.g. the value of $~x \,$'s rightmost
+%% $j-$th bit)
+%%\item[   $~$f.  ] 
+%%Min$(x,y)~~=~~x~-~(y-x)~~$
+%%(e.g. a quantity that represents the minimum of $x$ and $y$
+%%because our definition of ``integer subtraction'' 
+%%implies  $~y-x \,= \, 0 ~$ whenever $\, x \geq y$.$~~)$  
+%%
+\ennd
+
+\nvxs
+
+Let us say
+that
+ a function symbol $H(x_1,x_2...x_j)$
+is
+ {\bf Growth Permitting}
+iff for each integer $~k \geq 2~$ there exists a 
+``growth-tuple''
+$(a_1,a_2...a_j)$ 
+satisfying
+  $~H(a_1,a_2...a_j) \, >\, k~$
+and 
+also
+% simultaneously 
+having
+each $ \, a_i \leq k$ \, 
+It will  be necessary 
+%% for  us 
+to employ
+ either an infinite number
+of constant symbols or some Growth-Permitting function 
+so that an extension of the 
+language $L^G$ can construct the 
+full
+%infinite
+collection of
+ integers of $~3,4,5,6~....~$.
+
+
+
+
+%% One  awkward aspect of this notation is that 
+%% it provides
+%%  no guarantee
+%% that    integers larger than 2 will exist         without the 
+%% presence of some 
+%% further
+%% methodology for producing larger integers.
+
+
+% 
+\smallskip
+
+
+One method for resolving this problem was presented in \cite{wwapal}.
+% 
+% It employed  an infinite number of further constant symbols. The
+% latter's
+%  ISCE$(\beta)$ 
+%  system was
+% %   shown to be 
+% compatible with                     self-justification,
+% but such an infinite number of constant symbols clearly trespassed on
+% Hilbert's goal of using a 
+% %strictly 
+% finite-sized formalism.
+% 
+Its  ISCE$(\beta)$ axiom basis
+deployed  an infinite number of 
+% further 
+distinct
+constant symbols. It
+was
+compatible with   self-justification,
+but deviated from 
+%{\it very sharply from} 
+Hilbert's
+intended
+ goals 
+because it employed
+% by employing 
+%an  
+a {\it highly awkward}
+infinite number of 
+distinct
+ constant symbols.
+(The reader will 
+better
+appreciate this point when
+\textsection \ref{ss32}
+reviews 
+% the
+% properties of 
+ISCE's defining formalism.
+This  difficulty is fundamental 
+% to avoid 
+because
+the Invariant $++$'s generalization of the
+Second Incompleteness Theorem indicates that
+Type-S arithmetics are unable to confirm their
+own Hilbert consistency.)
+
+
+
+\medskip
+
+
+
+%% The 
+%% % self-justifying
+%% ``ISINF'' formalism 
+%% % in
+%% of 
+%% \cite{wwapal} 
+%% offered an alternate method for resolving this difficulty.
+%% %% in the context of a self-justifying logic. 
+%% It
+%% % required the use of
+%% used
+%%  {\it only
+%% three} constant symbols.  It   could prove analogs of all
+%% of Peano Arithmetic's
+%% $\Pi_1$ theorems, but almost all
+%% of
+%%  its proofs
+%% unfortunately
+%%  had lengths longer 
+%% than the number of atoms in the universe.
+%% Most other approaches, for resolving this dilemma, 
+%% % are 
+%% were
+%% also problematic
+%% because
+%% Example \ref{ex-2.3}'s
+%% invariant
+%%  $~++~$
+%% % 
+%% % which Example \ref{ex-2.3}
+%% % attributed to the joint work of 
+%% %   Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris,
+%% % 
+%% showed that essentially every Type-S arithmetic is unable to
+%% recognize its
+%% own consistency under a Hilbert-style deductive apparatus.
+%% 
+%\smallskip
+
+
+% The challenge posed by $++$ 
+% is, thus,
+% substantial.
+% % certainly formidable.
+
+Our goal in 
+%the current 
+this
+article
+will be to suggest 
+how
+%that 
+a Q-function primitive $F(x)$,
+that has
+% an extraordinarily 
+a deliberately
+ambiguous
+ function definition,
+can help overcome the constraints that $++$ imposes. 
+Such an
+% ultra-ambiguous defined 
+unusual
+primitive $~F~$ will have
+an  uncountable number
+of vectors, analogous to Line \eq{wow}, that are permitted 
+solutions to
+$F$'s
+definition.
+Our basic goal
+% in this article 
+will be to outline 
+how this unusual concept is
+likely  germane to the
+self-justifying
+axiom system satisfying
+{\it  a very diluted} but not
+immaterial subset of
+%% 
+%% why it is
+%% likely such Q-functions will enable one to build
+%% surprisingly efficient self-justifying logics that
+%% partially (but not fully) achieve the 
+%% %{\it  diluted portion} of the
+%% 
+%aspirations  
+%that
+Hilbert's and G\"{o}del's goals in 
+%expressed in 
+statements 
+$*$ and
+$**~$.
+
+\smallskip
+
+%\normalsize \baselineskip = 0.98 \normalbaselineskip
+%\gvxs
+
+
+%% \el{wow}'s 
+%% dizzying
+%% $\aleph_1$ distinct solutions.
+%% We will traverse in the opposite direction in this article
+%% because 
+%% Definition \ref{def-2.6}
+%% %\eq{wow} 
+%% formalizes 
+%% % fascinating 
+%% %  a possible
+%% an 
+%% avenue
+%% available
+%% for evading $++$'s generalization of the 
+%% %Second 
+%% Incompleteness
+%% Theorem.
+
+% {\it in at least a partial sense.}
+
+
+%%  
+%% %They 
+%% This level of multiplicity will be
+%% %These will turn out to be 
+%% useful in the present setting
+%% because
+%% %% 
+%% %% Our hypothesis is that such
+%% %% solutions, while awkward and  disadvantageous
+%% %% in many 
+%% %% evident
+%% %% %%%%%obvious 
+%% %% respects,
+%% %% should not be discarded.
+%% %% This is 
+%% %%  because 
+%% %% 
+%% it
+%% can
+%% formalize a type of 
+%% allowed
+%% growth-permitting function,
+%% that is
+%% not prohibited by $++$'s generalization of the Second Incompleteness
+%% Theorem. 
+%% 
+%% 
+
+%\smallskip
+
+\begin{deff}
+\label{def-3.1}
+\rm
+Let us say 
+% an integer-valued
+a function symbol
+$F(x_1,x_2,...x_j)$ is {\bf `` 1-Definitive ''} iff it has only one
+solution under its definition by an axiom system $\gamma$. 
+Let us call $~F~$  {\bf ``Indeterminate''} otherwise.
+%(The remainder of this article 
+(Mathematicians
+obviously typically avoid using 
+function definitions
+%%%%%%%%%, $~F~$ 
+with 
+%that have 
+%%%%  having
+even two solutions, not to
+speak of
+what will be
+\el{wow}'s 
+surprising
+%dizzying
+quantity
+of
+%potential
+%possible
+%% possibly $\aleph_1$ distinct 
+ potentially infinitely many distinct 
+% number of 
+solutions.
+This is
+because such objects are typically more of a burden than a benefit.
+Our conjecture
+% , in this paper, 
+will be that a special new
+Indeterminate function, called ``$~\theta~$'', will 
+be different and
+offer
+% an
+%%%a surprisingly
+%%% efficient means to eschew $++$'s prohibitions.)
+a 
+%surprisingly efficient 
+means to eschew $++$'s 
+%prohibitions.)
+prohibitions with pleasing levels of
+% quantitative
+efficency.)
+% efficency.)
+
+
+\end{deff}
+
+
+% considered in the current article.
+% Our 
+% % main
+% conjecture will be that this unconventional
+% approach is 
+% germane to the challenge posed
+% by  $++$'s 
+% %
+% % broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem
+% %
+% %This is because our conjecture will be that
+% %
+% because the  new
+% proposed $~\theta~$ 
+% primitive
+%  will represent
+% an efficient Indeterminate 
+% function
+% that eschews  $++$'s prohibitions.) 
+% \end{deff}
+
+% helpful when addressing 
+% the challenge 
+% We will traverse in 
+% an unconventional
+% %  the opposite 
+% direction in this article
+% because
+% our conjecture will be that
+% Indeterminate functions 
+% with $\aleph_1$ different 
+% allowed solutions for
+% \el{wow} 
+% formalize
+% %%  
+% %% Definition \ref{def-3.1}
+% %% % {def-2.6}
+% %% %\eq{wow} 
+% %% will formalize 
+% %% a 
+% %% possible
+% %% %feasible
+% %% %plausible
+% %% 
+% an
+% avenue
+% % available
+% for evading $++$'s 
+% broad-scale
+% generalization of the 
+% %Second 
+% Incompleteness
+% Theorem.)
+% 
+% (Our conjecture in this article will be that
+% %%
+% %%The next two sections
+% %%will explore
+% %%how 
+% %%
+% \el{wow}'s indeterminate ``Q-Function'' symbol $F$
+% %%%can clarify
+% is germane to
+% the
+% aspirations 
+% that 
+% Hilbert and G\"{o}del
+% % expressed
+% stated
+%  in 
+% % statements 
+% $*$ and
+% $** \, $, $\,$when
+% one employs an 
+% operator
+% $ \, F \, $ 
+% that
+% owns
+% % has
+% $\aleph_1$ distinct 
+% available
+% solutions). 
+% \end{deff}
+
+
+%%are viable,
+%%especially in an automated theorem proving setting,
+%%when Ambiguous function operatives are judiciously
+%%employed.
+%%
+
+%%!  
+%%! can
+%%! %%and germane about automated
+%%! %%deduction, will 
+%%! % theorem proving,
+%%! %computational deduction,
+%%! be satisfied by a self-justifying logic that employs
+%%! one 
+%%! %single $\aleph_1$ 
+%%!   Growth-Permitting function $F(x)$,
+%%! that is
+%%! % inherently 
+%%! ``ambiguous'',
+%%!  {\it accompanied 
+%%! by} a finite number of non-growth primitives
+%%! % {\it  non-growth}
+%%!  $G_1,~G_1,~... G_k$
+%%! that are ``unambiguous''.
+%%! 
+%%! It will also be explained how such results should have useful 
+%%! applications for automated theorem proving, even when they
+%%! employ 
+%%! only
+%%! diluted forms of  self-justifying logics.
+
+%% 
+%% \vspace*{- 1.0 em}
+%% 
+%% \subsection{Main Notation Conventions}
+%% % about Cantor's Paradise}
+%% % \large
+%% % \baselineskip = 1.8 \normalbaselineskip 
+%% 
+%% %\vspace*{- 0.7 em}
+
+%\gvxs
+
+{\bf More Notation:}
+$~$Let us say
+an axiom system $~\alpha~$ 
+has 
+{\bf Infinite Far Reach} iff
+it relies upon
+{\it only a  finite number} of
+axioms to
+% distinct constant symbols
+% (and/or axiom sentences) to 
+prove 
+for each $n$
+the
+\el{farreach}'s invariant.
+
+%for each particular integer $n$.
+
+\vspace*{- 0.8 em} 
+
+\beq
+%% \small
+\label{farreach}
+\exists ~~x~~~ \mbox{Pred}^n(x)~\geq ~1
+\enq
+
+%\newpage
+
+
+\nvxs
+
+\noindent
+The ``ISINF'' axiomatic       framework 
+from
+   \cite{wwapal} 
+was
+    a self-justifying
+system with Infinite Far Reach.
+%% 
+%% The opening paragraph of
+%% \cite{wwapal}'s Section 6
+%% %%% quite 
+%% %was frank 
+%% warned the reader
+%% about ISINF's limitations. 
+%% These arose because
+%% 
+%% 
+%% It
+%% used the word ``unnatural'' to describe 
+%% the ISINF system.
+%% Such caution
+%% % deliberately
+%% % self-deprecating term 
+%% was appropriate because
+%% 
+Unfortunately, this result was mostly useless because
+nearly all
+   theorem-proofs
+%of trivial theorems0000000 
+from ISINF 
+were
+longer than the number of atoms in the universe.
+
+\newpage
+\parskip 0pt
+
+The reason
+\cite{wwapal} defined ISINF,  
+%ISINF was worthy of mention,
+despite 
+its evident impractical  characteristics,
+% 
+% such
+% %%% these
+% % plainly 
+% %%%  obvious 
+% limitations,
+% 
+was 
+because
+ISINF 
+demonstrated some
+unusual
+ self-justifying logics,
+knowledgeable about their own Hilbert consistency,
+were
+{\it technically} 
+able to
+prove all of Peano Arithmetic's $\Pi_1$ theorems
+together with the
+existence of
+the infinite set of integers $ \, 1,2,3,... \,  \, $.
+This result 
+% is interesting because it casts 
+did cast
+% casts 
+a
+new 
+perspective
+%light 
+on  $\,++\,$'s  
+invariant
+% $++$  (appearing on \pag2)
+by showing how
+{\it  some  unusual}
+Type-NS
+forms of self-justifying arithmetics
+did
+escape $++$'s almost-ubiquitous 
+ reach
+by managing to possess infinite far reach
+ without taking
+% {\it without recognizing} 
+Successor as a total function.
+
+
+%%  of the current article.
+%% The latter result indicated that Type-S arithmetics, recognizing merely
+%% Successor as a total function, are unable to confirm their own
+%% Hilbert consistency. 
+%% Yet,
+%% %%  despite this fact, 
+%% ISINF was able to produce an 
+%% {\it  eye-squinting} caveat because it 
+%% supported the above ``Infinite Far Reach'' property
+%% without 
+%% needing
+%% %being able to prove
+%%  Line
+%% \eq{totdefxs}'s declaration that successor is a total function.
+
+%\smallskip
+
+We sent an advanced copy of \cite{wwapal}
+to
+Pudl\'{a}k.
+He
+appreciated the nature of the challenge we faced.
+% 
+% ,
+% concerning the delicate nature of self-justifying 
+% arithmetics that are
+% able to prove
+% % satisfy 
+% \eq{farreach}'s invariant
+% {\it for each fixed $~n~$} while 
+% being prohibited 
+% by $++$
+% from
+% recognizing  successor as a total
+% function.
+% % (due to $++$'s restrictions).
+% 
+% 
+Pudl\'{a}k's
+subsequent  
+%private 
+%His
+emailed communications
+\cite{Pupriv}
+suggested 
+that we look at 
+Ajtai's
+work 
+\cite{Aj94}
+about a
+%the 
+Pigeon-Hole function
+ $~ \glamb(x)~$ defined by the identities
+\eq{zm1} and \eq{zm2}.
+
+% \newpage
+\vspace*{- 1.2 em} 
+\beq
+%% \small
+\label{zm1}
+\forall ~~x~~~~~ \glamb(x)~ \neq ~ 0
+\enq
+
+\vspace*{- 1.2 em} 
+
+\beq
+\label{zm2}
+%% \small
+\forall ~~x~~~ \forall ~~y~~~~ x ~ \neq~ y ~~ \Rightarrow ~~
+\glamb(x)~ \neq ~\glamb(y)
+\enq
+The relevance of 
+$~\glamb~$ 
+% Pigeon-Hole functions
+can be
+best 
+%readily 
+appreciated
+% if 
+when
+%we let
+$~\glamb^n(x)~$ 
+ denotes
+% the 
+a
+term
+ $~\glamb(~\glamb(~ ... \glamb(x)))~$
+consisting of $~n~$ iterations of the $~\glamb~$ operator.
+Then the below
+% the
+%%% \el{DUMB1}'s composite
+term $~S_n~$
+% , defined below, shall
+will
+% then 
+satisfy
+Pred$^n(~S_n~)~\geq ~1.~$ 
+%% 
+%% An axiom system, employing the primitive 
+%% operation
+%%  $~ \glamb~,~$ 
+%% can thus 
+%% can easily
+%% prove
+%% Line \eq{farreach}'s 
+%% assertion.
+%% %claim.
+%% %under almost all conventional logics.
+%% 
+%% 
+\beq
+% \vspace*{- 0.5 em}
+\label{DUMB1}
+S_n~~~=~~~\mbox{Max}[~\glamb(0)~,~\glamb^2(0)~,~\glamb^3(0)~,~...~~\glamb^n(0)~]
+\enq
+Pudl\'{a}k
+observed
+that
+%the
+% Pigeon-Hole function 
+ $~ \glamb(x)~$  
+will
+grow too slowly
+(under well-defined non-standard models)
+% (in the worst case)
+for
+one to be able to 
+deduce
+successor is a total function
+from its properties.
+%%   
+%% % further observed that it is known 
+%%  \footnote{
+%% \tiny
+%%  \baselineskip = 0.94 \normalbaselineskip
+%%  The operation $\glamb(x)$ will grow
+%% at a slower rate than Successor, 
+%% if it equals $x+1$ for all standard
+%% numbers $~x~$ and if $\glamb(x)=x-1$ 
+%%  when $~x~$ is
+%% a non-standard integer. This seemingly minute detail
+%% implies  one cannot infer 
+%% Successor is a total function from
+%%  $\glamb$'s behavior.}.
+%% 
+%% 
+%% 
+%%  since the latter is contradicted by a
+%%  model  where
+%%  all  non-standard
+%% numbers have 
+%% %their 
+%% sizes bounded by some fixed  
+%% % non-standard 
+%% number B.
+%% (This 
+%% subtle 
+%% %detail, 
+%% raised by
+%% Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+%% it 
+%% %shows that
+%% raised the question about whether
+%%  a partial exception to
+%% Example \ref{ex-2.3}'s
+%% invariant $++$
+%% %% on \pag2,  
+%% might plausibly exist.)  }.
+%% 
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+%%% 
+%%% 
+%%% 
+%%%  (in the worst case)
+%%% for
+%%% one to be able to 
+%%% deduce
+%%% successor is a total function
+%%% from its properties  
+%%% % further observed that it is known 
+%%%  \footnote{
+%%% \tiny
+%%%  \baselineskip = 0.94 \normalbaselineskip
+%%%  The operation $\glamb(x)$ will grow
+%%% at a slower rate than Successor, 
+%%% if it equals $x+1$ for all standard
+%%% numbers $~x~$ and if $\glamb(x)=x-1$ 
+%%%  when $~x~$ is
+%%% a non-standard integer. This seemingly minute detail
+%%% implies  one cannot infer 
+%%% Successor is a total function from
+%%%  $\glamb$'s behavior.}.
+%% 
+%% 
+%%  since the latter is contradicted by a
+%%  model  where
+%%  all  non-standard
+%% numbers have 
+%% %their 
+%% sizes bounded by some fixed  
+%% % non-standard 
+%% number B.
+%% (This 
+%% subtle 
+%% %detail, 
+%% raised by
+%% Pudl\'{a}k's email \cite{Pupriv}, was fascinating because
+%% it 
+%% %shows that
+%% raised the question about whether
+%%  a partial exception to
+%% Example \ref{ex-2.3}'s
+%% invariant $++$
+%% %% on \pag2,  
+%% might plausibly exist.)  }.
+%% 
+%% 
+%% thus, 
+%% suggests the 
+%% Pudl\'{a}k-Solovay 
+%% version of the Second Incompleteness
+%% Theorem (stated on \pag2) 
+%% might
+%% %%%%%should
+%% allow for
+%% potential
+%%  exceptions 
+%% to it
+%% arising from the 
+%% %delicate 
+%% formal
+%% behaviour of
+%% some
+%% %% 
+%% %% presence of
+%% %% %some
+%% %% these permissible
+%% %% 
+%% %% 
+%%  non-standard 
+%% variants of
+%% % interpretations for 
+%% the  Pigeon-Hole function $\glamb$.  }.
+%% 
+%% 
+%that 
+%prove 
+His insightful email \cite{Pupriv} asked
+whether 
+the inequality 
+Pred$^n(~S_n~)~\geq ~1~$
+might
+%would,
+thus, 
+% still
+enable a formalism,
+% based around
+utilizing the
+ $\, \glamb \,$ operative, 
+to 
+somehow
+improve upon \cite{wwapal}'s results ?
+
+
+% our
+% formalisms could be 
+% revised
+% %modified 
+% so that 
+% % the  Pigeon-Hole function 
+%  $~ \glamb(x)~$ 
+%  could improve upon \cite{wwapal}'s results.
+
+%%
+%%(possibly using Ajtai's methodologies \cite{Aj-focs}).
+%%Sam Buss raised, interestingly,  a 
+%%partially
+%%similar 
+%%issue during an informal conversation
+%% \cite{Bu-priv} in 1977.
+%%
+%%\smallskip
+%%
+%%These questions
+%%% by
+%%%Pudl\'{a}k and Buss 
+%%were insightful because they isolated 
+%%an
+%%important juncture where $++$'s underlying methodology does not apply.
+%%A partial answer to these questions appeared in 
+%%\cite{wwapal}'s closing section, but a more comprehensive full
+%%answer  has always eluded us.
+
+%This is  because there always seemed to appear
+%one wrinkle of details that precluded a full proof.
+
+
+\smallskip
+
+
+It  was
+initially
+ unclear 
+%%%%% to us
+whether a positive answer to 
+Pudl\'{a}k's
+ probing
+ question would resolve ISINF's main difficulties.
+This is
+because
+% Expression 
+\eq{DUMB1}'s
+term
+$~S_n~$  requires $O(~n^2~)$ logic symbols to encode
+% essentially 
+an integer quantity
+greater than
+ $~n~$
+(since its term
+$~\glamb^j(0)~$ uses $O(j)$ logic symbols).
+%an integer quantity that exceeds the quantity $~n~$ in size.
+Thus once again, the quantity $~2^{100}~,~$ whose binary encoding
+requires 100 bits,  would require in excess of 
+ $~2^{100}~$ bits to encode. 
+Such large  quantities are obviously undesirable.
+
+
+% Such impractical quantities are obviously distant
+% from what is desired.
+
+
+% Such quantities, exceeding the number of atoms in the universe,
+% were troubling because our 
+% general
+% goal has been to 
+% construct self-justifying arithmetics
+% with  pragmatic features.
+
+
+%%  that 
+%%  possessed, at least,
+%% some
+%% partial  facets of
+%%  pragmatic value.
+
+
+% 
+% find a partial
+% answer to Hilbert's
+% Year-1900 Second
+% Problem  
+%  that would 
+%  possess, at least,
+% some
+% partial  facets of
+%  pragmatic value.
+% 
+
+\medskip
+\nvxs
+
+The remainder of this section will outline how a different type of
+Q-Function operator will 
+be
+%  much 
+better than
+ $~ \glamb~$ for meeting our needs.  
+During our discussion,
+Power$(x)$ will denote 
+a primitive specifying
+% that
+ $~x~$ is
+a power of
+$~2~$.  
+Its formal encoding
+in $L^G\,$'s language
+ is illustrated by \eq{wep2}. 
+%% 
+%% It is 
+%% %formally 
+%% encoded
+%% by 
+%% \eq{wep2} 
+%% because
+%% %under
+%% our Grounding language
+%% has
+%% ``Logarithm$(x) \,$''$ ~ = ~ \lfloor \,$Log$_2(x) \, \rfloor \,$.
+\beq
+\vspace*{- 0.6 em}
+\label{wep2} 
+%\small
+x=1 ~~~\vee ~~~ \mbox{Logarithm}(~x~)~\neq~\mbox{Logarithm}(~x-1~)
+\enq
+In this context,
+ $\zzthe(x)$ 
+will denote the analog of
+the $\glamb(x)$ function  
+%% haphazard
+that walks among the powers of 2 in a manner 
+similar to 
+$\glamb(x)$'s
+%  haphazard
+ walk through conventional
+integers. 
+It is 
+% formally
+defined by \eq{walk1}-\eq{walk4}.
+% 
+% It will thus satisfy
+% the axiomatic constraints below (which are 
+% $\zzthe(x)$'s analog of the more modest constraints given in
+% % sentences 
+% \eq{zm1} and \eq{zm2}).
+% The most important difference between these two constructs
+% is that axiom \eq{walk1} requires that 
+%  $\zzthe(x)$ maps power of 2 onto powers of 2.
+
+{
+%\small
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq} }
+
+\vspace*{- 1.2 em}
+\noindent
+{\it It needs to be emphasized} that 
+ \eq{walk1} -- \eq{walk4} will be the
+{\it only vehicle} our
+proposed formalisms will
+%self-justifying axioms 
+%will 
+have available to construct
+integers $\geq \, 3$. $~$These
+axioms will be called
+%They will be henceforth called
+the {\bf $~$Up-Walking$~$} axioms. 
+(The axiom \eq{walk4} 
+is
+% does not,
+%% , technically, 
+unnecessary
+ to construct any
+integer  $\geq \, 1\, $, but it is helpful 
+for 
+% because
+% it 
+% % allows us to 
+% formalizes 
+formalizing
+how our methodology will treat  integers
+which are not powers of 2.)
+
+% \gvxs
+% \svxs
+
+\smallskip
+
+
+Both
+$\glamb$ 
+ and $\zzthe$ are
+% unusual
+% complicated
+% entities 
+Q-functions
+%because they 
+that
+own
+% potentially
+ infinitely many distinct 
+% $\aleph_1$
+%  distinct 
+vectors, 
+%analogous 
+similar
+to 
+ Line \eq{wow},
+for representing them.
+We will soon see
+their 
+underlying
+% computational
+ complexities 
+% properties 
+are 
+%sharply 
+%quite
+surprisingly
+different.
+
+
+%% 
+%% they have sharply
+%% contrasting
+%% %%very
+%% %sharply 
+%% %% different
+%% computational
+%% % complexity 
+%% properties.
+%% 
+
+%% Both
+%% the 
+%% Q-functions 
+%% % the operators 
+%% $\glamb$ 
+%%  and $\zzthe$ are 
+%% %awkward 
+%% challenging
+%% to analyze
+%% %challenging
+%% %daunting
+%% because there are
+%% % a  
+%% % dizzying
+%% $\aleph_1$ distinct 
+%% vectors, analogous to 
+%%  Line \eq{wow}, 
+%% that are
+%% %where their definitions permits 
+%% representations of these functions.
+%% %% 
+%% %% Also, we may combine either operation with our 
+%% %% language $L^G$'s grounding function-primitives to formulate a term
+%% %% $~T_n~$ that defines any arbitrary integer $~n~$.
+%% %% 
+%% We will soon see that
+%% there is, however, a
+%% distinction
+%% %  major difference 
+%% between these
+%%  two concepts
+%% from a
+%% % computational 
+%% complexity perspective.
+
+\begin{definition}
+\label{defx-3.2}
+\rm
+Let $~L^Q~$ 
+and $~L^{Q^*}~$ 
+denote the 
+extensions
+of $~L^G\,$'s Grounding language that contain the
+respective
+additional 
+function symbols of  
+  $\zzthe$
+ and
+$\glamb$. Then 
+$~~L^Q~$ shall be called the {\bf  Q-Grounding} language, and 
+ $~~L^{Q^*}~$ 
+will be called the  {\bf  Q* Grounding} language. 
+\end{definition}
+
+\begin{propp}
+\label{th-3.3}
+In contrast to the
+Q* Grounding language 
+that requires $O(~n^2~)$ function symbols
+for defining a term $~T^*_n~$ for representing the integer
+$~n,~$ the    Q-Grounding language
+%% will need no more than 
+needs
+% uses 
+only
+$O(~$Log$^{ \, 3\,} \,n~)$ symbols to 
+encode
+%formalize 
+a term 
+$~T_n~$ representing
+$~n$.
+\end{propp}
+
+\vspace*{- 1.0 em}
+
+\begin{center}
+% \small
+% Our proof of \phx{th-3.3} 
+\phx{th-3.3}'s 
+proof
+will rely upon the following notation  convention:
+\end{center}
+
+\vspace*{- 0.8 em}
+
+\begin{definition}
+\label{def-3.3}
+\rm
+Let
+ $~\zzthe^j(x)~$
+denote the term
+ $~\zzthe(~\zzthe(~ ... \zzthe(x)))~$
+where there are 
+$~j~$ iterations of the 
+ $~\zzthe~$ operation.
+% Throughout this article,
+Then
+%for any  $~j \, \geq 1~,~$ 
+%the symbol 
+$~E_j~$ 
+will
+% shall
+% will 
+denote
+the quantity produced by
+\eq{ej-def}'s division operation: 
+
+\vspace*{- 0.6 em}
+
+\beq
+\small
+\label{ej-def}
+ \frac{~\mbox{Max}
+~[~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~~] }
+{~~\mbox{Half}^{\,j\,} ~ \{ ~\mbox{Max}~[
+ ~\zzthe^{\, j \,}(1)~,~~\zzthe^{\, j-1 \, }(1)~,~... ~\zzthe(1)~]~ 
+ ~ \}~~ } 
+%
+% \mbox{Max}(~\zzthe^j(1),~\zzthe^{j-1}(1),~... ~\zzthe^1(1)~ 
+\enq
+It is each to see
+$  E_j  =  2^j  $ for every
+$j   \geq 1$.
+This is
+ because \el{ej-def}'s
+twice-repeating
+ term
+% object  of
+``$\,  \mbox{Max}
+~[~\zzthe^j(1),  \zzthe^{j-1}(1),...\zzthe(1) \,]\,$''
+% is at least as large as $\, 2^j\,$.
+is a power of 2 exceeding  $\, 2^j\,$.
+%% 
+%% The definitions of the
+%% % Q-Grounding 
+%% functions of ``Half'', ``Max'' and
+%% ``$~\zzthe~$''   imply 
+%% $~E_j~=~2^j~$ for each
+%% $j \, \geq 1$. 
+%% 
+For the additional case where $~j=0~,~$
+we will 
+% formally
+define  $~E_0~=~1~$ (by 
+using the 
+%%
+%%setting it equal to 
+%%our
+%%%the
+%%
+built-in constant symbol
+of $~C_1~$).
+\end{definition}
+
+%% , which 
+%% is intended to
+%% %formally 
+%% represent the integer of ``1'').
+
+
+
+{\bf Proof of  \phx{th-3.3}:}
+%The justification of   \phx{th-3.3} is an 
+Easy consequence of
+\dfx{def-3.3}'s machinery. Thus if $~n~$ is a power of
+2 of the form $~2^j~$ then
+% the preceding
+% definition's
+expression $~E_j~$ is a term representing $~n \,$'s value
+that employs
+  $O(~$Log$^{ \, 2\,} \,n~)$ 
+logical
+symbols. On the other hand, if  
+ $~n~$ is not a power of
+2 then it can be defined 
+with   $O(~$Log$^{ \, 3\,} \,n~)$ symbols by 
+setting
+$~E_j~$ equal to the least power of 2 greater than $~n~$ and
+subtracting from $~E_j~$ those powers of 2 that are needed to
+produce $\,n\,$'s  value. 
+For example since $76~=~128~-~32~-~16~-~4~,\,$ it can
+be formalized as a term $T_{76}$ defined by
+$~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$.
+
+% $~~~~\Box$
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\begin{definition}
+\label{def-3.4}
+\rm
+A term in mathematical logic
+is defined to be a syntactic object,  built
+out of
+solely
+ symbols for representing 
+functions,
+constants and variables.
+% 
+% The nomenclature in
+% % classical
+% logic has 
+% %formally 
+% defined
+% a {\it ``term''} to be a syntactic object,  built
+% out of symbols for representing 
+% functions,
+% constants and variables.
+% 
+% 
+Such an object is called
+% either 
+a {\bf ``Ground Term''} 
+%% (or for precision a
+%%% {\bf ``Tree-Oriented Ground Term''} )
+when it  is built {\it out of solely}
+function and
+ constant symbols.
+For example in our Q-Grounding language (which 
+uses
+%owns only
+$  C_0  $, $  C_1  $ and $  C_2  $ 
+as its
+built-in
+ constants),
+% symbols), 
+the expression
+%of 
+``$\, C_2-  C_1\,$'' 
+is 
+% such 
+a Ground term.
+Two more complex 
+examples of 
+Ground terms are
+``Max$(   C_2 ,  C_1 - C_0)$'' 
+and ``Max$( ~\zzthe(C_1)~,~C_2 ~)$''.
+Also,
+expression $~E_j~$ 
+in Line \eq{ej-def}
+should be viewed as 
+a Ground term (when one 
+views
+its
+use of the
+symbol
+ ``1'' as an %informal 
+abbreviation
+for the
+constant
+ ``$~C_1~$''). 
+\end{definition}
+
+
+\begin{remm}
+\label{rem-def-3.4}
+{\bf (sharply improving upon 
+ \phx{th-3.3}'s result) : }
+\rm
+A longer version of this article
+will 
+%technically 
+distinguish between two
+kinds of Ground terms, which it calls the
+%
+%{\bf Comment of Definition \ref{def-3.4}'s Notation:} 
+%We will  distinguish  between two
+%kinds of Ground terms in Section \textsection \ref{ss6},
+%called its 
+{\bf ``Tree-Oriented''} and 
+{\bf ``Dag-Oriented''} formats.
+The latter will differ from a more 
+conventional tree structure
+by having a 
+Directed Acyclic Graph structure replace 
+a logic's
+usual 
+ tree format for defining its quantitative values.
+It will turn out that
+ Dag-Oriented Ground Terms
+will allow one to compress multiple repeating
+terms into single objects.
+This will
+%and thus 
+reduce the number of logical symbols in 
+ \phx{th-3.3}'s 
+$O(~$Log$^{ \, 3\,} \,n~)$ sized 
+%ground 
+terms to a 
+ more compact
+$O(~$Log$\, \,n~)$ quantity.
+(This is almost analogous to the 
+% (in a context where the pointers to our
+$O(~$Log$\, \,n~)$ 
+size of an integer's binary encoding,
+except that we will need  $O(~$LogLog$\, \,n~)$
+further bits to encode the pointers to 
+each 
+specified
+object.)
+ \end{remm}
+
+
+%% quantity
+%% $O(~$Log$\, \,n~)$ 
+%% objects will require  $O(~$LogLog$\, \,n~)$
+%% bits per pointer). 
+%% %(analogous to the classical binary encoding of an integer).
+
+% 
+%  {\it This is the same length
+% as would occur in a conventional
+% $O(~$Log$\, \,n~)$ sized binary encoding of an integer.}
+% We will refer to this improvement later in the current
+% article because it will suggest that 
+%  \phx{th-3.3}'s 
+% $O(~$Log$^{ \, 3\,} \,n~)$ sized ground terms attain a length
+% not too different from the binary encoding of an integer,
+% after further refinements are undertaken.
+% 
+
+
+
+\begin{definition}
+\label{def-3.5}
+\rm
+A ground term
+% $~T~$ will be 
+is
+called an 
+{\bf ``Observable''} 
+object iff it has a
+%{\it only one} 
+% an
+unique
+interpretation of its 
+quantified value in the
+%meaning  in our
+Q-Grounding language.
+It 
+%will be
+ is
+called an
+{\bf ``Unobservable''} iff it has multiple 
+%plausible 
+such
+interpretations
+due to $\zzthe$'s ``indeterminate'' definition
+(e.g. see Definition \ref{def-3.1}). 
+\end{definition}
+
+%%% (due to the 
+%%% %uncountably 
+%%% ambiguous nature of 
+%%% % our built-in function 
+%%% $~\zzthe~~$).
+%%% \end{definition}
+
+\begin{exx}
+\label{ex-3.6}
+\rm
+The previously mentioned ground term
+Max$( ~\zzthe(C_1)~,~C_2 ~)$ is an ``unobservable''
+because it can assume any of the plausible integer values
+of $~2 \, , \, 4 \, , \,8  \,  , \,16  \, 
+ \, ... ~$.
+On the other hand,
+
+\newpage
+ 
+\gvxs
+\nvxs
+\parskip 0pt
+
+\noindent
+\el{xoo} 
+%is 
+provides
+an 
+example of an
+``observable''
+that
+ represents
+ the integer value of ``3''.
+(This is because 
+its 
+twice-repeating
+term 
+``$~\mbox{Max}[ ~C_2~,  ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~$'' is bounded
+below by 4, causing the left and right sides of its subtraction
+operation to differ by 
+% an amount of 
+exactly 3.) 
+\beq
+%% \small
+\label{xoo}
+\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~] ~~-~~
+\mbox{Pred}^{\, 3 \,} \{~\mbox{Max}[ ~C_2~, ~\zzthe(C_2)~, ~\zzthe^{ \, 2 \,}(C_2)~]~\} 
+\enq
+Our notation 
+%thus 
+also
+implies that Line \eq{ej-def}'s
+ expression $~E_j~$ 
+is an
+ ``observable''. This implies, in turn, that 
+ \phx{th-3.3}'s term $~T_n~$ is an  ``observable''
+ (employing 
+conveniently
+no more than 
+ $O(~$Log$^{ \, 3\,} \,n~)$ 
+logical
+symbols).
+\end{exx}
+% 
+% 
+% For example since $76~=~128~-~32~-~16~-~4~,\,$ 
+% it follows that $~ T_{76}~$
+% corresponds to the term
+% $~E_7 \, - \,E_5 \, - \,E_4 \, - \,E_2 ~$,
+% $~$where each $~E_j~$ employs only 
+%  $O(~$Log$^{ \, 2\,} \,j~)$ symbols.
+% %%%%%\end{exx}
+
+
+\nvxs
+\hvxs
+
+
+Thus, \dfx{def-3.5} and Example \ref{ex-3.6} have illustrated
+%that 
+how
+the realm of ``observable'' objects is a 
+% very 
+broad and accessible world,
+of 
+potential usefulness.
+% 
+% non-trivial
+% %% pragmatic
+%  significance.
+% 
+It allows every integer $~n~$ to be represented by a
+% reasonably small 
+term $~T_n~$ with
+% an
+a tight
+ $O(~$Log$^{ \, 3\,} \,n~)$ length
+(in a context where
+Remark \ref{rem-def-3.4}'s refinement
+% Section \textsection \ref{ss6}'s more elaborate formalism
+%will allow us to 
+will 
+reduce this length
+almost to a more compact
+% 
+% nearly
+%  to a 
+% % far 
+% more
+% attractive  
+% 
+$O(~$Log$ \,n~)$ magnitude).
+
+
+The distinction between
+``Observables'' and ``Unobservables''
+% ground terms 
+will 
+%also 
+%% 
+%% cast a
+%% delightful
+%% 
+offer a
+ new perspective on
+the aspirations
+% that 
+which
+Hilbert and G\"{o}del 
+expressed
+in 
+their
+statements $*$ and $**$.
+%
+% under our proposed formalism.
+It
+ will suggest
+how the Second Incompleteness Theorem
+can
+%  remain to 
+be seen as a majestic result
+from a purist perspective, while
+a {\it well-defined fragment} of
+%their 
+what
+Hilbert and G\"{o}del
+sought in 
+ $*$ and $**$
+%aspirations
+%% in  statements $*$ and $**$
+can 
+likely
+%almost certainly
+be 
+%part-way
+satisfied (in at least a 
+% well-defined 
+limited sense).
+% 
+% 
+% 
+% 
+% \begin{remm}
+% \label{rem-3.7}
+% {\bf (explaining the goals of this paper):$~$}
+% \rm
+% Let us say 
+% that
+% a basis axiom system $~\alpha~$ owns
+% a {\it ``Finitized Perspective''}  of the Natural Numbers
+% if it requires only a 
+% {\it finite number} of proper axioms
+% to construct the full set of integers
+% $~0,1,2,3~... ~$. All conventional arithmetics have this property.
+% %It is useful to divide such 
+% Such
+% logics
+% %arithmetics
+% fall into two categories,
+% called {\it Single} and {\it Double-Formatted} systems.
+% %as defined below: 
+% They are defined below: 
+%  %These constructs are defined below:
+% \bed
+% \item[   a.  ]
+% %An axiom basis $~\alpha~$
+% %will be called
+% {\bf Single-Formatted Arithmetics} consist of
+% axiomatic basis systems
+% % $~\alpha~$
+% %all of 
+% whose 
+% %iff all its 
+% ground terms are 
+% all 
+% Observables.
+% (Most conventional arithmetics
+% %%%% will 
+% %fall into this 
+% lie in this
+% category
+% %when the 
+% because they
+% employ the
+%  growth
+% %  function
+% properties 
+%  of
+% the  Successor
+% operation
+% %function
+%  in
+% % a straightforward 
+% %the
+% %% a  conventional
+% the traditional
+% manner.)
+% %% since the 
+% %% the simple  growth function of Successor
+% %% easily
+% %% generates
+% %% all the natural numbers). 
+% %are {\it ``Single-Formatted Formatted''} logics. 
+% \item[   b.  ]
+% {\bf Double-Formatted Arithmetics} 
+% % representing 
+% represent
+% systems
+% %%%consisting of
+% %%%%axiomatic 
+% %%%logics
+% %%%%%basis systems
+% whose ground terms 
+% may be either
+%  Observables
+% or Unobservables.
+% (Axiomizations
+% for Q-Grounded logics 
+% %%  of
+% %% the
+% %% % our
+% %% Q-Grounding language
+% are
+% %%% will 
+% %obviously 
+% %%% be
+% ``Double-Formatted''
+% because they
+% allow $\theta$'s analog of 
+% \el{wow}'s function symbol $F$ 
+% to have
+% an uncountable number of 
+% different allowed
+% representations).
+% % 
+% % (Our
+% % Q-Grounding language 
+% % gives support to such a system.
+% % This is because it
+% % can  have its function primitives
+% % defined by a finite number of 
+% % proper axioms,)
+% % %axiom-sentences.) 
+% % \ennd
+% The distinction between 
+% categories
+% %Items 
+% (a) and (b) is
+% significant
+% % important
+% because 
+% Example \ref{ex-2.3}
+% %%%  \pag2 
+% %%% had 
+% already explained how
+%  statement $++$'s generalization
+% of the Second Incompleteness Theorem applies to 
+% any formalism recognizing Successor as a total function.
+% Thus, Item (b)'s Double-Formatted logics 
+% are useful, if one wishes to consider alternatives
+% to
+% %formalism that do not recognize
+% successor as a total function.
+% %More precisely, 
+% In this context,
+% 
+This will be because
+Hilbert's 
+%  famous 
+%Year-1900 
+Second 
+Open
+Problem
+can be viewed
+ as a  {\it  2-part question}, 
+composed of sub-queries Q-1 and Q-2:
+%%%%%
+%%%%% {\it  2-part question}.
+%%%%%The separation of Hilbert's question into two parts,
+%%%%%called Q-1 and Q-2, will allow 
+%%%%%%% 
+%%%%%%% This 
+%%%%%%% bipartite
+%%%%%%% distinction 
+%%%%%%% is useful because it
+%%%%%%% can enable 
+%%%%%%% 
+%%%%%the academic community to better
+%%%%%  with
+%%%%% what Hilbert and G\"{o}del were
+%%%%%seeking to accomplish
+%%%%%in 
+%%%%%their
+%%%%%statements
+%%%%%of $*$, $**$ and $***$.
+\bed
+\small
+\item[ {\bf Question Q-1$~~$}] {\it Are any axiom systems
+able to
+ prove 
+theorems 
+verifying
+ their own consistency in a robust sense?$~~$}
+The answer to Q-1 is clearly ``No'' because the combination
+ G\"{o}del's initial 1931 result \cite{Go31} with 
+%the 
+%further
+Hilbert-Bernays's result 
+\cite{HB39} 
+and the Pudl\'{a}k-Solovay invariant $++$
+(from Example \ref{ex-2.3})
+%% \pag2) 
+imply 
+arithmetics of ordinary strength cannot prove
+their own consistency in a robust sense.
+\item[ {\bf Question Q-2$~~$}]
+ {\it Can 
+logic systems
+%arithmetic logics
+%axiomatizations of Arithmetic
+%  , at least, 
+%somehow 
+``appreciate'' 
+% (not formally ``prove'')
+ their
+own consistency in some 
+{\bf REDUCED} sense, that is diluted
+but not fully immaterial?}
+$~~\,$The answer to 
+%question 
+Q-2 is 
+complex
+%%% more complex than Q-1
+%less clear-cut
+because 
+%several types of 
+some
+arithmetics,
+such as \cite{ww93,ww1,ww5,wwapal,ww9,ww14}'s paradigms,
+ can
+formalize
+% ``recognize''
+their 
+own consistency 
+using Example \ref{ex-2.5}'s
+% a 
+Fixed-Point {\it ``I am consistent''}
+axiom.
+Moreover,
+ Definition \ref{def-3.5}'s 
+% further 
+separation of
+the concepts  of ``Observables'' from ``Unobservables''
+%
+%the notions of Observable from Unobservable objects 
+%
+raises
+some
+% further
+% very
+subtle issues beyond these distinctions.
+\ennd
+
+
+%% %  sentence $\,\oplus\,$.
+%% %%Using
+%% %%%%%%%%%Under
+%% % Using the notation from 
+%% Under
+%% Lines
+%% \eq{totdefxs}--\eq{totdefxm}'s notation,
+%% these paradigms include:
+%% % both:
+%% \bee
+%% \small
+%% \baselineskip = .86 \normalbaselineskip 
+%% \item
+%%  Type-A arithmetics
+%% \cite{ww93,ww1,ww5,wwapal,ww9,ww14}
+%% %capable of 
+%% recognizing their self-consistencies under
+%% either the deductive mechanics of semantic tableaux or one
+%% of its cousins. (See especially \cite{ww14}'s
+%% recent Wollic-2014 paper.)
+%% \item
+%%  Type-NS arithmetics recognizing their Hilbert consistency,
+%% such as the formalisms of \cite{ww1,wwapal}
+%% %further 
+%% improved, possibly,
+%% with the added techniques introduced in
+%% this  article.
+%% \ene
+%% 
+%% \ennd
+
+One theme
+in 
+% the remainder  of 
+this article will be that
+the
+Second Incompleteness Theorem represents a
+$100 \, \% $ 
+full
+% comprehensive
+reply to question Q-1 
+but only a
+% and a
+ 90 \% 
+% adequate
+ reply to question Q-2.   
+Our
+tiny
+% only 
+%tiny
+ caveat to Q-2 will be related to Hilbert's 
+insistence that {\it some type of ``new formalism''} 
+% was needed
+will be needed
+to explain how
+% it is 
+humans
+motivate themselves to engage
+in cognition. 
+
+%The next section of this article will
+%  note 
+
+Our discussion will
+observe
+% that
+mathematicians 
+% had
+made no distinction between 
+Unobservable and  Observable ground terms during the
+% early 
+1930's.
+% We 
+It
+will suggest 
+a 
+% tiny new 
+revised
+interpretation can be assigned to the 
+% historic 
+%often-quoted
+statements $*$ and $** \,$
+of
+%by 
+Hilbert and G\"{o}del,
+when one 
+views
+them
+ from the perspective of 
+arithmetics that 
+rely upon indeterminate growth functions,
+similar to $\theta$.
+
+%employ $\theta-$like
+% growth functions.
+
+% logics that allow deploying unobservable  ground terms.
+
+ 
+% owns two types of ground terms.
+
+% looks more closely at these two types of
+% ground terms.
+
+
+%% where the remaining  5-10 \% fraction of
+%% unresolved issues is connected to the 
+%% fundamental
+%% distinction
+%%  separating
+%% % separation of
+%% Unobservable from  Observable ground terms.
+
+
+% that the final
+% tiny remaining
+%  5-10 \%
+% gap, pointed to
+% in
+% % by
+% Hilbert's and G\"{o}del's 
+% statements $*$ and $** \,$,  can be viewed as
+% being related to this 
+% fundamental
+% distinction.
+
+
+%% Some 
+%% %other insightful 
+%% different
+%% approaches to these dilemmas
+
+%Some other 
+
+Other insightful
+approaches
+to
+% the 
+Incompleteness paradigms
+are related to 
+ Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+% Kleene-like 
+{\it ``I am consistent''} axiom-sentences,
+similar to Example \ref{ex-2.5}'s
+``SelfRef'' statement.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a 
+% preferable 
+Hilbert-style 
+deductive 
+apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspectives  
+should
+% certainly,
+ be examined by researchers
+interested in the
+% Second 
+Incompleteness Theorem,
+but they are unrelated to 
+our 
+objectives
+%IQFS formalism.
+
+\smallskip
+
+
+%% During our 
+%% % description 
+%% investigation
+%% of IQFS, 
+
+Also during the next chapter,
+the reader should keep
+in mind that 
+Proposition \ref{th-3.3}'s
+% characterization of the 
+$O(~$Log$^3 \, n \,)$
+lengths for encoding $T_n$'s ground terms can be reduced to
+% an essentially 
+% a more compact
+%nearly an
+ essentially an
+  $O(~$Log$ \, n \,)$ complexity,
+when these terms are encoded 
+using
+%under
+ Remark \ref{rem-def-3.4}'s 
+more compressed 
+Directed Acyclic Graph
+% formalism.
+methodology.
+This fact will make IQFS's formalism look
+% much 
+% significantly more tempting.
+%%%% very
+quite
+  tempting.
+
+% cccc ddddddd 
+
+%% the next chapter's discussion.
+%% 
+%% 
+%% Their insights are important but 
+%% unrelated to 
+%% our particular
+%% % the next section's 
+%% %specific analysis of
+%% %%% type of
+%% Hilbert-styled self-justifying effects,
+%% explored in the next chapter.
+%% 
+
+
+
+\gvxs
+
+\vspace*{- 0.6 em}
+
+\section{Proposed New IQFS Formalism}
+
+ \label{ss32}
+\label{ss5}
+
+\vspace*{- 0.6 em}
+
+
+\nvxs
+
+% \rvxs
+
+
+\parskip 1 pt
+
+The only aspect of our prior research that will be 
+related to our proposed new IQFS formalism
+is the ISCE framework,
+defined in \cite{wwapal}'s Sections 3 \& 4.
+The next several paragraphs will review 
+\cite{wwapal}'s results, 
+so that a reader 
+can omit examining \cite{wwapal}. 
+
+
+% will not need to examine 
+% \cite{wwapal}'s 
+% formal treatment.
+%results.
+%%%%%%%%%%%%%%%for the reader's convenience.
+%% 
+%% This section will
+%% review \cite{wwapal}'s results in sufficient detail
+%% so that a reader need not examine \cite{wwapal}'s formal 
+%% text,
+%% 
+%% %%%%%definition of the ISCE axiom system. 
+%% 
+%% During our discussion, 
+%% 
+
+% \lvxs
+% \parskip 1 pt
+
+\smallskip
+
+During our 
+discussion,
+%review of \cite{wwapal}'s results,
+$~L^G~$ will  again denote
+our
+ ``Grounding-level'' 
+language 
+that 
+formalizes
+%employs
+\textsection \ref{seee3}'s
+six non-growth
+%  functions of
+operations of
+ Subtraction, Division,
+Maximum, Logarithm, Root and Count determination.
+%%% 
+%%%  the six ``Grounding-level'' 
+%%% % non-growth 
+%%% functions defined on Page 5.
+%  
+%  consisting
+%   of 
+%  the
+%  Subtraction, Division,
+%  Maximum, Logarithm, Root and Count operations.
+%  
+Also, $\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ will denote
+three constant symbols designating the
+integers 
+%values 
+of 
+``0'', ``1'' and  ``2''.
+In a context where Pred$(x)$ is an abbreviation for
+``$\,x \,- \, 1\,$'',
+%% (or more precisely ``$\,x \,- \, C_1\,$'' ), 
+the ISCE axiom system
+% from \cite{wwapal}
+used
+\eq{start}'s axiom
+ statement 
+to define
+ $\,C_0\,$, $\,C_1\,$ and $\,C_2~$:
+% these three constants:  
+\begin{equation}
+\label{start}
+\small
+\mbox{Pred}(    C_0    )  =  C_0~ \, \wedge ~ \,
+C_1 \neq C_0~ \, \wedge ~ \,
+\mbox{Pred}(    C_1    )  =  C_0 ~ \, \wedge ~ \,
+\mbox{Pred}(    C_2    )  =  C_1 
+\end{equation}
+%Also,
+The  challenge 
+\cite{wwapal} 
+faced was its formalism could
+not use any of the
+%  conventional 
+operations 
+%function-operations 
+of
+successor, addition or multiplication to infer the existence
+of larger integers from the initial constants of 
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$
+(without violating $++$'s generalization of the Second Incompleteness
+Theorem).
+% 
+%  This was because
+% the Pudl\'{a}k-Solovay result $++$ 
+% indicated
+%   the 
+% presumption  successor is a total function 
+% precludes   
+% most
+% %axiom 
+% systems
+% from recognizing their
+% own Hilbert
+% consistency.
+
+\smallskip
+
+
+Our article
+\cite{wwapal} 
+considered two 
+methods for achieving these tasks,
+%alternatives
+%to a conventional Successor 
+%function symbol 
+% for overcoming these difficulties,
+ called
+the  {\bf Additive} and  {\bf Multiplicative Naming}
+conventions.
+They defined 
+some
+further constant symbols $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+where 
+%respectively
+$~C_j~=~2^{j-1}~$ and $~C^*_j~=~2^{\,  2^{ \,j-2}}~$.
+
+\smallskip
+
+
+The definition of these
+% new 
+constants
+%  symbols
+is 
+easy
+%straightforward
+under $L^G\,$'s
+% Grounding-level 
+language.
+% called $L^G~,~$
+%all of whose function objects are non-growth primitives. 
+This is because
+Lines
+\eq{newadd}  and \eq{newmult}
+%had 
+specify how
+% that
+two 3-way predicates, called
+Add$(,x,y,z)$  and  Mult$(,x,y,z)\,$, 
+%can 
+%do
+encode the identities of
+% can be encoded to specify,  respectively,
+$x=y+z$ and $x*y=z$.
+Our additive and multiplicative 
+% naming 
+conventions
+can,
+%will,
+ then, define 
+ $~C_3,~C_4,~ C_5,~ ...~$
+and  $~C^*_3,~ C^*_4,~ C^*_5,~ ...~$
+%by  using
+via
+an infinite number of instances  of
+%utilizing respectively
+ \eq{addcov} and
+\eq{multcov}'s
+%{\it infinitely long}
+axioms:
+%%%%%%%%%% schemas:
+% two infinite schemas of  axiom-sentences:
+%% 
+%% that belong to our 
+%%  ``Additive'' and  ``Multiplicative
+%%  Naming Conventions'',
+%% then the values for 
+%% $~C_j~$ and 
+%% $~C^*_j~$ can be easily derived from $j-2$ instances of
+%% %respectively \eq{addcov} and \eq{multcov}'s 
+%% these
+%% schema:
+%% 
+
+
+{
+\small
+\beq
+\label{addcov}
+\mbox{Add}(~C_{j-1}~,~C_{j-1}~,~C_{j}~)
+\enq
+\beq
+\label{multcov}
+\mbox{Mult}(~C^*_{j-1}~,~C^*_{j-1}~,~C^*_{j}~)
+\enq}
+The methodology in
+ \cite{wwapal} 
+%% employed \eq{addcov}  and  \eq{multcov}'s schema in a context where it 
+presumed
+% assured
+%the Y of 
+the ``names'' for its constants $ C_j $ 
+and $ C^*_j $ 
+had nice compact encodings using $O(~Log(j)~)$ bits.  
+Its formalism calculated
+%, thereby, 
+the values of ``unnamed'' integers from 
+named entities via the {\it non-growth} Subtraction and
+Division primitives. For instance since $~20~=~32-8-4~,~$
+the quantity 20 
+can be encoded as $~C_6-C_4-C_3$.
+%%%%%%%%%%%%%%%% under \eq{addcov}'s naming convention.
+
+
+%% required
+%% $O(~Log(j)~)$ bits.  
+%% Thus, the length of these encodings was 
+%% much 
+%% smaller
+%% than the respective
+%% numbers
+%% % magnitudes of 
+%%  $2^{j-1}~$ and $2^{2^{j-2}}$ 
+%% %that 
+%% these constants represent.
+
+\smallskip
+
+
+The challenge \cite{wwapal} 
+faced was to determine whether 
+%it was possible to formulate 
+self-justification
+was possible
+under 
+%% either
+\eq{addcov}'s
+% ``Additive'' 
+or  \eq{multcov}'s 
+% ``Multiplicative'' 
+%% naming
+schema.
+It found 
+%that 
+ \eq{multcov}'s 
+multiplicative
+% naming 
+convention was  incompatible 
+with self-justification (due to its 
+%%very 
+speedy growth rate),
+but
+%In contrast,
+\eq{addcov}'s additive 
+% naming 
+schema did
+% conveniently,
+ permit self-justification. 
+
+\medskip
+
+Our new proposed IQFS
+axiom system is easiest to describe, if we first
+review \cite{wwapal}'s definition of ISCE
+and then
+explain how our IQFS framework 
+%% 
+%% will improve upon
+%% it (by not requiring
+%% the definition of an infinite number of separate
+%% constant symbols).
+%% 
+can  incrementally refine it.
+The extension of our base-language $~L^G~$
+that includes the Additive Naming Convention (ANC)'s
+additional constants 
+ $~C_3,~C_4,~ C_5,~ ...~$
+will be called
+an {\bf ANC-Based Language}.
+It will be denoted
+as  $~L^{ANC}~$. 
+Also if
+ $\, t \,$ denotes   any term in $\, L^{ANC} \,$'s   
+language, then 
+the quantifiers in 
+the two wffs of
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and
+$\exists ~ v \leq t~~ \Psi (v)$
+will be  called $\, L^{ANC} \,$'s   
+{\bf ``Bounded 
+Quantifiers''}.
+
+
+\begin{deff}
+\label{def-3.8}
+\rm
+The analogs of
+% a
+ conventional
+% arithmetic's
+$\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+formulae
+in the
+language $L^{ANC}$ will be denoted as
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$
+ and $\Sigma^{ANC}_n$.
+Thus,
+a formula will be defined to be
+$\Delta^{ANC}_0$  iff all its quantifiers are bounded.  
+The
+%%%%%%%%%  below 
+definitions 
+of  $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$
+formulae are
+%  also  
+quite conventional:
+\bee
+%\small
+\parskip -2 pt
+\baselineskip = 0.8 \normalbaselineskip 
+\item
+\small
+Every  
+$\Delta_0^{ANC}$ formula is considered to
+be 
+also 
+a
+$\Pi_0^{ANC}$  and 
+an
+$\Sigma_0^{ANC}  $ expression.
+%% 
+%% ``$~\Pi_0^{ANC}~ \,$''  and 
+%% %  also 
+%%  ``$~\Sigma_0^{ANC}~ \, $''. 
+%% 
+\item
+A
+formula
+is  called
+ $ \,\Pi_n^{ANC} \,$
+when it
+% is 
+can be
+encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  
+where
+%with
+$\Phi$ is  $\Sigma_{n-1}^{ANC}$
+\item
+A formula
+is  called
+ $\Sigma_n^{ANC}$
+when it can be encoded as 
+$\exists v_1~ ...~ \exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{ANC}$.
+\ene
+\end{deff}
+
+
+
+%%\begin{deff}
+%%\label{def3.9}
+%%\rm
+
+ \parskip 0pt
+
+
+Given an initial axiom system $\beta,$
+the Theorem 3 of \cite{wwapal} defined a 
+self-justifying logic, called
+ISCE$(\beta)$ 
+that could prove all 
+$~\beta\,$'s $\Pi_1^{ANC}$ theorems and 
+verify its own consistency under a Hilbert-style deductive
+apparatus. It consisted of the following four
+groups of axioms:
+%
+%  \newpage
+\begin{description}
+\small
+ \parskip 2pt
+\item
+{\bf GROUP-ZERO:} 
+This 
+schema
+%  axiom group 
+will
+use \el{start}'s axiom to define the constants of
+$\,C_0\,$, $\,C_1\,$ and $\,C_2\,$ 
+and 
+%employed
+%an infinite number of instances of
+\el{addcov}'s Additive Naming 
+schema
+%convention
+to define 
+ the further constants
+ $    C_3,  C_4,  C_5,  ...   $  
+\item
+{\bf GROUP-1:}
+It is convenient  to
+define
+ ISCE's Group-1 and Group-2
+axioms using a notation that 
+will support \cite{wwapal}'s Theorem 3
+in a 
+% slightly
+ more 
+general sense than
+appeared in \cite{wwapal},
+%%% under a slightly different notation convention,
+% is transparently equivalent
+% (but slightly different) from \cite{wwapal}'s counterpart,
+so our
+% a 
+new
+% proposed
+%%%% second
+ ``IQFS''
+formalism
+(appearing later in this section)
+% shall
+will
+%proposal shall
+% framework will 
+be easier to define.
+Let us
+therefore
+ say  a $\Pi_1^{ANC} $  sentence is {\bf Simple}
+iff the only built-in constants it employs are
+$\,C_0\,$, $\,C_1\,$ and $\,C_2$.
+Then ISCE's Group-1 scheme will
+allowed to
+ be any finite set of 
+simple $\Pi_1^{ANC} $  axioms, called $~S~,~$
+that is consistent with Group-zero schema and
+which
+ has
+the following 
+two
+properties:
+\bee
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+axioms
+%will be capable of proving 
+%do 
+will
+prove
+all true $\Delta^{ANC}_0 $
+sentences.  
+%
+% statements which
+% are true.
+\item
+  The union of $~S~$ with ISCE's Group-Zero
+scheme
+%will be capable of proving 
+will
+% do
+ prove
+that
+the ``=" and ``$\leq$" predicates
+% own 
+support
+their conventional
+transitivity, reflexivity, symmetry and total ordering
+properties.
+\ene
+Any finite set 
+$\Pi_1^{ANC} $  axioms
+with the above properties can be used to define $~S~$
+and 
+support
+%prove 
+an analog of
+\cite{wwapal}'s Theorem 3,
+by a trivial generalization
+of
+\cite{wwapal}'s results.
+% 
+% \footnote{A formal proof of this generalization of
+% \cite{wwapal}'s results is 
+% %absolutely 
+% entirely
+% routine.
+% %  and omitted here for the sake of brevity.}
+% For the sake of brevity, it  is
+% omitted.} 
+% 
+%  of
+% the methodologies from Sections 3 and 4 of
+% \cite{wwapal}. (Thus,
+% any such finite set $~S~$ supporting Conditions (1) and (2)
+% can  be employed
+% by
+% ISCE's
+% Group-1 part.)
+%%% 
+%%% and it is unimportant which
+%%% particular defining
+%%%  set is used.
+% 
+% BBB111
+%  
+% This
+% % schema
+% axiom group 
+% consisted of a finite
+% set of 
+% $\Pi_1^{ANC} $ axioms
+% %, CALLED $F$,
+%  defining ISCE's
+% Grounding  function primitives. 
+% %This means that
+% For each  such function $G$ and set of numbers 
+% $    {k},   {k_1},    {k_2}, ...    {k_m}$,
+% %the combination of 
+% the Group-Zero and Group-1 axioms 
+% %must
+% will
+%  imply 
+% $ G(    {k_1},    {k_2}, ...    {k_m}) \,=\,    {k}   $ when
+% this sentence is true
+% \footnote{ \f55 
+% Our
+%  $\Pi^{ANC}_1$
+% encoding for the
+% Group-1 scheme  needs,
+% technically, 
+% % employ
+%  only
+% employ
+%  the three constant symbol $C_0$, $C_1$ and $C_2$ for the
+% union of all 
+% the
+%  Group-Zero and Group-1 axioms
+% to satisfy 
+% their
+% %its
+% %the 
+% above requirements.} .
+% The Group-1 schema
+% of \cite{wwapal} 
+% will also
+% assign the ``=" and ``$<$" predicates
+% their conventional
+% %  logical
+% properties.
+% %footnoted property.}
+% %%
+% %%(Any finite 
+% %%set of  $\Pi_1^{ANC} $
+% %%sentences  meeting these conditions is
+% %%suitable.)
+% %%
+\item
+{\bf GROUP-2:}
+Let
+$\ulcorner \, \Phi \, \urcorner$ denote $\Phi$'s G\"{o}del number, and
+$\mbox{HilbPrf}_{ \beta  }(x,y)$
+denote a 
+%%%%%%%%%%%  $\Delta _0^{ANC+}$ 
+$\Delta _0^{ANC}$ 
+formula indicating  $y$ is a
+Hilbert-styled
+proof
+from axiom system $\beta  $ of the theorem 
+$x$.
+%
+% Suppose that
+%$~\beta~$ uses the same Grounding function symbols as
+%ISCE$^{ANC}(\beta)$,
+%and it therefore generates
+%a set of 
+%% $\Pi_1^{ANC+} $ theorems.
+% $\Pi_1^{ANC} $ 
+%theorems.
+%
+For each 
+%$\Pi_1^{ANC+} $
+$\Pi_1^{ANC} $
+ sentence  $\Phi$,
+the Group-2 schema
+for ISCE$(\beta)$ 
+%
+%was defined in  \cite{wwapal} 
+%did 
+will 
+contain
+% an 
+one
+axiom of the form:
+%% 
+%% \begin{equation}
+%% \small
+%% \label{group2nold}
+%% \forall ~x~\forall ~y~
+%% ~~\{~~[~~ \sigma_{~ \ulcorner \, \Phi \, \urcorner
+%%  ~}(x)~\wedge~
+%% \{~ \mbox{HilbPrf}~_\beta
+%% ~(~ x ~,~y~)~~]~~
+%% \Rightarrow ~~ \Phi~~ \}
+%% \end{equation}
+%% % {\bf IMPORTANT CLARIFICATION:} 
+%% %{\small 
+%% %%{{\bf DECIPHERING LINE \eq{group2nold}:$~$}
+%% {{\bf Clarification:$~$ }
+%%  \el{group2nold} is {\it helpful}
+%% because   ISCE(\beta)$  can             infer
+%% \eq{group2old}'s {\it simpler statement}
+%% directly
+%% from the combination of
+%% \eq{group2nold},
+%% % it,
+%% the Group-1 schema and \el{deltf}'s definition of
+%% ``$~\sigma~$''.}
+%% 
+\begin{equation}
+% \small
+\label{group2old}
+\forall ~y~~~\{~ \mbox{HilbPrf}~_\beta
+~(~ \ulcorner \Phi \urcorner ~,~y~)~~
+\Rightarrow ~~ \Phi~~\}
+\end{equation}
+\item
+{\bf GROUP-3:}
+This last part of
+%%%%%%%%%%%%%%%  \cite{wwapal}'s
+ISCE$(\beta)$
+% formalism 
+was
+ a single 
+self-referencing
+$\Pi_1^{ANC}$
+sentence  
+stating:
+ %% essentially declaring:
+\begin{quote}
+% \small
+%%%%%%%%%%%%% $ \oplus ~ \oplus ~~~$
+$ \oplus  \oplus ~~~$
+ ``There 
+%is 
+exists
+no
+Hilbert-style proof of 0=1 from the union of the Group-0, 1 and 2
+axioms  with {\it THIS  SENTENCE} (referring to itself)''.
+\end{quote}
+\end{description}
+%{\bf CLARIFICATION:}
+{\bf Clarifying $ \oplus  \oplus$'s Meaning:}
+ $~$Several of our articles
+\cite{ww1,ww5,wwapal,ww9}
+employed
+self-referential
+  $\Pi_1^{ANC}$ constructions,
+similar 
+to 
+%%%%%%%%%the sentence
+ $ \oplus  \oplus \,$,
+as Example \ref{ex-2.5} had mentioned.
+%% 
+%% whose 
+%% % precise implications were outlined in
+%% significance was explained by
+%% %formalized by 
+%% Example \ref{ex-2.5}.
+%% 
+A reader can find
+several
+%detailed
+slightly different
+ illustrations about how 
+$~ \oplus  \oplus ~ $
+%  $\, \oplus  \oplus $'s 
+%   self-referential statement 
+is encoded in  these articles.
+
+
+% 
+% Each of these articles provide examples of
+% how analogs for
+% $\, \oplus  \oplus $'s 
+% self-referential
+% statement
+% are encoded.
+
+
+ 
+% If the reader wishes to see
+% a formal encoding  for
+% $\, \oplus  \oplus $'s 
+% %self-referential
+% % Fixed-Point 
+% statement,
+% %it 
+% one such example
+% is provided by  
+% \cite{wwapal}'s 
+%  Lemma 1.
+% 
+
+
+\begin{deff}
+\label{def-3.9x10}
+\rm
+Let $~I(~\bullet~)~$ denote
+an operation that maps
+an initial axiom basis $\, \beta \,$ onto an alternate
+system  $\,I(\beta)\, $.
+(One example of
+such an operation is the
+  ISCE$( \, \bullet \, )$ 
+framework,
+that maps 
+an initial axiom basis of 
+  $~\beta~$ onto 
+the alternate formalism of
+ ISCE$(\beta).~)~$ 
+Such an operation  $~I(~\bullet~)~$
+is  called {\bf Consistency Preserving}
+iff  $\,I(\beta)\, $ is consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemas is
+consistent.
+\end{deff}
+
+
+%Most of our research in 
+% \cite{ww93}-\cite{ww14}
+% has 
+
+Several of our research projects 
+%centered around
+%had 
+employed
+ \dfx{def-3.9x10}'s
+framework.
+For instance, 
+%% 
+%% the 
+%% 
+%% 
+%% Its
+%% %%%  main
+%% % central 
+%% focus in
+\cite{wwapal} 
+demonstrated
+%consisted of showing
+ the   ISCE$( \, \bullet \, )$ 
+mapping was consistency preserving.
+Thus if PA+ denotes the extension of
+Peano Arithmetic that 
+includes
+PA's traditional  Addition and Multiplication
+functions
+%% 
+%% 1n addition to the conventional
+%% functions of addition and multiplication
+%% contains 
+%% 
+%% 
+plus $L^G\,$'s six
+added
+%previously mentions
+ Grounding-level function 
+primitives,
+%functions,
+then
+ ISCE$( \, $PA+$ \, )$
+will 
+be automatically
+%be
+ consistent
+(because  PA+ was consistent).
+% consistent whenever PA+ is consistent.
+Hence while Peano Arithmetic is unable to
+verify its own consistency,
+%  (on account of G\"{o}del's
+% seminal 1931 discovery), 
+it is sufficiently agile to
+prove the following relative-consistency statement:
+\begin{center}
+%% \small
+$\#~~~$ If PA is consistent then 
+ ISCE$( \, $PA+$ \, )$ is  
+ self-justifying.
+ \end{center}
+This
+%The above
+% statement
+ relative-consistency statement
+%does offer 
+provides
+a partial 
+positive
+answer to
+the
+Q-2 version of Hilbert's Second  Question.
+It 
+captures
+% Brad change encapsulizes 
+one
+% positive 
+respect
+in which
+%such as  
+ISCE$( \, $PA+$ \, )$
+can {\it appreciate} its own consistency.
+% 
+% \newpage
+% 
+% \svxs
+% 
+% \noindent
+%This is because it formalizes one respect 
+This respect is, obviously,
+only
+of a limited nature
+because $++$'s generalization of the Second
+Incompleteness Theorem indicates 
+that
+no Type-S arithmetic
+can
+% simultaneously 
+recognize 
+% {\it both} 
+its Hilbert consistency and
+take
+successor 
+to be
+ a total function.
+%The consistency-preservation property of 
+% ISCE$( \, \bullet \, )$
+%dies, however,
+It does,  however,
+ raise the following
+enticing
+ question:
+\newpage
+
+\lvxs
+\parskip 0pt
+
+\begin{quote}
+$\# \, \#~ $
+\small
+Can the infinite number of
+distinct
+ constant symbols, employed by
+ISCE's Group-Zero schema, be reduced to a finite size
+by a Type-NS Self-Justifying Logic,
+without resorting to \cite{wwapal}'s inefficient
+``ISINF'' 
+methodology (which requires 
+a proof 
+having an expensive
+ $\Omega(N)$ length for constructing integers $N$
+whose binary encoding uses $O(~$Log$(N)~)$ bits) ?
+\end{quote}
+The remainder of this section will outline how an encouraging
+answer to 
+$\, \# \, \# \, $'s query
+is likely to
+%%%should,
+% conveniently
+arrive,
+%be plausible
+when one 
+% carefully
+%delicately 
+modifies ISCE's formalism
+with  the Q-function operative of $~\zzthe~$.
+
+\begin{deff}
+\label{def-3.10}
+\rm
+Let $L^Q$ 
+% once 
+again denote the extension of
+$~L^G\,$'s Grounding language that includes
+the
+% further
+ Q-function symbol of $\, \theta $.
+Then
+$\Delta^Q_0$,
+ $\Pi^Q_n$ and $\Sigma^Q_n$
+will,
+intuitively,
+%similarly
+ denote the
+%  1-to-1 
+analogs of
+\dfx{def-3.8}'s
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ and $\Sigma^{ANC}_n$'s
+formulae
+in $~L^G\,$'s  language.
+In particular, if $~\Phi~$ 
+is one of an
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+formula,
+then
+% the formula 
+$~\Phi^Q~$
+will be called
+% respectively
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+when
+%if 
+it
+differs from $~\Phi~$ 
+only
+by
+ replacing each constant $~C_J~$
+from the set $~C_3,C_4,C_5...~$ 
+with Line \eq{ej-def}'s 
+% mathematically equivalent term of
+term  $~E_{J-1}~$. 
+\end{deff}
+
+\parskip 2pt
+
+%% 444444444444444
+
+\begin{example}
+\label{ex-3.11}
+\rm
+Suppose $~\Phi$ 
+is one of a 
+$\Delta^{ANC}_0$,
+ $\Pi^{ANC}_n$ or $\Sigma^{ANC}_n$
+sentence that employs the three constant symbols
+of $C_4$,   $C_6$ and  $C_{10}\,$
+for
+ representing the
+three numbers
+of 8, 32 and 512. 
+Let us recall 
+that
+ $E_3$,   $E_5$ and  $E_9\,$
+%  do
+formulate these three quantities
+under  Line \eq{ej-def}'s notation.
+Then $~\Phi^Q$  will have an 
+identical definition as
+ $~\Phi$ 
+except each $C_j$ is replaced by
+$E_{j-1}$.
+
+
+A formula is,
+moreover,
+ defined to lie in one
+of the 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+classes 
+{\it only if} it is constructed in such a manner.
+This fact
+% brad assures 
+ensures
+that all the terms employed in these
+three classes of sentences are 
+{\it ``Observable''} terms.
+Hence ``Unobservable'' ground terms are allowed in
+$~L^Q\,$'s language, but {\it they are excluded}
+from occurring in the 
+{\it ``end-product''} 
+$\Delta^Q_0$,
+ $\Pi^Q_n$ or $\Sigma^Q_n$
+theorems 
+that 
+%%will now be discussed.
+%it proves !
+do
+encapsulate
+%formalize
+ the {\it intended use of 
+%its
+this 
+formalism.}
+\end{example}
+
+
+\begin{deff}
+  \label{def-3.12}
+\rm
+The term {\bf IQFS($ ~\bullet~$)$~$}
+will  refer
+to the self-justifying analog of
+  ISCE($ ~\bullet~$)$~$
+%that will be employed 
+under  $L^Q\,$'s
+language.
+(The acronym ``IQFS'' stands for 
+``Introspective Q-Function Semantics''.)
+In a context where $~\beta~$ is 
+%some i
+an initial axiom
+system that proves theorems 
+%under
+in 
+the 
+language  $L^Q$, the 
+system
+%formalism 
+ IQFS($ \, \beta\,$)
+%$~$
+will 
+be defined as a
+ 4-part 
+formalism,
+analogous to  ISCE($\beta$),
+except for the following 
+%relatively modest 
+three 
+relatively
+modest
+changes: 
+\bed
+\small
+\parskip 0pt
+\item[  a.  ]
+The Group-Zero schema of
+ IQFS will
+differ from ISCE's analog
+by replacing 
+\el{addcov}'s ``Additive Naming'' schema with  
+the
+Up-Walking axioms,
+given in Lines  \eq{walk1}--\eq{walk4}.
+(This is because
+the language  $L^Q$ differs from
+ $L^{ANC}$ by 
+having the 
+ Q-function operator of $~\zzthe~$
+define the formal quantities that are represented by
+the constant symbols
+of $~C_3,C_4,C_5~~....~$ 
+under  $L^{ANC}.~~)$ 
+%% 
+%% Otherwise both
+%% these
+%% Group-Zero 
+%% schemes will be 
+%% identical.
+%% Thus,
+%%  they 
+%% will 
+%% both 
+%% use \el{start}'s axiom to define the 
+%% three initial constants of
+%% $\,C_0\,$, $\,C_1\,$ and $\,C_2\,~$.
+%% 
+\item[  b.  ]
+All the $\Pi_1^Q$ axioms lying in IQFS's
+Group-1 and Group-2 schemes will be
+% identical
+analogous to their counterparts
+under ISCE, except  they
+will
+ employ 
+\dfx{def-3.10}'s machinery for translating 
+ $ \,\Pi_1^{ANC} \,$ 
+sentences into
+essentially their
+% equivalent
+ $ \,\Pi_1^Q \,$ counterparts. 
+\item[  c.  ]
+The Group-3 axiom of  IQFS
+will be similar to ISCE's Group-3 
+{\it ``I am consistent''} 
+axiom-statement, except 
+the latter's notion of ``I'' will reflect the above
+changes in the Groups 0, 1 and 2 schemes. 
+It
+%Thus, the new
+%Group-3 axiom 
+will,
+thus,
+be a $\Pi_1^Q$ sentence declaring that
+{\it ``There is no 
+Hilbert-style
+proof of 0=1 from the union of the preceding axioms
+with THIS SENTENCE (looking at itself)''.}
+\ennd
+\end{deff}
+
+%\noindent
+
+\bvxs
+\parskip 2pt
+
+{\bf REVISITING THE Q-2 VERSION OF
+  HILBERT'S SECOND OPEN
+QUESTION FROM THE PERSPECTIVE OF  ``IQFS'' .}
+$~$
+Let us recall
+% that 
+\textsection \ref{ss4} indicated 
+that   Hilbert's Second Open Problem could be
+divided into two sub-queries, 
+that were
+called Q-1 and Q-2.
+The former query asked whether axiom systems could
+verify their own consistency in a robust sense, and the latter
+inquired whether some 
+{\it weaker but non-trivial} forms of
+self-justification might exist.
+The 
+Q-1
+paradigm
+% 
+% former query
+% addressed the larger part of Hilbert's
+% open question. It
+% % We already noted that the Q-1 version of this query
+% 
+was definitively resolved in a negative direction
+by the combination of G\"{o}del's initial Second Incompleteness
+Effect,
+its
+generalization
+appearing
+% documented 
+in the 
+Hilbert-Bernays textbook \cite{HB39} and  the
+Result $++$ due to the combined work of
+ combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87}.
+In contrast,
+we noted in
+\textsection \ref{ss4} 
+ that there
+were other
+issues raised by the Q-2 version of Hilbert's
+question that were 
+not yet
+fully
+resolved.
+
+% \gvxs
+ 
+\smallskip
+
+It is within 
+this
+context where 
+Definition \ref{def-3.12}'s IQFS framework is helpful.
+The strong similarity between the definitions of ISCE and IQFS,
+{\it by itself,} 
+suggests that IQFS is likely to satisfy a
+consistency-preservation property analogous to ISCE.
+Moreover, all the techniques that were used to prove
+either $++$'s generalization of the Second Incompleteness
+Theorem
+% 
+%  (due to the combined 
+% work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+% \cite{Ne86,Pu85,So94,WP87}) 
+% 
+or the related subsequent 
+results of
+% 
+% generalizations
+% %% of the Second Incompleteness Effect 
+% that
+% Example \ref{ex-2.3} attributed
+% to 
+% 
+Buss-Ignjatovic,
+H\'{a}jek, 
+\v{S}vejdar 
+and Willard 
+ \cite{BI95,Ha7,Sv7,ww1}
+% \cite{BI95,Ha7,Sv7,ww1,wwlogos}
+lose their relevance in IQFS's context.
+
+\medskip
+
+This is because a longer version of the current article
+demonstrates
+the Groups 0, 1 and 2 axioms of IQFS
+{\it  are unable}
+to prove 
+% an invariant implying
+successor is a total function, 
+while the 
+incompleteness results
+of   \cite{BI95,Ha7,Ne86,Pu85,So94,Sv7,WP87,ww1}
+require taking
+successor as a total function.
+
+% is
+% precisely what is needed for these formalisms to 
+% become applicable
+%  to IQFS.
+
+% (because they require a counterpart of a successor
+% function operation).
+
+
+
+\smallskip
+
+The properties of IQFS
+ are
+interesting
+% especially intriguing 
+from a 
+% Computer Science 
+Complexity perspective
+because
+\phx{th-3.3} showed 
+% that 
+every integer $\,n\,$ 
+can
+%could 
+be encoded
+%under it by
+by
+%via
+a
+ term $T_n$ that has an $O\{~[~$Log$(n)~]^3~\}$ length.
+This is
+unlike the 
+%much 
+%%%%%%%%%% far worse 
+asymptote  $\Omega(n^2)$ that results
+when the 
+$~\zzthe$ primitive 
+(from Lines  \eq{walk1}-\eq{walk4})
+is replaced by 
+Lines \ref{zm1} and  \ref{zm2}'s
+less efficient
+primitive of 
+$~\glamb~$.  
+Moreover, Remark \ref{rem-def-3.4}
+indicated
+% our
+that these
+ $O\{~[~$Log$(n)~]^3~\}$ lengths
+could be reduced
+to almost an  $O\{~$Log$(n)~\}$ size if 
+$L^Q\,$'s Ground terms were encoded as 
+Directed Acyclic Graphs (rather than as tree-like objects).
+
+We consider it 99 \% likely that
+{\it BOTH} 
+Definition \ref{def-3.12}'s
+%precise
+specified
+formulation of 
+% the
+ IQFS($ ~\bullet~)$
+%construct 
+and its 
+% more compressed Dag 
+implied Dag
+refinement
+(using  Remark \ref{rem-def-3.4}'s additional machinery)
+%modification
+will satisfy 
+a consistency preservation property analogous to ISCE.
+If this 
+2-part
+conjecture is correct, it will support our
+hypothesis that the Q-2 version of Hilbert's Second Open
+Question 
+% would 
+does
+support some 
+{\it fragmentary}
+positive results,
+in the context of
+% with regards to
+Hilbert-styled deductive methods using the
+$~\zzthe$ primitive for formulating growth among integers.
+
+Moreover, a reader should not be
+especially
+ concerned that the Group-2
+axiom schemas for ISCE and IQFS involve employing
+an infinite number of separate incarnations
+of \el{group2old}'s axiom schema.
+This is because these
+Group-2 schemas
+can be
+% should be able to be
+nicely
+ reduced to a 
+purely
+finite size, with almost no loss 
+in
+%of useful
+information. This was done in \cite{ww14} 
+for the Group-2
+scheme
+of its IS$_D(\beta)$ formalism, 
+with the latter
+%%%%%%%%%%%%%%%% still 
+%where the
+% 
+% germane Group-2 scheme was
+% reduced to one
+% single  axiom sentence 
+% while the resulting 
+% 
+% latter
+%formalism still
+% produced
+producing
+isomorphic counterparts
+of all of $~\beta \,$'s
+full set of
+ $\Pi_1$ theorems
+(e.g. see 
+%Sections 5 and 6 
+\textsection $\,$5
+of    \cite{ww14}).
+The same methods will
+% trivially 
+%routinely
+easily
+generalize for
+%%%% the
+% 
+% Analogs of the techniques from Sections 5 and 6 of 
+% \cite{ww14} 
+% % will easily 
+% apply to each of the 
+% 
+ IQFS, 
+% axiom framework, 
+if it does satisfy
+Definition \ref{def-3.9x10}'s
+Consistency Preservation property (as we
+conjecture it does).
+
+% it does).
+
+\vspace*{- 1.0 em}
+
+%\section{Broader Perspectives Produced by These Results}
+
+
+\section{Concluding Remarks}
+
+\vspace*{- 0.8 em}
+
+\lvxs
+
+\label{nnnew}
+
+%\Large
+% \baselineskip = 1.8 \normalbaselineskip 
+
+There is no question that the 
+% Second 
+Incompleteness Theorem
+%does imply
+% demonstrates
+illustrates 
+that 
+90-95 \% of the initial objectives of
+Hilbert's Consistency Program were overly ambitious.
+It would, nevertheless, be of interest if
+some 5-10 \%
+fragment of 
+Hilbert's
+% initially 
+intended 
+goals
+% objectives 
+were
+partially
+ achieved.
+
+%% bbbbbb
+
+\smallskip
+
+
+This is because it is difficult to fathom how humans
+can
+maintain
+their psychological motive to engage in cognition without owning some
+type of
+% qualified 
+instinctive faith in their own consistency.
+%% 
+%% Moreover, the close similarity between the defining structures of the
+%% ISCE and IQFS frameworks strongly suggests
+%% \cite{wwapal}'s proof of ISCE's consistency preservation property
+%% should generalize for both 
+%% IQFS  and 
+%%  IQFS$^*$ under a more elaborate
+%% % and sophisticated
+%% inductive machinery.
+%% 
+Moreover, it is fascinating that 
+the distinction between
+Unobservable and Observable ground terms, using 
+Proposition \ref{th-3.3}'s  and Remark \ref{rem-def-3.4}'s
+ $\, \theta \, $ operator,
+% 
+% whose $O(~$Log$^3\,n )$ and $O(~$Log$~n )$ complexities
+% are characterized
+% by Proposition \ref{th-3.3}  and Remark \ref{rem-def-3.4},
+% 
+%%%%%%
+$\,$does 
+seem to 
+lend credibility to a
+% fraction 
+{\it  partial  subset}
+% {\it fragment}
+of the goals
+that
+Hilbert and G\"{o}del 
+advanced
+%% were seeking 
+in
+% aspiring to in
+their
+statements $*$ and $**~$.
+
+\smallskip
+
+\nvxs
+
+Also, the last five minutes of a YouTube lecture
+by Harvey Friedman,
+entitled
+% the 
+{\it ``The Blessing and Curse of Kurt  G\"{o}del''},
+raised the question 
+\cite{Fr14}
+of whether 
+some type of 
+{\it sharply  circumscribed} boundary-case exception
+to 
+the Second Incompleteness
+Theorem 
+might be possible.
+
+
+% 
+% could be evaded with some
+% % new
+% non-recursive  function symbol
+% (which under
+% \cite{Fr14}'s
+% % Friedman's
+% hypothetical
+%  example
+% involved deploying the laws of Physics
+% instead of
+% %  rather than 
+% Lines  \eq{walk1}-\eq{walk4}'s
+% indeterminate definition).
+
+
+
+% (in a context where the broader ambitions of these
+% two statements 
+% are clearly untenable).
+
+%% infeasible
+
+
+%Thus,
+
+% \medskip
+
+
+Our proposed IQFS 
+% axiom system 
+framework
+is intended to
+be no full remedy,
+when the
+traditional growth properties of the  addition,
+multiplication and successor function operations are replaced
+by an
+alternative  $~\theta~$ function symbol.
+It is only a partial solution, similar to our
+alternative class of 
+partially
+positive
+ results in
+\cite{ww93,ww1,ww5,ww14},
+involving 
+%axiom systems 
+arithmetics
+that
+ sacrifice
+%      sacrificed 
+their
+understanding that multiplication is a total function
+for the sake of gaining an appreciation of their 
+semantic
+tableaux consistency.
+
+\smallskip
+
+Neither of these
+% results 
+formalisms
+are perfect, and 
+imperfections 
+will 
+% be ever-present
+always be 
+ present
+%result
+%be inevitable
+when one
+%  explores
+considers
+ the
+%  tight
+ dilemma posed by the
+% 
+% must
+%  always
+% % have to 
+% be tolerated
+% 
+% when examining the  dilemma posed by the
+Second Incompleteness 
+Effect. 
+It is within such a  context that
+{\it a well-defined fragment} of 
+%what 
+the goals
+% which
+that
+Hilbert and 
+G\"{o}del sought in
+$*$ and $**$ 
+should be
+%% 
+%% looks 
+%% %part-way 
+%% like it is probably
+%% 
+%         realistically feasible 
+possible to 
+reach
+%realize
+under
+%  some
+certain
+% % might be plausibly 
+% %possible 
+% should be 
+% possible 
+% to obtain
+{\it meticulously defined weak-logic settings$\,$,}
+if IQFS satisfies an
+analog of ISCE's
+ consistency-preservation
+property (as we conjecture it 
+will almost certainly
+ do). 
+
+%\textsection \ref{ss5}'s conjectures about IQFS do
+%old to be ture.
+
+
+%  special
+% % broad
+% % potential 
+% interest.
+% 
+
+
+   \medskip
+
+{\bf Acknowledgments:}
+%%%%As  several Sections 1-4, 
+%\textsection \ref{ss2}, 
+I am
+% much
+%very 
+grateful to
+%was influenced by an emailed letter from 
+Pavel Pudl\'{a}k
+for suggesting 
+\cite {Pupriv}
+I investigate how to apply
+% an analog of 
+Ajtai's study
+\cite{Aj94} of Pigeon-Hole effects 
+for
+refining my prior results about self-justifying logics.
+(The combination of
+ Pudl\'{a}k's
+insightful             suggestion
+% \cite {Pupriv}
+and our
+subsequent
+% further 
+        distinguishing
+between the
+$~\glamb   ~$ and $~\theta~$ operators
+has led to the 
+conjectured
+ improvement of 
+\cite{wwapal}'s ISCE formalism.)
+% I am very grateful to Pudl\'{a}k for making this
+% suggestion. 
+I also thank Bradley Armour-Garb
+and Seth Chaiken
+ for
+%  many
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%several
+comments
+%  about how to improve 
+that improved 
+ the 
+%this paper's
+presentation.
+
+
+\footnotesize
+% \tiny
+\parskip -3 pt
+
+%\baselineskip =  0.92 \normalbaselineskip 
+\baselineskip =  0.5 \normalbaselineskip 
+%\baselineskip =  0.65 \normalbaselineskip 
+\bibliographystyle{abbrv}
+\bibliography{aa}
+
+ \end{document}
+\newpage
+
+rrrrrrrrrrrrrr
+
+
+\Large
+
+
+On How the Introducing of a New $~\theta~$ Function Symbol Into Arithmetic's
+Formalism Is Germane to Devising Axiom Systems that Can Appreciate Fragments
+of Their Own Hilbert Consistency
+
+
+Why a Small Fragment of Hilbert's Consistency Program
+Ought to Be Feasible
+for Hilbert-like Deductive Methods
+After A New $~\theta~$ Function Primitive
+Is Added to Arithmetic's Formalism
+
+AFTER A NEW ``$~\theta~$'' Function Primitive
+
+
+ \baselineskip = 1.5 \normalbaselineskip 
+
+It is known that the combined work of Pudlak and Solovay, enhanced by some
+added techniques of Nelson and Wilkie-Paris, implies no reasonable axiom
+system can verify its own Hilbert consistency, when it recognizes Successor as
+a total function and treats addition and multiplication as 3-way relations.
+These considerations will lead us to examine unconventional axiomatizations
+for arithmetic that continue to view addition and multiplication as 3-way
+relations, but which replace the successor function symbol with an entirely
+new operator, called the $\theta$ primitive.
+
+
+
+It is likely that this paradigm can be combined with the prior results in our
+APAL 2006 paper to construct axiom systems that are seriously diluted but able
+to verify their Hilbert-style consistency in some interesting fragmentary
+respects.
+
+
+
+%% This operator will allow us to encode any integer  n  by a term $T_n$ whose
+%% length will exceed the O(Log n) length of a binary encoding by only the
+%% relatively small magnitudes formalized by our Proposition 3.3 and Remark 3.6.
+%% It is likely that this paradigm can be combined with the prior results in our
+%% APAL-2006 paper to construct axiom systems that are seriously diluted but able
+%% to verify their Hilbert-style consistency in certain interesting respects.
+
+
+
+{\bf Keywords:}
+
+G\"{o}del's Second Incompleteness Theorem, 
+Hilbert's Second Open Question,
+Bounded Arithmetic, 
+Distinction Between Semantic Tableaux and Hilbert Deduction, 
+Weak Arithmetics.
+
+\end{document}
+
+% \parskip 2pt
+
+The 
+significance
+% roles 
+of
+% Observations
+(a) and (b)
+%in our research
+%from the current example, 
+will become
+% more 
+evident
+as this article progresses. 
+Essentially, our
+prior research
+% ,
+% best summarized in \cite{ww14},
+%  has 
+%
+% had 
+focused
+% mostly 
+on Type-A arithmetics
+that could verify their consistency under
+% either 
+semantic tableaux deduction 
+and/or its near cousins.
+%% 
+%% or some near-cousin of this concept
+%% (e.g. see \cite{ww14}'s  summary
+%% of \cite{ww93}-\cite{ww9}'s results).
+%% The 
+%% 
+(A 15-page summary of this research appears in
+\cite{ww14},
+ but 
+it
+%the latter
+does need to be examined.)
+%%% 
+%%%  is 
+%%% not
+%%% % unnecessary to examine  as 
+%%% a prerequisite for reading this paper.
+%%% 
+%%% 
+%%%  but this technical
+%%% material does not need to be examined.) 
+%%% does hot need to examine
+%%% this material for the 
+%%% 
+%%% 
+Our
+% new
+$~\theta~$ operator, defined in the next section, will
+raise the question about whether a
+% surprising
+% powerful 
+new
+class of 
+%new 
+Type-NS systems 
+will
+%may 
+satisfy an analogous
+property in the context of
+Definition \ref{def-2.2}'s more pristine
+Hilbert-style methodology for deduction. 
+
+\medskip
+
+% have
+% a similar property.(The article \cite{ww14} offers a nice 16-page summary
+% of our prior results 
+% \cite{ww93}-\cite{ww9}
+% about Type-A arithmetics, but 
+% none of these results will
+% be
+%  needed
+% %  to be examined 
+% during our current article's exploration of 
+% the properties of the new % 
+% $~\theta~$ operator.)
+
+% It will be unnecessary for a reader to examine any of our
+
+ 
+%% % year-2014 
+%% Wollic-2014 paper \cite{ww14}
+%% summarized and extended our
+%% results about
+%% semantic tableaux consistency.
+%% % and this 
+%% The current 
+%% new 
+%% year-2015
+%% paper 
+%% will, now,
+%% explore whether
+%%  systems  can
+%% also corroborate
+%% their Hilbert-styled consistency
+%% under certain well-defined circumstances.
+
+%% (and seek to explore the restrictions $++$ imposes upon
+%% Hilbert-styled deduction).
+
+% 
+% (The latter topic
+% % is 
+% %very 
+% %entirely
+% %different 
+% differs
+% from the former
+% because
+% constraint  $++$ 
+% applies only 
+% to its
+% particular
+%  domain.)
+
+%in the second context.)
+
+
+%% The constraints imposed by $++$ 
+%% are challenging 
+%% because Type-NS arithmetics 
+
+This
+topic
+% subject 
+is 
+interesting
+%challenging 
+because 
+%% essentially all
+Type-S arithmetics
+are forbidden by $++$ from
+verifying the consistency
+of their own Hilbert-styled deductions
+%%%%%%%%%%%%%%%%%%% (and conventional 
+%forms of
+(while
+Type-NS formalisms are
+%typically 
+ usually 
+quite weak).
+%% 
+%% (Thus, our efforts
+%% to design
+%% self-verifying systems
+%%  must focus on
+%% Type-NS arithmetics).
+%% 
+Our new 
+$~\theta~$ operator,
+together with
+Proposition \ref{th-3.3}
+and Remark \ref{rem-def-3.4}, 
+% and \ref{th-6.1}
+will suggest a 
+%possible
+% plausible 
+partial
+solution to this 
+problem by
+% daunting challenge by
+illustrating how
+an {\it unusual class} of 
+Type-NS arithmetics can efficiently construct the
+full set of integers $~0,1,2,3,...~$
+% by finite means 
+{\it without using}
+any of the successor, addition or multiplication
+% functional 
+operations.
+
+% function symbols. 
+
+As a result, we will suggest 
+a
+%  a {\it part-way} 
+% that an interesting
+%%% non-trivial (although diluted)
+{\it small fragment}
+of what Hilbert
+and G\"{o}del
+% sought 
+% referred to
+did seek
+%sought
+in 
+statements 
+$*$ and $**$ 
+% will
+%  be formally  achieved 
+% become tempting 
+is likely
+% be 
+viable
+under Definition \ref{def-2.4}'s formalism.
+
+
+
+% 
+% This
+% topic
+% % subject 
+% is challenging because 
+% $++$'s 
+% Type-NS arithmetics 
+% %  obviously 
+% have sharply circumscribed powers
+% (demonstrating the 
+% broad reach
+% % ubiquitous  nature 
+% of
+% %the Second Incompleteness Theorem's reach).
+% G\"{o}del's 
+% second theorem).
+% %%  
+% %% The current article will
+% %% show, however,  that 
+% %% some Type-NS arithmetics are 
+% %% substantially
+% %%  stronger than previously 
+% %% anticipated
+% %% (and they will have useful applications 
+% %% in
+% %% computer science settings).
+% %% 
+% %% Thus in a context where 
+% %% the power of both G\"{o}del's initial
+% %% Second Incompleteness Theorem and $++$'s strengthening of it
+% %% are stunning
+% %% and
+% %% have pervasive implications,
+% %% we will show that a 
+% %% {\it partial-and-much-less-than-full}
+% %% fragment 
+% %% of what Hilbert
+% %% desired
+% %%  in statements $*$ and $**$ an be
+% %% positively  achieved.
+% %% 
+% The current article will
+% %show, however, 
+% suggest,
+% however, 
+% % that 
+% some Type-NS arithmetics are
+% % , however,  
+% % significantly 
+% %% substantially
+% more far-reaching
+% than 
+% previously 
+% anticipated. Thus,  a 
+% % well-defined 
+%   {\it
+% partial but non-trivial} fragment of what Hilbert
+% and G\"{o}del
+% % sought 
+% % referred to
+%  anticipated
+% in 
+% statements 
+% $*$ and $**$ will
+% %  be formally  achieved 
+% % become tempting 
+% look
+% % be 
+% viable
+% under Definition \ref{def-2.4}'s formalism.
+
+
+
+\end{document}
+
+% \textsextion
+
+%\setlength{\textwidth}{5.0 in}
+
+\gvxs
+
+Line 1
+
+
+Line 1
+
+
+Line 1
+
+
+Line 1
+
+
+%% eeeeee
+
+\newpage
+
+A theme of this article will be that
+% distinction 
+the distinguishing
+between questions Q-1 and Q-2 and 
+the separation of Observables from
+Unobservables
+is 
+related
+% likely central 
+to the mystery
+% that has enshrouded 
+enshrouding
+the Second Incompleteness Theorem.
+This is
+%is germane to the aspirations of automated theorem proving
+%will be germane to this article 
+because there 
+%is no doubt
+can be no doubt that
+% can be no question 
+%%%%%%%% that 
+the Second
+Incompleteness Theorem is  fully
+robust
+% result 
+from a purist 
+%pristine 
+mathematical perspective.
+Yet,
+it is still problematic to fully
+% 
+%  simultaneously
+% % at the same time,
+% it is 
+% hard to 
+% entirely
+% 
+dismiss
+ Hilbert's 1926
+suggestion that 
+ some 
+specialized forms of logics should
+%declaration 
+%% 
+%% concerns
+%% in $\,*\,$ 
+%% that 
+%% {\it ``the honor of human understanding''}
+%% requires
+%% examining
+%% % explaining 
+%% % considering
+%% how  logic systems can
+%% 
+possess
+a type of well-defined 
+ knowledge about their
+own 
+internal
+consistency.
+(This is because it is
+highly
+ awkward to explain how and why
+human beings 
+are able to
+%can 
+%manage to 
+motivate 
+their 
+%cogitations,
+cognitive process,
+% themselves to think,
+ if they do not own
+some type of 
+% instinctive
+internal
+knowledge about their own
+ consistency.)
+
+% sufficient
+% % enough 
+%  knowledge about their
+% % own 
+% internal
+% consistency
+% to motivate 
+% cognition.
+
+% Bad change above
+%cogitation.
+% themselves to cogitate. 
+
+%%It is also
+%%especially 
+%%% very 
+%%tempting 
+%%to divide Hilbert's Year-1900
+%%Open Question into its Q-1 and Q-2 separate parts
+%% during the 21st century,
+%%as computers share with humans cogitative abilities.
+%%
+%%Maybe DELETE above sentence ??? 
+%\end{remm}
+
+% \baselineskip = 1.8 \normalbaselineskip 
+
+\smallskip
+
+The next 
+section will 
+formalize our
+% new
+ proposed IQFS formalism.
+% 
+% describe our 
+% 2-part
+% conjecture about how 
+% %a
+% Double-Formatted Logics are 
+% likely to 
+% %produce some 
+% cast
+% new perspectives
+% on 
+% this topic.
+% %the nature of the Second Incompleteness Theorem.
+% 
+Before starting this subject, it should be mentioned
+that other unusual interpretations of the Second Incompleteness
+Theorem have followed
+from Gentzen's perspectives about
+transfinite induction 
+under his $\epsilon_0$ ordinal
+\cite{Ge36,Ta87}, the 
+%% 
+%% 
+%% explore
+%% how \cite{wwapal}'s results for a Single-Formatted logic
+%% can be revised
+%% % with our new $~\zzthe~$ function
+%% under a
+%% 
+%% Before 
+%% broaching
+%%  this topic it should be mentioned that
+%% %0fascinating
+%% other approaches to
+%% %efforts to partially 
+%%  the Second Incompleteness Theorem 
+%% % do
+%% have centered around
+%% 
+ Kreisel-Takeuti's    ``CFA''
+system \cite{KT74}
+and also
+the {\it interpretational frameworks} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to 
+our 
+%% main
+%\cite{ww93}--\cite{ww14}'s 
+methods.
+%approach.
+They
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Also,
+they
+%apply to 
+employ
+``cut-free'' logics
+(rather 
+than
+a 
+% preferable 
+Hilbert-style 
+deductive 
+apparatus).
+%that 
+%%%%%%%%%%% explored 
+%%%%%%%%%%% in 
+%%%%%%%%%%% \textsection \ref{ss32} ).
+%%%we are considering).
+%%
+%%Instead, CFA uses the 
+%%special
+%%properties of ``second order'' generalizations of Gentzen's
+%%{\it cut-free}
+%%Sequent Calculus, 
+%%and 
+%%the
+%%interpretational approach
+%%formalizes how some systems 
+%%recognize their
+%% Herbrand consistency 
+%%on localized sets of integers,
+%%which 
+%%unbeknownst to 
+%%themselves,
+%%includes all
+%%integers.
+%%
+%%%These
+%   alternate 
+%%%approaches 
+Their 
+%alternate 
+% very
+ fascinating
+perspective  
+should
+% certainly,
+ be examined by researchers
+interested in the
+Second 
+Incompleteness Theorem,
+although 
+%but
+it is
+%% 
+% they are 
+unrelated to 
+our particular
+% the next section's 
+%specific analysis of
+%%% type of
+Hilbert-styled self-justifying effects,
+studied in the current article.
+
+
+%% systems 
+%% formalizing
+%% %verifying 
+%% their
+%% own consistency
+%% %%%%%Definition \ref{def-2.2}'s
+%% %%% approximate 
+%% under
+%% Hilbert-styled 
+%% deduction.
+
+
+%deduction.
+%  Hilbert deduction.
+
+%methods.
+%formalism.
+
+
+%% It is,
+%% % They
+%% %are, 
+%% however,  not germane to the next section's
+%% perspective.
+
+%methodology.
+%main formalisms.
+%methods.
+%results.
+
+ % \baselineskip = 1.8 \normalbaselineskip 
+
+%\section{
+%\small 
+%Improving \cite{wwapal}'s Results with a 
+%``Double-Formatted'' Logic  }
+
+\newpage
+xxxxxxxxxxx
+
diff --git a/nachlass/collected_dew_materials/2011-2019/wollic-2019.bak b/nachlass/collected_dew_materials/2011-2019/wollic-2019.bak
new file mode 100644
index 0000000..30cb459
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/wollic-2019.bak
@@ -0,0 +1,2622 @@
+% 2019 feb24 8am while listeing to jazz
+% 1019 feb22  1am 1-word change
+%  2018 feb-17 11.30 pm while listenning to Bobby Dee
+
+%% seth and I agrred to meet on thursday 3pm
+
+
+\documentclass{llncs}
+\usepackage{amssymb}
+
+
+\newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+
+
+
+
+
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\begin{document}
+
+
+ \title{On the Breadth and
+Also
+ Limitations of the
+ Second Incompletenss
+Theorem}
+
+
+\def\aaa{\beta}
+\def\ccc{Class}
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+\author{  Dan E.Willard}
+
+
+\institute{University  at Albany Computer Science and Mathematics Departments}
+
+
+\maketitle
+\pagestyle{plain}
+
+%\bigskip
+
+
+\normalsize  
+
+\begin{abstract}
+%%% large \baselineskip =  2.4 \normalbaselineskip 
+This article will investigate several quite similar-looking
+ axiom systems that seek to confirm
+their own consistency via the use of
+self-referencing axiomatic
+statements, roughly declaring that
+{\it ``I am consistent''}.
+Surprisingly although our
+two main
+ investigated formalisms will differ only
+slightly in the fine print of their particular
+ definitions,
+one among these  systems will be consistent, while
+the other
+is inconsistent.
+This result will  clarify 
+both
+the breadth and 
+limitations
+ of the Second Incompleteness Theorem.  
+It will also have practical consequences.
+
+\end{abstract}
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\hgskip{ \medskip }
+
+\def\njp{\newpage}
+\def\njp{ }
+
+\def\nskip{\bigskip}
+
+
+%\Large
+%\baselineskip =  2.07 \normalbaselineskip 
+
+ \large \baselineskip =  1.5 \normalbaselineskip 
+% \large \baselineskip =  2.3 \normalbaselineskip 
+%\Large \baselineskip =  1.9 \normalbaselineskip 
+%% \LARGE
+\normalsize  \baselineskip =  2.6 \normalbaselineskip 
+\Large \baselineskip =  2.6 \normalbaselineskip 
+
+\large \baselineskip =  2.6 \normalbaselineskip 
+
+
+ \normalsize \baselineskip =  1.0 \normalbaselineskip 
+\parskip 1pt
+
+% \normalsize \baselineskip =  3.0 \normalbaselineskip 
+%%% \normalsize \baselineskip =  3.4 \normalbaselineskip 
+
+% \large \baselineskip =  3.0 \normalbaselineskip 
+  \normalsize \baselineskip =  2.22 \normalbaselineskip 
+% \large \baselineskip =  2.22 \normalbaselineskip 
+\normalsize \baselineskip =  1.77 \normalbaselineskip 
+\normalsize \baselineskip =  1.0 \normalbaselineskip 
+
+
+\section{Introduction}
+%11111
+\setcounter{section}{1}
+
+
+
+We have published a series of papers about generalizations and
+boundary-case exceptions to the Second Incompleteness
+during the last 25 years
+% cite {????},
+\cite{ww93,ww1,ww2,ww5,wwapal,ww6,ww7,ww9}
+including six papers that have appeared in the JSL and APAL.
+Our goal in the current article will be to focus on
+\cite{ww5}'s
+ IS$_D(\beta)$
+ formalism.
+It was  shown in
+ \cite{ww5} 
+that if $\beta$ denotes any consistent extension of 
+Peano Arithmetic and  if $D$ denotes semantic tableaux deduction,
+then  IS$_D(\beta)$ 
+will be a consistent axiom system that
+can simultaneously prove isomorphic analogs of all of
+Peano Arithmetic's $\Pi_1$ theorems while also corroborating
+its own consistency under semantic tableaux deduction.
+The current article will show, surprisingly, that if one changes the
+definition of $D$ in a relatively minor manner, corresponding
+to what we shall call {\it ``Extended Semantic Tableaux'',} then 
+the resulting modification of 
+ IS$_D(\beta)$ will 
+become inconsistent (and thus
+ useless).
+
+This result
+% does
+will
+ not lessen 
+the significance of 
+ \cite{ww5}'s prior result.
+It 
+%does,
+will,
+ however,
+% will 
+clarify 
+its
+basic
+ theoretical meaning.
+% of  \cite{ww5}. 
+Most of 
+our current discussion
+will
+% shall
+ be addressed to an audience
+of theoretical logicians, but
+% an important short
+a brief
+3-page passage 
+ (in Section \ref{sect5}) 
+will explain
+how some related
+future
+ AI software could
+plausibly relieve
+% section
+% two  pages
+ global warming.
+It
+%This latter discussion
+ will suggest the contrast
+between our positive and negative results,
+together with
+future
+results
+developed
+by other
+ logicians,
+ should help
+significantly
+% fine-tune
+refine
+ future artificially intelligent software
+systems
+that seek to ameliorate the
+harmful
+ effects from global warming.
+
+
+The author of this article has now reached the retirement age of 70.
+We 
+hope
+ this article 
+will be  written
+in a style that  encourages
+other
+logicians to investigate 
+its  subject further.
+%With such a goal in mind,
+We will strive to make this paper
+comprehensible to a 
+broad audience, and we
+shall
+ briefly discuss its
+pragmatic significance.
+
+\section{General Perspective}
+\label{sect2}
+
+%%22222
+
+It is well known that
+G\"{o}del's Incompleteness Theorem 
+is a 2-part result.
+Its
+first half  specifies no decision
+procedure can identify all
+arithmetic's
+ true statements.
+On the other hand, the 
+ ``Second Incompleteness Theorem'' 
+ assures that
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his
+%%!! historic 
+paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+meet some success:
+\begin{quote} 
+$*~$ {\it ``It must be 
+expressly 
+noted 
+Proposition XI''
+{\rm (e.g. the Second Incompleteness Theorem)} 
+``represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialectica interpretation.
+On the other hand,
+%%d the Stanford Encyclopedia's
+%%d entry about  G\"{o}del
+%%d quotes him,
+%%d in its
+%%d  Section 2.2.4,
+%%d stating
+%%d he was hesitant to
+%%d view     the
+%%d Second Incompleteness Theorem
+%%d  as    
+%%d fully
+%%d ubiquitous, until 
+%%d learning 
+%%d of Turing's
+%%d work.
+%%d Moreover,
+%
+Yourgrau's biography of
+G\"{o}del \cite{Yo5}
+indicates
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+had initially
+presumed his
+second theorem
+was false, before
+he had
+ proved his stunning 
+result.
+
+It is, of course, obvious that 
+the Second Incompleteness Theorem has forced 
+the objectives of Hilbert's Consistency Program
+to be radically reorganized. For instance,
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+Yet
+despite this fact,
+ Gerald Sacks (who 
+interacted
+% extensively 
+with
+G\"{o}del,
+ during the early 1960's
+ at the Institute for Advanced Studies) 
+recalls \cite{YouSa14}
+G\"{o}del declaring,
+{\it thirty years after  \cite{Go31}'s  publication,}
+that a type of 
+partial
+revival of Hilbert's Consistency
+Program, using at least Gentzen's approach,
+should be considered (see the below footnote 
+\footnote{\label{f2}
+Some quotes from Sacks's
+YouTube talk
+\cite{YouSa14}
+ are that G\"{o}del
+ {\it ``did not  think''}
+the objectives of Hilbert's Consistency Program 
+{\it ``were erased''} 
+by
+the Incompleteness Theorem, and
+G\"{o}del believed (according to Sacks) 
+that  it left
+  Hilbert's program 
+{\it ``very much alive and
+even more interesting than it initially was''}. 
+} 
+for a summary of 
+ Sacks's  observations). 
+
+The research that has followed
+ G\"{o}del's  seminal  1931 discovery has 
+ focused mostly
+% mainly
+on studying generalizations of the Second Incompleteness
+Theorem
+(instead of also viewing its
+ boundary-case
+exceptions). Several generalizations 
+of the Second Incompleteness Theorem 
+\cite{AB1,AZ1,Be17,Be14,BS76,Bu86,BI95,Fe60,Fr79a,Ha11,HP91,KT74,Pa71,PD83,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,ww2,wwapal}  
+are quite beautiful. The author of this paper has been
+            especially impressed by a generalization of the
+Second Incompleteness Effect, arrived at by the 
+combined work of Pudl\'{a}k and  Solovay,
+abetted by
+the research of
+ Nelson and Wilkie-Paris\cite{Ne86,Pu85,So94,WP87}.
+These results, which 
+  also have been more recently
+discussed 
+ in \cite{BI95,Ha7,Sv7,ww1},
+have noted the Second Incompleteness Theorem
+does not require the presence of the Principle of Induction
+to apply to most formalisms that use a Hilbert-style form of 
+deduction. (The 
+next chapter will offer a detailed summary of this 
+important generalization
+ of
+the Second Incompleteness Theorem in
+its
+% the paragraph called
+ Remark \ref{nremm-2.5}.) 
+
+
+Our research, during the last 25 years has 
+had a different focus,
+exploring 
+Boundary-Case exceptions to the Second Incompleteness Theorem
+with equal intensity as
+its generalizations.
+It would be natural for many readers to ask why such
+exceptions should also be studied
+with 
+%such
+fully
+equal intensity?
+
+One reason is that while generalizations of the Second Incompleteness
+Theorem are pure
+%%seth  form 
+from
+a mathematical standpoint, it must
+not be forgotten that
+our civilization
+must confront
+ecological issues
+ in the future, more important 
+ than how to devise short proofs for the existence
+of a
+large number
+that  cannot be easily encoded in a binary format,
+ such as
+a  googolplex $= \,2^{2^{100}}$.
+
+
+The current paper will explore
+G\"{o}del's 
+% underlying
+motivation for making his
+ statement $*$ and
+Hilbert's
+% similar
+closely
+related famous
+  declaration
+    $**$ from
+ \cite{Hil26}: 
+
+\begin{quote}
+$**~$
+{\it ``
+Let us admit that the situation in which we presently
+find ourselves with respect to paradoxes is in the long
+run intolerable. Just think: in mathematics, this paragon of
+reliability and truth, the very notions and inferences,
+as everyone learns, teaches, and uses them, lead to absurdities.
+And 
+where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?''}
+\end{quote}
+A surprising facet of both $*$ and $**$ is that our short 3-page
+discussion 
+ (in  \textsection \ref{sect5}), 
+about how to ameliorate the effects of global warming,
+will be related
+to Hilbert's and G\"{o}del's
+ predictions, about the
+underlying
+ importance of  
+boundary-case exceptions to the Second Incompleteness
+Theorem.                   
+
+
+\section{ Notation for Introducing  Main Formalism}
+%%  333333   }
+
+\label{nnn2}
+
+Let us 
+call an
+ordered pair $(\alpha,D)$ a
+    {\bf Generalized Arithmetic Configuration}
+(abbreviated as a {\bf ``GenAC'' })
+when  its 
+first and second 
+components 
+are 
+defined 
+as 
+follows:
+\bee
+\item
+The {\bf Axiom Basis} ``$~\alpha~$'' 
+for a 
+ GenAC
+%ssss  Generalized Arithmetic
+will be defined as
+the set of 
+ proper axioms 
+it employs.
+\item
+The second component ``$~D~$'' of a 
+ GenAC
+%ssss  Generalized Arithmetic
+will represent
+the 
+{\it combination} of its formal rules of inference
+with 
+%its
+the
+ logical axioms ``$~L_D~$'' it employs.
+The
+term  {\bf ``Deductive Apparatus''} will be often
+used to refer to $D$. 
+\ene
+
+% \cite{End,Fit,HP91,Mend}
+
+
+\begin{example}
+\label{nex-2.1}
+\rm
+This notation 
+allows us to
+ conveniently separate  the logical axioms
+$~L_D~,~$ associated with  $(  \alpha  , D  )~$, from 
+its
+``basis axioms'' $\, \alpha \,$.
+It also allows one to  compare
+the various
+deductive apparatus techniques
+that  
+have
+appeared in the literature.
+For instance,
+the
+  $~D_E~$ apparatus,
+introduced
+ in 
+\textsection
+ 2.4 of  Enderton's textbook \cite{End}, 
+has
+ used only  modus ponens
+as a rule of inference,
+combined with a 
+complicated
+4-part  schema of logical axioms.
+This differs from
+the  $~D_M~$ ,  $~D_H~$ and  $~D_F~$ approaches of
+Mendelson \cite{Mend}, 
+H\'{a}jek-Pudl\'{a}k \cite{HP91}
+and Fitting \cite{Fi96}.
+The former two textbooks
+employ a simpler set of logical axioms
+than $\, D_E \,$,
+but they require
+two rules of inference
+(modus ponens and generalization).
+The  $~D_F~$ apparatus, appearing in Fitting's textbook \cite{Fi96},
+as well as its predecessor due to Smullyan \cite{Sm95},
+actually  employ {\it no logical axioms.}
+Instead,
+\cite{Fi96,Sm95}
+ rely upon a
+``tableaux style'' method for generating a 
+%consequently
+%complicated
+larger number of
+rules of
+inference.
+\end{example}
+
+
+\begin{definition}
+\label{ndef-2.2}
+\rm
+Let
+$ \, \alpha \, $ again 
+denote an axiom basis, 
+and $ \, D \, $ 
+designate
+ a
+deduction apparatus.
+Then 
+the  GenAC of
+ $(  \alpha  , D  )$
+will
+% shall
+be called  {\bf Self Justifying}
+ when
+\begin{description}
+% \xxitch
+% \small
+  \item[  i.   ] one of  $~(  \alpha  , D  )$'s  theorems
+(or possibly one of $\alpha$'s axioms)
+does
+%will
+state that the deduction method $ \, d, \, $ applied to the
+basis
+system $ \, \alpha, \, $ 
+%will 
+produces a consistent set of theorems, and
+\item[  ii.   ]
+     the GenAC formalism $ \,( \alpha,D)  \, $ is, in fact,
+actually consistent.
+\end{description}
+\end{definition}
+
+
+\begin{example}
+\label{nex-2.3}
+\label{ex2}
+%%%%%%%%%%%%%%%%%%%  OLD \label{ex-2.5}
+\rm
+Using 
+Definition \ref{ndef-2.2}'s 
+ notation, our
+prior
+ research  
+% in
+\cite{ww93,ww1,ww5,wwapal,ww9}
+% has
+constructed
+%developed
+GenAC pairs  
+% arithmetics
+$~(  \alpha  , D  )$
+that 
+ were
+``Self Justifying''.
+We
+also 
+proved
+the Incompleteness Theorem 
+implies specific
+limits beyond which 
+self-justifying
+formalisms
+simply
+ cannot transgress.
+For any  $\,(\alpha,D) \,$, 
+it is 
+thus
+easy
+%almost trivial 
+to construct a 
+system $ \, \alpha^D \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition
+(in an isolated context {\it where the Part-ii condition is
+ not also
+satisfied}).
+For instance,  $ \, \alpha^D \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+the added {\bf $\,$``SelfRef$(\alpha,D)$''$\,$} sentence,
+defined below: 
+%% as stating:
+\begin{quote} 
+% \small
+$\oplus~~~$ 
+There is no proof 
+(using 
+$D$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,D) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+showed \cite{Kl38} 
+how
+to
+encode
+%  rough
+ analogs of 
+the above  ``SelfRef$(\alpha,D) \,$'' statement,
+which we often call an 
+ {\bf $\,$``I AM CONSISTENT'' 
+%axiomatic
+ declarative axiom.}
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71},
+however,
+ emphasized
+% that
+$\alpha ^D$ 
+may
+be inconsistent
+(e.g.  violating Part-ii of   self-justification's
+definition),
+{\it despite SelfRef$(\alpha,D)$'s 
+formal
+assertion.}
+This is because if the 
+ pair $(\alpha,D)$ is too strong
+then a
+quite conventional
+G\"{o}del-style diagonalization argument can
+be applied to the axiom basis of
+$\alpha^D~~=~~ \alpha \, + \, $ SelfRef$(\alpha,D), ~$
+where the added presence of the statement 
+SelfRef$(\alpha,D)$ 
+will cause this extended version of 
+$\, \alpha\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, an
+encoding for
+``SelfRef$(\alpha,D)$'' is relatively easy,
+via an application of the Fixed Point Theorem,
+but this sentence
+ is, typically
+{\it
+utterly
+useless!} 
+\end{example}
+
+% \bigskip
+
+\parskip 0pt
+
+
+\begin{definition}
+\label{ndef-2.4}
+\rm
+Let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote two 3-way predicates,  specifying 
+ $x+y=z$ and $x*y=z$,
+for which the
+associative, commutative, identity and distributive 
+properties have $\Pi_1$ style encodings under an
+axiom system of $\alpha$.
+Then we will say
+that $~\alpha~$
+{\bf recognizes} successor,  addition  and multiplication
+as {\bf Total Functions} iff 
+it can  prove all of
+\eq{totdefxs} - \eq{totdefxm}
+as theorems:
+\end{definition}
+% \newpage
+{ \small
+\baselineskip =  .9 \normalbaselineskip 
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+%%% \vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+
+\noindent
+Furthermore, we will call the GenAC system $(\alpha,D)$ a
+% $\alpha$ will be  called 
+{\bf Type-M} 
+formalism
+iff it includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems, {\bf Type-A} if it includes 
+only \eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+ $(\alpha,D)$ 
+ will be 
+% is
+called 
+{\bf Type-NS} iff it  can prove
+none of these theorems.
+
+\parskip 2pt
+
+%\bigskip
+
+\begin{remark}
+\label{nremm-2.5}
+%% 
+\rm
+The separation of GenAC systems into the four 
+categories of Type$-$NS, Type-S, Type-A and Type-M systems
+will enable us to nicely summarize the prior literature
+about generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem. This is because:
+\bed
+\item[   $~~~~$i.$~~$  ]
+The combined research of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as is formalized by Theorem $\, ++ \,$,
+implies no
+natural Type$-$S  GenAC system $(\alpha,D)$
+can recognize  its own  consistency
+when $D$ is one of 
+Example \ref{nex-2.1}'s three
+examples of
+  Hilbert-style 
+deduction operators of 
+$\, D_E \,$, $\, D_H \,$  
+or
+$\, D_M ~~$. In particular,
+it establishes the following result:
+\medskip
+\begin{quote}
+\normalsize \baselineskip = 1.0 \normalbaselineskip 
+{\bf ++ }
+{\it 
+(Solovay's  
+modification
+\cite{So94}
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+using some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let 
+$ \, (\alpha,D) \, $ 
+denote 
+a GenAC system
+supporting
+% which contains 
+the
+\el{totdefxs}'s
+Type-S statement
+and 
+assuring
+the successor operation
+will
+satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then
+$ \, (\alpha,D  ) \, $  
+%%%%$~\alpha~$
+cannot verify its own
+%will be unable to recognize its
+%own  
+consistency
+whenever
+simultaneously
+ $D$ is some type of
+a Frege-Hilbert
+deductive
+apparatus and
+%whenever
+$~\alpha~$
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+their usual % identity,
+associative, commutative 
+ distributive 
+and identity 
+%  axiomatic properties.
+axioms.
+%   -axiom
+% properties.
+\end{quote}
+\medskip
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of theorem $++$.
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+ by  Buss- Ignjatovi\'{c},
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k's initial 1985 article  \cite{Pu85} 
+% implicitly
+captured
+% , notably, 
+the majority 
+%most
+%%%  much 
+of $++$'s 
+essence, chronologically before Solovay's observations,
+%Also,
+ and
+Friedman did
+% some
+related work
+ in
+\cite{Fr79a}.
+
+\medskip
+
+\item[   $~~$ii.$~~$  ]
+Part of what makes  $++$ interesting is that 
+\cite{ww1,ww5,wwapal}
+explored two methods for 
+GenAC systems
+to confirm their own consistency, whose
+natural hybridizations  are precluded by $++$.
+Specifically,  these results involve using
+Example \ref{nex-2.3}'s 
+self-referencing  {\it ``I am consistent''}
+ axiom (from its
+statement  $\oplus$ ). 
+They will enable
+some (not all)
+  Type-NS  
+systems \cite{ww1,wwapal}
+to verify their   own consistency under 
+a Hilbert-style deductive apparatus
+\footnote{ The Example \ref{nex-2.1} had
+provided
+three examples of
+  Hilbert-style 
+deduction operators, called 
+$\, D_E \,$, $\, D_H \,$ , 
+and
+ $\, D_M ~~$. It explained how these 
+ deductive operators differ from a tableaux-style
+deductive apparatus by containing a modus ponens rule.},
+or alternatively allow 
+some (not all)
+ Type-A 
+ systems \cite{ww93,ww1,ww5,ww6} to 
+corroborate
+their 
+self-consistency
+under a more restricted semantic
+tableaux style deductive apparatus.
+Also, we observed in  \cite{ww2,ww7} how one could
+refine $++$ with Adamowicz-Zbierski's
+methodology \cite{AZ1} to show 
+ most Type-M  systems
+cannot recognize their own semantic tableaux style consistencies.
+\ennd
+\end{remark}
+
+\begin{remark}
+\label{rem2}
+Several of our articles have
+used
+Example \ref{ex2}'s
+ {\it ``I am consistent''} 
+axiomatic statement
+$~\oplus~$
+% as a vehicle 
+ to evade the Second Incompleteness Theorem.
+This methodology is 
+unrelated to alternate 
+techniques for evading G\"{o}del's Theorem
+that have been explored by 
+Gentzen in \cite{Ge36},
+Kreisel-Takeuti 
+in
+% the analysis of
+ their    ``CFA''
+formalism \cite{KT74}
+and to
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+It turns out these latter
+systems will not make as broad statements
+about their consistency because they do not
+make use of
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+On the other hand,
+they will possess a
+% much
+ different
+style
+% kind 
+of advantage
+because they will provide a more detailed proof
+of their consistency than would follow from 
+statement $~\oplus\,$'s
+single-sentence  {\it ``I am consistent''} declaration.
+\end{remark}
+
+
+\section{More  Notation and Main Formalism}
+%%mmm  
+
+%%% 444444
+
+\label{sect4}
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1 \ldots a_j) 
+\leq   Maximum(a_1 \ldots a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+$Maximum(x,y),$
+$ Logarithm(x)$
+$\,Root(x,y) \, =  \, \lceil  \, x^{1/y} \,  \rceil$ and
+$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+that included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+A wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+%{\it Also throughout this article,}
+ Also,$~$
+ the sentence $\Psi$ will be called a$~$ {\bf  Rank-1* } $~$statement iff it
+can be encoded in either a  $\Pi_1^*$
+or  $\Sigma_1^*$ format.
+
+Our article \cite{ww5} used the symbol $~D~$ to denote
+a deduction method. 
+There will be three variants of deduction methods
+% that we will compare 
+compared
+in this article. The first will be
+{\it semantic tableaux},
+ which will receive the abbreviated name of
+ {\bf Tab}. It will be similar to 
+%%% its analog
+% appearing 
+the tableaux formalism
+in Fitting's textbook  \cite{Fi96}.
+Thus a Tableaux proof of a theorem $~\Psi$
+ from an axiom basis
+% a set of  proper axioms 
+$~\alpha~$ will be  tree-like
+structure that begins with the sentence
+ $~\neg ~ \Psi$ 
+ stored in the tree's root and whose every root$-$to$-$leaf
+path establishes a contradiction by containing some pair of contradictory
+nodes  that ``closes'' its path. The rules for generating internal
+nodes along each
+ root$-$to$-$leaf
+path will be that each node
+must be either a
+% formal 
+proper axiom of $\alpha$ or a deduction
+from an ancestor node via one of the 
+``elimination''
+rules associated with
+our logic symbols of $\wedge$, $\vee$, $\rightarrow$, $\neg$,
+$\forall$, or $\exists$. (A precise definition of these six rules
+and other details
+appears in the attached appendix.) 
+
+\smallskip
+
+Our second 
+explored
+ deductive apparatus
+% in this article, 
+will be called {\it Extended Tableaux}, abbreviated as
+{\bf Xtab}. Its definition 
+will be
+ identical to the prior
+paragraph's {\it Tab-}deduction, {\it  except that} for any sentence
+$\phi$
+in our  language $L^*$, the sentence $\phi \, \vee \, \neg \phi$
+can be permissibly 
+  stored 
+% inserted 
+as an axiom
+inside
+ any internal node of our proof tree.
+(In other words,    {\it Xtab-}deduction
+will differ from  {\it Tab-}deduction by allowing all instances
+of the Law of Excluded Middle to appear as a logical schema of axioms.
+In contrast,  {\it Tab-}deduction will treat the infinite schema of
+instances of the Law of Excluded Middle as
+%  permissibly
+{\it  derived theorems,}
+{\bf BUT NOT ALSO} as
+instances of
+  logical axioms.
+
+% \medskip
+
+Our third
+% studied
+ deductive apparatus
+will be called {\bf Tab-1}.
+It will be a
+% type of
+  compromise between Tab and Xtab,
+% deduction,
+where  a ``Tab-1'' proof of $\Psi$ from
+an axiom basis
+$\alpha$  is defined as a set of
+ordered pairs $ \, (p_1,\phi_1), \, (p_2,\phi_2), \, .. (p_k,\phi_k) \, $
+where
+\bee
+\item $ ~ \phi_k ~  = ~  \Psi \,$ 
+\item
+Each  $~p_j~$ is a Tab-proof of
+what we have called
+ a Rank-1* sentence
+$~\phi_j~$ from the union of $~\alpha~$ with some further Rank-1* axioms
+of   $~\phi_1,~\phi_2,~..~\phi_{j-1}~$.
+\ene 
+{\bf We emphasize} that  Tab-1 is 
+{\it  less efficient}
+ than Xtab 
+deduction
+because {\it the$ \,$former
+requires} $\phi_j$ be a Rank-1* sentence, while Xtab
+{\it  does not impose}
+ a similar Rank-1* constraint upon
+$\, \phi \,$, when
+ it
+% uses
+invokes
+  an  $~\phi \, \vee \, \neg \phi~$ axiom.
+
+
+Let us say that an axiom system $\alpha$ owns a
+{\bf Level-1} appreciation of its own self-consistency
+(under a deductive apparatus $D$)
+iff it can verify 
+$D$ produces
+ no two simultaneous
+proofs of a 
+ $\Pi_1^*$ 
+sentence and 
+its negation.
+Within this
+ context, where $~\aaa~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(\aaa)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$\aaa$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms will define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+(They will  enable our formalism to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.)
+
+\item[Group-1:] 
+This axiom group  will consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5} had  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_\aaa(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $\aaa$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema will contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(\aaa)$ can trivially prove
+ all $\aaa$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_\aaa(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+The final part of  IS$_D(\aaa)$ 
+% shall
+will 
+be a
+self-referencing
+$\Pi_1^*$
+axiom,
+that indicates
+IS$_D(\aaa)$
+is
+``Level-1 consistent'' 
+under
+ $D$'s deductive method.
+It 
+is, 
+thus,  the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+\smallskip
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+had appeared
+ in 
+\cite{ww5}.
+Thus, 
+\eq{group3} 
+is a
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+$~$when:  
+$~~1)~~ \, \mbox{Prf} \, _{\mbox{IS}_D(\aaa)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(\aaa)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and $~~2)~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+%%%%% \parskip 2pt
+%\smallskip
+
+\begin{definition}
+\label{def3} \rm
+Let $~D~$ denote 
+any one of 
+ the {\it Tab,  Xtab} or
+{\it Tab-1}
+ deductive methodologies.
+We will  say that 
+the mapping  $\mbox{IS}_D(~\bullet~)$ 
+ is
+{\bf Consistency Preserving}
+ iff
+ $\mbox{IS}_D(\aaa)$ 
+is guaranteed to be consistent 
+under $D$'s deductive apparatus
+whenever all the axioms of $\aaa$ hold
+true under the standard model of the natural numbers. 
+\end{definition}
+
+%\smallskip
+
+
+
+%%%%% 66666 NEW PASSAGE
+
+\begin{theorem}
+\label{th1} \rm
+The     $\mbox{IS}_{\it Tab}(~\bullet~)$ 
+and   $\mbox{IS}_{\it Tab-1}(~\bullet~)$ 
+mappings 
+are
+consistency preserving, but the similar-looking  
+   $\mbox{IS}_{\it Xtab}(~\bullet~)$ mapping 
+is actually not{\it $~~$. This 
+statement
+ is 
+ formalized  by the
+%   formalized, more  precisely,  by the
+Items (a) and (b) below:}
+\bed
+\item[  a.  ]
+ $\mbox{IS}_{Tab}(\aaa)$ 
+and
+$\mbox{IS}_{Tab-1}(\aaa)$ 
+are always consistent 
+whenever all the axioms of $\aaa$ hold
+true under the standard model
+ of the natural numbers. 
+
+\smallskip
+
+\item[  b.  ]
+In contrast,
+ $\mbox{IS}_{Xtab}(\aaa)$ 
+is {\it automatically inconsistent}
+whenever
+$~\beta~$
+proves 
+the
+%some
+ conventional $\Pi_1^*$ theorems
+showing 
+%that
+ addition and multiplication
+satisfy 
+their
+associative, commutative, 
+ distributive 
+and identity 
+properties.
+% principles.
+\ennd
+\end{theorem}
+
+{\it Proof Summary:}
+The two halves of Theorem \ref{th1}'s
+proof would be quite lengthy, if
+% we used
+%%% had to rely on 
+first principles
+were used
+ to justify
+them.
+Fortunately, there is an easier method to 
+justify
+(a) and (b), 
+% Theorem \ref{th1}'s two them
+by relying upon the earlier literature.
+
+% \parskip 2pt
+
+Thus, Part (a)  follows
+from  our
+ main 
+result
+in  \cite{ww5}.
+ It 
+indicated   $\mbox{IS}_{\it Tab-1}(~\bullet~)$ 
+was a consistency preserving
+%%%%  mapping, and this implies
+ mapping. This implies
+that  $\mbox{IS}_{\it Tab}(~\bullet~)$
+is also  consistency preserving
+because Tab-deduction is weaker than Tab-1.
+
+
+Let us now turn our attention to 
+Theorem  \ref{th1}'s second
+claim (b). 
+Its proof is more complex but analogous to the
+justification for the Invariant
+ ++ , which
+Remark \ref{nremm-2.5} 
+%   had
+attributed
+mostly  to
+ the work 
+of Pudl\'{a}k. Solovay, Nelson
+and Wilkie-Paris. The
+crucial aspect of the Frege-Hilbert methodologies
+employed by ++ is that modus ponens assures that
+a proof of a theorem 
+$\psi$ 
+from an axiom system $\alpha$ has a length no
+greater than the sum of the proof-lengths needed to derive
+ $\phi$ and $~\phi \rightarrow \psi~$ from
+$\alpha$. This 
+{\bf ``Linear-Sum Effect''
+}
+ does not apply,
+%%  technically, 
+actually,
+also to
+{\it Tab-}deduction because it owns no analog of
+a modus ponens rule (for assuring  that  
+$\psi$'s proof-length is bounded by the sum of the 
+lengths for the proofs of 
+ $\phi$ and $~\phi \rightarrow \psi~$) .
+
+The {\it Xtab}
+%%deductive 
+methodology, however, 
+ differs
+from
+{\it Tab-}deduction 
+by allowing any node of its proof-tree
+to store a sentence of the form $~ \phi \, \vee \, \neg ~\phi~$,
+as an application of its allowed use of the Law of Excluded
+Middle.
+This added feature allows an
+ {\it Xtab} proof for
+$\psi$ to have a length
+%% be bounded approximately  
+proportional to
+ the sum of
+the proof lengths for 
+ $\phi$ and $~\phi \rightarrow \psi~$. In particular,
+the relevant 
+ {\it Xtab} proof for 
+$\psi$ 
+can be summarized as having the following 4-part structure:
+\bee
+\item
+The root of the proof of $\psi$ 
+will be the usual temporary negated hypothesis of
+ $~\neg \psi~$ (which the remainder of the proof tree will
+show is impossible to hold). 
+\item
+The child of this 
+root
+node will be an allowed invocation of the
+Law of the Excluded Middle of the 
+ form $~ \phi \, \vee \, \neg ~\phi~$.
+\item Our tableaux proof tree will next employ
+the Appendix's branching rule for allowing the two
+sibling nodes of
+ $~ \phi ~$
+ and $~ \neg ~\phi~$ to descend from
+Item 2's node.
+\item 
+Finally, our tableaux proof will insert below
+(3)'s  left sibling  node
+of  $~ \phi ~$ a subtree structure that is no longer than
+the proof of   $~\phi \rightarrow \psi~$,
+and likewise
+insert
+ a   proof for $~\phi~$ 
+below (3)'s right sibling
+%%%%%%%  node  
+of  $~ \neg ~\phi~$.
+\ene
+The point is that the last step of our 4-part proof
+has a length no greater than the sum of the two
+ proof lengths for 
+ $\phi$ and $~\phi \rightarrow \psi~$,
+and its first three steps have inconsequential lengths
+that will increase 
+its
+ overall proof length by no
+more than an unimportant
+amount proportional to the length of the 
+sentence ``$~\phi ~ \rightarrow ~\psi~$''.
+
+%%%8888888888888
+
+%%{sect2}
+
+We can apply
+the preceeding 
+ ``Linear-Sum Effect''
+to construct  an analog 
+of Remark \ref{nremm-2.5} 's
+earlier  Theorem $\, ++ \,$ 
+germane to
+ Extended Tableaux  deduction (which is also called ``Xtab''). 
+Saving 
+several
+%the 
+details
+for a longer paper,
+the   intuition
+behind this analog is that modus ponens
+ {\it is the only
+ rule of inference} used
+ by 
+the classic
+ \cite{End}'s  textbook-style
+% classic
+first-order logic  system,
+ and
+Xtab can
+%  that Xtab
+% can
+% thus
+ use its 
+ Linear-Sum Effect
+  to simulate modus ponens.
+This natural analog of  
+ $++$  will
+%%%% then
+%% thus
+assure 
+%% that 
+any axiom system $~\cal{A}~$ is
+% then
+ {\it  automatically inconsistent} whenever:
+\begin{enumerate}
+\item
+$\cal{A}$
+ can verify Successor is a total function (as formalized
+by \el{totdefxs} ),   
+\item 
+$\cal{A}$
+can prove
+%%% 
+%%%  proves 
+%%%  $\Pi_1^*$ theorems
+%%% %%% showing 
+%%% that show
+%%% 
+ addition and multiplication
+satisfy 
+their usual 
+associative, commutative, 
+ distributive 
+and identity axioms. 
+\item  
+$\cal{A} \,$
+proves 
+an added
+ theorem (which turns out to be false) affirming its own
+consistency when the
+Xtab
+ deductive apparatus is used.
+\end{enumerate}
+The preceding observation completes 
+Theorem
+%%%  \ref{th1}'s
+1-b's
+ proof because 
+ $\mbox{IS}_{Xtab}(\aaa)$ 
+meets
+% satisfies
+all three of these
+requirements.
+%% ?????? (see footnote).
+$~~~\Box$
+ 
+
+
+\begin{remark}
+\label{rem3} \rm
+We 
+have
+deliberately resisted the temptation of providing a more
+elaborate proof of Theorem \ref{th1},
+beyond the above 
+summary
+in this
+% current
+ extended abstract.
+This is because 
+it is desirable
+ to reserve
+%%% three pages of space
+page space
+ for explaining
+how Theorem \ref{th1} (and other aspects of symbolic logic)
+can help
+ biologists, chemists, physicists and mathematicians
+develop formal  Artificial Intelligence mechanisms
+for
+%  alleviating 
+relieving$~$the harmful effects
+from global warming.
+\end{remark}
+
+
+\parskip 1 pt
+
+
+The contrast between the positive and negative results
+of Theorem \ref{th1}'s two claims will 
+be central to the next section's  discussion.
+Although the topic of global warming was
+essentially
+ unknown
+to the world of the  1920's and 1930's,
+the next section will show
+that
+ the often quoted statements
+$*$ and $**$, of G\"{o}del and
+Hilbert,
+%%%%%%%%% Hilbert
+do gain
+some
+new
+interpretations.
+
+%END OF NEW PASSAGE
+
+
+
+ 
+
+\section{An Application of Theorem  \ref{th1}
+Germane to Diminishing the Damage
+Caused by Global Warming} 
+
+% \large \baselineskip =  1.5 \normalbaselineskip 
+%%%555555555555555
+\label{sect5}
+
+We 
+%did become
+became
+ aware about 
+how the hard-core sciences of
+ biology, chemistry, and  physics could use concepts
+from Symbolic Logic (such as Self-Justification) 
+to alleviate the effects of global warming
+after  
+reading several
+% quite
+alarming
+ Year-2017 interviews of 
+the late physicist Stephen Hawking
+\cite{Ha17a,Ha17b}. 
+In those interviews, Hawking
+% has
+%joined 
+ did join
+a growing team of scientists
+concerned that if current trends continue, then
+global warming could cause the planet Earth to become too hot
+for mammals to survive, within one or two centuries.
+(Hawking actually states 
+%that 
+the extinction of all mammal species could
+ occur
+as early as
+% even
+ within ``100 years'',
+but we would prefer to
+optimistically
+ hope that there is, at least,
+an available  200-year window in  time before the most tragic results
+%will 
+occur, if indeed they cannot (?) be 
+prevented.)
+%avoided.) 
+
+Hawking has 
+%also 
+hinted that computers, employing various forms
+of Artificial Intelligence (AI), could be
+one part of
+% the 
+a 
+reply
+ to global warming's 
+emerging challenge
+(if AI can {\it somehow}   be handled  (?)  adeptly).
+The
+ two advantages of computers
+over humans are that computers
+can produce
+deductions more quickly, and they can 
+be engineered to
+ survive
+in environments that are inhospitable for humans.
+For example, computers do not need Oxygen as a vital energy
+supply. They could, thus, rely upon  solar power when residing
+on the Moon or Mars, and they could
+possibly survive
+ underneath the Earth's oceans
+(assuming the latter do not evaporate under intense heat). 
+
+% \parskip 0pt
+
+We want to keep this chapter brief and avoid 
+too much speculation.
+The main point is
+% to suggest
+ that
+AI-oriented
+ computers could possibly offer a
+partially positive outcome to a global warming
+crisis by employing a 2-part strategy where:
+\bee
+\item
+AI-based computers 
+will
+first take complete control over 
+the planet's
+destiny,
+if 
+(?) 
+humans are rendered 
+temporarily
+extinct by global over-heating.
+\item
+AI-based computers
+could then
+ subsequently restore human and 
+perhaps
+other 
+higher
+life forms on Earth after the planet returns 
+\footnote{This
+footnote is a 
+tentative  remark, but bacteria are known to be a  much
+more durable species  than mammals, which are capable of 
+surviving under very high temperatures and also able to
+undergo
+astonishingly
+speedy evolutionary changes and adaptations.  
+A new species of bacteria can  hopefully 
+(?) be
+engineered, 
+{\it perhaps with the assistance of computers,} that
+can hopefully pull greenhouse gasses out of the atmosphere
+and lower the temperature on Earth over 
+a spam of  perhaps
+%1,000 years. 
+several decades 
+Frozen samples of primate embryos and accompanying stored
+DNA molecules, could then be used to restore human life
+on Earth.}
+to a cooler state and stored frozen embryos and DNA samples
+%were
+have been
+ previously  (?) saved.
+\ene
+This methodology will be called a {\bf 2-Prong Strategy.}
+%We have no doubts its 
+%The author of this article has no doubt that
+Its two parts
+will, clearly, make
+some readers cringe with 
+discomfort. One should, however,
+%  take some measures to
+prepare for
+% plausible
+ worst-case
+scenarios, {\it  
+ in case
+such
+paradigms
+% circumstances
+ arise? }
+The next two pages
+% , thus,
+will 
+% explain particular
+focus on 
+ its 
+% very
+special
+% details germane to 
+implications for
+symbolic logic.
+
+
+
+\begin{example}
+\label{main-ex} 
+\rm
+%%%% eeeee
+%During our current discussion,
+%let us have
+Let
+ $\, \beta \,$ denote
+some
+% particular 
+consistent formal  axiom 
+basis
+% formalism
+% that has been selected to 
+that can
+prove a substantially
+broader
+ set of $\Pi_1^*$ theorems than some 
+% fixed
+conventional axiom system
+(such as
+% perhaps 
+Peano Arithmetic or ZF Set Theory).
+Let us consider 
+using Artificially Intelligent (AI)
+% systems 
+computers
+to combine the formalisms
+of the  preceding  2-Prong 
+methodologies
+ with
+% that of 
+Theorem \ref{th1}. 
+Then
+% the
+a
+% preceding paragraph's  
+2-Prong Strategy may
+potentially use any one of 
+ $\mbox{IS}_{Xtab}(\aaa)$,
+ $\mbox{IS}_{Tab}(\aaa)$,
+ $\mbox{IS}_{Tab-1}(\aaa)$ or say Peano Arithmetic (PA)
+as  its core invoked AI decision mechanism.
+A 
+comparison between these four
+% quite  different AI
+ variations of 
+ 2-Prong methods
+% does
+will
+  reveal that:
+\bed
+\item[   a. ]
+Although the  $\mbox{IS}_{Xtab}(\aaa)$ formalism may look
+superficially similar to 
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$, 
+it is
+actually
+% entirely 
+unacceptable as a
+%formal
+ core invoked AI 
+% mechanism, 
+formalism
+used by a  2-Prong strategy. This is 
+because Theorem 1-b
+indicates
+%  that
+  $\mbox{IS}_{Xtab}(\aaa)$ 
+is inconsistent.
+It will thus cause a 2-Prong strategy to make
+unacceptably false decisions 
+\footnote{ A surprising aspect of 
+ $\mbox{IS}_{Xtab}(\aaa)$'s faulty behavior is that its Group 0, 1 and 2 axioms
+are identical to those of 
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$.
+Thus,
+ $\mbox{IS}_{Xtab}(\aaa)$'s unacceptable behavior
+is solely because of its Group-3 {\it ``I am consistent''}
+axiom. \smallskip} 
+when it uses
+$\mbox{IS}_{Xtab}(\aaa)$ as its core AI mechanism.
+
+\smallskip
+
+\item[   b. ]
+In a context where $\beta$ has been chosen to be an axiom basis that
+proves a richer set of $\Pi_1^*$ 
+theorems than Peano Arithmetic,  PA
+is 
+substantially
+ less desirable 
+as a 2-Prong Strategy's
+%core
+main
+ AI 
+decision
+system than either  
+ $\mbox{IS}_{Tab}(\aaa)$ or
+ $\mbox{IS}_{Tab-1}(\aaa)$
+would be.
+This is
+partly
+  because PA produces a weaker set of
+ $\Pi_1^*$ theorems than the latter 
+ two 
+systems.
+From our perspective,
+a more important drawback of PA
+is that it is 
+substantially
+less adept when it is
+{\it formally  unable} to confirm its own consistency.
+
+\smallskip
+
+\item[   c. ]
+Although  both the
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$ formalisms 
+may be used as the core AI mechanism for
+our  2-Prong Strategy, 
+there are important distinctions
+between these two mechanisms. This is because
+ $\mbox{IS}_{Tab-1}(\aaa)$ 
+uses a more sophisticated form
+of a Self-Justifying 
+ Group-3 {\it ``I am consistent''}
+axiom than 
+ $\mbox{IS}_{Tab}(\aaa)$.
+(More generally,
+there is probably no
+% maximally
+ best form of Group-3 axiom
+that a 2-Prong Strategy can
+gainfully
+employ. 
+For instance,
+ it is 
+evident
+ (see footnote 
+\footnote{For instance,  
+ $\mbox{IS}_{Tab-1}$ can be beneficially hybridized with
+\cite{ww1}'s
+Tangibility
+  reflection principle, but we won't go into the
+%  germane
+details here. 
+} )
+that
+there do exist partially
+stronger forms of
+ Group-3 {\it ``I am consistent''}
+axioms 
+than those
+used 
+% applied besides those invoked
+ by
+ $\mbox{IS}_{Tab-1}(\aaa).~~)$
+\ennd
+\end{example}
+
+\parskip  0pt
+
+\begin{remark}
+\label{rem4}
+\rm
+The central issue raised by this chapter's
+2-Prong Strategy and the preceeding Example \ref{main-ex} 
+is 
+that
+ the dangers
+posed by global warming
+{\it  would be  tragic
+but NICELY only temporary,}
+%{\it temporary inconvenience}, 
+if robots and computers can reverse
+global warming after a period of 
+several
+ thousand years.  In contrast,
+the implications of global warming would be
+% much
+far
+ more
+severe,
+% tragic,
+if either it cannot be reversed or no frozen embryos
+%% of mammals (and primates)
+are saved so that the 4-billion-year cycle of life
+on Earth can 
+resume
+% restored
+after global  warming subsides.
+(We do not dismiss 
+the hopeful 
+prospect
+%point
+ that a theoretical temporary
+over-heating of the planet can possibly be avoided,
+but one
+ {\it  should  also not overlook} the 
+possibility
+ that some variant of a
+2-Prong Strategy may become necessary, if Mankind does not
+act  quickly enough ? )
+This article has, thus, been written to encourage a 
+more direct
+dialog
+% to take place
+between various subfields of Logic 
+with the hard-core
+sciences of 
+ biology, chemistry,   physics and mathematics.
+Its goal
+will be
+to discover what kinds of AI mechanisms can best reply to the
+challenges posed by global warming.
+% {\it if (?) they are needed !}
+\end{remark}
+
+\begin{remark}
+\label{rem5}
+Lastly, let us
+% should  
+%address 
+consider
+how G\"{o}del's Second Incompleteness
+Theorem
+should be 
+% likely
+viewed
+ by a broad 
+community
+         of scientists.
+% working in a 
+% diversity of different fields. 
+This topic is 
+% quite
+ subtle because the Second
+Incompleteness
+Theorem is 
+undoubtedly
+ a seminal result that was further strengthened
+by many of  G\"{o}del's 
+% subsequent 
+successors.
+But yet like many
+other
+ mathematical
+results, the Second Incompleteness Theorem also owns 
+{\it some types of 
+partial
+limitations?}
+%disadvantages? }
+% limitations?} 
+....Thus, we should
+  remind ourselves that
+ G\"{o}del
+% had
+% firmly
+held out the possibility 
+in
+% his  historic paper
+ \cite{Go31} 
+that some
+type of partial fulfillment of Hilbert's goals
+ would 
+%   ultimately 
+be achieved
+(see again G\"{o}del's quoted
+statement of$~*~ ).$
+Moreover,
+ Gerald Sacks in his 
+ Year-2014
+YouTube lecture
+(see
+\cite{YouSa14}
+and  our
+earlier
+%inserted
+% particular
+ footnote \ref{f2} ) 
+has recalled hearing
+G\"{o}del {\it repeating}
+% related
+similar
+ comments
+about the 
+importance 
+ of
+ continuing 
+ Hilbert's program
+ during the
+%  early
+1960's.
+%%% (This was {\it thirty years} after  \cite{Go31}'s seminal publication,)  
+ The main point is 
+that
+ a 2-Prong methodology, 
+permitting
+ an AI-based
+computer 
+%system 
+% has taken,
+% takes 
+to take temporary
+%temporarily, complete 
+control over 
+%the planet's
+planet Earth's
+destiny, {\it should
+%inherently 
+%substantially
+ultimately  be
+more efficient when
+it
+%  such
+%  a formalism 
+is allowed
+to presume its own consistency}.
+%
+%(similar to what humans commonly do).
+%
+%%  (as humans 
+%% seem to
+%% have
+%% % currently
+%%  informally
+%% done during their intuitive  thought processes).
+%% 
+These observations
+will
+ lead to
+significantly
+ new interpretations of
+G\"{o}del and Hilbert's statements $*$ and $**$.  
+\end{remark}
+
+%%% \parskip 4pt
+
+% \LARGE \baselineskip =  2.0 \normalbaselineskip 
+
+%6666666
+
+%999999
+
+In essence,
+several  research projects in 
+ symbolic logic should
+be undertaken that interact with
+the  core sciences
+of  biology, chemistry and  physics 
+%and Mathematics
+to collectively  address the
+% fundamental 
+challenges
+ posed by global warming.
+As a 
+logician (who was never
+% formally
+ trained in the experimental sciences), this author is
+%of this
+%article is
+% simply 
+ unable to assess how likely it is
+% that
+ that
+global warming will  need
+a 2-Prong response
+% necessary
+(where
+humans
+% will 
+do concede
+temporary control over the planet's destiny to AI-based computers).
+However, it is safe to presume
+that  such a loss of control would be
+only temporary 
+if  trustworthy self-justifying axiom systems
+are deployed that can simultaneously be:
+{\it 
+$\, 1)~$consistent,
+$\, 2)~$aware of their own consistency and
+$\, 3)~$capable of proving a rich set of $\Pi_1^*$ theorems.}
+
+
+
+
+% \parskip 1pt
+
+This article is  written in a context where its author
+has  reached the age of 70 and is retiring from teaching.
+Our hope is that the preceeding
+% above
+%Example \ref{main-ex} and Remarks \ref{rem4} and \ref{rem5} 
+discussion
+will encourage several logicians to  join  projects
+that   enhance  interactions 
+between the 
+logic community with the 
+allied research in
+ biology, chemistry,   physics and mathematics.
+A 
+tempting
+% nice 
+and 
+ new form of artificial intelligence
+ would
+% should 
+then
+%%%%% be likely 
+likely be
+born and 
+% nicely
+ nicely
+  prosper.
+
+
+\smallskip
+
+%ggggggg
+
+This article is dedicated to the
+memory of both
+my parents, 
+%%%Alfred and Ruth, 
+who
+% had  
+narrowly
+escaped
+the Holocaust.
+{\it $~$Let us pray that$\,$} 
+ global warming 
+% does
+ will
+%shall
+ not
+% cause
+% engulf
+impose
+% a much wider and extreme
+a
+% much
+ more wide-spread
+% wide-spreading
+calamity
+upon
+% all
+the 
+rest
+%remainder
+%
+% on the rest of
+ of humanity.
+We also hope
+%% Our hope is that  
+the
+% technical
+ mathematical 
+issues,
+raised in Sections 1-4, shall
+help future generations of researchers resolve the
+global warming crisis.... {\it  before it has become
+%sadly
+ too late ...  } 
+
+
+%% We felt  Section 5 of the current article
+%% should have a less mathematical tone than its Sections 1-4
+%% because the subject of global warming, together with its potentially
+%% required   (?) 
+%% % computerized 
+%% 2-Prong resolution, 
+%% %was sufficiently urgent to mention.
+%% should receive at least some attention
+
+
+%%  response to it,
+%% cannot
+%% %simply should not
+%% be ignored.
+
+%% 
+%% We suspect that logicians 
+%% can significantly
+%%  help
+%% biologists, chemists and physicists 
+%% resolve the global warming  crisis
+%% in the future.
+
+%hhhh 
+
+
+\parskip 0pt
+\normalsize \baselineskip =  0.96 \normalbaselineskip 
+
+\section*{APPENDIX  Reviewing
+the
+ Definition of a Tableaux Proof}
+
+Our definition of a semantic tableaux proof 
+is similar to its analog appearing in the textbooks by
+Fitting and Smullyan 
+ \cite{Fi96,Sm95}.
+A tableaux proof  of a theorem $\Psi$ from a set of proper
+axioms, denoted as $~\alpha~$, will  be a tree structure whose 
+root contains the temporary contradictory assumption of $~\neg \, \Psi~$
+and whose every descending root-to-leaf branch affirms a contradiction
+by containing both some sentence $\phi$ and its negation of $\neg \, \phi$.
+Each internal node in this tree will be either a proper axiom of
+$~\alpha~$ or a deduction
+from
+a higher 
+ancestor
+node in this tree using one of
+the
+ following
+six % deduction 
+elimination
+rules for the logical connective symbols of
+$~\wedge~$,  $~\vee~$,   
+$~ \rightarrow ~$, $~\neg~$, $~\forall~$ and $~\exists~$.
+These six rules are described below in a context where
+the 
+expression
+``{\bf $ \,  $A$  ~  \Longrightarrow  ~  $B$  \, $}''
+is an abbreviation for the sentence
+ {\bf $ \,   $B$  \, $} 
+being
+ an allowed deduction
+from its ancestor of {\bf $ \,  $A$  \, $}.
+\begin{enumerate}
+%\itemsep 5pt
+%note \small
+%%corebl \baselineskip = 1.05 \normalbaselineskip
+\item $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Gamma ~$ .
+$~~~$
+\item $ \,  \neg  \,\neg \, \Upsilon  \,  \Longrightarrow  \,  \Upsilon. \, $  
+Other rules for
+the ``$ \, \neg \,$'' symbol  are:
+$ \, \neg ( \Upsilon \vee \Gamma )  \,  \Longrightarrow  \,  \neg \Upsilon
+\wedge \neg \Gamma$,
+%\newline
+$ \, \neg ( \Upsilon \rightarrow \Gamma )  \,  \Longrightarrow  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \Longrightarrow  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $
+ and $ ~~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed
+when their ancestor is
+$~\Upsilon \, \vee \, \Gamma.~$
+\item A pair of sibling nodes $ \,  \neg \Upsilon  \, $ and $ \,  \Gamma  \, $ is
+allowed
+when their ancestor is
+$ \, \Upsilon \, \rightarrow \, \Gamma$.
+\item $\forall v \, \Upsilon  (v)  \,  \Longrightarrow    \, \Upsilon(t)  \, $
+where $t$  may denote any 
+term.
+%%  (These terms may be any one of
+%% a constant symbol,
+%% a  parameter symbol 
+%% or a more complex object in our language $L^*$ that combines
+%% its function symbols, constant symbols and parameter symbols
+%% in a routine manner)
+\item $~ \exists v \, \Upsilon  (v) ~ \Longrightarrow ~ \, \Upsilon(p) ~$
+where $\,p \,$ is a newly introduced parameter symbol. 
+\end{enumerate}
+A minor additional comment about our notation is that we treat 
+ ``$~ \forall~ v \leq s~~~ \Phi(v)~$''
+as an abbreviation for
+ $~ \forall  v  ~~ \{ ~ v \leq s~~ \rightarrow ~~\Phi(v)~ \}~$
+and likewise
+ ``$~ \exists~ v \leq s~~~ \Phi(v)~$''
+as an abbreviation for
+ $~ \exists  v  ~~ \{ ~ v \leq s~~ \wedge ~~\Phi(v)~ \}~$.
+In our year-2005 article \cite{ww5}, we thus applied
+Rules  5 and 6 
+to derive the following further hybrid rules
+for processing bounded universal and
+ bounded
+ existential quantifiers:
+\begin{description}
+\item[  a. ]  
+ $\forall v \leq s  \, \Upsilon  (v) ~~ \Longrightarrow ~~
+t \leq s \, \rightarrow \, \Upsilon(t) $ 
+where $\,t \,$ may be any arithmetic term.
+\item[  b. ] 
+ $~ \exists v \leq s ~ \, \Upsilon  (v) ~~ \Longrightarrow ~ ~
+u \leq s ~ \wedge~ \Upsilon(p) ~$
+where $\,p \,$ is a new parameter symbol. 
+\end{description}
+
+
+% \smallskip
+
+{\bf ACKNOWLEDGMENT:} I thank Seth Chaiken for several helpful comments about
+how to
+improve the presentation.
+
+
+\bibliographystyle{abbrv}
+
+\bibliography{bb}
+
+
+\end{document}
+	
+
+
+
diff --git a/nachlass/collected_dew_materials/2011-2019/wollic-2019.tex b/nachlass/collected_dew_materials/2011-2019/wollic-2019.tex
new file mode 100644
index 0000000..30cb459
--- /dev/null
+++ b/nachlass/collected_dew_materials/2011-2019/wollic-2019.tex
@@ -0,0 +1,2622 @@
+% 2019 feb24 8am while listeing to jazz
+% 1019 feb22  1am 1-word change
+%  2018 feb-17 11.30 pm while listenning to Bobby Dee
+
+%% seth and I agrred to meet on thursday 3pm
+
+
+\documentclass{llncs}
+\usepackage{amssymb}
+
+
+\newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+
+
+
+
+
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\begin{document}
+
+
+ \title{On the Breadth and
+Also
+ Limitations of the
+ Second Incompletenss
+Theorem}
+
+
+\def\aaa{\beta}
+\def\ccc{Class}
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+\author{  Dan E.Willard}
+
+
+\institute{University  at Albany Computer Science and Mathematics Departments}
+
+
+\maketitle
+\pagestyle{plain}
+
+%\bigskip
+
+
+\normalsize  
+
+\begin{abstract}
+%%% large \baselineskip =  2.4 \normalbaselineskip 
+This article will investigate several quite similar-looking
+ axiom systems that seek to confirm
+their own consistency via the use of
+self-referencing axiomatic
+statements, roughly declaring that
+{\it ``I am consistent''}.
+Surprisingly although our
+two main
+ investigated formalisms will differ only
+slightly in the fine print of their particular
+ definitions,
+one among these  systems will be consistent, while
+the other
+is inconsistent.
+This result will  clarify 
+both
+the breadth and 
+limitations
+ of the Second Incompleteness Theorem.  
+It will also have practical consequences.
+
+\end{abstract}
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\hgskip{ \medskip }
+
+\def\njp{\newpage}
+\def\njp{ }
+
+\def\nskip{\bigskip}
+
+
+%\Large
+%\baselineskip =  2.07 \normalbaselineskip 
+
+ \large \baselineskip =  1.5 \normalbaselineskip 
+% \large \baselineskip =  2.3 \normalbaselineskip 
+%\Large \baselineskip =  1.9 \normalbaselineskip 
+%% \LARGE
+\normalsize  \baselineskip =  2.6 \normalbaselineskip 
+\Large \baselineskip =  2.6 \normalbaselineskip 
+
+\large \baselineskip =  2.6 \normalbaselineskip 
+
+
+ \normalsize \baselineskip =  1.0 \normalbaselineskip 
+\parskip 1pt
+
+% \normalsize \baselineskip =  3.0 \normalbaselineskip 
+%%% \normalsize \baselineskip =  3.4 \normalbaselineskip 
+
+% \large \baselineskip =  3.0 \normalbaselineskip 
+  \normalsize \baselineskip =  2.22 \normalbaselineskip 
+% \large \baselineskip =  2.22 \normalbaselineskip 
+\normalsize \baselineskip =  1.77 \normalbaselineskip 
+\normalsize \baselineskip =  1.0 \normalbaselineskip 
+
+
+\section{Introduction}
+%11111
+\setcounter{section}{1}
+
+
+
+We have published a series of papers about generalizations and
+boundary-case exceptions to the Second Incompleteness
+during the last 25 years
+% cite {????},
+\cite{ww93,ww1,ww2,ww5,wwapal,ww6,ww7,ww9}
+including six papers that have appeared in the JSL and APAL.
+Our goal in the current article will be to focus on
+\cite{ww5}'s
+ IS$_D(\beta)$
+ formalism.
+It was  shown in
+ \cite{ww5} 
+that if $\beta$ denotes any consistent extension of 
+Peano Arithmetic and  if $D$ denotes semantic tableaux deduction,
+then  IS$_D(\beta)$ 
+will be a consistent axiom system that
+can simultaneously prove isomorphic analogs of all of
+Peano Arithmetic's $\Pi_1$ theorems while also corroborating
+its own consistency under semantic tableaux deduction.
+The current article will show, surprisingly, that if one changes the
+definition of $D$ in a relatively minor manner, corresponding
+to what we shall call {\it ``Extended Semantic Tableaux'',} then 
+the resulting modification of 
+ IS$_D(\beta)$ will 
+become inconsistent (and thus
+ useless).
+
+This result
+% does
+will
+ not lessen 
+the significance of 
+ \cite{ww5}'s prior result.
+It 
+%does,
+will,
+ however,
+% will 
+clarify 
+its
+basic
+ theoretical meaning.
+% of  \cite{ww5}. 
+Most of 
+our current discussion
+will
+% shall
+ be addressed to an audience
+of theoretical logicians, but
+% an important short
+a brief
+3-page passage 
+ (in Section \ref{sect5}) 
+will explain
+how some related
+future
+ AI software could
+plausibly relieve
+% section
+% two  pages
+ global warming.
+It
+%This latter discussion
+ will suggest the contrast
+between our positive and negative results,
+together with
+future
+results
+developed
+by other
+ logicians,
+ should help
+significantly
+% fine-tune
+refine
+ future artificially intelligent software
+systems
+that seek to ameliorate the
+harmful
+ effects from global warming.
+
+
+The author of this article has now reached the retirement age of 70.
+We 
+hope
+ this article 
+will be  written
+in a style that  encourages
+other
+logicians to investigate 
+its  subject further.
+%With such a goal in mind,
+We will strive to make this paper
+comprehensible to a 
+broad audience, and we
+shall
+ briefly discuss its
+pragmatic significance.
+
+\section{General Perspective}
+\label{sect2}
+
+%%22222
+
+It is well known that
+G\"{o}del's Incompleteness Theorem 
+is a 2-part result.
+Its
+first half  specifies no decision
+procedure can identify all
+arithmetic's
+ true statements.
+On the other hand, the 
+ ``Second Incompleteness Theorem'' 
+ assures that
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his
+%%!! historic 
+paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+meet some success:
+\begin{quote} 
+$*~$ {\it ``It must be 
+expressly 
+noted 
+Proposition XI''
+{\rm (e.g. the Second Incompleteness Theorem)} 
+``represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialectica interpretation.
+On the other hand,
+%%d the Stanford Encyclopedia's
+%%d entry about  G\"{o}del
+%%d quotes him,
+%%d in its
+%%d  Section 2.2.4,
+%%d stating
+%%d he was hesitant to
+%%d view     the
+%%d Second Incompleteness Theorem
+%%d  as    
+%%d fully
+%%d ubiquitous, until 
+%%d learning 
+%%d of Turing's
+%%d work.
+%%d Moreover,
+%
+Yourgrau's biography of
+G\"{o}del \cite{Yo5}
+indicates
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+had initially
+presumed his
+second theorem
+was false, before
+he had
+ proved his stunning 
+result.
+
+It is, of course, obvious that 
+the Second Incompleteness Theorem has forced 
+the objectives of Hilbert's Consistency Program
+to be radically reorganized. For instance,
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+Yet
+despite this fact,
+ Gerald Sacks (who 
+interacted
+% extensively 
+with
+G\"{o}del,
+ during the early 1960's
+ at the Institute for Advanced Studies) 
+recalls \cite{YouSa14}
+G\"{o}del declaring,
+{\it thirty years after  \cite{Go31}'s  publication,}
+that a type of 
+partial
+revival of Hilbert's Consistency
+Program, using at least Gentzen's approach,
+should be considered (see the below footnote 
+\footnote{\label{f2}
+Some quotes from Sacks's
+YouTube talk
+\cite{YouSa14}
+ are that G\"{o}del
+ {\it ``did not  think''}
+the objectives of Hilbert's Consistency Program 
+{\it ``were erased''} 
+by
+the Incompleteness Theorem, and
+G\"{o}del believed (according to Sacks) 
+that  it left
+  Hilbert's program 
+{\it ``very much alive and
+even more interesting than it initially was''}. 
+} 
+for a summary of 
+ Sacks's  observations). 
+
+The research that has followed
+ G\"{o}del's  seminal  1931 discovery has 
+ focused mostly
+% mainly
+on studying generalizations of the Second Incompleteness
+Theorem
+(instead of also viewing its
+ boundary-case
+exceptions). Several generalizations 
+of the Second Incompleteness Theorem 
+\cite{AB1,AZ1,Be17,Be14,BS76,Bu86,BI95,Fe60,Fr79a,Ha11,HP91,KT74,Pa71,PD83,Pa72,Pu85,Pu96,So94,Sv7,Vi5,WP87,ww1,ww2,wwapal}  
+are quite beautiful. The author of this paper has been
+            especially impressed by a generalization of the
+Second Incompleteness Effect, arrived at by the 
+combined work of Pudl\'{a}k and  Solovay,
+abetted by
+the research of
+ Nelson and Wilkie-Paris\cite{Ne86,Pu85,So94,WP87}.
+These results, which 
+  also have been more recently
+discussed 
+ in \cite{BI95,Ha7,Sv7,ww1},
+have noted the Second Incompleteness Theorem
+does not require the presence of the Principle of Induction
+to apply to most formalisms that use a Hilbert-style form of 
+deduction. (The 
+next chapter will offer a detailed summary of this 
+important generalization
+ of
+the Second Incompleteness Theorem in
+its
+% the paragraph called
+ Remark \ref{nremm-2.5}.) 
+
+
+Our research, during the last 25 years has 
+had a different focus,
+exploring 
+Boundary-Case exceptions to the Second Incompleteness Theorem
+with equal intensity as
+its generalizations.
+It would be natural for many readers to ask why such
+exceptions should also be studied
+with 
+%such
+fully
+equal intensity?
+
+One reason is that while generalizations of the Second Incompleteness
+Theorem are pure
+%%seth  form 
+from
+a mathematical standpoint, it must
+not be forgotten that
+our civilization
+must confront
+ecological issues
+ in the future, more important 
+ than how to devise short proofs for the existence
+of a
+large number
+that  cannot be easily encoded in a binary format,
+ such as
+a  googolplex $= \,2^{2^{100}}$.
+
+
+The current paper will explore
+G\"{o}del's 
+% underlying
+motivation for making his
+ statement $*$ and
+Hilbert's
+% similar
+closely
+related famous
+  declaration
+    $**$ from
+ \cite{Hil26}: 
+
+\begin{quote}
+$**~$
+{\it ``
+Let us admit that the situation in which we presently
+find ourselves with respect to paradoxes is in the long
+run intolerable. Just think: in mathematics, this paragon of
+reliability and truth, the very notions and inferences,
+as everyone learns, teaches, and uses them, lead to absurdities.
+And 
+where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?''}
+\end{quote}
+A surprising facet of both $*$ and $**$ is that our short 3-page
+discussion 
+ (in  \textsection \ref{sect5}), 
+about how to ameliorate the effects of global warming,
+will be related
+to Hilbert's and G\"{o}del's
+ predictions, about the
+underlying
+ importance of  
+boundary-case exceptions to the Second Incompleteness
+Theorem.                   
+
+
+\section{ Notation for Introducing  Main Formalism}
+%%  333333   }
+
+\label{nnn2}
+
+Let us 
+call an
+ordered pair $(\alpha,D)$ a
+    {\bf Generalized Arithmetic Configuration}
+(abbreviated as a {\bf ``GenAC'' })
+when  its 
+first and second 
+components 
+are 
+defined 
+as 
+follows:
+\bee
+\item
+The {\bf Axiom Basis} ``$~\alpha~$'' 
+for a 
+ GenAC
+%ssss  Generalized Arithmetic
+will be defined as
+the set of 
+ proper axioms 
+it employs.
+\item
+The second component ``$~D~$'' of a 
+ GenAC
+%ssss  Generalized Arithmetic
+will represent
+the 
+{\it combination} of its formal rules of inference
+with 
+%its
+the
+ logical axioms ``$~L_D~$'' it employs.
+The
+term  {\bf ``Deductive Apparatus''} will be often
+used to refer to $D$. 
+\ene
+
+% \cite{End,Fit,HP91,Mend}
+
+
+\begin{example}
+\label{nex-2.1}
+\rm
+This notation 
+allows us to
+ conveniently separate  the logical axioms
+$~L_D~,~$ associated with  $(  \alpha  , D  )~$, from 
+its
+``basis axioms'' $\, \alpha \,$.
+It also allows one to  compare
+the various
+deductive apparatus techniques
+that  
+have
+appeared in the literature.
+For instance,
+the
+  $~D_E~$ apparatus,
+introduced
+ in 
+\textsection
+ 2.4 of  Enderton's textbook \cite{End}, 
+has
+ used only  modus ponens
+as a rule of inference,
+combined with a 
+complicated
+4-part  schema of logical axioms.
+This differs from
+the  $~D_M~$ ,  $~D_H~$ and  $~D_F~$ approaches of
+Mendelson \cite{Mend}, 
+H\'{a}jek-Pudl\'{a}k \cite{HP91}
+and Fitting \cite{Fi96}.
+The former two textbooks
+employ a simpler set of logical axioms
+than $\, D_E \,$,
+but they require
+two rules of inference
+(modus ponens and generalization).
+The  $~D_F~$ apparatus, appearing in Fitting's textbook \cite{Fi96},
+as well as its predecessor due to Smullyan \cite{Sm95},
+actually  employ {\it no logical axioms.}
+Instead,
+\cite{Fi96,Sm95}
+ rely upon a
+``tableaux style'' method for generating a 
+%consequently
+%complicated
+larger number of
+rules of
+inference.
+\end{example}
+
+
+\begin{definition}
+\label{ndef-2.2}
+\rm
+Let
+$ \, \alpha \, $ again 
+denote an axiom basis, 
+and $ \, D \, $ 
+designate
+ a
+deduction apparatus.
+Then 
+the  GenAC of
+ $(  \alpha  , D  )$
+will
+% shall
+be called  {\bf Self Justifying}
+ when
+\begin{description}
+% \xxitch
+% \small
+  \item[  i.   ] one of  $~(  \alpha  , D  )$'s  theorems
+(or possibly one of $\alpha$'s axioms)
+does
+%will
+state that the deduction method $ \, d, \, $ applied to the
+basis
+system $ \, \alpha, \, $ 
+%will 
+produces a consistent set of theorems, and
+\item[  ii.   ]
+     the GenAC formalism $ \,( \alpha,D)  \, $ is, in fact,
+actually consistent.
+\end{description}
+\end{definition}
+
+
+\begin{example}
+\label{nex-2.3}
+\label{ex2}
+%%%%%%%%%%%%%%%%%%%  OLD \label{ex-2.5}
+\rm
+Using 
+Definition \ref{ndef-2.2}'s 
+ notation, our
+prior
+ research  
+% in
+\cite{ww93,ww1,ww5,wwapal,ww9}
+% has
+constructed
+%developed
+GenAC pairs  
+% arithmetics
+$~(  \alpha  , D  )$
+that 
+ were
+``Self Justifying''.
+We
+also 
+proved
+the Incompleteness Theorem 
+implies specific
+limits beyond which 
+self-justifying
+formalisms
+simply
+ cannot transgress.
+For any  $\,(\alpha,D) \,$, 
+it is 
+thus
+easy
+%almost trivial 
+to construct a 
+system $ \, \alpha^D \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition
+(in an isolated context {\it where the Part-ii condition is
+ not also
+satisfied}).
+For instance,  $ \, \alpha^D \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+the added {\bf $\,$``SelfRef$(\alpha,D)$''$\,$} sentence,
+defined below: 
+%% as stating:
+\begin{quote} 
+% \small
+$\oplus~~~$ 
+There is no proof 
+(using 
+$D$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,D) \,$'' (looking at itself).
+\end{quote}
+Kleene 
+showed \cite{Kl38} 
+how
+to
+encode
+%  rough
+ analogs of 
+the above  ``SelfRef$(\alpha,D) \,$'' statement,
+which we often call an 
+ {\bf $\,$``I AM CONSISTENT'' 
+%axiomatic
+ declarative axiom.}
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71},
+however,
+ emphasized
+% that
+$\alpha ^D$ 
+may
+be inconsistent
+(e.g.  violating Part-ii of   self-justification's
+definition),
+{\it despite SelfRef$(\alpha,D)$'s 
+formal
+assertion.}
+This is because if the 
+ pair $(\alpha,D)$ is too strong
+then a
+quite conventional
+G\"{o}del-style diagonalization argument can
+be applied to the axiom basis of
+$\alpha^D~~=~~ \alpha \, + \, $ SelfRef$(\alpha,D), ~$
+where the added presence of the statement 
+SelfRef$(\alpha,D)$ 
+will cause this extended version of 
+$\, \alpha\,$, ironically,
+ to
+ become automatically inconsistent.
+Thus, an
+encoding for
+``SelfRef$(\alpha,D)$'' is relatively easy,
+via an application of the Fixed Point Theorem,
+but this sentence
+ is, typically
+{\it
+utterly
+useless!} 
+\end{example}
+
+% \bigskip
+
+\parskip 0pt
+
+
+\begin{definition}
+\label{ndef-2.4}
+\rm
+Let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote two 3-way predicates,  specifying 
+ $x+y=z$ and $x*y=z$,
+for which the
+associative, commutative, identity and distributive 
+properties have $\Pi_1$ style encodings under an
+axiom system of $\alpha$.
+Then we will say
+that $~\alpha~$
+{\bf recognizes} successor,  addition  and multiplication
+as {\bf Total Functions} iff 
+it can  prove all of
+\eq{totdefxs} - \eq{totdefxm}
+as theorems:
+\end{definition}
+% \newpage
+{ \small
+\baselineskip =  .9 \normalbaselineskip 
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+%%% \vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+
+\noindent
+Furthermore, we will call the GenAC system $(\alpha,D)$ a
+% $\alpha$ will be  called 
+{\bf Type-M} 
+formalism
+iff it includes
+\eq{totdefxs} - \eq{totdefxm}
+as theorems, {\bf Type-A} if it includes 
+only \eq{totdefxs} and \eq{totdefxa} as theorems,
+and {\bf Type-S} if it contains
+only \eq{totdefxs} as a
+ theorem. 
+%Moreover, 
+Also,
+ $(\alpha,D)$ 
+ will be 
+% is
+called 
+{\bf Type-NS} iff it  can prove
+none of these theorems.
+
+\parskip 2pt
+
+%\bigskip
+
+\begin{remark}
+\label{nremm-2.5}
+%% 
+\rm
+The separation of GenAC systems into the four 
+categories of Type$-$NS, Type-S, Type-A and Type-M systems
+will enable us to nicely summarize the prior literature
+about generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem. This is because:
+\bed
+\item[   $~~~~$i.$~~$  ]
+The combined research of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87},
+as is formalized by Theorem $\, ++ \,$,
+implies no
+natural Type$-$S  GenAC system $(\alpha,D)$
+can recognize  its own  consistency
+when $D$ is one of 
+Example \ref{nex-2.1}'s three
+examples of
+  Hilbert-style 
+deduction operators of 
+$\, D_E \,$, $\, D_H \,$  
+or
+$\, D_M ~~$. In particular,
+it establishes the following result:
+\medskip
+\begin{quote}
+\normalsize \baselineskip = 1.0 \normalbaselineskip 
+{\bf ++ }
+{\it 
+(Solovay's  
+modification
+\cite{So94}
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+using some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let 
+$ \, (\alpha,D) \, $ 
+denote 
+a GenAC system
+supporting
+% which contains 
+the
+\el{totdefxs}'s
+Type-S statement
+and 
+assuring
+the successor operation
+will
+satisfy
+both 
+% the axioms of 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then
+$ \, (\alpha,D  ) \, $  
+%%%%$~\alpha~$
+cannot verify its own
+%will be unable to recognize its
+%own  
+consistency
+whenever
+simultaneously
+ $D$ is some type of
+a Frege-Hilbert
+deductive
+apparatus and
+%whenever
+$~\alpha~$
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+their usual % identity,
+associative, commutative 
+ distributive 
+and identity 
+%  axiomatic properties.
+axioms.
+%   -axiom
+% properties.
+\end{quote}
+\medskip
+Essentially, Solovay \cite{So94} 
+privately communicated 
+to us 
+in 1994
+%to us
+an analog of theorem $++$.
+%but 
+Many authors
+have noted Solovay
+ has 
+been
+%often
+reluctant to publish
+% several of 
+his 
+nice 
+privately communicated
+results 
+on many occasions
+%in several contexts
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}. 
+Thus,
+%polished
+approximate  analogs of 
+%statement
+ $++$
+ were  explored 
+subsequently
+ by  Buss- Ignjatovi\'{c},
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha7,Sv7},
+as well as in Appendix A of 
+our paper
+\cite{ww1}.
+Also, 
+Pudl\'{a}k's initial 1985 article  \cite{Pu85} 
+% implicitly
+captured
+% , notably, 
+the majority 
+%most
+%%%  much 
+of $++$'s 
+essence, chronologically before Solovay's observations,
+%Also,
+ and
+Friedman did
+% some
+related work
+ in
+\cite{Fr79a}.
+
+\medskip
+
+\item[   $~~$ii.$~~$  ]
+Part of what makes  $++$ interesting is that 
+\cite{ww1,ww5,wwapal}
+explored two methods for 
+GenAC systems
+to confirm their own consistency, whose
+natural hybridizations  are precluded by $++$.
+Specifically,  these results involve using
+Example \ref{nex-2.3}'s 
+self-referencing  {\it ``I am consistent''}
+ axiom (from its
+statement  $\oplus$ ). 
+They will enable
+some (not all)
+  Type-NS  
+systems \cite{ww1,wwapal}
+to verify their   own consistency under 
+a Hilbert-style deductive apparatus
+\footnote{ The Example \ref{nex-2.1} had
+provided
+three examples of
+  Hilbert-style 
+deduction operators, called 
+$\, D_E \,$, $\, D_H \,$ , 
+and
+ $\, D_M ~~$. It explained how these 
+ deductive operators differ from a tableaux-style
+deductive apparatus by containing a modus ponens rule.},
+or alternatively allow 
+some (not all)
+ Type-A 
+ systems \cite{ww93,ww1,ww5,ww6} to 
+corroborate
+their 
+self-consistency
+under a more restricted semantic
+tableaux style deductive apparatus.
+Also, we observed in  \cite{ww2,ww7} how one could
+refine $++$ with Adamowicz-Zbierski's
+methodology \cite{AZ1} to show 
+ most Type-M  systems
+cannot recognize their own semantic tableaux style consistencies.
+\ennd
+\end{remark}
+
+\begin{remark}
+\label{rem2}
+Several of our articles have
+used
+Example \ref{ex2}'s
+ {\it ``I am consistent''} 
+axiomatic statement
+$~\oplus~$
+% as a vehicle 
+ to evade the Second Incompleteness Theorem.
+This methodology is 
+unrelated to alternate 
+techniques for evading G\"{o}del's Theorem
+that have been explored by 
+Gentzen in \cite{Ge36},
+Kreisel-Takeuti 
+in
+% the analysis of
+ their    ``CFA''
+formalism \cite{KT74}
+and to
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+It turns out these latter
+systems will not make as broad statements
+about their consistency because they do not
+make use of
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+On the other hand,
+they will possess a
+% much
+ different
+style
+% kind 
+of advantage
+because they will provide a more detailed proof
+of their consistency than would follow from 
+statement $~\oplus\,$'s
+single-sentence  {\it ``I am consistent''} declaration.
+\end{remark}
+
+
+\section{More  Notation and Main Formalism}
+%%mmm  
+
+%%% 444444
+
+\label{sect4}
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1 \ldots a_j) 
+\leq   Maximum(a_1 \ldots a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+$Maximum(x,y),$
+$ Logarithm(x)$
+$\,Root(x,y) \, =  \, \lceil  \, x^{1/y} \,  \rceil$ and
+$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+that included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+A wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+%{\it Also throughout this article,}
+ Also,$~$
+ the sentence $\Psi$ will be called a$~$ {\bf  Rank-1* } $~$statement iff it
+can be encoded in either a  $\Pi_1^*$
+or  $\Sigma_1^*$ format.
+
+Our article \cite{ww5} used the symbol $~D~$ to denote
+a deduction method. 
+There will be three variants of deduction methods
+% that we will compare 
+compared
+in this article. The first will be
+{\it semantic tableaux},
+ which will receive the abbreviated name of
+ {\bf Tab}. It will be similar to 
+%%% its analog
+% appearing 
+the tableaux formalism
+in Fitting's textbook  \cite{Fi96}.
+Thus a Tableaux proof of a theorem $~\Psi$
+ from an axiom basis
+% a set of  proper axioms 
+$~\alpha~$ will be  tree-like
+structure that begins with the sentence
+ $~\neg ~ \Psi$ 
+ stored in the tree's root and whose every root$-$to$-$leaf
+path establishes a contradiction by containing some pair of contradictory
+nodes  that ``closes'' its path. The rules for generating internal
+nodes along each
+ root$-$to$-$leaf
+path will be that each node
+must be either a
+% formal 
+proper axiom of $\alpha$ or a deduction
+from an ancestor node via one of the 
+``elimination''
+rules associated with
+our logic symbols of $\wedge$, $\vee$, $\rightarrow$, $\neg$,
+$\forall$, or $\exists$. (A precise definition of these six rules
+and other details
+appears in the attached appendix.) 
+
+\smallskip
+
+Our second 
+explored
+ deductive apparatus
+% in this article, 
+will be called {\it Extended Tableaux}, abbreviated as
+{\bf Xtab}. Its definition 
+will be
+ identical to the prior
+paragraph's {\it Tab-}deduction, {\it  except that} for any sentence
+$\phi$
+in our  language $L^*$, the sentence $\phi \, \vee \, \neg \phi$
+can be permissibly 
+  stored 
+% inserted 
+as an axiom
+inside
+ any internal node of our proof tree.
+(In other words,    {\it Xtab-}deduction
+will differ from  {\it Tab-}deduction by allowing all instances
+of the Law of Excluded Middle to appear as a logical schema of axioms.
+In contrast,  {\it Tab-}deduction will treat the infinite schema of
+instances of the Law of Excluded Middle as
+%  permissibly
+{\it  derived theorems,}
+{\bf BUT NOT ALSO} as
+instances of
+  logical axioms.
+
+% \medskip
+
+Our third
+% studied
+ deductive apparatus
+will be called {\bf Tab-1}.
+It will be a
+% type of
+  compromise between Tab and Xtab,
+% deduction,
+where  a ``Tab-1'' proof of $\Psi$ from
+an axiom basis
+$\alpha$  is defined as a set of
+ordered pairs $ \, (p_1,\phi_1), \, (p_2,\phi_2), \, .. (p_k,\phi_k) \, $
+where
+\bee
+\item $ ~ \phi_k ~  = ~  \Psi \,$ 
+\item
+Each  $~p_j~$ is a Tab-proof of
+what we have called
+ a Rank-1* sentence
+$~\phi_j~$ from the union of $~\alpha~$ with some further Rank-1* axioms
+of   $~\phi_1,~\phi_2,~..~\phi_{j-1}~$.
+\ene 
+{\bf We emphasize} that  Tab-1 is 
+{\it  less efficient}
+ than Xtab 
+deduction
+because {\it the$ \,$former
+requires} $\phi_j$ be a Rank-1* sentence, while Xtab
+{\it  does not impose}
+ a similar Rank-1* constraint upon
+$\, \phi \,$, when
+ it
+% uses
+invokes
+  an  $~\phi \, \vee \, \neg \phi~$ axiom.
+
+
+Let us say that an axiom system $\alpha$ owns a
+{\bf Level-1} appreciation of its own self-consistency
+(under a deductive apparatus $D$)
+iff it can verify 
+$D$ produces
+ no two simultaneous
+proofs of a 
+ $\Pi_1^*$ 
+sentence and 
+its negation.
+Within this
+ context, where $~\aaa~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(\aaa)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$\aaa$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms will define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+(They will  enable our formalism to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.)
+
+\item[Group-1:] 
+This axiom group  will consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5} had  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_\aaa(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $\aaa$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema will contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(\aaa)$ can trivially prove
+ all $\aaa$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_\aaa(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+The final part of  IS$_D(\aaa)$ 
+% shall
+will 
+be a
+self-referencing
+$\Pi_1^*$
+axiom,
+that indicates
+IS$_D(\aaa)$
+is
+``Level-1 consistent'' 
+under
+ $D$'s deductive method.
+It 
+is, 
+thus,  the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+\smallskip
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+had appeared
+ in 
+\cite{ww5}.
+Thus, 
+\eq{group3} 
+is a
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+$~$when:  
+$~~1)~~ \, \mbox{Prf} \, _{\mbox{IS}_D(\aaa)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(\aaa)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and $~~2)~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+%%%%% \parskip 2pt
+%\smallskip
+
+\begin{definition}
+\label{def3} \rm
+Let $~D~$ denote 
+any one of 
+ the {\it Tab,  Xtab} or
+{\it Tab-1}
+ deductive methodologies.
+We will  say that 
+the mapping  $\mbox{IS}_D(~\bullet~)$ 
+ is
+{\bf Consistency Preserving}
+ iff
+ $\mbox{IS}_D(\aaa)$ 
+is guaranteed to be consistent 
+under $D$'s deductive apparatus
+whenever all the axioms of $\aaa$ hold
+true under the standard model of the natural numbers. 
+\end{definition}
+
+%\smallskip
+
+
+
+%%%%% 66666 NEW PASSAGE
+
+\begin{theorem}
+\label{th1} \rm
+The     $\mbox{IS}_{\it Tab}(~\bullet~)$ 
+and   $\mbox{IS}_{\it Tab-1}(~\bullet~)$ 
+mappings 
+are
+consistency preserving, but the similar-looking  
+   $\mbox{IS}_{\it Xtab}(~\bullet~)$ mapping 
+is actually not{\it $~~$. This 
+statement
+ is 
+ formalized  by the
+%   formalized, more  precisely,  by the
+Items (a) and (b) below:}
+\bed
+\item[  a.  ]
+ $\mbox{IS}_{Tab}(\aaa)$ 
+and
+$\mbox{IS}_{Tab-1}(\aaa)$ 
+are always consistent 
+whenever all the axioms of $\aaa$ hold
+true under the standard model
+ of the natural numbers. 
+
+\smallskip
+
+\item[  b.  ]
+In contrast,
+ $\mbox{IS}_{Xtab}(\aaa)$ 
+is {\it automatically inconsistent}
+whenever
+$~\beta~$
+proves 
+the
+%some
+ conventional $\Pi_1^*$ theorems
+showing 
+%that
+ addition and multiplication
+satisfy 
+their
+associative, commutative, 
+ distributive 
+and identity 
+properties.
+% principles.
+\ennd
+\end{theorem}
+
+{\it Proof Summary:}
+The two halves of Theorem \ref{th1}'s
+proof would be quite lengthy, if
+% we used
+%%% had to rely on 
+first principles
+were used
+ to justify
+them.
+Fortunately, there is an easier method to 
+justify
+(a) and (b), 
+% Theorem \ref{th1}'s two them
+by relying upon the earlier literature.
+
+% \parskip 2pt
+
+Thus, Part (a)  follows
+from  our
+ main 
+result
+in  \cite{ww5}.
+ It 
+indicated   $\mbox{IS}_{\it Tab-1}(~\bullet~)$ 
+was a consistency preserving
+%%%%  mapping, and this implies
+ mapping. This implies
+that  $\mbox{IS}_{\it Tab}(~\bullet~)$
+is also  consistency preserving
+because Tab-deduction is weaker than Tab-1.
+
+
+Let us now turn our attention to 
+Theorem  \ref{th1}'s second
+claim (b). 
+Its proof is more complex but analogous to the
+justification for the Invariant
+ ++ , which
+Remark \ref{nremm-2.5} 
+%   had
+attributed
+mostly  to
+ the work 
+of Pudl\'{a}k. Solovay, Nelson
+and Wilkie-Paris. The
+crucial aspect of the Frege-Hilbert methodologies
+employed by ++ is that modus ponens assures that
+a proof of a theorem 
+$\psi$ 
+from an axiom system $\alpha$ has a length no
+greater than the sum of the proof-lengths needed to derive
+ $\phi$ and $~\phi \rightarrow \psi~$ from
+$\alpha$. This 
+{\bf ``Linear-Sum Effect''
+}
+ does not apply,
+%%  technically, 
+actually,
+also to
+{\it Tab-}deduction because it owns no analog of
+a modus ponens rule (for assuring  that  
+$\psi$'s proof-length is bounded by the sum of the 
+lengths for the proofs of 
+ $\phi$ and $~\phi \rightarrow \psi~$) .
+
+The {\it Xtab}
+%%deductive 
+methodology, however, 
+ differs
+from
+{\it Tab-}deduction 
+by allowing any node of its proof-tree
+to store a sentence of the form $~ \phi \, \vee \, \neg ~\phi~$,
+as an application of its allowed use of the Law of Excluded
+Middle.
+This added feature allows an
+ {\it Xtab} proof for
+$\psi$ to have a length
+%% be bounded approximately  
+proportional to
+ the sum of
+the proof lengths for 
+ $\phi$ and $~\phi \rightarrow \psi~$. In particular,
+the relevant 
+ {\it Xtab} proof for 
+$\psi$ 
+can be summarized as having the following 4-part structure:
+\bee
+\item
+The root of the proof of $\psi$ 
+will be the usual temporary negated hypothesis of
+ $~\neg \psi~$ (which the remainder of the proof tree will
+show is impossible to hold). 
+\item
+The child of this 
+root
+node will be an allowed invocation of the
+Law of the Excluded Middle of the 
+ form $~ \phi \, \vee \, \neg ~\phi~$.
+\item Our tableaux proof tree will next employ
+the Appendix's branching rule for allowing the two
+sibling nodes of
+ $~ \phi ~$
+ and $~ \neg ~\phi~$ to descend from
+Item 2's node.
+\item 
+Finally, our tableaux proof will insert below
+(3)'s  left sibling  node
+of  $~ \phi ~$ a subtree structure that is no longer than
+the proof of   $~\phi \rightarrow \psi~$,
+and likewise
+insert
+ a   proof for $~\phi~$ 
+below (3)'s right sibling
+%%%%%%%  node  
+of  $~ \neg ~\phi~$.
+\ene
+The point is that the last step of our 4-part proof
+has a length no greater than the sum of the two
+ proof lengths for 
+ $\phi$ and $~\phi \rightarrow \psi~$,
+and its first three steps have inconsequential lengths
+that will increase 
+its
+ overall proof length by no
+more than an unimportant
+amount proportional to the length of the 
+sentence ``$~\phi ~ \rightarrow ~\psi~$''.
+
+%%%8888888888888
+
+%%{sect2}
+
+We can apply
+the preceeding 
+ ``Linear-Sum Effect''
+to construct  an analog 
+of Remark \ref{nremm-2.5} 's
+earlier  Theorem $\, ++ \,$ 
+germane to
+ Extended Tableaux  deduction (which is also called ``Xtab''). 
+Saving 
+several
+%the 
+details
+for a longer paper,
+the   intuition
+behind this analog is that modus ponens
+ {\it is the only
+ rule of inference} used
+ by 
+the classic
+ \cite{End}'s  textbook-style
+% classic
+first-order logic  system,
+ and
+Xtab can
+%  that Xtab
+% can
+% thus
+ use its 
+ Linear-Sum Effect
+  to simulate modus ponens.
+This natural analog of  
+ $++$  will
+%%%% then
+%% thus
+assure 
+%% that 
+any axiom system $~\cal{A}~$ is
+% then
+ {\it  automatically inconsistent} whenever:
+\begin{enumerate}
+\item
+$\cal{A}$
+ can verify Successor is a total function (as formalized
+by \el{totdefxs} ),   
+\item 
+$\cal{A}$
+can prove
+%%% 
+%%%  proves 
+%%%  $\Pi_1^*$ theorems
+%%% %%% showing 
+%%% that show
+%%% 
+ addition and multiplication
+satisfy 
+their usual 
+associative, commutative, 
+ distributive 
+and identity axioms. 
+\item  
+$\cal{A} \,$
+proves 
+an added
+ theorem (which turns out to be false) affirming its own
+consistency when the
+Xtab
+ deductive apparatus is used.
+\end{enumerate}
+The preceding observation completes 
+Theorem
+%%%  \ref{th1}'s
+1-b's
+ proof because 
+ $\mbox{IS}_{Xtab}(\aaa)$ 
+meets
+% satisfies
+all three of these
+requirements.
+%% ?????? (see footnote).
+$~~~\Box$
+ 
+
+
+\begin{remark}
+\label{rem3} \rm
+We 
+have
+deliberately resisted the temptation of providing a more
+elaborate proof of Theorem \ref{th1},
+beyond the above 
+summary
+in this
+% current
+ extended abstract.
+This is because 
+it is desirable
+ to reserve
+%%% three pages of space
+page space
+ for explaining
+how Theorem \ref{th1} (and other aspects of symbolic logic)
+can help
+ biologists, chemists, physicists and mathematicians
+develop formal  Artificial Intelligence mechanisms
+for
+%  alleviating 
+relieving$~$the harmful effects
+from global warming.
+\end{remark}
+
+
+\parskip 1 pt
+
+
+The contrast between the positive and negative results
+of Theorem \ref{th1}'s two claims will 
+be central to the next section's  discussion.
+Although the topic of global warming was
+essentially
+ unknown
+to the world of the  1920's and 1930's,
+the next section will show
+that
+ the often quoted statements
+$*$ and $**$, of G\"{o}del and
+Hilbert,
+%%%%%%%%% Hilbert
+do gain
+some
+new
+interpretations.
+
+%END OF NEW PASSAGE
+
+
+
+ 
+
+\section{An Application of Theorem  \ref{th1}
+Germane to Diminishing the Damage
+Caused by Global Warming} 
+
+% \large \baselineskip =  1.5 \normalbaselineskip 
+%%%555555555555555
+\label{sect5}
+
+We 
+%did become
+became
+ aware about 
+how the hard-core sciences of
+ biology, chemistry, and  physics could use concepts
+from Symbolic Logic (such as Self-Justification) 
+to alleviate the effects of global warming
+after  
+reading several
+% quite
+alarming
+ Year-2017 interviews of 
+the late physicist Stephen Hawking
+\cite{Ha17a,Ha17b}. 
+In those interviews, Hawking
+% has
+%joined 
+ did join
+a growing team of scientists
+concerned that if current trends continue, then
+global warming could cause the planet Earth to become too hot
+for mammals to survive, within one or two centuries.
+(Hawking actually states 
+%that 
+the extinction of all mammal species could
+ occur
+as early as
+% even
+ within ``100 years'',
+but we would prefer to
+optimistically
+ hope that there is, at least,
+an available  200-year window in  time before the most tragic results
+%will 
+occur, if indeed they cannot (?) be 
+prevented.)
+%avoided.) 
+
+Hawking has 
+%also 
+hinted that computers, employing various forms
+of Artificial Intelligence (AI), could be
+one part of
+% the 
+a 
+reply
+ to global warming's 
+emerging challenge
+(if AI can {\it somehow}   be handled  (?)  adeptly).
+The
+ two advantages of computers
+over humans are that computers
+can produce
+deductions more quickly, and they can 
+be engineered to
+ survive
+in environments that are inhospitable for humans.
+For example, computers do not need Oxygen as a vital energy
+supply. They could, thus, rely upon  solar power when residing
+on the Moon or Mars, and they could
+possibly survive
+ underneath the Earth's oceans
+(assuming the latter do not evaporate under intense heat). 
+
+% \parskip 0pt
+
+We want to keep this chapter brief and avoid 
+too much speculation.
+The main point is
+% to suggest
+ that
+AI-oriented
+ computers could possibly offer a
+partially positive outcome to a global warming
+crisis by employing a 2-part strategy where:
+\bee
+\item
+AI-based computers 
+will
+first take complete control over 
+the planet's
+destiny,
+if 
+(?) 
+humans are rendered 
+temporarily
+extinct by global over-heating.
+\item
+AI-based computers
+could then
+ subsequently restore human and 
+perhaps
+other 
+higher
+life forms on Earth after the planet returns 
+\footnote{This
+footnote is a 
+tentative  remark, but bacteria are known to be a  much
+more durable species  than mammals, which are capable of 
+surviving under very high temperatures and also able to
+undergo
+astonishingly
+speedy evolutionary changes and adaptations.  
+A new species of bacteria can  hopefully 
+(?) be
+engineered, 
+{\it perhaps with the assistance of computers,} that
+can hopefully pull greenhouse gasses out of the atmosphere
+and lower the temperature on Earth over 
+a spam of  perhaps
+%1,000 years. 
+several decades 
+Frozen samples of primate embryos and accompanying stored
+DNA molecules, could then be used to restore human life
+on Earth.}
+to a cooler state and stored frozen embryos and DNA samples
+%were
+have been
+ previously  (?) saved.
+\ene
+This methodology will be called a {\bf 2-Prong Strategy.}
+%We have no doubts its 
+%The author of this article has no doubt that
+Its two parts
+will, clearly, make
+some readers cringe with 
+discomfort. One should, however,
+%  take some measures to
+prepare for
+% plausible
+ worst-case
+scenarios, {\it  
+ in case
+such
+paradigms
+% circumstances
+ arise? }
+The next two pages
+% , thus,
+will 
+% explain particular
+focus on 
+ its 
+% very
+special
+% details germane to 
+implications for
+symbolic logic.
+
+
+
+\begin{example}
+\label{main-ex} 
+\rm
+%%%% eeeee
+%During our current discussion,
+%let us have
+Let
+ $\, \beta \,$ denote
+some
+% particular 
+consistent formal  axiom 
+basis
+% formalism
+% that has been selected to 
+that can
+prove a substantially
+broader
+ set of $\Pi_1^*$ theorems than some 
+% fixed
+conventional axiom system
+(such as
+% perhaps 
+Peano Arithmetic or ZF Set Theory).
+Let us consider 
+using Artificially Intelligent (AI)
+% systems 
+computers
+to combine the formalisms
+of the  preceding  2-Prong 
+methodologies
+ with
+% that of 
+Theorem \ref{th1}. 
+Then
+% the
+a
+% preceding paragraph's  
+2-Prong Strategy may
+potentially use any one of 
+ $\mbox{IS}_{Xtab}(\aaa)$,
+ $\mbox{IS}_{Tab}(\aaa)$,
+ $\mbox{IS}_{Tab-1}(\aaa)$ or say Peano Arithmetic (PA)
+as  its core invoked AI decision mechanism.
+A 
+comparison between these four
+% quite  different AI
+ variations of 
+ 2-Prong methods
+% does
+will
+  reveal that:
+\bed
+\item[   a. ]
+Although the  $\mbox{IS}_{Xtab}(\aaa)$ formalism may look
+superficially similar to 
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$, 
+it is
+actually
+% entirely 
+unacceptable as a
+%formal
+ core invoked AI 
+% mechanism, 
+formalism
+used by a  2-Prong strategy. This is 
+because Theorem 1-b
+indicates
+%  that
+  $\mbox{IS}_{Xtab}(\aaa)$ 
+is inconsistent.
+It will thus cause a 2-Prong strategy to make
+unacceptably false decisions 
+\footnote{ A surprising aspect of 
+ $\mbox{IS}_{Xtab}(\aaa)$'s faulty behavior is that its Group 0, 1 and 2 axioms
+are identical to those of 
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$.
+Thus,
+ $\mbox{IS}_{Xtab}(\aaa)$'s unacceptable behavior
+is solely because of its Group-3 {\it ``I am consistent''}
+axiom. \smallskip} 
+when it uses
+$\mbox{IS}_{Xtab}(\aaa)$ as its core AI mechanism.
+
+\smallskip
+
+\item[   b. ]
+In a context where $\beta$ has been chosen to be an axiom basis that
+proves a richer set of $\Pi_1^*$ 
+theorems than Peano Arithmetic,  PA
+is 
+substantially
+ less desirable 
+as a 2-Prong Strategy's
+%core
+main
+ AI 
+decision
+system than either  
+ $\mbox{IS}_{Tab}(\aaa)$ or
+ $\mbox{IS}_{Tab-1}(\aaa)$
+would be.
+This is
+partly
+  because PA produces a weaker set of
+ $\Pi_1^*$ theorems than the latter 
+ two 
+systems.
+From our perspective,
+a more important drawback of PA
+is that it is 
+substantially
+less adept when it is
+{\it formally  unable} to confirm its own consistency.
+
+\smallskip
+
+\item[   c. ]
+Although  both the
+ $\mbox{IS}_{Tab}(\aaa)$ and
+ $\mbox{IS}_{Tab-1}(\aaa)$ formalisms 
+may be used as the core AI mechanism for
+our  2-Prong Strategy, 
+there are important distinctions
+between these two mechanisms. This is because
+ $\mbox{IS}_{Tab-1}(\aaa)$ 
+uses a more sophisticated form
+of a Self-Justifying 
+ Group-3 {\it ``I am consistent''}
+axiom than 
+ $\mbox{IS}_{Tab}(\aaa)$.
+(More generally,
+there is probably no
+% maximally
+ best form of Group-3 axiom
+that a 2-Prong Strategy can
+gainfully
+employ. 
+For instance,
+ it is 
+evident
+ (see footnote 
+\footnote{For instance,  
+ $\mbox{IS}_{Tab-1}$ can be beneficially hybridized with
+\cite{ww1}'s
+Tangibility
+  reflection principle, but we won't go into the
+%  germane
+details here. 
+} )
+that
+there do exist partially
+stronger forms of
+ Group-3 {\it ``I am consistent''}
+axioms 
+than those
+used 
+% applied besides those invoked
+ by
+ $\mbox{IS}_{Tab-1}(\aaa).~~)$
+\ennd
+\end{example}
+
+\parskip  0pt
+
+\begin{remark}
+\label{rem4}
+\rm
+The central issue raised by this chapter's
+2-Prong Strategy and the preceeding Example \ref{main-ex} 
+is 
+that
+ the dangers
+posed by global warming
+{\it  would be  tragic
+but NICELY only temporary,}
+%{\it temporary inconvenience}, 
+if robots and computers can reverse
+global warming after a period of 
+several
+ thousand years.  In contrast,
+the implications of global warming would be
+% much
+far
+ more
+severe,
+% tragic,
+if either it cannot be reversed or no frozen embryos
+%% of mammals (and primates)
+are saved so that the 4-billion-year cycle of life
+on Earth can 
+resume
+% restored
+after global  warming subsides.
+(We do not dismiss 
+the hopeful 
+prospect
+%point
+ that a theoretical temporary
+over-heating of the planet can possibly be avoided,
+but one
+ {\it  should  also not overlook} the 
+possibility
+ that some variant of a
+2-Prong Strategy may become necessary, if Mankind does not
+act  quickly enough ? )
+This article has, thus, been written to encourage a 
+more direct
+dialog
+% to take place
+between various subfields of Logic 
+with the hard-core
+sciences of 
+ biology, chemistry,   physics and mathematics.
+Its goal
+will be
+to discover what kinds of AI mechanisms can best reply to the
+challenges posed by global warming.
+% {\it if (?) they are needed !}
+\end{remark}
+
+\begin{remark}
+\label{rem5}
+Lastly, let us
+% should  
+%address 
+consider
+how G\"{o}del's Second Incompleteness
+Theorem
+should be 
+% likely
+viewed
+ by a broad 
+community
+         of scientists.
+% working in a 
+% diversity of different fields. 
+This topic is 
+% quite
+ subtle because the Second
+Incompleteness
+Theorem is 
+undoubtedly
+ a seminal result that was further strengthened
+by many of  G\"{o}del's 
+% subsequent 
+successors.
+But yet like many
+other
+ mathematical
+results, the Second Incompleteness Theorem also owns 
+{\it some types of 
+partial
+limitations?}
+%disadvantages? }
+% limitations?} 
+....Thus, we should
+  remind ourselves that
+ G\"{o}del
+% had
+% firmly
+held out the possibility 
+in
+% his  historic paper
+ \cite{Go31} 
+that some
+type of partial fulfillment of Hilbert's goals
+ would 
+%   ultimately 
+be achieved
+(see again G\"{o}del's quoted
+statement of$~*~ ).$
+Moreover,
+ Gerald Sacks in his 
+ Year-2014
+YouTube lecture
+(see
+\cite{YouSa14}
+and  our
+earlier
+%inserted
+% particular
+ footnote \ref{f2} ) 
+has recalled hearing
+G\"{o}del {\it repeating}
+% related
+similar
+ comments
+about the 
+importance 
+ of
+ continuing 
+ Hilbert's program
+ during the
+%  early
+1960's.
+%%% (This was {\it thirty years} after  \cite{Go31}'s seminal publication,)  
+ The main point is 
+that
+ a 2-Prong methodology, 
+permitting
+ an AI-based
+computer 
+%system 
+% has taken,
+% takes 
+to take temporary
+%temporarily, complete 
+control over 
+%the planet's
+planet Earth's
+destiny, {\it should
+%inherently 
+%substantially
+ultimately  be
+more efficient when
+it
+%  such
+%  a formalism 
+is allowed
+to presume its own consistency}.
+%
+%(similar to what humans commonly do).
+%
+%%  (as humans 
+%% seem to
+%% have
+%% % currently
+%%  informally
+%% done during their intuitive  thought processes).
+%% 
+These observations
+will
+ lead to
+significantly
+ new interpretations of
+G\"{o}del and Hilbert's statements $*$ and $**$.  
+\end{remark}
+
+%%% \parskip 4pt
+
+% \LARGE \baselineskip =  2.0 \normalbaselineskip 
+
+%6666666
+
+%999999
+
+In essence,
+several  research projects in 
+ symbolic logic should
+be undertaken that interact with
+the  core sciences
+of  biology, chemistry and  physics 
+%and Mathematics
+to collectively  address the
+% fundamental 
+challenges
+ posed by global warming.
+As a 
+logician (who was never
+% formally
+ trained in the experimental sciences), this author is
+%of this
+%article is
+% simply 
+ unable to assess how likely it is
+% that
+ that
+global warming will  need
+a 2-Prong response
+% necessary
+(where
+humans
+% will 
+do concede
+temporary control over the planet's destiny to AI-based computers).
+However, it is safe to presume
+that  such a loss of control would be
+only temporary 
+if  trustworthy self-justifying axiom systems
+are deployed that can simultaneously be:
+{\it 
+$\, 1)~$consistent,
+$\, 2)~$aware of their own consistency and
+$\, 3)~$capable of proving a rich set of $\Pi_1^*$ theorems.}
+
+
+
+
+% \parskip 1pt
+
+This article is  written in a context where its author
+has  reached the age of 70 and is retiring from teaching.
+Our hope is that the preceeding
+% above
+%Example \ref{main-ex} and Remarks \ref{rem4} and \ref{rem5} 
+discussion
+will encourage several logicians to  join  projects
+that   enhance  interactions 
+between the 
+logic community with the 
+allied research in
+ biology, chemistry,   physics and mathematics.
+A 
+tempting
+% nice 
+and 
+ new form of artificial intelligence
+ would
+% should 
+then
+%%%%% be likely 
+likely be
+born and 
+% nicely
+ nicely
+  prosper.
+
+
+\smallskip
+
+%ggggggg
+
+This article is dedicated to the
+memory of both
+my parents, 
+%%%Alfred and Ruth, 
+who
+% had  
+narrowly
+escaped
+the Holocaust.
+{\it $~$Let us pray that$\,$} 
+ global warming 
+% does
+ will
+%shall
+ not
+% cause
+% engulf
+impose
+% a much wider and extreme
+a
+% much
+ more wide-spread
+% wide-spreading
+calamity
+upon
+% all
+the 
+rest
+%remainder
+%
+% on the rest of
+ of humanity.
+We also hope
+%% Our hope is that  
+the
+% technical
+ mathematical 
+issues,
+raised in Sections 1-4, shall
+help future generations of researchers resolve the
+global warming crisis.... {\it  before it has become
+%sadly
+ too late ...  } 
+
+
+%% We felt  Section 5 of the current article
+%% should have a less mathematical tone than its Sections 1-4
+%% because the subject of global warming, together with its potentially
+%% required   (?) 
+%% % computerized 
+%% 2-Prong resolution, 
+%% %was sufficiently urgent to mention.
+%% should receive at least some attention
+
+
+%%  response to it,
+%% cannot
+%% %simply should not
+%% be ignored.
+
+%% 
+%% We suspect that logicians 
+%% can significantly
+%%  help
+%% biologists, chemists and physicists 
+%% resolve the global warming  crisis
+%% in the future.
+
+%hhhh 
+
+
+\parskip 0pt
+\normalsize \baselineskip =  0.96 \normalbaselineskip 
+
+\section*{APPENDIX  Reviewing
+the
+ Definition of a Tableaux Proof}
+
+Our definition of a semantic tableaux proof 
+is similar to its analog appearing in the textbooks by
+Fitting and Smullyan 
+ \cite{Fi96,Sm95}.
+A tableaux proof  of a theorem $\Psi$ from a set of proper
+axioms, denoted as $~\alpha~$, will  be a tree structure whose 
+root contains the temporary contradictory assumption of $~\neg \, \Psi~$
+and whose every descending root-to-leaf branch affirms a contradiction
+by containing both some sentence $\phi$ and its negation of $\neg \, \phi$.
+Each internal node in this tree will be either a proper axiom of
+$~\alpha~$ or a deduction
+from
+a higher 
+ancestor
+node in this tree using one of
+the
+ following
+six % deduction 
+elimination
+rules for the logical connective symbols of
+$~\wedge~$,  $~\vee~$,   
+$~ \rightarrow ~$, $~\neg~$, $~\forall~$ and $~\exists~$.
+These six rules are described below in a context where
+the 
+expression
+``{\bf $ \,  $A$  ~  \Longrightarrow  ~  $B$  \, $}''
+is an abbreviation for the sentence
+ {\bf $ \,   $B$  \, $} 
+being
+ an allowed deduction
+from its ancestor of {\bf $ \,  $A$  \, $}.
+\begin{enumerate}
+%\itemsep 5pt
+%note \small
+%%corebl \baselineskip = 1.05 \normalbaselineskip
+\item $~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \Longrightarrow ~ \, \Gamma ~$ .
+$~~~$
+\item $ \,  \neg  \,\neg \, \Upsilon  \,  \Longrightarrow  \,  \Upsilon. \, $  
+Other rules for
+the ``$ \, \neg \,$'' symbol  are:
+$ \, \neg ( \Upsilon \vee \Gamma )  \,  \Longrightarrow  \,  \neg \Upsilon
+\wedge \neg \Gamma$,
+%\newline
+$ \, \neg ( \Upsilon \rightarrow \Gamma )  \,  \Longrightarrow  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \Longrightarrow  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $
+ and $ ~~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \Longrightarrow  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed
+when their ancestor is
+$~\Upsilon \, \vee \, \Gamma.~$
+\item A pair of sibling nodes $ \,  \neg \Upsilon  \, $ and $ \,  \Gamma  \, $ is
+allowed
+when their ancestor is
+$ \, \Upsilon \, \rightarrow \, \Gamma$.
+\item $\forall v \, \Upsilon  (v)  \,  \Longrightarrow    \, \Upsilon(t)  \, $
+where $t$  may denote any 
+term.
+%%  (These terms may be any one of
+%% a constant symbol,
+%% a  parameter symbol 
+%% or a more complex object in our language $L^*$ that combines
+%% its function symbols, constant symbols and parameter symbols
+%% in a routine manner)
+\item $~ \exists v \, \Upsilon  (v) ~ \Longrightarrow ~ \, \Upsilon(p) ~$
+where $\,p \,$ is a newly introduced parameter symbol. 
+\end{enumerate}
+A minor additional comment about our notation is that we treat 
+ ``$~ \forall~ v \leq s~~~ \Phi(v)~$''
+as an abbreviation for
+ $~ \forall  v  ~~ \{ ~ v \leq s~~ \rightarrow ~~\Phi(v)~ \}~$
+and likewise
+ ``$~ \exists~ v \leq s~~~ \Phi(v)~$''
+as an abbreviation for
+ $~ \exists  v  ~~ \{ ~ v \leq s~~ \wedge ~~\Phi(v)~ \}~$.
+In our year-2005 article \cite{ww5}, we thus applied
+Rules  5 and 6 
+to derive the following further hybrid rules
+for processing bounded universal and
+ bounded
+ existential quantifiers:
+\begin{description}
+\item[  a. ]  
+ $\forall v \leq s  \, \Upsilon  (v) ~~ \Longrightarrow ~~
+t \leq s \, \rightarrow \, \Upsilon(t) $ 
+where $\,t \,$ may be any arithmetic term.
+\item[  b. ] 
+ $~ \exists v \leq s ~ \, \Upsilon  (v) ~~ \Longrightarrow ~ ~
+u \leq s ~ \wedge~ \Upsilon(p) ~$
+where $\,p \,$ is a new parameter symbol. 
+\end{description}
+
+
+% \smallskip
+
+{\bf ACKNOWLEDGMENT:} I thank Seth Chaiken for several helpful comments about
+how to
+improve the presentation.
+
+
+\bibliographystyle{abbrv}
+
+\bibliography{bb}
+
+
+\end{document}
+	
+
+
+
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@@ -0,0 +1,24 @@
+MEMO TO karen blessing   Email= blessing.17@osu.edu
+
+Subject: NEED CITATION INFORMATION for 2 articles by HARVEY FRIEDMAN 
+
+I was told to contact Karen Blessing by Christian Shaw. I have
+
+already cited three articles by emiritus Professor Harvey Friedman in
+    
+a paper that has been invited by Oxford's Journal of Logic
+
+and Computation. This article has been accepted for publication,
+
+but the type-setting office would like more precise citation information
+
+about Freidman's Ohio State technical reports. I learned about these
+
+Freidman results in a technical report by Pudlak, which did not give
+
+exact citation information. Could you please provide me with
+
+additional information about Ohio State technical report numbers
+
+for the following two papers:
+
diff --git a/nachlass/collected_dew_materials/NYTines-Jan2017.pdf b/nachlass/collected_dew_materials/NYTines-Jan2017.pdf
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+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/#n.tex#
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/b15.bib b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/b15.bib
new file mode 100644
index 0000000..6f575cd
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/b15.bib
@@ -0,0 +1,1062 @@
+        
+%% 2014 home feb 9  -3pm 
+
+%%  home 2015 jan5 same as suny copy and old ``may14''
+
+%% 2014 may 14 5.1 pm 
+
+
+\bibliography
+\bibliographystyle
+@string{notre = " Notre Dame Journal on Formal Logic"}
+@string{zeit = "Zeitscrfit fur Math Logic"}
+@string{carol =  "Com. Math. Univ. Carol"}
+@string{arch = "Archive for Mathematical Logic"}
+@string{spv = "Springer LNCS"}.
+@string{spm = "Springer Lecture Notes in Mathematics"}.
+@string{apal = "Annals Pure and Applied Logic"}.
+@string{jsl = "Jour. Symb. Logic"}
+@string{sicomp = "Siam Journal on Computing"}
+@string{fund = " Fundamenta Mathematicae"}
+@string{ljipl = "Logic Journal of the IPL"}
+@string{icomp = "Information and Computation"}
+@string{ma26 = "Mathematische Annalen"}
+
+
+
+@article{Ad2,
+        author =        {Z. Adamowicz},
+        title =         {Herbrand Consistency and Bounded
+Arithmetic}, 
+        journal =       fund,
+        year =          2002,
+         volume =       171,
+        number = 3, 
+        pages =        {279-292}
+}
+
+
+
+@article{AB1,
+        author =        {Z. Adamowicz and T. Bigorajska},
+        title =         {Existentially Closed Structures and \mbox{G\"{o}del's} Second Incompleteness Theorem},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+         pages =        {349-356},
+}
+
+
+
+@book{AZ96,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {The Logic of Mathematics: A Modern Course in Classical Logic}, 
+        publisher = {John Wiley and Sons},
+        year =      1997
+}
+
+
+@article{AZ1,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {On  \mbox{Herbrand} consistency in weak theories},
+        journal =       arch,
+        year =          2001,
+         volume =       40,
+        number = 6,
+      pages =        {399-413}
+}
+
+
+
+
+@article{Ar90,
+        author =        {T. Arai},
+        title =         {Derivability Conditions on  \mbox{Rosser's} Proof Predicates},
+        journal =       notre,
+        year =          1990,
+         volume =       31,
+         pages =        {487-497}
+}
+
+@phdthesis{Be62,
+        author =        {J. Benett},
+               school = {Princeton University},
+               year =          1962,
+        note = {A detailed summary of Benett's main theorem can
+be found on pages 299--303 and 406
+of the  H\'{a}jek-Pudl\'{a}k textbook \cite{HP91}}
+}
+
+@article{BS76,
+        author =        {A. Bezboruah and J. C. Shepherdson},
+        title =         {G\"{o}del's Second Incompleteness Theorem for \mbox{Q}},
+        journal =       jsl,
+        year =          1976,
+         volume =       41,
+         number = 2,
+         pages =        {503-512}
+}
+
+
+
+
+@misc{Br94,
+        author =        {S. Bringsford},
+        note = {Private  
+conversation,  1994,  helpfully suggesting 
+I place my mathematical and philosophical results
+in distinctly separate articles.}
+
+}
+
+@book{Bu86,
+        author =        {S. R.  Buss},
+        title =         { Bounded Arithmetic},
+        publisher = {Studies in Proof Theory, Lecture Notes 3, published
+by  Bibliopolis},
+        year =       1986,
+        note = {(Revised version of Ph. D. Thesis.)}
+}
+
+
+
+@article{BI95,
+        author =        {S. R.  Buss and A. Ignjatovic},
+        title =         {Unprovability of Consistency Statements in Fragments of Bounded Arithmetic},
+        journal =       apal,
+        year =          1995,
+         volume =       74,
+        number = 3,
+         pages =        {221-244},
+}
+
+
+
+@book{Da97,
+        author =        {J W Dawson},
+        title =         {Logical Dilemmas the life and work of
+Kurt G\"{o}del},
+        publisher = {AKPeters},
+        year =      1997
+}
+
+
+
+@article{D89,
+        author =        {C. Dimitracopoulos},
+        title =         {Overspill and Fragments of Arithmetic},
+        journal =       arch,
+        year =          1989,
+         volume =       28,
+         pages =        {173-179},
+}
+
+@article{Fe60,
+        author =        {S. Feferman},
+        title =    {Arithmetization of Metamathematics in a General Setting},
+        journal =       fund,
+        year =          1960,
+         volume =       49,
+         pages =        { 35-92},
+}
+
+
+@article{FKO60,
+        author =        {S. Feferman and G. Kriesel and S. Orey},
+        title =         {1-Consistency and Faithful Interpretations},
+        journal =       arch,
+        year =          1962,
+         volume =       6,
+         pages =        {52-63},
+}
+
+
+
+
+@book{Fi90,
+        author =        {M. Fitting},
+        title =         { First Order Logic and Automated Theorem Proving},
+        publisher =  {Springer},
+        year =      1996
+}
+
+@techreport{Fr79a,
+        author =        {H. M.  Friedman},
+        title =         {On the Consistency, Completeness and Correctness
+Problems},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+
+
+@techreport{Fr79b,
+        author =        {H. M.  Friedman},
+        title =         {Translatability and Relative Consistency},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+@article{Go31,
+        author =        {K. G\"{o}del},
+        title =         {{\"{U}ber formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandter Systeme I}},
+        journal =       { Monatshefte f\"{u}r Math. Phys.},
+        year =          1931,
+         volume =       38,
+         pages =        {173-198}
+}
+
+
+
+@inproceedings{Go33,    
+        author =        {K. G\"{o}del}, 
+  title =       {The present situation in the foundations of
+mathematics},
+  booktitle =   {Collected Works Volume III: Unpublished Essays and Lectures},
+        year =          {2004},
+        pages =         {45--53},
+ editor = {S. Feferman and J. W. Dawson   and W. Goldfarb and C. Parson and R. Solovay}, 
+         publisher =     {Oxford University Press},
+     note = {Our quotes from this 1933 lecture come from its page 52.}
+}
+
+
+@book{Go5,
+        author =        {R. Goldstein},
+    title =         {Incompleteness The Proof and Paradox of Kurt G\"{o}del},
+        publisher = {Norton},
+        year =      2005
+}
+
+
+
+@article{Ha71,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part I}},
+        journal =       carol,
+        year =          1971,
+         volume =       12,
+         pages =        {73--79}
+}
+
+
+@article{Ha72,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part II}},
+        journal =       carol,
+        year =          1972,
+         volume =       13,
+         pages =        { 445-455}
+}
+
+
+@article{Ha81,
+        author =        {P. H\'{a}jek},
+        title =         {Interpretability in Theories Containing Arithmetic},
+        journal =       carol,
+        year =          1981,
+         volume =       22,
+         pages =        {225-234}
+}
+
+
+
+@article{Ha7,
+        author =        {P. H\'{a}jek},
+        title =         {Mathematical fuzzy logic and natural numbers},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {155-163}
+}
+
+
+@article{Ha11,
+        author =        {P. H\'{a}jek},
+        title =         {Towards metamathematics of weak arithmetics over
+fuzzy},
+        journal =       ljipl,
+        year =          2011,
+         volume =       19,        
+        number = 3,
+         pages =        {467-475}
+}
+
+
+
+@book{HP91,
+        author =        {P. H\'{a}jek and P. Pudl\'{a}k},
+        title =         { Metamathematics of First Order Arithmetic},
+        publisher =  {Springer Verlag},
+        year =  1991     
+}
+
+
+@article{Hil26,
+        author =        {D. Hilbert},
+        title =         {{\"{U}ber das Unendliche}},
+        journal =       ma26,
+        year =          1926,
+         volume =       95,
+         pages =        {161-191}
+}
+
+
+
+
+@book{HB39,
+        author =        {D. Hilbert and P. Bernays},
+        title =         { Grundlagen der Mathematik (Vol II)}, 
+        publisher = {Springer},
+        year =      1939
+}
+
+
+@article{Je71,
+        author =        {R. G. Jeroslow},
+        title =         {Consistency Statements in Formal Theories},
+        journal =       fund,
+        year =          1971,
+         volume =       72,
+         pages =        { 17-40},
+}
+
+
+@book{Ka91,
+        author =        {R. Kaye},
+        title =         { Models of Peano Arithmetic},
+        publisher = {Oxford University Press},
+        year =          1991
+}
+
+
+@article{Kl38,
+        author =        {S. C. Kleene},
+        title =         {On  Notation for Ordinal Numbers},
+        journal =       jsl,
+        year =          1938,
+         volume =       3,
+         number =     1,
+         pages =        {150-156},
+}
+
+
+
+@article{Ko6,
+        author =        {L. A. Ko{\l}odziejczyk},
+        title =         {On the 
+\mbox{Herbrand} Notion of Consistency for Finitely
+Axiomatizable Fragments of Bounded Arithmetic Theories},
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 2,
+         pages =        {624-638}
+}
+
+
+
+@article{Kr87,
+        author =        { J. Kraj\'{i}cek},
+        title =         {A Note on Proofs of Falsehood},
+        journal =       arch,
+        year =          26,
+         volume =       1987,
+         pages =        {169-176}
+}
+
+
+@book{Kr95,
+        author =        { J. Kraj\'{i}cek},
+        title =         {Bounded Propositional Logic and Complexity Theory},
+        publisher =   {Cambridge University Press}, 
+        year =          1995
+}
+
+
+
+
+
+
+@article{KT74,
+        author =        {G. Kreisel and G. Takeuti},
+        title =         {Formally Self-Referential Propositions in Cut-Free Classical Analysis and Related Systems},
+        journal =       {Dissertationes Mathematicae},
+        year =          1974,
+         volume =       118,
+         pages =        {1--50}
+}
+
+
+@incollection{Li84,
+        author =        {P. Lindstr\"{o}m},
+        title =         {On faithful Interpretability},
+        booktitle =      {Computation and Proof Theory},
+        publisher = spm,
+        volume = 1104,
+        year =          1984,
+         pages =        {279-288}
+}
+
+@article{Lo55,
+        author =        { M. H. L\"{o}b},
+        title =         {A Solution of a Problem of \mbox{Leon Henkin}},
+        journal =       jsl,
+        year =          1955,
+         volume =       20,
+        number = 2,
+         pages =        {115-118},
+}
+
+
+
+@book{Mend,
+        author =        {E. Mendelson},
+        title =         {Introduction to Mathematical Logic},
+        publisher =     { Chapman Hall},
+        year = 2010
+}
+
+
+@article{Mo58,
+        author =        {R. Montague},
+        title =         {The Continuum of Relative Interpretability},
+        journal =       jsl,
+        year =          23,
+         volume =       1958,
+         pages =        {494-511},
+}
+
+
+
+@article{Mo62,
+        author =        {R. Montague},
+        title =         {Theories Incomparable with Respect to Interpretability},
+        journal =       jsl,
+        year =          1962,
+         volume =       27,
+         pages =        {195-211},
+}
+
+
+@article{My62,
+        author =        {J. Mycieslski},
+        title =         {A Lattice Connected with 
+ Relative Interpretability},
+        journal =        {Notices of AMS},
+        year =          1962,
+         volume =       9,
+         pages =        {407-408},
+}
+
+
+
+@book{Ne86,
+        author =        {E. Nelson},
+        title =         {Predicative Arithmetic},
+        publisher =     { Math Notes, Princeton Univ Press},
+        year = 1986 
+}
+
+
+
+@article{Or61,
+        author =        {S. Orey},
+        title =         {Relative Interpretations},
+        journal =       zeit,
+        year =          1961,
+         volume =       7,
+         pages =        {146-153.},
+}
+
+
+
+@article{Pa71,
+        author =        {R. Parikh},
+        title =         {Existence and Feasibility in Arithmetic},
+        journal =       jsl,
+        year =          1971,
+         volume =       36,
+        number = 3,
+         pages =        {494-508},
+}
+
+
+
+@article{PD83,
+        author =        {J. B. Paris and C. Dimitracopoulos},
+        title =         {A Note on the Undefinability of Cuts},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {564-569},
+}
+
+
+@inproceedings{PW81,
+        author =        {J. B. Paris and A. J. Wilkie},
+        title =         {\mbox{$\Delta_0$} Sets and Induction},
+        booktitle =     {Proceedings of the  \mbox{Jadswin} Logic Conference},
+        year =          1981,
+        publisher = {Leeds University Press},
+         pages =        {237-248}
+}
+
+
+@article{Pa72,
+        author =        {C. H. Parsons},
+        title =         {On $n-$Quantifier Induction},
+        journal =       jsl,
+        year =          1972,
+         volume =       37,
+        number = 3,
+         pages =        {466-482},
+}
+
+
+@article{Pu83,
+        author =        {P. Pudl\'{a}k},
+        title =         {Some Prime Elements in the Lattice of Interpretability},
+        journal =       {Transactions of the  AMS},
+        year =          1983,
+         volume =       280,
+         pages =        {255-275},
+}
+
+
+@incollection{Pu84,
+        author =        {P. Pudl\'{a}k},
+        title =         {On Lengths of Proofs of Finisitic Consistency Statements in First order Theories},
+        booktitle =      {Logic Colloquium 84},
+        publisher = {North Holland},
+        year =          1984,
+         pages =        {165-196.}
+}
+
+
+
+@article{Pu85,
+        author =        {P. Pudl\'{a}k},
+        title =         {Cuts, Consistency Statements and Interpretations},
+        journal =       jsl,
+        year =          1985,
+         volume =       50,
+        number = 2,
+         pages =        {423-442}
+}
+
+
+
+@article{Pu87,
+        author =        {P. Pudl\'{a}k},
+        title =         {Improved Bounds on the Lengths of Proofs of Finistic Consistency Statements},
+        journal =   {AMS Contemporary Mathematics Series},     
+        year =          1987,
+         volume =       65,
+         pages =        {309-331},
+}
+
+@incollection{Pu96,
+        author =        {P. Pudl\'{a}k},
+        title =         {On the Lengths of Proofs of Consistency},
+        booktitle =  {Collegium Logicum: Annals of the Kurt G\"{o}del},
+        year =        1996,
+         publisher = {Springer-Wien},
+         volume =      2,
+         pages =        { 65-86},
+}
+
+
+
+
+@incollection{Pu98,
+        author =        {P. Pudl\'{a}k},
+        title =         {The Lengths of Proofs},
+        booktitle =       {The Handbook of Proof Theory} ,
+        year =          1998,
+         publisher =       {North Holland},
+         pages =        {547-636}
+}
+
+
+
+
+
+@article{Ra93,
+        author =        {Z. Ratajczyk},
+        title =         {Subsystems of True Arithmetic and Hierarchies of
+Functions},
+        journal =       apal,
+        year =          1993,
+         volume =       64,
+         pages =        { 95--152},
+}
+
+
+
+@book{Ro67,
+        author =        {H. A. Rogers },
+        title =         {Theory of Recursive Functions and Effective Compatibility}, 
+        publisher =      {McGraw Hill},
+        year =          {1967}
+}
+
+
+
+@article{Ro36,
+        author =        {J. B. Rosser},
+        title =         {Extensions of Some Earlier Theorems by  \mbox{ G\"{o}del and Church} },
+        journal =       jsl,
+        year =          1936,
+         volume =       1,
+         pages =        {87-91},
+}
+
+@phdthesis{Sa1,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency in Arithmetics with Bounded Induction},
+        school = {Polish Academy},
+               year =          2001
+}
+
+
+@article{Sa12,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency of Some Arithmetical Theories},
+        journal =       jsl,
+        year =          2012,
+         volume =       77,
+        number = 3,
+         pages =        {807-827}
+}
+
+
+@incollection{Sm77,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {The Incompleteness Theorem},
+        booktitle =     { Handbook on Mathematical Logic},
+        year =          1977,
+         publisher =       {North Holland},
+         pages =        {821--865}
+}
+
+
+
+@incollection{Sm85,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {Non-standard Models and Related Developments},
+        booktitle =     {Harvey Friedman's Research in the Foundations of Mathematics},
+        year =          1985,
+         publisher =       {North Holland},
+         pages =        {179--220}
+}
+
+
+
+
+@book{Smul,
+        author =        {R. M. Smullyan},
+        title =         {  First Order Logic},
+        publisher =      {Diver},
+        year =          {1995}
+}
+
+
+
+@article{So88,
+        author =        {R. M Solovay},
+        title =         {Injecting Inconsistencies into Models of \mbox{PA}},
+        journal =       apal,
+        year =          1988,
+         volume =       44,
+         pages =        {102-132}
+}
+
+
+@misc{So94,
+        author =        {R. M. Solovay},
+        note = {Telephone  
+conversation
+in 1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using 
+some 
+methods
+of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+(The Appendix A of 
+\cite{ww1} offers a 
+4-page summary of
+this conversation.)}
+}
+
+
+@article{Sv78,
+        author =        {V. Svejdar},
+        title =         {Degrees of Interpretability},
+        journal =       carol,
+        year =          1978,
+         volume =       19,
+         pages =        {783-813}
+}
+
+
+
+@article{Sv83,
+        author =        {V. Svejdar},
+        title =         {Modal Analysis of Generalized
+\mbox{Rosser} Sentences},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {986-999}
+}
+
+
+
+
+@article{Sv7,
+        author =        {V. Svejdar},
+        title =         {{An interpretation of Robinson arithmetic in its
+Grzegorczjk's weaker variant}},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {347-354}
+}
+
+
+
+
+
+@book{Ta87,
+        author =        {G. Takeuti},
+        title =         {Proof Theory},
+        publisher =     {North Holland},
+        year =          {1987}
+}
+
+
+
+
+@article{Ta0,
+        author =        {G. Takeuti},
+        title =         {G\"{o}del Sentences of Bounded Arithmetic},
+        journal =       jsl,
+        year =          2000,
+         volume =       65,
+         pages =        {1338-1346}
+}
+
+@book{TMR53,
+        author =        {A. Tarski and A. Mostowski and R.  Robinson},
+        title =         {Undecidable Theories},
+        publisher =     { North Holland Press}, 
+        year =          1953,
+}
+
+
+@inproceedings{Vi90,
+        author =        {A. Visser},
+        title =         {Interpretability Logic},
+        booktitle =     {Mathematical \mbox{Logic: Proceedings of the Heyting Summer School}},
+        year =         1988,
+                 pages =        {175-208},
+}
+
+
+@article{Vi92,
+        author =        {A. Visser},
+        title =         {An Inside View of Exp},
+        journal =       jsl,
+        year =         1992,
+         volume =       57,
+         pages =        {131--165},
+}
+
+
+
+@article{Vi93,
+        author =        {A. Visser},
+        title =         {The Unprovability of Small Inconsistency},
+        journal =       arch,
+         year =         1993,
+         volume =       32,
+         pages =        {131--165},
+}
+
+@article{Vi5,
+        author =        {A. Visser},
+        title =         {Faith and Falsity},
+        journal =       apal,
+        year =          2005,
+         volume =       131,
+  number = 1,
+              pages =        {103--131}
+}
+
+@article{VH73,
+        author =        {P. Vop\v{e}nka and P. H\'{a}jek},
+        title =         {Existence of  a Generalized Semantic Model of
+\mbox{G\"{o}del-Bernays} Set Theory},
+        journal =       { Bulletin de l'Academie Polonaise des Sciences,Mathmatiques, Astromiques et Physiques},
+        year =         1973,
+         volume =       12,
+         pages =        {1079-1086},
+}
+12 (1973) pp.1079-1086.
+
+@article{WP87,
+        author =        {A. J. Wilkie and J. B. Paris},
+        title =         {On the Scheme of Induction for Bounded
+Arithmetic},
+        journal =       apal,
+        year =          1987,
+         volume =       35,
+         pages =        {261-302},
+}
+
+
+
+
+@article{ww93,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying Axiom Systems},
+        journal =       spv,
+        year =          1993,
+         volume =       713,
+         pages =        {325-336},
+          note =  {Proceedings of  Third Kurt  G\"{o}del Symposium}
+         }
+
+@article{sp0,
+        author =        {D. E. Willard},
+        title =         {The semantic tableaux version of the second
+incompleteness theorem extends almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+                journal =       spv,
+        year =          2000,
+         volume =       1847, 
+         pages =        {415-430},
+         note = {Proceedings of ``Tableaux-2000'' Conference }
+       }
+
+
+
+@article{ww1,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying  Systems, the Incompleteness
+Theorem and the
+Tangibility Reflection Principle},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+        number = 2,
+         pages =        {536-596}
+}
+
+
+
+
+@article{ww2,
+        author =        {D. E. Willard},
+        title =         {How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem Almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+        journal =       jsl,
+        year =          2002,
+         volume =       67,
+        number = 1,
+         pages =        {465-496}
+         }
+
+
+@article{tab2,
+        author =        {D. E. Willard},
+        title =         {Some New Exceptions for
+the Semantic Tableaux Version of the
+Second Incompleteness Theorem},
+        journal =       spv,
+        year =          2002,
+         volume =       2381, 
+         pages =        {281-297},
+         note = { (``Tableaux-2002'' Conference Proceedings)}
+         }
+
+
+
+
+@inproceedings{wwlogos,    
+        author =        {D. E. Willard}, 
+        title =         {A Version of the
+                         Second Incompleteness Theorem For Axiom
+                         Systems that Recognize Addition 
+                         But Not Multiplication as a Total Function},
+        booktitle =     {First Order Logic Revisited},
+        year =          {2004},
+        address =       {Berlin},
+        pages =         {337--368},
+ editor = {V. Hendricks and F. Neuhaus and S. A. Pederson and U. Scheffler and H. Wansing},
+         publisher =     {Logos Verlag}
+}
+
+
+
+
+@misc{wwconf,
+        author =        {D. E. Willard},
+        title =         {On Two Partial (and not Full) Respects 
+Where an Axiom System Can Recognize Its Own Consistency
+and Multiplication as a Total Function},
+        year =          2005,
+        note = {a presented talk at the
+summer ASL-2005 conference in Athens whose
+300-word abstract will be publised in the {\it Bulletin of
+Symbolic Logic} and which is described in further detail in an
+University of Albany technical report.}
+
+ }
+
+
+
+@article{wwpete,
+        author =        {D. E. Willard},
+        title =         {A New Variant of \mbox{Hilbert} Styled Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+        note = {A 200-word abstract summarizing
+the contents of this 
+forthcoming invited article can be found on the
+web-site of {\it The 2nd St. Petersburg Conference on Logic and
+Computability (2003)}, i.e. http://logic.pdmi.ras.ru/2ndDays
+or in the Atlas Mathematical Conference Abstracts.},
+ 
+         }
+
+
+
+@article{ww5,
+        author =        {D. E. Willard},
+        title =          {An Exploration of the Partial Respects 
+in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency},
+        journal =       jsl,
+        year =          2005,
+         volume =       70,
+        number = 4,
+         pages =        {1171-1209},
+}
+
+
+@article{sp5,
+        author =        {D. E. Willard},
+        title =    {On the Partial Respects in which a
+Real Valued  Arithmetic System Can Verify its Tableaux Consistency},
+        journal =       spv,
+        year =          2005,
+         volume =       3702,
+         pages =        {292-306},
+}
+
+
+@article{wwapal,
+        author =        {D. E. Willard},
+        title =         {A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+         volume =       141,
+        number = 3,
+         pages =        {472-496}
+         }
+
+
+
+
+
+@article{ww6,
+        author =        {D. E. Willard},
+        title =          {On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency}, 
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 4,
+         pages =        {1189-1199}
+}
+
+
+
+@article{ww7,
+        author =        {D. E. Willard},
+        title =         {{Passive induction and a solution to a Paris-Wilkie 
+open question}},
+        journal =       apal,
+        year =          2007,
+         volume =       146,
+        number = 2,
+         pages =        {124-149}
+         }
+
+
+
+
+@article{ww9,
+        author =        {D. E. Willard},
+        title =         {{Some 
+specially formulated
+axiomizations for I$\Sigma_0$
+manage to
+evade
+the Herbrandized version of the second incompleteness theorem}},
+        journal =       icomp,
+        year =          2009,
+         volume =       207,
+        number = 10,
+         pages  =        {1078-1093}
+         }
+
+
+@article{ww14,
+        author =        {D. E. Willard},
+        title =         {On the Broader Epistemological
+Significance of Self-Justiying Axiom Systems},
+        journal =       spv,
+        year =          2014,
+         volume =       8652,
+         pages  =        {221-236},
+         note = {(an earlier more abbreviated
+         version of this current article that had appeared in
+         the Proceedings of 21st Wollic Conference)}
+         }
+
+
+
+@article{Wr78,
+        author =        {C. Wrathall},
+        title =         {Rudimentary Predicates and Relative Computation},
+        journal =       sicomp,
+        year =          1978,
+         volume =       7,
+         pages =        {194-209}
+}
+
+
+@book{Yo5,
+        author =        {P. Yourgrau},
+        title =         {A World Without Time: The Forgotten Legacy of
+G\"{o}del and Einstein},
+        publisher = {Basic Books},
+        year =      2005,
+        note = {See page 58 for the passages we have quoted}
+}
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup.bib b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup.bib
new file mode 100644
index 0000000..6f575cd
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup.bib
@@ -0,0 +1,1062 @@
+        
+%% 2014 home feb 9  -3pm 
+
+%%  home 2015 jan5 same as suny copy and old ``may14''
+
+%% 2014 may 14 5.1 pm 
+
+
+\bibliography
+\bibliographystyle
+@string{notre = " Notre Dame Journal on Formal Logic"}
+@string{zeit = "Zeitscrfit fur Math Logic"}
+@string{carol =  "Com. Math. Univ. Carol"}
+@string{arch = "Archive for Mathematical Logic"}
+@string{spv = "Springer LNCS"}.
+@string{spm = "Springer Lecture Notes in Mathematics"}.
+@string{apal = "Annals Pure and Applied Logic"}.
+@string{jsl = "Jour. Symb. Logic"}
+@string{sicomp = "Siam Journal on Computing"}
+@string{fund = " Fundamenta Mathematicae"}
+@string{ljipl = "Logic Journal of the IPL"}
+@string{icomp = "Information and Computation"}
+@string{ma26 = "Mathematische Annalen"}
+
+
+
+@article{Ad2,
+        author =        {Z. Adamowicz},
+        title =         {Herbrand Consistency and Bounded
+Arithmetic}, 
+        journal =       fund,
+        year =          2002,
+         volume =       171,
+        number = 3, 
+        pages =        {279-292}
+}
+
+
+
+@article{AB1,
+        author =        {Z. Adamowicz and T. Bigorajska},
+        title =         {Existentially Closed Structures and \mbox{G\"{o}del's} Second Incompleteness Theorem},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+         pages =        {349-356},
+}
+
+
+
+@book{AZ96,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {The Logic of Mathematics: A Modern Course in Classical Logic}, 
+        publisher = {John Wiley and Sons},
+        year =      1997
+}
+
+
+@article{AZ1,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {On  \mbox{Herbrand} consistency in weak theories},
+        journal =       arch,
+        year =          2001,
+         volume =       40,
+        number = 6,
+      pages =        {399-413}
+}
+
+
+
+
+@article{Ar90,
+        author =        {T. Arai},
+        title =         {Derivability Conditions on  \mbox{Rosser's} Proof Predicates},
+        journal =       notre,
+        year =          1990,
+         volume =       31,
+         pages =        {487-497}
+}
+
+@phdthesis{Be62,
+        author =        {J. Benett},
+               school = {Princeton University},
+               year =          1962,
+        note = {A detailed summary of Benett's main theorem can
+be found on pages 299--303 and 406
+of the  H\'{a}jek-Pudl\'{a}k textbook \cite{HP91}}
+}
+
+@article{BS76,
+        author =        {A. Bezboruah and J. C. Shepherdson},
+        title =         {G\"{o}del's Second Incompleteness Theorem for \mbox{Q}},
+        journal =       jsl,
+        year =          1976,
+         volume =       41,
+         number = 2,
+         pages =        {503-512}
+}
+
+
+
+
+@misc{Br94,
+        author =        {S. Bringsford},
+        note = {Private  
+conversation,  1994,  helpfully suggesting 
+I place my mathematical and philosophical results
+in distinctly separate articles.}
+
+}
+
+@book{Bu86,
+        author =        {S. R.  Buss},
+        title =         { Bounded Arithmetic},
+        publisher = {Studies in Proof Theory, Lecture Notes 3, published
+by  Bibliopolis},
+        year =       1986,
+        note = {(Revised version of Ph. D. Thesis.)}
+}
+
+
+
+@article{BI95,
+        author =        {S. R.  Buss and A. Ignjatovic},
+        title =         {Unprovability of Consistency Statements in Fragments of Bounded Arithmetic},
+        journal =       apal,
+        year =          1995,
+         volume =       74,
+        number = 3,
+         pages =        {221-244},
+}
+
+
+
+@book{Da97,
+        author =        {J W Dawson},
+        title =         {Logical Dilemmas the life and work of
+Kurt G\"{o}del},
+        publisher = {AKPeters},
+        year =      1997
+}
+
+
+
+@article{D89,
+        author =        {C. Dimitracopoulos},
+        title =         {Overspill and Fragments of Arithmetic},
+        journal =       arch,
+        year =          1989,
+         volume =       28,
+         pages =        {173-179},
+}
+
+@article{Fe60,
+        author =        {S. Feferman},
+        title =    {Arithmetization of Metamathematics in a General Setting},
+        journal =       fund,
+        year =          1960,
+         volume =       49,
+         pages =        { 35-92},
+}
+
+
+@article{FKO60,
+        author =        {S. Feferman and G. Kriesel and S. Orey},
+        title =         {1-Consistency and Faithful Interpretations},
+        journal =       arch,
+        year =          1962,
+         volume =       6,
+         pages =        {52-63},
+}
+
+
+
+
+@book{Fi90,
+        author =        {M. Fitting},
+        title =         { First Order Logic and Automated Theorem Proving},
+        publisher =  {Springer},
+        year =      1996
+}
+
+@techreport{Fr79a,
+        author =        {H. M.  Friedman},
+        title =         {On the Consistency, Completeness and Correctness
+Problems},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+
+
+@techreport{Fr79b,
+        author =        {H. M.  Friedman},
+        title =         {Translatability and Relative Consistency},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+@article{Go31,
+        author =        {K. G\"{o}del},
+        title =         {{\"{U}ber formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandter Systeme I}},
+        journal =       { Monatshefte f\"{u}r Math. Phys.},
+        year =          1931,
+         volume =       38,
+         pages =        {173-198}
+}
+
+
+
+@inproceedings{Go33,    
+        author =        {K. G\"{o}del}, 
+  title =       {The present situation in the foundations of
+mathematics},
+  booktitle =   {Collected Works Volume III: Unpublished Essays and Lectures},
+        year =          {2004},
+        pages =         {45--53},
+ editor = {S. Feferman and J. W. Dawson   and W. Goldfarb and C. Parson and R. Solovay}, 
+         publisher =     {Oxford University Press},
+     note = {Our quotes from this 1933 lecture come from its page 52.}
+}
+
+
+@book{Go5,
+        author =        {R. Goldstein},
+    title =         {Incompleteness The Proof and Paradox of Kurt G\"{o}del},
+        publisher = {Norton},
+        year =      2005
+}
+
+
+
+@article{Ha71,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part I}},
+        journal =       carol,
+        year =          1971,
+         volume =       12,
+         pages =        {73--79}
+}
+
+
+@article{Ha72,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part II}},
+        journal =       carol,
+        year =          1972,
+         volume =       13,
+         pages =        { 445-455}
+}
+
+
+@article{Ha81,
+        author =        {P. H\'{a}jek},
+        title =         {Interpretability in Theories Containing Arithmetic},
+        journal =       carol,
+        year =          1981,
+         volume =       22,
+         pages =        {225-234}
+}
+
+
+
+@article{Ha7,
+        author =        {P. H\'{a}jek},
+        title =         {Mathematical fuzzy logic and natural numbers},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {155-163}
+}
+
+
+@article{Ha11,
+        author =        {P. H\'{a}jek},
+        title =         {Towards metamathematics of weak arithmetics over
+fuzzy},
+        journal =       ljipl,
+        year =          2011,
+         volume =       19,        
+        number = 3,
+         pages =        {467-475}
+}
+
+
+
+@book{HP91,
+        author =        {P. H\'{a}jek and P. Pudl\'{a}k},
+        title =         { Metamathematics of First Order Arithmetic},
+        publisher =  {Springer Verlag},
+        year =  1991     
+}
+
+
+@article{Hil26,
+        author =        {D. Hilbert},
+        title =         {{\"{U}ber das Unendliche}},
+        journal =       ma26,
+        year =          1926,
+         volume =       95,
+         pages =        {161-191}
+}
+
+
+
+
+@book{HB39,
+        author =        {D. Hilbert and P. Bernays},
+        title =         { Grundlagen der Mathematik (Vol II)}, 
+        publisher = {Springer},
+        year =      1939
+}
+
+
+@article{Je71,
+        author =        {R. G. Jeroslow},
+        title =         {Consistency Statements in Formal Theories},
+        journal =       fund,
+        year =          1971,
+         volume =       72,
+         pages =        { 17-40},
+}
+
+
+@book{Ka91,
+        author =        {R. Kaye},
+        title =         { Models of Peano Arithmetic},
+        publisher = {Oxford University Press},
+        year =          1991
+}
+
+
+@article{Kl38,
+        author =        {S. C. Kleene},
+        title =         {On  Notation for Ordinal Numbers},
+        journal =       jsl,
+        year =          1938,
+         volume =       3,
+         number =     1,
+         pages =        {150-156},
+}
+
+
+
+@article{Ko6,
+        author =        {L. A. Ko{\l}odziejczyk},
+        title =         {On the 
+\mbox{Herbrand} Notion of Consistency for Finitely
+Axiomatizable Fragments of Bounded Arithmetic Theories},
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 2,
+         pages =        {624-638}
+}
+
+
+
+@article{Kr87,
+        author =        { J. Kraj\'{i}cek},
+        title =         {A Note on Proofs of Falsehood},
+        journal =       arch,
+        year =          26,
+         volume =       1987,
+         pages =        {169-176}
+}
+
+
+@book{Kr95,
+        author =        { J. Kraj\'{i}cek},
+        title =         {Bounded Propositional Logic and Complexity Theory},
+        publisher =   {Cambridge University Press}, 
+        year =          1995
+}
+
+
+
+
+
+
+@article{KT74,
+        author =        {G. Kreisel and G. Takeuti},
+        title =         {Formally Self-Referential Propositions in Cut-Free Classical Analysis and Related Systems},
+        journal =       {Dissertationes Mathematicae},
+        year =          1974,
+         volume =       118,
+         pages =        {1--50}
+}
+
+
+@incollection{Li84,
+        author =        {P. Lindstr\"{o}m},
+        title =         {On faithful Interpretability},
+        booktitle =      {Computation and Proof Theory},
+        publisher = spm,
+        volume = 1104,
+        year =          1984,
+         pages =        {279-288}
+}
+
+@article{Lo55,
+        author =        { M. H. L\"{o}b},
+        title =         {A Solution of a Problem of \mbox{Leon Henkin}},
+        journal =       jsl,
+        year =          1955,
+         volume =       20,
+        number = 2,
+         pages =        {115-118},
+}
+
+
+
+@book{Mend,
+        author =        {E. Mendelson},
+        title =         {Introduction to Mathematical Logic},
+        publisher =     { Chapman Hall},
+        year = 2010
+}
+
+
+@article{Mo58,
+        author =        {R. Montague},
+        title =         {The Continuum of Relative Interpretability},
+        journal =       jsl,
+        year =          23,
+         volume =       1958,
+         pages =        {494-511},
+}
+
+
+
+@article{Mo62,
+        author =        {R. Montague},
+        title =         {Theories Incomparable with Respect to Interpretability},
+        journal =       jsl,
+        year =          1962,
+         volume =       27,
+         pages =        {195-211},
+}
+
+
+@article{My62,
+        author =        {J. Mycieslski},
+        title =         {A Lattice Connected with 
+ Relative Interpretability},
+        journal =        {Notices of AMS},
+        year =          1962,
+         volume =       9,
+         pages =        {407-408},
+}
+
+
+
+@book{Ne86,
+        author =        {E. Nelson},
+        title =         {Predicative Arithmetic},
+        publisher =     { Math Notes, Princeton Univ Press},
+        year = 1986 
+}
+
+
+
+@article{Or61,
+        author =        {S. Orey},
+        title =         {Relative Interpretations},
+        journal =       zeit,
+        year =          1961,
+         volume =       7,
+         pages =        {146-153.},
+}
+
+
+
+@article{Pa71,
+        author =        {R. Parikh},
+        title =         {Existence and Feasibility in Arithmetic},
+        journal =       jsl,
+        year =          1971,
+         volume =       36,
+        number = 3,
+         pages =        {494-508},
+}
+
+
+
+@article{PD83,
+        author =        {J. B. Paris and C. Dimitracopoulos},
+        title =         {A Note on the Undefinability of Cuts},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {564-569},
+}
+
+
+@inproceedings{PW81,
+        author =        {J. B. Paris and A. J. Wilkie},
+        title =         {\mbox{$\Delta_0$} Sets and Induction},
+        booktitle =     {Proceedings of the  \mbox{Jadswin} Logic Conference},
+        year =          1981,
+        publisher = {Leeds University Press},
+         pages =        {237-248}
+}
+
+
+@article{Pa72,
+        author =        {C. H. Parsons},
+        title =         {On $n-$Quantifier Induction},
+        journal =       jsl,
+        year =          1972,
+         volume =       37,
+        number = 3,
+         pages =        {466-482},
+}
+
+
+@article{Pu83,
+        author =        {P. Pudl\'{a}k},
+        title =         {Some Prime Elements in the Lattice of Interpretability},
+        journal =       {Transactions of the  AMS},
+        year =          1983,
+         volume =       280,
+         pages =        {255-275},
+}
+
+
+@incollection{Pu84,
+        author =        {P. Pudl\'{a}k},
+        title =         {On Lengths of Proofs of Finisitic Consistency Statements in First order Theories},
+        booktitle =      {Logic Colloquium 84},
+        publisher = {North Holland},
+        year =          1984,
+         pages =        {165-196.}
+}
+
+
+
+@article{Pu85,
+        author =        {P. Pudl\'{a}k},
+        title =         {Cuts, Consistency Statements and Interpretations},
+        journal =       jsl,
+        year =          1985,
+         volume =       50,
+        number = 2,
+         pages =        {423-442}
+}
+
+
+
+@article{Pu87,
+        author =        {P. Pudl\'{a}k},
+        title =         {Improved Bounds on the Lengths of Proofs of Finistic Consistency Statements},
+        journal =   {AMS Contemporary Mathematics Series},     
+        year =          1987,
+         volume =       65,
+         pages =        {309-331},
+}
+
+@incollection{Pu96,
+        author =        {P. Pudl\'{a}k},
+        title =         {On the Lengths of Proofs of Consistency},
+        booktitle =  {Collegium Logicum: Annals of the Kurt G\"{o}del},
+        year =        1996,
+         publisher = {Springer-Wien},
+         volume =      2,
+         pages =        { 65-86},
+}
+
+
+
+
+@incollection{Pu98,
+        author =        {P. Pudl\'{a}k},
+        title =         {The Lengths of Proofs},
+        booktitle =       {The Handbook of Proof Theory} ,
+        year =          1998,
+         publisher =       {North Holland},
+         pages =        {547-636}
+}
+
+
+
+
+
+@article{Ra93,
+        author =        {Z. Ratajczyk},
+        title =         {Subsystems of True Arithmetic and Hierarchies of
+Functions},
+        journal =       apal,
+        year =          1993,
+         volume =       64,
+         pages =        { 95--152},
+}
+
+
+
+@book{Ro67,
+        author =        {H. A. Rogers },
+        title =         {Theory of Recursive Functions and Effective Compatibility}, 
+        publisher =      {McGraw Hill},
+        year =          {1967}
+}
+
+
+
+@article{Ro36,
+        author =        {J. B. Rosser},
+        title =         {Extensions of Some Earlier Theorems by  \mbox{ G\"{o}del and Church} },
+        journal =       jsl,
+        year =          1936,
+         volume =       1,
+         pages =        {87-91},
+}
+
+@phdthesis{Sa1,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency in Arithmetics with Bounded Induction},
+        school = {Polish Academy},
+               year =          2001
+}
+
+
+@article{Sa12,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency of Some Arithmetical Theories},
+        journal =       jsl,
+        year =          2012,
+         volume =       77,
+        number = 3,
+         pages =        {807-827}
+}
+
+
+@incollection{Sm77,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {The Incompleteness Theorem},
+        booktitle =     { Handbook on Mathematical Logic},
+        year =          1977,
+         publisher =       {North Holland},
+         pages =        {821--865}
+}
+
+
+
+@incollection{Sm85,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {Non-standard Models and Related Developments},
+        booktitle =     {Harvey Friedman's Research in the Foundations of Mathematics},
+        year =          1985,
+         publisher =       {North Holland},
+         pages =        {179--220}
+}
+
+
+
+
+@book{Smul,
+        author =        {R. M. Smullyan},
+        title =         {  First Order Logic},
+        publisher =      {Diver},
+        year =          {1995}
+}
+
+
+
+@article{So88,
+        author =        {R. M Solovay},
+        title =         {Injecting Inconsistencies into Models of \mbox{PA}},
+        journal =       apal,
+        year =          1988,
+         volume =       44,
+         pages =        {102-132}
+}
+
+
+@misc{So94,
+        author =        {R. M. Solovay},
+        note = {Telephone  
+conversation
+in 1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using 
+some 
+methods
+of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+(The Appendix A of 
+\cite{ww1} offers a 
+4-page summary of
+this conversation.)}
+}
+
+
+@article{Sv78,
+        author =        {V. Svejdar},
+        title =         {Degrees of Interpretability},
+        journal =       carol,
+        year =          1978,
+         volume =       19,
+         pages =        {783-813}
+}
+
+
+
+@article{Sv83,
+        author =        {V. Svejdar},
+        title =         {Modal Analysis of Generalized
+\mbox{Rosser} Sentences},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {986-999}
+}
+
+
+
+
+@article{Sv7,
+        author =        {V. Svejdar},
+        title =         {{An interpretation of Robinson arithmetic in its
+Grzegorczjk's weaker variant}},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {347-354}
+}
+
+
+
+
+
+@book{Ta87,
+        author =        {G. Takeuti},
+        title =         {Proof Theory},
+        publisher =     {North Holland},
+        year =          {1987}
+}
+
+
+
+
+@article{Ta0,
+        author =        {G. Takeuti},
+        title =         {G\"{o}del Sentences of Bounded Arithmetic},
+        journal =       jsl,
+        year =          2000,
+         volume =       65,
+         pages =        {1338-1346}
+}
+
+@book{TMR53,
+        author =        {A. Tarski and A. Mostowski and R.  Robinson},
+        title =         {Undecidable Theories},
+        publisher =     { North Holland Press}, 
+        year =          1953,
+}
+
+
+@inproceedings{Vi90,
+        author =        {A. Visser},
+        title =         {Interpretability Logic},
+        booktitle =     {Mathematical \mbox{Logic: Proceedings of the Heyting Summer School}},
+        year =         1988,
+                 pages =        {175-208},
+}
+
+
+@article{Vi92,
+        author =        {A. Visser},
+        title =         {An Inside View of Exp},
+        journal =       jsl,
+        year =         1992,
+         volume =       57,
+         pages =        {131--165},
+}
+
+
+
+@article{Vi93,
+        author =        {A. Visser},
+        title =         {The Unprovability of Small Inconsistency},
+        journal =       arch,
+         year =         1993,
+         volume =       32,
+         pages =        {131--165},
+}
+
+@article{Vi5,
+        author =        {A. Visser},
+        title =         {Faith and Falsity},
+        journal =       apal,
+        year =          2005,
+         volume =       131,
+  number = 1,
+              pages =        {103--131}
+}
+
+@article{VH73,
+        author =        {P. Vop\v{e}nka and P. H\'{a}jek},
+        title =         {Existence of  a Generalized Semantic Model of
+\mbox{G\"{o}del-Bernays} Set Theory},
+        journal =       { Bulletin de l'Academie Polonaise des Sciences,Mathmatiques, Astromiques et Physiques},
+        year =         1973,
+         volume =       12,
+         pages =        {1079-1086},
+}
+12 (1973) pp.1079-1086.
+
+@article{WP87,
+        author =        {A. J. Wilkie and J. B. Paris},
+        title =         {On the Scheme of Induction for Bounded
+Arithmetic},
+        journal =       apal,
+        year =          1987,
+         volume =       35,
+         pages =        {261-302},
+}
+
+
+
+
+@article{ww93,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying Axiom Systems},
+        journal =       spv,
+        year =          1993,
+         volume =       713,
+         pages =        {325-336},
+          note =  {Proceedings of  Third Kurt  G\"{o}del Symposium}
+         }
+
+@article{sp0,
+        author =        {D. E. Willard},
+        title =         {The semantic tableaux version of the second
+incompleteness theorem extends almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+                journal =       spv,
+        year =          2000,
+         volume =       1847, 
+         pages =        {415-430},
+         note = {Proceedings of ``Tableaux-2000'' Conference }
+       }
+
+
+
+@article{ww1,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying  Systems, the Incompleteness
+Theorem and the
+Tangibility Reflection Principle},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+        number = 2,
+         pages =        {536-596}
+}
+
+
+
+
+@article{ww2,
+        author =        {D. E. Willard},
+        title =         {How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem Almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+        journal =       jsl,
+        year =          2002,
+         volume =       67,
+        number = 1,
+         pages =        {465-496}
+         }
+
+
+@article{tab2,
+        author =        {D. E. Willard},
+        title =         {Some New Exceptions for
+the Semantic Tableaux Version of the
+Second Incompleteness Theorem},
+        journal =       spv,
+        year =          2002,
+         volume =       2381, 
+         pages =        {281-297},
+         note = { (``Tableaux-2002'' Conference Proceedings)}
+         }
+
+
+
+
+@inproceedings{wwlogos,    
+        author =        {D. E. Willard}, 
+        title =         {A Version of the
+                         Second Incompleteness Theorem For Axiom
+                         Systems that Recognize Addition 
+                         But Not Multiplication as a Total Function},
+        booktitle =     {First Order Logic Revisited},
+        year =          {2004},
+        address =       {Berlin},
+        pages =         {337--368},
+ editor = {V. Hendricks and F. Neuhaus and S. A. Pederson and U. Scheffler and H. Wansing},
+         publisher =     {Logos Verlag}
+}
+
+
+
+
+@misc{wwconf,
+        author =        {D. E. Willard},
+        title =         {On Two Partial (and not Full) Respects 
+Where an Axiom System Can Recognize Its Own Consistency
+and Multiplication as a Total Function},
+        year =          2005,
+        note = {a presented talk at the
+summer ASL-2005 conference in Athens whose
+300-word abstract will be publised in the {\it Bulletin of
+Symbolic Logic} and which is described in further detail in an
+University of Albany technical report.}
+
+ }
+
+
+
+@article{wwpete,
+        author =        {D. E. Willard},
+        title =         {A New Variant of \mbox{Hilbert} Styled Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+        note = {A 200-word abstract summarizing
+the contents of this 
+forthcoming invited article can be found on the
+web-site of {\it The 2nd St. Petersburg Conference on Logic and
+Computability (2003)}, i.e. http://logic.pdmi.ras.ru/2ndDays
+or in the Atlas Mathematical Conference Abstracts.},
+ 
+         }
+
+
+
+@article{ww5,
+        author =        {D. E. Willard},
+        title =          {An Exploration of the Partial Respects 
+in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency},
+        journal =       jsl,
+        year =          2005,
+         volume =       70,
+        number = 4,
+         pages =        {1171-1209},
+}
+
+
+@article{sp5,
+        author =        {D. E. Willard},
+        title =    {On the Partial Respects in which a
+Real Valued  Arithmetic System Can Verify its Tableaux Consistency},
+        journal =       spv,
+        year =          2005,
+         volume =       3702,
+         pages =        {292-306},
+}
+
+
+@article{wwapal,
+        author =        {D. E. Willard},
+        title =         {A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+         volume =       141,
+        number = 3,
+         pages =        {472-496}
+         }
+
+
+
+
+
+@article{ww6,
+        author =        {D. E. Willard},
+        title =          {On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency}, 
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 4,
+         pages =        {1189-1199}
+}
+
+
+
+@article{ww7,
+        author =        {D. E. Willard},
+        title =         {{Passive induction and a solution to a Paris-Wilkie 
+open question}},
+        journal =       apal,
+        year =          2007,
+         volume =       146,
+        number = 2,
+         pages =        {124-149}
+         }
+
+
+
+
+@article{ww9,
+        author =        {D. E. Willard},
+        title =         {{Some 
+specially formulated
+axiomizations for I$\Sigma_0$
+manage to
+evade
+the Herbrandized version of the second incompleteness theorem}},
+        journal =       icomp,
+        year =          2009,
+         volume =       207,
+        number = 10,
+         pages  =        {1078-1093}
+         }
+
+
+@article{ww14,
+        author =        {D. E. Willard},
+        title =         {On the Broader Epistemological
+Significance of Self-Justiying Axiom Systems},
+        journal =       spv,
+        year =          2014,
+         volume =       8652,
+         pages  =        {221-236},
+         note = {(an earlier more abbreviated
+         version of this current article that had appeared in
+         the Proceedings of 21st Wollic Conference)}
+         }
+
+
+
+@article{Wr78,
+        author =        {C. Wrathall},
+        title =         {Rudimentary Predicates and Relative Computation},
+        journal =       sicomp,
+        year =          1978,
+         volume =       7,
+         pages =        {194-209}
+}
+
+
+@book{Yo5,
+        author =        {P. Yourgrau},
+        title =         {A World Without Time: The Forgotten Legacy of
+G\"{o}del and Einstein},
+        publisher = {Basic Books},
+        year =      2005,
+        note = {See page 58 for the passages we have quoted}
+}
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup2.bib b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup2.bib
new file mode 100644
index 0000000..6f575cd
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/backup2.bib
@@ -0,0 +1,1062 @@
+        
+%% 2014 home feb 9  -3pm 
+
+%%  home 2015 jan5 same as suny copy and old ``may14''
+
+%% 2014 may 14 5.1 pm 
+
+
+\bibliography
+\bibliographystyle
+@string{notre = " Notre Dame Journal on Formal Logic"}
+@string{zeit = "Zeitscrfit fur Math Logic"}
+@string{carol =  "Com. Math. Univ. Carol"}
+@string{arch = "Archive for Mathematical Logic"}
+@string{spv = "Springer LNCS"}.
+@string{spm = "Springer Lecture Notes in Mathematics"}.
+@string{apal = "Annals Pure and Applied Logic"}.
+@string{jsl = "Jour. Symb. Logic"}
+@string{sicomp = "Siam Journal on Computing"}
+@string{fund = " Fundamenta Mathematicae"}
+@string{ljipl = "Logic Journal of the IPL"}
+@string{icomp = "Information and Computation"}
+@string{ma26 = "Mathematische Annalen"}
+
+
+
+@article{Ad2,
+        author =        {Z. Adamowicz},
+        title =         {Herbrand Consistency and Bounded
+Arithmetic}, 
+        journal =       fund,
+        year =          2002,
+         volume =       171,
+        number = 3, 
+        pages =        {279-292}
+}
+
+
+
+@article{AB1,
+        author =        {Z. Adamowicz and T. Bigorajska},
+        title =         {Existentially Closed Structures and \mbox{G\"{o}del's} Second Incompleteness Theorem},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+         pages =        {349-356},
+}
+
+
+
+@book{AZ96,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {The Logic of Mathematics: A Modern Course in Classical Logic}, 
+        publisher = {John Wiley and Sons},
+        year =      1997
+}
+
+
+@article{AZ1,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {On  \mbox{Herbrand} consistency in weak theories},
+        journal =       arch,
+        year =          2001,
+         volume =       40,
+        number = 6,
+      pages =        {399-413}
+}
+
+
+
+
+@article{Ar90,
+        author =        {T. Arai},
+        title =         {Derivability Conditions on  \mbox{Rosser's} Proof Predicates},
+        journal =       notre,
+        year =          1990,
+         volume =       31,
+         pages =        {487-497}
+}
+
+@phdthesis{Be62,
+        author =        {J. Benett},
+               school = {Princeton University},
+               year =          1962,
+        note = {A detailed summary of Benett's main theorem can
+be found on pages 299--303 and 406
+of the  H\'{a}jek-Pudl\'{a}k textbook \cite{HP91}}
+}
+
+@article{BS76,
+        author =        {A. Bezboruah and J. C. Shepherdson},
+        title =         {G\"{o}del's Second Incompleteness Theorem for \mbox{Q}},
+        journal =       jsl,
+        year =          1976,
+         volume =       41,
+         number = 2,
+         pages =        {503-512}
+}
+
+
+
+
+@misc{Br94,
+        author =        {S. Bringsford},
+        note = {Private  
+conversation,  1994,  helpfully suggesting 
+I place my mathematical and philosophical results
+in distinctly separate articles.}
+
+}
+
+@book{Bu86,
+        author =        {S. R.  Buss},
+        title =         { Bounded Arithmetic},
+        publisher = {Studies in Proof Theory, Lecture Notes 3, published
+by  Bibliopolis},
+        year =       1986,
+        note = {(Revised version of Ph. D. Thesis.)}
+}
+
+
+
+@article{BI95,
+        author =        {S. R.  Buss and A. Ignjatovic},
+        title =         {Unprovability of Consistency Statements in Fragments of Bounded Arithmetic},
+        journal =       apal,
+        year =          1995,
+         volume =       74,
+        number = 3,
+         pages =        {221-244},
+}
+
+
+
+@book{Da97,
+        author =        {J W Dawson},
+        title =         {Logical Dilemmas the life and work of
+Kurt G\"{o}del},
+        publisher = {AKPeters},
+        year =      1997
+}
+
+
+
+@article{D89,
+        author =        {C. Dimitracopoulos},
+        title =         {Overspill and Fragments of Arithmetic},
+        journal =       arch,
+        year =          1989,
+         volume =       28,
+         pages =        {173-179},
+}
+
+@article{Fe60,
+        author =        {S. Feferman},
+        title =    {Arithmetization of Metamathematics in a General Setting},
+        journal =       fund,
+        year =          1960,
+         volume =       49,
+         pages =        { 35-92},
+}
+
+
+@article{FKO60,
+        author =        {S. Feferman and G. Kriesel and S. Orey},
+        title =         {1-Consistency and Faithful Interpretations},
+        journal =       arch,
+        year =          1962,
+         volume =       6,
+         pages =        {52-63},
+}
+
+
+
+
+@book{Fi90,
+        author =        {M. Fitting},
+        title =         { First Order Logic and Automated Theorem Proving},
+        publisher =  {Springer},
+        year =      1996
+}
+
+@techreport{Fr79a,
+        author =        {H. M.  Friedman},
+        title =         {On the Consistency, Completeness and Correctness
+Problems},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+
+
+@techreport{Fr79b,
+        author =        {H. M.  Friedman},
+        title =         {Translatability and Relative Consistency},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+@article{Go31,
+        author =        {K. G\"{o}del},
+        title =         {{\"{U}ber formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandter Systeme I}},
+        journal =       { Monatshefte f\"{u}r Math. Phys.},
+        year =          1931,
+         volume =       38,
+         pages =        {173-198}
+}
+
+
+
+@inproceedings{Go33,    
+        author =        {K. G\"{o}del}, 
+  title =       {The present situation in the foundations of
+mathematics},
+  booktitle =   {Collected Works Volume III: Unpublished Essays and Lectures},
+        year =          {2004},
+        pages =         {45--53},
+ editor = {S. Feferman and J. W. Dawson   and W. Goldfarb and C. Parson and R. Solovay}, 
+         publisher =     {Oxford University Press},
+     note = {Our quotes from this 1933 lecture come from its page 52.}
+}
+
+
+@book{Go5,
+        author =        {R. Goldstein},
+    title =         {Incompleteness The Proof and Paradox of Kurt G\"{o}del},
+        publisher = {Norton},
+        year =      2005
+}
+
+
+
+@article{Ha71,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part I}},
+        journal =       carol,
+        year =          1971,
+         volume =       12,
+         pages =        {73--79}
+}
+
+
+@article{Ha72,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part II}},
+        journal =       carol,
+        year =          1972,
+         volume =       13,
+         pages =        { 445-455}
+}
+
+
+@article{Ha81,
+        author =        {P. H\'{a}jek},
+        title =         {Interpretability in Theories Containing Arithmetic},
+        journal =       carol,
+        year =          1981,
+         volume =       22,
+         pages =        {225-234}
+}
+
+
+
+@article{Ha7,
+        author =        {P. H\'{a}jek},
+        title =         {Mathematical fuzzy logic and natural numbers},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {155-163}
+}
+
+
+@article{Ha11,
+        author =        {P. H\'{a}jek},
+        title =         {Towards metamathematics of weak arithmetics over
+fuzzy},
+        journal =       ljipl,
+        year =          2011,
+         volume =       19,        
+        number = 3,
+         pages =        {467-475}
+}
+
+
+
+@book{HP91,
+        author =        {P. H\'{a}jek and P. Pudl\'{a}k},
+        title =         { Metamathematics of First Order Arithmetic},
+        publisher =  {Springer Verlag},
+        year =  1991     
+}
+
+
+@article{Hil26,
+        author =        {D. Hilbert},
+        title =         {{\"{U}ber das Unendliche}},
+        journal =       ma26,
+        year =          1926,
+         volume =       95,
+         pages =        {161-191}
+}
+
+
+
+
+@book{HB39,
+        author =        {D. Hilbert and P. Bernays},
+        title =         { Grundlagen der Mathematik (Vol II)}, 
+        publisher = {Springer},
+        year =      1939
+}
+
+
+@article{Je71,
+        author =        {R. G. Jeroslow},
+        title =         {Consistency Statements in Formal Theories},
+        journal =       fund,
+        year =          1971,
+         volume =       72,
+         pages =        { 17-40},
+}
+
+
+@book{Ka91,
+        author =        {R. Kaye},
+        title =         { Models of Peano Arithmetic},
+        publisher = {Oxford University Press},
+        year =          1991
+}
+
+
+@article{Kl38,
+        author =        {S. C. Kleene},
+        title =         {On  Notation for Ordinal Numbers},
+        journal =       jsl,
+        year =          1938,
+         volume =       3,
+         number =     1,
+         pages =        {150-156},
+}
+
+
+
+@article{Ko6,
+        author =        {L. A. Ko{\l}odziejczyk},
+        title =         {On the 
+\mbox{Herbrand} Notion of Consistency for Finitely
+Axiomatizable Fragments of Bounded Arithmetic Theories},
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 2,
+         pages =        {624-638}
+}
+
+
+
+@article{Kr87,
+        author =        { J. Kraj\'{i}cek},
+        title =         {A Note on Proofs of Falsehood},
+        journal =       arch,
+        year =          26,
+         volume =       1987,
+         pages =        {169-176}
+}
+
+
+@book{Kr95,
+        author =        { J. Kraj\'{i}cek},
+        title =         {Bounded Propositional Logic and Complexity Theory},
+        publisher =   {Cambridge University Press}, 
+        year =          1995
+}
+
+
+
+
+
+
+@article{KT74,
+        author =        {G. Kreisel and G. Takeuti},
+        title =         {Formally Self-Referential Propositions in Cut-Free Classical Analysis and Related Systems},
+        journal =       {Dissertationes Mathematicae},
+        year =          1974,
+         volume =       118,
+         pages =        {1--50}
+}
+
+
+@incollection{Li84,
+        author =        {P. Lindstr\"{o}m},
+        title =         {On faithful Interpretability},
+        booktitle =      {Computation and Proof Theory},
+        publisher = spm,
+        volume = 1104,
+        year =          1984,
+         pages =        {279-288}
+}
+
+@article{Lo55,
+        author =        { M. H. L\"{o}b},
+        title =         {A Solution of a Problem of \mbox{Leon Henkin}},
+        journal =       jsl,
+        year =          1955,
+         volume =       20,
+        number = 2,
+         pages =        {115-118},
+}
+
+
+
+@book{Mend,
+        author =        {E. Mendelson},
+        title =         {Introduction to Mathematical Logic},
+        publisher =     { Chapman Hall},
+        year = 2010
+}
+
+
+@article{Mo58,
+        author =        {R. Montague},
+        title =         {The Continuum of Relative Interpretability},
+        journal =       jsl,
+        year =          23,
+         volume =       1958,
+         pages =        {494-511},
+}
+
+
+
+@article{Mo62,
+        author =        {R. Montague},
+        title =         {Theories Incomparable with Respect to Interpretability},
+        journal =       jsl,
+        year =          1962,
+         volume =       27,
+         pages =        {195-211},
+}
+
+
+@article{My62,
+        author =        {J. Mycieslski},
+        title =         {A Lattice Connected with 
+ Relative Interpretability},
+        journal =        {Notices of AMS},
+        year =          1962,
+         volume =       9,
+         pages =        {407-408},
+}
+
+
+
+@book{Ne86,
+        author =        {E. Nelson},
+        title =         {Predicative Arithmetic},
+        publisher =     { Math Notes, Princeton Univ Press},
+        year = 1986 
+}
+
+
+
+@article{Or61,
+        author =        {S. Orey},
+        title =         {Relative Interpretations},
+        journal =       zeit,
+        year =          1961,
+         volume =       7,
+         pages =        {146-153.},
+}
+
+
+
+@article{Pa71,
+        author =        {R. Parikh},
+        title =         {Existence and Feasibility in Arithmetic},
+        journal =       jsl,
+        year =          1971,
+         volume =       36,
+        number = 3,
+         pages =        {494-508},
+}
+
+
+
+@article{PD83,
+        author =        {J. B. Paris and C. Dimitracopoulos},
+        title =         {A Note on the Undefinability of Cuts},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {564-569},
+}
+
+
+@inproceedings{PW81,
+        author =        {J. B. Paris and A. J. Wilkie},
+        title =         {\mbox{$\Delta_0$} Sets and Induction},
+        booktitle =     {Proceedings of the  \mbox{Jadswin} Logic Conference},
+        year =          1981,
+        publisher = {Leeds University Press},
+         pages =        {237-248}
+}
+
+
+@article{Pa72,
+        author =        {C. H. Parsons},
+        title =         {On $n-$Quantifier Induction},
+        journal =       jsl,
+        year =          1972,
+         volume =       37,
+        number = 3,
+         pages =        {466-482},
+}
+
+
+@article{Pu83,
+        author =        {P. Pudl\'{a}k},
+        title =         {Some Prime Elements in the Lattice of Interpretability},
+        journal =       {Transactions of the  AMS},
+        year =          1983,
+         volume =       280,
+         pages =        {255-275},
+}
+
+
+@incollection{Pu84,
+        author =        {P. Pudl\'{a}k},
+        title =         {On Lengths of Proofs of Finisitic Consistency Statements in First order Theories},
+        booktitle =      {Logic Colloquium 84},
+        publisher = {North Holland},
+        year =          1984,
+         pages =        {165-196.}
+}
+
+
+
+@article{Pu85,
+        author =        {P. Pudl\'{a}k},
+        title =         {Cuts, Consistency Statements and Interpretations},
+        journal =       jsl,
+        year =          1985,
+         volume =       50,
+        number = 2,
+         pages =        {423-442}
+}
+
+
+
+@article{Pu87,
+        author =        {P. Pudl\'{a}k},
+        title =         {Improved Bounds on the Lengths of Proofs of Finistic Consistency Statements},
+        journal =   {AMS Contemporary Mathematics Series},     
+        year =          1987,
+         volume =       65,
+         pages =        {309-331},
+}
+
+@incollection{Pu96,
+        author =        {P. Pudl\'{a}k},
+        title =         {On the Lengths of Proofs of Consistency},
+        booktitle =  {Collegium Logicum: Annals of the Kurt G\"{o}del},
+        year =        1996,
+         publisher = {Springer-Wien},
+         volume =      2,
+         pages =        { 65-86},
+}
+
+
+
+
+@incollection{Pu98,
+        author =        {P. Pudl\'{a}k},
+        title =         {The Lengths of Proofs},
+        booktitle =       {The Handbook of Proof Theory} ,
+        year =          1998,
+         publisher =       {North Holland},
+         pages =        {547-636}
+}
+
+
+
+
+
+@article{Ra93,
+        author =        {Z. Ratajczyk},
+        title =         {Subsystems of True Arithmetic and Hierarchies of
+Functions},
+        journal =       apal,
+        year =          1993,
+         volume =       64,
+         pages =        { 95--152},
+}
+
+
+
+@book{Ro67,
+        author =        {H. A. Rogers },
+        title =         {Theory of Recursive Functions and Effective Compatibility}, 
+        publisher =      {McGraw Hill},
+        year =          {1967}
+}
+
+
+
+@article{Ro36,
+        author =        {J. B. Rosser},
+        title =         {Extensions of Some Earlier Theorems by  \mbox{ G\"{o}del and Church} },
+        journal =       jsl,
+        year =          1936,
+         volume =       1,
+         pages =        {87-91},
+}
+
+@phdthesis{Sa1,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency in Arithmetics with Bounded Induction},
+        school = {Polish Academy},
+               year =          2001
+}
+
+
+@article{Sa12,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency of Some Arithmetical Theories},
+        journal =       jsl,
+        year =          2012,
+         volume =       77,
+        number = 3,
+         pages =        {807-827}
+}
+
+
+@incollection{Sm77,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {The Incompleteness Theorem},
+        booktitle =     { Handbook on Mathematical Logic},
+        year =          1977,
+         publisher =       {North Holland},
+         pages =        {821--865}
+}
+
+
+
+@incollection{Sm85,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {Non-standard Models and Related Developments},
+        booktitle =     {Harvey Friedman's Research in the Foundations of Mathematics},
+        year =          1985,
+         publisher =       {North Holland},
+         pages =        {179--220}
+}
+
+
+
+
+@book{Smul,
+        author =        {R. M. Smullyan},
+        title =         {  First Order Logic},
+        publisher =      {Diver},
+        year =          {1995}
+}
+
+
+
+@article{So88,
+        author =        {R. M Solovay},
+        title =         {Injecting Inconsistencies into Models of \mbox{PA}},
+        journal =       apal,
+        year =          1988,
+         volume =       44,
+         pages =        {102-132}
+}
+
+
+@misc{So94,
+        author =        {R. M. Solovay},
+        note = {Telephone  
+conversation
+in 1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using 
+some 
+methods
+of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+(The Appendix A of 
+\cite{ww1} offers a 
+4-page summary of
+this conversation.)}
+}
+
+
+@article{Sv78,
+        author =        {V. Svejdar},
+        title =         {Degrees of Interpretability},
+        journal =       carol,
+        year =          1978,
+         volume =       19,
+         pages =        {783-813}
+}
+
+
+
+@article{Sv83,
+        author =        {V. Svejdar},
+        title =         {Modal Analysis of Generalized
+\mbox{Rosser} Sentences},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {986-999}
+}
+
+
+
+
+@article{Sv7,
+        author =        {V. Svejdar},
+        title =         {{An interpretation of Robinson arithmetic in its
+Grzegorczjk's weaker variant}},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {347-354}
+}
+
+
+
+
+
+@book{Ta87,
+        author =        {G. Takeuti},
+        title =         {Proof Theory},
+        publisher =     {North Holland},
+        year =          {1987}
+}
+
+
+
+
+@article{Ta0,
+        author =        {G. Takeuti},
+        title =         {G\"{o}del Sentences of Bounded Arithmetic},
+        journal =       jsl,
+        year =          2000,
+         volume =       65,
+         pages =        {1338-1346}
+}
+
+@book{TMR53,
+        author =        {A. Tarski and A. Mostowski and R.  Robinson},
+        title =         {Undecidable Theories},
+        publisher =     { North Holland Press}, 
+        year =          1953,
+}
+
+
+@inproceedings{Vi90,
+        author =        {A. Visser},
+        title =         {Interpretability Logic},
+        booktitle =     {Mathematical \mbox{Logic: Proceedings of the Heyting Summer School}},
+        year =         1988,
+                 pages =        {175-208},
+}
+
+
+@article{Vi92,
+        author =        {A. Visser},
+        title =         {An Inside View of Exp},
+        journal =       jsl,
+        year =         1992,
+         volume =       57,
+         pages =        {131--165},
+}
+
+
+
+@article{Vi93,
+        author =        {A. Visser},
+        title =         {The Unprovability of Small Inconsistency},
+        journal =       arch,
+         year =         1993,
+         volume =       32,
+         pages =        {131--165},
+}
+
+@article{Vi5,
+        author =        {A. Visser},
+        title =         {Faith and Falsity},
+        journal =       apal,
+        year =          2005,
+         volume =       131,
+  number = 1,
+              pages =        {103--131}
+}
+
+@article{VH73,
+        author =        {P. Vop\v{e}nka and P. H\'{a}jek},
+        title =         {Existence of  a Generalized Semantic Model of
+\mbox{G\"{o}del-Bernays} Set Theory},
+        journal =       { Bulletin de l'Academie Polonaise des Sciences,Mathmatiques, Astromiques et Physiques},
+        year =         1973,
+         volume =       12,
+         pages =        {1079-1086},
+}
+12 (1973) pp.1079-1086.
+
+@article{WP87,
+        author =        {A. J. Wilkie and J. B. Paris},
+        title =         {On the Scheme of Induction for Bounded
+Arithmetic},
+        journal =       apal,
+        year =          1987,
+         volume =       35,
+         pages =        {261-302},
+}
+
+
+
+
+@article{ww93,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying Axiom Systems},
+        journal =       spv,
+        year =          1993,
+         volume =       713,
+         pages =        {325-336},
+          note =  {Proceedings of  Third Kurt  G\"{o}del Symposium}
+         }
+
+@article{sp0,
+        author =        {D. E. Willard},
+        title =         {The semantic tableaux version of the second
+incompleteness theorem extends almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+                journal =       spv,
+        year =          2000,
+         volume =       1847, 
+         pages =        {415-430},
+         note = {Proceedings of ``Tableaux-2000'' Conference }
+       }
+
+
+
+@article{ww1,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying  Systems, the Incompleteness
+Theorem and the
+Tangibility Reflection Principle},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+        number = 2,
+         pages =        {536-596}
+}
+
+
+
+
+@article{ww2,
+        author =        {D. E. Willard},
+        title =         {How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem Almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+        journal =       jsl,
+        year =          2002,
+         volume =       67,
+        number = 1,
+         pages =        {465-496}
+         }
+
+
+@article{tab2,
+        author =        {D. E. Willard},
+        title =         {Some New Exceptions for
+the Semantic Tableaux Version of the
+Second Incompleteness Theorem},
+        journal =       spv,
+        year =          2002,
+         volume =       2381, 
+         pages =        {281-297},
+         note = { (``Tableaux-2002'' Conference Proceedings)}
+         }
+
+
+
+
+@inproceedings{wwlogos,    
+        author =        {D. E. Willard}, 
+        title =         {A Version of the
+                         Second Incompleteness Theorem For Axiom
+                         Systems that Recognize Addition 
+                         But Not Multiplication as a Total Function},
+        booktitle =     {First Order Logic Revisited},
+        year =          {2004},
+        address =       {Berlin},
+        pages =         {337--368},
+ editor = {V. Hendricks and F. Neuhaus and S. A. Pederson and U. Scheffler and H. Wansing},
+         publisher =     {Logos Verlag}
+}
+
+
+
+
+@misc{wwconf,
+        author =        {D. E. Willard},
+        title =         {On Two Partial (and not Full) Respects 
+Where an Axiom System Can Recognize Its Own Consistency
+and Multiplication as a Total Function},
+        year =          2005,
+        note = {a presented talk at the
+summer ASL-2005 conference in Athens whose
+300-word abstract will be publised in the {\it Bulletin of
+Symbolic Logic} and which is described in further detail in an
+University of Albany technical report.}
+
+ }
+
+
+
+@article{wwpete,
+        author =        {D. E. Willard},
+        title =         {A New Variant of \mbox{Hilbert} Styled Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+        note = {A 200-word abstract summarizing
+the contents of this 
+forthcoming invited article can be found on the
+web-site of {\it The 2nd St. Petersburg Conference on Logic and
+Computability (2003)}, i.e. http://logic.pdmi.ras.ru/2ndDays
+or in the Atlas Mathematical Conference Abstracts.},
+ 
+         }
+
+
+
+@article{ww5,
+        author =        {D. E. Willard},
+        title =          {An Exploration of the Partial Respects 
+in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency},
+        journal =       jsl,
+        year =          2005,
+         volume =       70,
+        number = 4,
+         pages =        {1171-1209},
+}
+
+
+@article{sp5,
+        author =        {D. E. Willard},
+        title =    {On the Partial Respects in which a
+Real Valued  Arithmetic System Can Verify its Tableaux Consistency},
+        journal =       spv,
+        year =          2005,
+         volume =       3702,
+         pages =        {292-306},
+}
+
+
+@article{wwapal,
+        author =        {D. E. Willard},
+        title =         {A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+         volume =       141,
+        number = 3,
+         pages =        {472-496}
+         }
+
+
+
+
+
+@article{ww6,
+        author =        {D. E. Willard},
+        title =          {On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency}, 
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 4,
+         pages =        {1189-1199}
+}
+
+
+
+@article{ww7,
+        author =        {D. E. Willard},
+        title =         {{Passive induction and a solution to a Paris-Wilkie 
+open question}},
+        journal =       apal,
+        year =          2007,
+         volume =       146,
+        number = 2,
+         pages =        {124-149}
+         }
+
+
+
+
+@article{ww9,
+        author =        {D. E. Willard},
+        title =         {{Some 
+specially formulated
+axiomizations for I$\Sigma_0$
+manage to
+evade
+the Herbrandized version of the second incompleteness theorem}},
+        journal =       icomp,
+        year =          2009,
+         volume =       207,
+        number = 10,
+         pages  =        {1078-1093}
+         }
+
+
+@article{ww14,
+        author =        {D. E. Willard},
+        title =         {On the Broader Epistemological
+Significance of Self-Justiying Axiom Systems},
+        journal =       spv,
+        year =          2014,
+         volume =       8652,
+         pages  =        {221-236},
+         note = {(an earlier more abbreviated
+         version of this current article that had appeared in
+         the Proceedings of 21st Wollic Conference)}
+         }
+
+
+
+@article{Wr78,
+        author =        {C. Wrathall},
+        title =         {Rudimentary Predicates and Relative Computation},
+        journal =       sicomp,
+        year =          1978,
+         volume =       7,
+         pages =        {194-209}
+}
+
+
+@book{Yo5,
+        author =        {P. Yourgrau},
+        title =         {A World Without Time: The Forgotten Legacy of
+G\"{o}del and Einstein},
+        publisher = {Basic Books},
+        year =      2005,
+        note = {See page 58 for the passages we have quoted}
+}
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/bb.bib b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/bb.bib
new file mode 100644
index 0000000..6f575cd
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/bb.bib
@@ -0,0 +1,1062 @@
+        
+%% 2014 home feb 9  -3pm 
+
+%%  home 2015 jan5 same as suny copy and old ``may14''
+
+%% 2014 may 14 5.1 pm 
+
+
+\bibliography
+\bibliographystyle
+@string{notre = " Notre Dame Journal on Formal Logic"}
+@string{zeit = "Zeitscrfit fur Math Logic"}
+@string{carol =  "Com. Math. Univ. Carol"}
+@string{arch = "Archive for Mathematical Logic"}
+@string{spv = "Springer LNCS"}.
+@string{spm = "Springer Lecture Notes in Mathematics"}.
+@string{apal = "Annals Pure and Applied Logic"}.
+@string{jsl = "Jour. Symb. Logic"}
+@string{sicomp = "Siam Journal on Computing"}
+@string{fund = " Fundamenta Mathematicae"}
+@string{ljipl = "Logic Journal of the IPL"}
+@string{icomp = "Information and Computation"}
+@string{ma26 = "Mathematische Annalen"}
+
+
+
+@article{Ad2,
+        author =        {Z. Adamowicz},
+        title =         {Herbrand Consistency and Bounded
+Arithmetic}, 
+        journal =       fund,
+        year =          2002,
+         volume =       171,
+        number = 3, 
+        pages =        {279-292}
+}
+
+
+
+@article{AB1,
+        author =        {Z. Adamowicz and T. Bigorajska},
+        title =         {Existentially Closed Structures and \mbox{G\"{o}del's} Second Incompleteness Theorem},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+         pages =        {349-356},
+}
+
+
+
+@book{AZ96,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {The Logic of Mathematics: A Modern Course in Classical Logic}, 
+        publisher = {John Wiley and Sons},
+        year =      1997
+}
+
+
+@article{AZ1,
+        author =        {Z. Adamowicz and P. Zbierski},
+        title =         {On  \mbox{Herbrand} consistency in weak theories},
+        journal =       arch,
+        year =          2001,
+         volume =       40,
+        number = 6,
+      pages =        {399-413}
+}
+
+
+
+
+@article{Ar90,
+        author =        {T. Arai},
+        title =         {Derivability Conditions on  \mbox{Rosser's} Proof Predicates},
+        journal =       notre,
+        year =          1990,
+         volume =       31,
+         pages =        {487-497}
+}
+
+@phdthesis{Be62,
+        author =        {J. Benett},
+               school = {Princeton University},
+               year =          1962,
+        note = {A detailed summary of Benett's main theorem can
+be found on pages 299--303 and 406
+of the  H\'{a}jek-Pudl\'{a}k textbook \cite{HP91}}
+}
+
+@article{BS76,
+        author =        {A. Bezboruah and J. C. Shepherdson},
+        title =         {G\"{o}del's Second Incompleteness Theorem for \mbox{Q}},
+        journal =       jsl,
+        year =          1976,
+         volume =       41,
+         number = 2,
+         pages =        {503-512}
+}
+
+
+
+
+@misc{Br94,
+        author =        {S. Bringsford},
+        note = {Private  
+conversation,  1994,  helpfully suggesting 
+I place my mathematical and philosophical results
+in distinctly separate articles.}
+
+}
+
+@book{Bu86,
+        author =        {S. R.  Buss},
+        title =         { Bounded Arithmetic},
+        publisher = {Studies in Proof Theory, Lecture Notes 3, published
+by  Bibliopolis},
+        year =       1986,
+        note = {(Revised version of Ph. D. Thesis.)}
+}
+
+
+
+@article{BI95,
+        author =        {S. R.  Buss and A. Ignjatovic},
+        title =         {Unprovability of Consistency Statements in Fragments of Bounded Arithmetic},
+        journal =       apal,
+        year =          1995,
+         volume =       74,
+        number = 3,
+         pages =        {221-244},
+}
+
+
+
+@book{Da97,
+        author =        {J W Dawson},
+        title =         {Logical Dilemmas the life and work of
+Kurt G\"{o}del},
+        publisher = {AKPeters},
+        year =      1997
+}
+
+
+
+@article{D89,
+        author =        {C. Dimitracopoulos},
+        title =         {Overspill and Fragments of Arithmetic},
+        journal =       arch,
+        year =          1989,
+         volume =       28,
+         pages =        {173-179},
+}
+
+@article{Fe60,
+        author =        {S. Feferman},
+        title =    {Arithmetization of Metamathematics in a General Setting},
+        journal =       fund,
+        year =          1960,
+         volume =       49,
+         pages =        { 35-92},
+}
+
+
+@article{FKO60,
+        author =        {S. Feferman and G. Kriesel and S. Orey},
+        title =         {1-Consistency and Faithful Interpretations},
+        journal =       arch,
+        year =          1962,
+         volume =       6,
+         pages =        {52-63},
+}
+
+
+
+
+@book{Fi90,
+        author =        {M. Fitting},
+        title =         { First Order Logic and Automated Theorem Proving},
+        publisher =  {Springer},
+        year =      1996
+}
+
+@techreport{Fr79a,
+        author =        {H. M.  Friedman},
+        title =         {On the Consistency, Completeness and Correctness
+Problems},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+
+
+@techreport{Fr79b,
+        author =        {H. M.  Friedman},
+        title =         {Translatability and Relative Consistency},
+        year =          1979,
+         institution = {Ohio State Univ},
+     note = {See Pudl\'{a}k \cite{Pu96}'s summary of this result}
+}
+
+@article{Go31,
+        author =        {K. G\"{o}del},
+        title =         {{\"{U}ber formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandter Systeme I}},
+        journal =       { Monatshefte f\"{u}r Math. Phys.},
+        year =          1931,
+         volume =       38,
+         pages =        {173-198}
+}
+
+
+
+@inproceedings{Go33,    
+        author =        {K. G\"{o}del}, 
+  title =       {The present situation in the foundations of
+mathematics},
+  booktitle =   {Collected Works Volume III: Unpublished Essays and Lectures},
+        year =          {2004},
+        pages =         {45--53},
+ editor = {S. Feferman and J. W. Dawson   and W. Goldfarb and C. Parson and R. Solovay}, 
+         publisher =     {Oxford University Press},
+     note = {Our quotes from this 1933 lecture come from its page 52.}
+}
+
+
+@book{Go5,
+        author =        {R. Goldstein},
+    title =         {Incompleteness The Proof and Paradox of Kurt G\"{o}del},
+        publisher = {Norton},
+        year =      2005
+}
+
+
+
+@article{Ha71,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part I}},
+        journal =       carol,
+        year =          1971,
+         volume =       12,
+         pages =        {73--79}
+}
+
+
+@article{Ha72,
+        author =        {P. H\'{a}jek},
+        title =         {On  Interpretability in Set Theory \mbox{Part II}},
+        journal =       carol,
+        year =          1972,
+         volume =       13,
+         pages =        { 445-455}
+}
+
+
+@article{Ha81,
+        author =        {P. H\'{a}jek},
+        title =         {Interpretability in Theories Containing Arithmetic},
+        journal =       carol,
+        year =          1981,
+         volume =       22,
+         pages =        {225-234}
+}
+
+
+
+@article{Ha7,
+        author =        {P. H\'{a}jek},
+        title =         {Mathematical fuzzy logic and natural numbers},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {155-163}
+}
+
+
+@article{Ha11,
+        author =        {P. H\'{a}jek},
+        title =         {Towards metamathematics of weak arithmetics over
+fuzzy},
+        journal =       ljipl,
+        year =          2011,
+         volume =       19,        
+        number = 3,
+         pages =        {467-475}
+}
+
+
+
+@book{HP91,
+        author =        {P. H\'{a}jek and P. Pudl\'{a}k},
+        title =         { Metamathematics of First Order Arithmetic},
+        publisher =  {Springer Verlag},
+        year =  1991     
+}
+
+
+@article{Hil26,
+        author =        {D. Hilbert},
+        title =         {{\"{U}ber das Unendliche}},
+        journal =       ma26,
+        year =          1926,
+         volume =       95,
+         pages =        {161-191}
+}
+
+
+
+
+@book{HB39,
+        author =        {D. Hilbert and P. Bernays},
+        title =         { Grundlagen der Mathematik (Vol II)}, 
+        publisher = {Springer},
+        year =      1939
+}
+
+
+@article{Je71,
+        author =        {R. G. Jeroslow},
+        title =         {Consistency Statements in Formal Theories},
+        journal =       fund,
+        year =          1971,
+         volume =       72,
+         pages =        { 17-40},
+}
+
+
+@book{Ka91,
+        author =        {R. Kaye},
+        title =         { Models of Peano Arithmetic},
+        publisher = {Oxford University Press},
+        year =          1991
+}
+
+
+@article{Kl38,
+        author =        {S. C. Kleene},
+        title =         {On  Notation for Ordinal Numbers},
+        journal =       jsl,
+        year =          1938,
+         volume =       3,
+         number =     1,
+         pages =        {150-156},
+}
+
+
+
+@article{Ko6,
+        author =        {L. A. Ko{\l}odziejczyk},
+        title =         {On the 
+\mbox{Herbrand} Notion of Consistency for Finitely
+Axiomatizable Fragments of Bounded Arithmetic Theories},
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 2,
+         pages =        {624-638}
+}
+
+
+
+@article{Kr87,
+        author =        { J. Kraj\'{i}cek},
+        title =         {A Note on Proofs of Falsehood},
+        journal =       arch,
+        year =          26,
+         volume =       1987,
+         pages =        {169-176}
+}
+
+
+@book{Kr95,
+        author =        { J. Kraj\'{i}cek},
+        title =         {Bounded Propositional Logic and Complexity Theory},
+        publisher =   {Cambridge University Press}, 
+        year =          1995
+}
+
+
+
+
+
+
+@article{KT74,
+        author =        {G. Kreisel and G. Takeuti},
+        title =         {Formally Self-Referential Propositions in Cut-Free Classical Analysis and Related Systems},
+        journal =       {Dissertationes Mathematicae},
+        year =          1974,
+         volume =       118,
+         pages =        {1--50}
+}
+
+
+@incollection{Li84,
+        author =        {P. Lindstr\"{o}m},
+        title =         {On faithful Interpretability},
+        booktitle =      {Computation and Proof Theory},
+        publisher = spm,
+        volume = 1104,
+        year =          1984,
+         pages =        {279-288}
+}
+
+@article{Lo55,
+        author =        { M. H. L\"{o}b},
+        title =         {A Solution of a Problem of \mbox{Leon Henkin}},
+        journal =       jsl,
+        year =          1955,
+         volume =       20,
+        number = 2,
+         pages =        {115-118},
+}
+
+
+
+@book{Mend,
+        author =        {E. Mendelson},
+        title =         {Introduction to Mathematical Logic},
+        publisher =     { Chapman Hall},
+        year = 2010
+}
+
+
+@article{Mo58,
+        author =        {R. Montague},
+        title =         {The Continuum of Relative Interpretability},
+        journal =       jsl,
+        year =          23,
+         volume =       1958,
+         pages =        {494-511},
+}
+
+
+
+@article{Mo62,
+        author =        {R. Montague},
+        title =         {Theories Incomparable with Respect to Interpretability},
+        journal =       jsl,
+        year =          1962,
+         volume =       27,
+         pages =        {195-211},
+}
+
+
+@article{My62,
+        author =        {J. Mycieslski},
+        title =         {A Lattice Connected with 
+ Relative Interpretability},
+        journal =        {Notices of AMS},
+        year =          1962,
+         volume =       9,
+         pages =        {407-408},
+}
+
+
+
+@book{Ne86,
+        author =        {E. Nelson},
+        title =         {Predicative Arithmetic},
+        publisher =     { Math Notes, Princeton Univ Press},
+        year = 1986 
+}
+
+
+
+@article{Or61,
+        author =        {S. Orey},
+        title =         {Relative Interpretations},
+        journal =       zeit,
+        year =          1961,
+         volume =       7,
+         pages =        {146-153.},
+}
+
+
+
+@article{Pa71,
+        author =        {R. Parikh},
+        title =         {Existence and Feasibility in Arithmetic},
+        journal =       jsl,
+        year =          1971,
+         volume =       36,
+        number = 3,
+         pages =        {494-508},
+}
+
+
+
+@article{PD83,
+        author =        {J. B. Paris and C. Dimitracopoulos},
+        title =         {A Note on the Undefinability of Cuts},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {564-569},
+}
+
+
+@inproceedings{PW81,
+        author =        {J. B. Paris and A. J. Wilkie},
+        title =         {\mbox{$\Delta_0$} Sets and Induction},
+        booktitle =     {Proceedings of the  \mbox{Jadswin} Logic Conference},
+        year =          1981,
+        publisher = {Leeds University Press},
+         pages =        {237-248}
+}
+
+
+@article{Pa72,
+        author =        {C. H. Parsons},
+        title =         {On $n-$Quantifier Induction},
+        journal =       jsl,
+        year =          1972,
+         volume =       37,
+        number = 3,
+         pages =        {466-482},
+}
+
+
+@article{Pu83,
+        author =        {P. Pudl\'{a}k},
+        title =         {Some Prime Elements in the Lattice of Interpretability},
+        journal =       {Transactions of the  AMS},
+        year =          1983,
+         volume =       280,
+         pages =        {255-275},
+}
+
+
+@incollection{Pu84,
+        author =        {P. Pudl\'{a}k},
+        title =         {On Lengths of Proofs of Finisitic Consistency Statements in First order Theories},
+        booktitle =      {Logic Colloquium 84},
+        publisher = {North Holland},
+        year =          1984,
+         pages =        {165-196.}
+}
+
+
+
+@article{Pu85,
+        author =        {P. Pudl\'{a}k},
+        title =         {Cuts, Consistency Statements and Interpretations},
+        journal =       jsl,
+        year =          1985,
+         volume =       50,
+        number = 2,
+         pages =        {423-442}
+}
+
+
+
+@article{Pu87,
+        author =        {P. Pudl\'{a}k},
+        title =         {Improved Bounds on the Lengths of Proofs of Finistic Consistency Statements},
+        journal =   {AMS Contemporary Mathematics Series},     
+        year =          1987,
+         volume =       65,
+         pages =        {309-331},
+}
+
+@incollection{Pu96,
+        author =        {P. Pudl\'{a}k},
+        title =         {On the Lengths of Proofs of Consistency},
+        booktitle =  {Collegium Logicum: Annals of the Kurt G\"{o}del},
+        year =        1996,
+         publisher = {Springer-Wien},
+         volume =      2,
+         pages =        { 65-86},
+}
+
+
+
+
+@incollection{Pu98,
+        author =        {P. Pudl\'{a}k},
+        title =         {The Lengths of Proofs},
+        booktitle =       {The Handbook of Proof Theory} ,
+        year =          1998,
+         publisher =       {North Holland},
+         pages =        {547-636}
+}
+
+
+
+
+
+@article{Ra93,
+        author =        {Z. Ratajczyk},
+        title =         {Subsystems of True Arithmetic and Hierarchies of
+Functions},
+        journal =       apal,
+        year =          1993,
+         volume =       64,
+         pages =        { 95--152},
+}
+
+
+
+@book{Ro67,
+        author =        {H. A. Rogers },
+        title =         {Theory of Recursive Functions and Effective Compatibility}, 
+        publisher =      {McGraw Hill},
+        year =          {1967}
+}
+
+
+
+@article{Ro36,
+        author =        {J. B. Rosser},
+        title =         {Extensions of Some Earlier Theorems by  \mbox{ G\"{o}del and Church} },
+        journal =       jsl,
+        year =          1936,
+         volume =       1,
+         pages =        {87-91},
+}
+
+@phdthesis{Sa1,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency in Arithmetics with Bounded Induction},
+        school = {Polish Academy},
+               year =          2001
+}
+
+
+@article{Sa12,
+        author =        {S. Salehi},
+        title =         {Herbrand Consistency of Some Arithmetical Theories},
+        journal =       jsl,
+        year =          2012,
+         volume =       77,
+        number = 3,
+         pages =        {807-827}
+}
+
+
+@incollection{Sm77,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {The Incompleteness Theorem},
+        booktitle =     { Handbook on Mathematical Logic},
+        year =          1977,
+         publisher =       {North Holland},
+         pages =        {821--865}
+}
+
+
+
+@incollection{Sm85,
+        author =        {C. A. Smory\'{n}ski},
+        title =         {Non-standard Models and Related Developments},
+        booktitle =     {Harvey Friedman's Research in the Foundations of Mathematics},
+        year =          1985,
+         publisher =       {North Holland},
+         pages =        {179--220}
+}
+
+
+
+
+@book{Smul,
+        author =        {R. M. Smullyan},
+        title =         {  First Order Logic},
+        publisher =      {Diver},
+        year =          {1995}
+}
+
+
+
+@article{So88,
+        author =        {R. M Solovay},
+        title =         {Injecting Inconsistencies into Models of \mbox{PA}},
+        journal =       apal,
+        year =          1988,
+         volume =       44,
+         pages =        {102-132}
+}
+
+
+@misc{So94,
+        author =        {R. M. Solovay},
+        note = {Telephone  
+conversation
+in 1994
+describing Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using 
+some 
+methods
+of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+(The Appendix A of 
+\cite{ww1} offers a 
+4-page summary of
+this conversation.)}
+}
+
+
+@article{Sv78,
+        author =        {V. Svejdar},
+        title =         {Degrees of Interpretability},
+        journal =       carol,
+        year =          1978,
+         volume =       19,
+         pages =        {783-813}
+}
+
+
+
+@article{Sv83,
+        author =        {V. Svejdar},
+        title =         {Modal Analysis of Generalized
+\mbox{Rosser} Sentences},
+        journal =       jsl,
+        year =          1983,
+         volume =       48,
+         pages =        {986-999}
+}
+
+
+
+
+@article{Sv7,
+        author =        {V. Svejdar},
+        title =         {{An interpretation of Robinson arithmetic in its
+Grzegorczjk's weaker variant}},
+        journal =       fund,
+        year =          2007,
+         volume =       81,
+         pages =        {347-354}
+}
+
+
+
+
+
+@book{Ta87,
+        author =        {G. Takeuti},
+        title =         {Proof Theory},
+        publisher =     {North Holland},
+        year =          {1987}
+}
+
+
+
+
+@article{Ta0,
+        author =        {G. Takeuti},
+        title =         {G\"{o}del Sentences of Bounded Arithmetic},
+        journal =       jsl,
+        year =          2000,
+         volume =       65,
+         pages =        {1338-1346}
+}
+
+@book{TMR53,
+        author =        {A. Tarski and A. Mostowski and R.  Robinson},
+        title =         {Undecidable Theories},
+        publisher =     { North Holland Press}, 
+        year =          1953,
+}
+
+
+@inproceedings{Vi90,
+        author =        {A. Visser},
+        title =         {Interpretability Logic},
+        booktitle =     {Mathematical \mbox{Logic: Proceedings of the Heyting Summer School}},
+        year =         1988,
+                 pages =        {175-208},
+}
+
+
+@article{Vi92,
+        author =        {A. Visser},
+        title =         {An Inside View of Exp},
+        journal =       jsl,
+        year =         1992,
+         volume =       57,
+         pages =        {131--165},
+}
+
+
+
+@article{Vi93,
+        author =        {A. Visser},
+        title =         {The Unprovability of Small Inconsistency},
+        journal =       arch,
+         year =         1993,
+         volume =       32,
+         pages =        {131--165},
+}
+
+@article{Vi5,
+        author =        {A. Visser},
+        title =         {Faith and Falsity},
+        journal =       apal,
+        year =          2005,
+         volume =       131,
+  number = 1,
+              pages =        {103--131}
+}
+
+@article{VH73,
+        author =        {P. Vop\v{e}nka and P. H\'{a}jek},
+        title =         {Existence of  a Generalized Semantic Model of
+\mbox{G\"{o}del-Bernays} Set Theory},
+        journal =       { Bulletin de l'Academie Polonaise des Sciences,Mathmatiques, Astromiques et Physiques},
+        year =         1973,
+         volume =       12,
+         pages =        {1079-1086},
+}
+12 (1973) pp.1079-1086.
+
+@article{WP87,
+        author =        {A. J. Wilkie and J. B. Paris},
+        title =         {On the Scheme of Induction for Bounded
+Arithmetic},
+        journal =       apal,
+        year =          1987,
+         volume =       35,
+         pages =        {261-302},
+}
+
+
+
+
+@article{ww93,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying Axiom Systems},
+        journal =       spv,
+        year =          1993,
+         volume =       713,
+         pages =        {325-336},
+          note =  {Proceedings of  Third Kurt  G\"{o}del Symposium}
+         }
+
+@article{sp0,
+        author =        {D. E. Willard},
+        title =         {The semantic tableaux version of the second
+incompleteness theorem extends almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+                journal =       spv,
+        year =          2000,
+         volume =       1847, 
+         pages =        {415-430},
+         note = {Proceedings of ``Tableaux-2000'' Conference }
+       }
+
+
+
+@article{ww1,
+        author =        {D. E. Willard},
+        title =         {Self-Verifying  Systems, the Incompleteness
+Theorem and the
+Tangibility Reflection Principle},
+        journal =       jsl,
+        year =          2001,
+         volume =       66,
+        number = 2,
+         pages =        {536-596}
+}
+
+
+
+
+@article{ww2,
+        author =        {D. E. Willard},
+        title =         {How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem Almost to 
+\mbox{Robinson's} 
+arithmetic \mbox{Q} } ,
+        journal =       jsl,
+        year =          2002,
+         volume =       67,
+        number = 1,
+         pages =        {465-496}
+         }
+
+
+@article{tab2,
+        author =        {D. E. Willard},
+        title =         {Some New Exceptions for
+the Semantic Tableaux Version of the
+Second Incompleteness Theorem},
+        journal =       spv,
+        year =          2002,
+         volume =       2381, 
+         pages =        {281-297},
+         note = { (``Tableaux-2002'' Conference Proceedings)}
+         }
+
+
+
+
+@inproceedings{wwlogos,    
+        author =        {D. E. Willard}, 
+        title =         {A Version of the
+                         Second Incompleteness Theorem For Axiom
+                         Systems that Recognize Addition 
+                         But Not Multiplication as a Total Function},
+        booktitle =     {First Order Logic Revisited},
+        year =          {2004},
+        address =       {Berlin},
+        pages =         {337--368},
+ editor = {V. Hendricks and F. Neuhaus and S. A. Pederson and U. Scheffler and H. Wansing},
+         publisher =     {Logos Verlag}
+}
+
+
+
+
+@misc{wwconf,
+        author =        {D. E. Willard},
+        title =         {On Two Partial (and not Full) Respects 
+Where an Axiom System Can Recognize Its Own Consistency
+and Multiplication as a Total Function},
+        year =          2005,
+        note = {a presented talk at the
+summer ASL-2005 conference in Athens whose
+300-word abstract will be publised in the {\it Bulletin of
+Symbolic Logic} and which is described in further detail in an
+University of Albany technical report.}
+
+ }
+
+
+
+@article{wwpete,
+        author =        {D. E. Willard},
+        title =         {A New Variant of \mbox{Hilbert} Styled Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+        note = {A 200-word abstract summarizing
+the contents of this 
+forthcoming invited article can be found on the
+web-site of {\it The 2nd St. Petersburg Conference on Logic and
+Computability (2003)}, i.e. http://logic.pdmi.ras.ru/2ndDays
+or in the Atlas Mathematical Conference Abstracts.},
+ 
+         }
+
+
+
+@article{ww5,
+        author =        {D. E. Willard},
+        title =          {An Exploration of the Partial Respects 
+in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency},
+        journal =       jsl,
+        year =          2005,
+         volume =       70,
+        number = 4,
+         pages =        {1171-1209},
+}
+
+
+@article{sp5,
+        author =        {D. E. Willard},
+        title =    {On the Partial Respects in which a
+Real Valued  Arithmetic System Can Verify its Tableaux Consistency},
+        journal =       spv,
+        year =          2005,
+         volume =       3702,
+         pages =        {292-306},
+}
+
+
+@article{wwapal,
+        author =        {D. E. Willard},
+        title =         {A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It},
+        journal =       apal,
+        year =          2006,
+         volume =       141,
+        number = 3,
+         pages =        {472-496}
+         }
+
+
+
+
+
+@article{ww6,
+        author =        {D. E. Willard},
+        title =          {On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency}, 
+        journal =       jsl,
+        year =          2006,
+         volume =       71,
+        number = 4,
+         pages =        {1189-1199}
+}
+
+
+
+@article{ww7,
+        author =        {D. E. Willard},
+        title =         {{Passive induction and a solution to a Paris-Wilkie 
+open question}},
+        journal =       apal,
+        year =          2007,
+         volume =       146,
+        number = 2,
+         pages =        {124-149}
+         }
+
+
+
+
+@article{ww9,
+        author =        {D. E. Willard},
+        title =         {{Some 
+specially formulated
+axiomizations for I$\Sigma_0$
+manage to
+evade
+the Herbrandized version of the second incompleteness theorem}},
+        journal =       icomp,
+        year =          2009,
+         volume =       207,
+        number = 10,
+         pages  =        {1078-1093}
+         }
+
+
+@article{ww14,
+        author =        {D. E. Willard},
+        title =         {On the Broader Epistemological
+Significance of Self-Justiying Axiom Systems},
+        journal =       spv,
+        year =          2014,
+         volume =       8652,
+         pages  =        {221-236},
+         note = {(an earlier more abbreviated
+         version of this current article that had appeared in
+         the Proceedings of 21st Wollic Conference)}
+         }
+
+
+
+@article{Wr78,
+        author =        {C. Wrathall},
+        title =         {Rudimentary Predicates and Relative Computation},
+        journal =       sicomp,
+        year =          1978,
+         volume =       7,
+         pages =        {194-209}
+}
+
+
+@book{Yo5,
+        author =        {P. Yourgrau},
+        title =         {A World Without Time: The Forgotten Legacy of
+G\"{o}del and Einstein},
+        publisher = {Basic Books},
+        year =      2005,
+        note = {See page 58 for the passages we have quoted}
+}
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.aux b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.aux
new file mode 100644
index 0000000..91e79d6
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.aux
@@ -0,0 +1,305 @@
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+\bibcite{Ko6}{21}
+\bibcite{Kr95}{22}
+\bibcite{KT74}{23}
+\bibcite{Lo55}{24}
+\bibcite{Mend}{25}
+\bibcite{Ne86}{26}
+\bibcite{Pa71}{27}
+\bibcite{Pa72}{28}
+\bibcite{Pu85}{29}
+\bibcite{Pu96}{30}
+\bibcite{Ro67}{31}
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+\bibcite{wwapal}{45}
+\bibcite{ww6}{46}
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+\bibcite{ww9}{48}
+\bibcite{ww14}{49}
+\bibcite{Wr78}{50}
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diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.bbl b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.bbl
new file mode 100644
index 0000000..acb8207
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.bbl
@@ -0,0 +1,294 @@
+\begin{thebibliography}{10}
+
+\bibitem{Ad2}
+Z.~Adamowicz.
+\newblock Herbrand consistency and bounded arithmetic.
+\newblock {\em Fundamenta Mathematicae}, 171(3):279--292, 2002.
+
+\bibitem{AZ1}
+Z.~Adamowicz and P.~Zbierski.
+\newblock On \mbox{Herbrand} consistency in weak theories.
+\newblock {\em Archive for Mathematical Logic}, 40(6):399--413, 2001.
+
+\bibitem{BS76}
+A.~Bezboruah and J.~C. Shepherdson.
+\newblock G\"{o}del's second incompleteness theorem for \mbox{Q}.
+\newblock {\em Jour. Symb. Logic}, 41(2):503--512, 1976.
+
+\bibitem{Bu86}
+S.~R. Buss.
+\newblock {\em Bounded Arithmetic}.
+\newblock Studies in Proof Theory, Lecture Notes 3, published by Bibliopolis,
+  1986.
+\newblock (Revised version of Ph. D. Thesis.).
+
+\bibitem{BI95}
+S.~R. Buss and A.~Ignjatovic.
+\newblock Unprovability of consistency statements in fragments of bounded
+  arithmetic.
+\newblock {\em Annals Pure and Applied Logic}, 74(3):221--244, 1995.
+
+\bibitem{Da97}
+J.~W. Dawson.
+\newblock {\em Logical Dilemmas the life and work of Kurt G\"{o}del}.
+\newblock AKPeters, 1997.
+
+\bibitem{Fe60}
+S.~Feferman.
+\newblock Arithmetization of metamathematics in a general setting.
+\newblock {\em Fundamenta Mathematicae}, 49:35--92, 1960.
+
+\bibitem{Fi90}
+M.~Fitting.
+\newblock {\em First Order Logic and Automated Theorem Proving}.
+\newblock Springer, 1996.
+
+\bibitem{Fr79a}
+H.~M. Friedman.
+\newblock On the consistency, completeness and correctness problems.
+\newblock Technical report, Ohio State Univ, 1979.
+\newblock See Pudl\'{a}k \cite{Pu96}'s summary of this result.
+
+\bibitem{Fr79b}
+H.~M. Friedman.
+\newblock Translatability and relative consistency.
+\newblock Technical report, Ohio State Univ, 1979.
+\newblock See Pudl\'{a}k \cite{Pu96}'s summary of this result.
+
+\bibitem{Go31}
+K.~G\"{o}del.
+\newblock {\"{U}ber formal unentscheidbare S\"{a}tze der Principia Mathematica
+  und verwandter Systeme I}.
+\newblock {\em Monatshefte f\"{u}r Math. Phys.}, 38:173--198, 1931.
+
+\bibitem{Go33}
+K.~G\"{o}del.
+\newblock The present situation in the foundations of mathematics.
+\newblock In S.~Feferman, J.~W. Dawson, W.~Goldfarb, C.~Parson, and R.~Solovay,
+  editors, {\em Collected Works Volume III: Unpublished Essays and Lectures},
+  pages 45--53. Oxford University Press, 2004.
+\newblock Our quotes from this 1933 lecture come from its page 52.
+
+\bibitem{Go5}
+R.~Goldstein.
+\newblock {\em Incompleteness The Proof and Paradox of Kurt G\"{o}del}.
+\newblock Norton, 2005.
+
+\bibitem{Ha7}
+P.~H\'{a}jek.
+\newblock Mathematical fuzzy logic and natural numbers.
+\newblock {\em Fundamenta Mathematicae}, 81:155--163, 2007.
+
+\bibitem{Ha11}
+P.~H\'{a}jek.
+\newblock Towards metamathematics of weak arithmetics over fuzzy.
+\newblock {\em Logic Journal of the IPL}, 19(3):467--475, 2011.
+
+\bibitem{HP91}
+P.~H\'{a}jek and P.~Pudl\'{a}k.
+\newblock {\em Metamathematics of First Order Arithmetic}.
+\newblock Springer Verlag, 1991.
+
+\bibitem{Hil26}
+D.~Hilbert.
+\newblock {\"{U}ber das Unendliche}.
+\newblock {\em Mathematische Annalen}, 95:161--191, 1926.
+
+\bibitem{HB39}
+D.~Hilbert and P.~Bernays.
+\newblock {\em Grundlagen der Mathematik (Vol II)}.
+\newblock Springer, 1939.
+
+\bibitem{Je71}
+R.~G. Jeroslow.
+\newblock Consistency statements in formal theories.
+\newblock {\em Fundamenta Mathematicae}, 72:17--40, 1971.
+
+\bibitem{Kl38}
+S.~C. Kleene.
+\newblock On notation for ordinal numbers.
+\newblock {\em Jour. Symb. Logic}, 3(1):150--156, 1938.
+
+\bibitem{Ko6}
+L.~A. Ko{\l}odziejczyk.
+\newblock On the \mbox{Herbrand} notion of consistency for finitely
+  axiomatizable fragments of bounded arithmetic theories.
+\newblock {\em Jour. Symb. Logic}, 71(2):624--638, 2006.
+
+\bibitem{Kr95}
+J.~Kraj\'{i}cek.
+\newblock {\em Bounded Propositional Logic and Complexity Theory}.
+\newblock Cambridge University Press, 1995.
+
+\bibitem{KT74}
+G.~Kreisel and G.~Takeuti.
+\newblock Formally self-referential propositions in cut-free classical analysis
+  and related systems.
+\newblock {\em Dissertationes Mathematicae}, 118:1--50, 1974.
+
+\bibitem{Lo55}
+M.~H. L\"{o}b.
+\newblock A solution of a problem of \mbox{Leon Henkin}.
+\newblock {\em Jour. Symb. Logic}, 20(2):115--118, 1955.
+
+\bibitem{Mend}
+E.~Mendelson.
+\newblock {\em Introduction to Mathematical Logic}.
+\newblock Chapman Hall, 2010.
+
+\bibitem{Ne86}
+E.~Nelson.
+\newblock {\em Predicative Arithmetic}.
+\newblock Math Notes, Princeton Univ Press, 1986.
+
+\bibitem{Pa71}
+R.~Parikh.
+\newblock Existence and feasibility in arithmetic.
+\newblock {\em Jour. Symb. Logic}, 36(3):494--508, 1971.
+
+\bibitem{Pa72}
+C.~H. Parsons.
+\newblock On $n-$quantifier induction.
+\newblock {\em Jour. Symb. Logic}, 37(3):466--482, 1972.
+
+\bibitem{Pu85}
+P.~Pudl\'{a}k.
+\newblock Cuts, consistency statements and interpretations.
+\newblock {\em Jour. Symb. Logic}, 50(2):423--442, 1985.
+
+\bibitem{Pu96}
+P.~Pudl\'{a}k.
+\newblock On the lengths of proofs of consistency.
+\newblock In {\em Collegium Logicum: Annals of the Kurt G\"{o}del}, volume~2,
+  pages 65--86. Springer-Wien, 1996.
+
+\bibitem{Ro67}
+H.~A. Rogers.
+\newblock {\em Theory of Recursive Functions and Effective Compatibility}.
+\newblock McGraw Hill, 1967.
+
+\bibitem{Sa12}
+S.~Salehi.
+\newblock Herbrand consistency of some arithmetical theories.
+\newblock {\em Jour. Symb. Logic}, 77(3):807--827, 2012.
+
+\bibitem{Smul}
+R.~M. Smullyan.
+\newblock {\em First Order Logic}.
+\newblock Diver, 1995.
+
+\bibitem{So94}
+R.~M. Solovay.
+\newblock Telephone conversation in 1994 describing Solovay's generalization of
+  one of Pudl\'{a}k's theorems \cite{Pu85}, using some methods of Nelson and
+  Wilkie-Paris \cite{Ne86,WP87}. (The Appendix A of \cite{ww1} offers a 4-page
+  summary of this conversation.).
+
+\bibitem{Sv7}
+V.~Svejdar.
+\newblock {An interpretation of Robinson arithmetic in its Grzegorczjk's weaker
+  variant}.
+\newblock {\em Fundamenta Mathematicae}, 81:347--354, 2007.
+
+\bibitem{Vi5}
+A.~Visser.
+\newblock Faith and falsity.
+\newblock {\em Annals Pure and Applied Logic}, 131(1):103--131, 2005.
+
+\bibitem{WP87}
+A.~J. Wilkie and J.~B. Paris.
+\newblock On the scheme of induction for bounded arithmetic.
+\newblock {\em Annals Pure and Applied Logic}, 35:261--302, 1987.
+
+\bibitem{ww93}
+D.~E. Willard.
+\newblock Self-verifying axiom systems.
+\newblock {\em Springer LNCS}, 713:325--336, 1993.
+\newblock Proceedings of Third Kurt G\"{o}del Symposium.
+
+\bibitem{sp0}
+D.~E. Willard.
+\newblock The semantic tableaux version of the second incompleteness theorem
+  extends almost to \mbox{Robinson's} arithmetic \mbox{Q}.
+\newblock {\em Springer LNCS}, 1847:415--430, 2000.
+\newblock Proceedings of ``Tableaux-2000'' Conference.
+
+\bibitem{ww1}
+D.~E. Willard.
+\newblock Self-verifying systems, the incompleteness theorem and the
+  tangibility reflection principle.
+\newblock {\em Jour. Symb. Logic}, 66(2):536--596, 2001.
+
+\bibitem{ww2}
+D.~E. Willard.
+\newblock How to extend the semantic tableaux and cut-free versions of the
+  second incompleteness theorem almost to \mbox{Robinson's} arithmetic
+  \mbox{Q}.
+\newblock {\em Jour. Symb. Logic}, 67(1):465--496, 2002.
+
+\bibitem{tab2}
+D.~E. Willard.
+\newblock Some new exceptions for the semantic tableaux version of the second
+  incompleteness theorem.
+\newblock {\em Springer LNCS}, 2381:281--297, 2002.
+\newblock (``Tableaux-2002'' Conference Proceedings).
+
+\bibitem{wwlogos}
+D.~E. Willard.
+\newblock A version of the second incompleteness theorem for axiom systems that
+  recognize addition but not multiplication as a total function.
+\newblock In V.~Hendricks, F.~Neuhaus, S.~A. Pederson, U.~Scheffler, and
+  H.~Wansing, editors, {\em First Order Logic Revisited}, pages 337--368,
+  Berlin, 2004. Logos Verlag.
+
+\bibitem{ww5}
+D.~E. Willard.
+\newblock An exploration of the partial respects in which an axiom system
+  recognizing solely addition as a total function can verify its own
+  consistency.
+\newblock {\em Jour. Symb. Logic}, 70(4):1171--1209, 2005.
+
+\bibitem{wwapal}
+D.~E. Willard.
+\newblock A generalization of the second incompleteness theorem and some
+  exceptions to it.
+\newblock {\em Annals Pure and Applied Logic}, 141(3):472--496, 2006.
+
+\bibitem{ww6}
+D.~E. Willard.
+\newblock On the available partial respects in which an axiomatization for real
+  valued arithmetic can recognize its consistency.
+\newblock {\em Jour. Symb. Logic}, 71(4):1189--1199, 2006.
+
+\bibitem{ww7}
+D.~E. Willard.
+\newblock {Passive induction and a solution to a Paris-Wilkie open question}.
+\newblock {\em Annals Pure and Applied Logic}, 146(2):124--149, 2007.
+
+\bibitem{ww9}
+D.~E. Willard.
+\newblock {Some specially formulated axiomizations for I$\Sigma_0$ manage to
+  evade the Herbrandized version of the second incompleteness theorem}.
+\newblock {\em Information and Computation}, 207(10):1078--1093, 2009.
+
+\bibitem{ww14}
+D.~E. Willard.
+\newblock On the broader epistemological significance of self-justiying axiom
+  systems.
+\newblock {\em Springer LNCS}, 8652:221--236, 2014.
+\newblock (an earlier more abbreviated version of this current article that had
+  appeared in the Proceedings of 21st Wollic Conference).
+
+\bibitem{Wr78}
+C.~Wrathall.
+\newblock Rudimentary predicates and relative computation.
+\newblock {\em Siam Journal on Computing}, 7:194--209, 1978.
+
+\bibitem{Yo5}
+P.~Yourgrau.
+\newblock {\em A World Without Time: The Forgotten Legacy of G\"{o}del and
+  Einstein}.
+\newblock Basic Books, 2005.
+\newblock See page 58 for the passages we have quoted.
+
+\end{thebibliography}
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.blg b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.blg
new file mode 100644
index 0000000..cfa1f12
--- /dev/null
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diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.pdf b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.pdf
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diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.tex b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.tex
new file mode 100644
index 0000000..38afa22
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/n.tex
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/o.tex b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/o.tex
new file mode 100644
index 0000000..38afa22
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/o.tex
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/r.tex b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/r.tex
new file mode 100644
index 0000000..38afa22
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+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/r.tex
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejected-wolic b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejected-wolic
new file mode 100644
index 0000000..38afa22
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejected-wolic
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejecter-wolic b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejecter-wolic
new file mode 100644
index 0000000..38afa22
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/rejecter-wolic
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/wolinf b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/wolinf
new file mode 100644
index 0000000..38afa22
--- /dev/null
+++ b/nachlass/collected_dew_materials/REJECTED-2014-WOLLIC/wolinf
@@ -0,0 +1,4931 @@
+%% suny feb 11 noon removed bib
+
+% home 2014 Feb 9 9.6 -3pm old title  with key words and bibliog added
+
+%% NEED to do SPELL
+
+%% godel t0 goedel and spell 
+
+%%% home jan17 8.31  am 
+
+%%% suny jannary11 spell 6pm
+
+% home 2015 january 10 7 am -minor amendment while listening Sinatra
+
+%   home 2015 january 4 1.1 pm
+
+%   home 2015 january 3  2.3 pm abstract and new-bib; jan4 3,1am reformat 
+
+
+%% 2014 home march 29  8.5  pm
+%% AFTER PAPER SUBMITTED CHANGED LAST paragraph 
+
+%% 2014 home march 28, 4.1 am  suny 10.1 am changed 7 -10 to  6 -10
+
+%IMPORTANT REMINDER Long Paper should prove Theorem 3 for D= sem tab
+
+%\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+%\documentclass[11pt]{article}
+\documentclass[11pt]{article}
+
+
+
+
+
+ 
+
+
+\usepackage{amssymb}
+
+
+
+\addtolength{\oddsidemargin}{-0.9in}
+
+\setlength{\textheight}{9.0 in}
+
+
+\setlength{\textwidth}{6.5 in}
+\setlength{\textwidth}{6.6 in}
+\setlength{\textwidth}{6.4 in}
+
+
+
+%  \addtolength{\topmargin}{-.5in}
+%  \addtolength{\topmargin}{-.9in}
+  \addtolength{\topmargin}{-.6in}
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+\newtheorem{corollary}{Corollary}
+\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+
+%%% changed to double numbers
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+\def\fffx{Fact \#}
+\def\zhz{H }
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+
+\def\iii{IS$_D(\aaa)$}
+\def\I2{IS$^{\#}_D(\beta)$}
+\def\ik3{IS$^{\#}_D(\beta_{A,i})$}
+
+\def\js{IS$_D(A^*)$}
+\def\ns{IS$^{\#}_D(\beta^*)$}
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\phx}[1]{Theorem \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+%% 
+%%  \title{
+%% %\Large 
+%% On the 
+%% %Broader
+%% Epistemological 
+%% Significance of 
+%% Self-Justifying Axiom Systems
+%% from a Semantic Tableaux Perspective}
+%% 
+
+
+
+
+% old title is
+
+ \title{
+%\Large 
+On the Broader
+Epistemological 
+Significance of 
+Self-Justifying Axiom Systems}
+% from the Perspective of Analytic Tableaux}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\def\aaa{A}
+\def\ccc{Class}
+
+
+
+
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard\thanks{\normalsize This research 
+was partially supported
+by the NSF Grant CCR  0956495.
+\newline
+Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+ \setcounter{page}{0}
+ \thispagestyle{empty}
+
+
+
+\begin{abstract}
+\large
+\baselineskip = 1.5 \normalbaselineskip 
+This article will be a continuation of our 
+research into self-justifying 
+systems.
+It will introduce 
+several 
+new theorems
+(one of which
+will transform our previous infinite-sized
+self-verifying
+logics 
+into formalisms 
+or purely finite size). 
+It will explain how self-justification
+is useful, even when the Incompleteness
+Theorem 
+clearly
+does sharply 
+limit its 
+scope.
+\end{abstract}
+
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+\bigskip
+\bigskip
+\bigskip
+
+{\large
+{\bf Keywords and Phrases:}
+G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+Open Question, Semantic Tableaux Deduction,  
+  Consistency.}
+
+
+%% 
+%% \begin{quote}
+%% %{\bf $~~~~$ Detailed Abstract (as requested by Call for Papers):}
+%% {\bf $~~~~ $     Abstract:}
+%%  $~$
+%% This article will be a continuation of our research into self-justifying
+%% systems.  It will introduce several new theorems and then explore their
+%% philosophical significance.  Its two specific goals will be to:
+%% \bed
+%% \item[   A. ]
+%%       Explain how to transform our prior results about infinite-sized
+%%       self-verifying axiom systems into tighter results about axiom 
+%%       systems   of purely finite cardinality.
+%% \item[   B. ]
+%%      Explain how self-justifying axiom systems are useful {\it even when
+%%      the Second Incompleteness Theorem specifies limits for their reach.}
+%%      In particular, this second part of our 
+%% research
+%% %results
+%% discourse 
+%% will explain how
+%%      self-justification is related to open questions and conjectures that
+%%      G\"{o}del and Hilbert raised in 1926 and 1931.
+%% \ennd
+%% \end{quote}
+
+%% 
+%% Our discussion will have a more philosophical and easier-to-comprehend tone
+%% than the more mathematically styled presentation in our prior published
+%% papers.  
+%% %
+%% %Our discussion will have a more philosophical and easier-to-comprehend tone
+%% %than the more mathematically styled  in our prior published papers. 
+%% %% 
+%% %% The discussion in this article will have a more philosophical and
+%% %% easier-to-comprehend tone than the mostly mathematical discourse in our
+%% %% prior published papers.  Its 
+%% %% 
+%% Its
+%% concluding section will offer a new
+%% interpretation of the Second Incompleteness Theorem, where G\"{o}del's
+%% historic result is taken as being {\it robust and ubiquitous} from a purist
+%% theoretical perspective, while 
+%% % still 
+%% permitting enough wiggle room to
+%% explain how humans gain the {\it psychological motive} to cogitate in
+%% applications-oriented engineering-style environments.
+
+
+
+
+\normalsize
+
+
+
+
+\baselineskip = 1.5\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\fgf {\baselineskip = 1.3 \normalbaselineskip }
+
+
+
+\bbint
+
+
+
+
+\normalsize  
+%% \LARGE\baselineskip = 1.1\normalbaselineskip
+\baselineskip = 1.2\normalbaselineskip
+
+%\vspace*{- 3.0 em}
+
+\newpage
+
+
+\def\J1{IS$_D(~\cdot ~)$}
+
+
+
+\def\K1{IS$_D(~\cdot ~)$}
+\def\J2{IS$^{\#}_D(~\cdot ~)$}
+
+
+%%% ssssssssssssss
+%% TEXT IS HERE
+
+ \parskip 5 pt
+
+%%%%%\large
+ \baselineskip = 1.235 \normalbaselineskip 
+
+\large
+
+\baselineskip = 1.6 \normalbaselineskip 
+\baselineskip = 2.0 \normalbaselineskip 
+\normalsize \baselineskip = 1.22 \normalbaselineskip 
+ 
+\def\ssspace{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+% \def\ssspace{\normalsize \baselineskip = 2.1 \normalbaselineskip }
+
+\ssspace
+
+ \parskip 5 pt
+
+\section{Introduction}
+\label{pppp1}
+
+
+G\"{o}del's Incompleteness Theorem 
+has two parts.
+Its
+first half indicates no decision
+procedure can identify 
+all of
+arithmetic's
+ true statements.
+Its
+ ``Second Incompleteness'' 
+result
+ specifies
+sufficiently strong 
+logics
+{\it cannot} verify their own consistency.
+G\"{o}del 
+was  careful to insert 
+a
+ caveat
+into
+his historic paper 
+\cite{Go31}, 
+indicating  
+a
+{\it
+diluted} 
+form
+of Hilbert's Consistency Program 
+might 
+% have some success:
+reach some
+levels of 
+partial
+ success:
+\begin{quote} 
+$*~$ 
+%  (G\"{o}del  \cite{Go31} 1931):
+{\it ``It must be 
+expressly 
+noted 
+Proposition XI 
+(e.g. G\"{o}del's
+``Second'' Incompleteness
+Result)
+represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and { there might
+conceivably be finite proofs} which cannot
+be stated in P or in ...''} 
+\end{quote}
+Some 
+scholars
+have interpreted
+$\,*\,$
+as, possibly,
+anticipating attempts
+to confirm Peano Arithmetic's consistency,
+via 
+either
+Gentzen's formalism or 
+ G\"{o}del's Dialetica interpretation.
+On the other hand, 
+the Stanford's Encyclopedia's
+entry about  G\"{o}del
+quotes him,
+in its
+ Section 2.2.4,
+stating
+he was hesitant to
+view     the
+Second Incompleteness Theorem
+ as    
+fully
+ubiquitous, until 
+learning 
+of Turing's
+work.
+Moreover,
+Yourgrau \cite{Yo5}
+states
+von Neumann 
+{\it ``argued 
+against G\"{o}del 
+himself''}
+in the early 1930's,
+ about the definitive termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+Also, it is known
+ \cite{Da97,Go5,Yo5} 
+that  
+G\"{o}del
+ initially
+presumed the
+second theorem
+was false, before proving his stunning 
+result.
+%hhhh
+
+
+
+\smallskip
+
+
+
+In any case  
+several
+ year after he wrote    $*$'s
+initial
+ statement, 
+G\"{o}del gave a
+1933
+ lecture \cite{Go33},
+where he 
+told his audience
+that
+Hilbert's
+initial
+1926 objectives, summarized 
+formally
+by
+ $**$ below,
+had
+ {\it ``unfortunately''}
+no
+{\it 
+ ``hope of succeeding along''}
+its originally intended plans.
+\begin{quote} 
+$**~$ (Hilbert  \cite{Hil26} 1926):
+{\it ``Where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?   The definitive nature
+of the infinite has become necessary,  not merely for the special
+interests of individual sciences, but rather { for the
+honor} of human understanding itself.''}
+\end{quote}
+
+Our research, in both the current article
+and 
+the
+prior papers
+\cite{ww93}-\cite{ww14}
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9},
+was stimulated by the prospect that we find $**$ enticing,
+even though  the Second Incompleteness
+Theorem 
+{\it unequivocally}
+ demonstrates that logics
+{\it cannot} recognize
+their own consistency
+{\it in a robust sense.}
+Accordingly, we have studied
+{\it both} generalizations and boundary-case exceptions
+for the Second Incompleteness Theorem 
+in
+\cite{ww93}-\cite{ww14}.
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+The current article will  seek to
+{\it both} strengthen these prior
+results,
+in the context of axiom systems 
+with
+{\it 
+ strictly finite cardinalities},
+and to also provide a more intuitive explanation of the
+meaning 
+behind
+\cite{ww93}-\cite{ww14}'s
+% \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}'s
+results.
+
+The thesis of this article will be delicate
+because there can be no doubt that 
+ the Second Incompleteness
+Theorem is
+sharply robust,
+when  viewed
+from a 
+ conventional 
+purist 
+mathematical
+ perspective.
+On the other hand, we will argue that there are certain facets
+of a ``Self-Justifying Logics'', that are tempting
+under a hard-nosed
+engineering perspective,
+contemplating
+ sharply
+ {\it  curtailed forms} of Hilbert's goals.
+These results will be
+         fragile 
+{\it but
+not
+fully
+immaterial.}
+
+
+%bbbb
+In other words, this
+article  will offer a somewhat complicated
+2-part interpretation of the Second Incompleteness Theorem
+where:
+\bee
+\item
+The Second Incompleteness Theorem is seen as
+being 100 \% 
+robust from a mathematical perspective
+because of  the
+% ubiquitous and 
+widely
+encompassing nature of the 1939 
+Hilbert-Bernays analysis \cite{HB39} (centering around
+their three 
+well-known
+``Derivability Conditions'' \cite{Mend} ).
+\item
+On the other hand, our discourse
+will partially
+appreciate Hilbert's reluctance
+to fully embrace the Second Incompleteness Theorem,
+despite his 
+joint
+work with Bernays \cite{HB39}
+generalizing the Second Incompleteness Effect.
+(This is
+because it is awkward to explain how human beings can
+% undeniable 
+acquire the mental energy 
+for motivating themselves to cogitate,
+without possessing some type of instinctive faith  
+in their own self-consistency.)
+\ene
+%It is in the context where 
+Thus,
+the current article
+ will seek to
+separate a {\it ``mathematical''} from  
+what perhaps  should be
+{\it ``engineering-style''} 
+ appreciation
+of one's 
+internal consistency. We will seek to define and explore the
+latter
+%nature of this 
+%engineering  notion in the current article
+(with the hope that it will help formalize how future
+21st century computers can benefit from its engineering-style
+%% notion 
+perspective,
+while still respecting 
+%%% at the same time
+the strict prohibitions formalized by
+G\"{o}del's millennial result.)
+
+
+As the reader examines this paper, it should be kept in mind
+that 
+it does
+focus on 
+% the properties of 
+semantic tableaux
+deduction (similar to the earlier 
+% more abbreviated 
+discussion that had
+appeared in \cite{ww14}'s more abbreviated 
+conference-style summary of our results).
+A second paper, currently under preparation, 
+will examine Hilbert-style deductive systems (whose 
+self-justification properties
+are partially analogous and partly 
+quite
+different from
+% our
+tableaux-style systems). 
+The combination of these two results will formally
+define both the potential of self-justifying logics
+and the limitations which the Second Incompleteness
+Theorem imposes upon them.
+
+
+%% 
+%% In other words, the theme of this article will be that conventional
+%% interpretations of the Second Incompleteness Theorem are 
+%% certainly 100 \%
+%% correct from a mathematical perspective.
+%% as foreseen very rigorously
+%% as early as 1939
+%%  by Hilbert-Bernays \cite{HB39}.
+%% This is because
+%% no formalism can
+%% recognize its own consistency in a very robust
+%% strictly
+%% %purely
+%%  mathematical
+%% respect. 
+%% On the other hand, it also
+%% seems 
+%% evident
+%% %% appears apparent
+%% % undeniable 
+%% that
+%% human beings 
+%% will
+%% %would 
+%% find it awkward
+%% %be unable 
+%% to acquire the mental energy 
+%% for motivating themselves to cogitate,
+%% without possessing some type of instinctive faith  
+%% in their own self-consistency.
+%% This perhaps  should be
+%% called an
+%% % {\it quasi-
+%% {\it engineering=style appreciation} of one's 
+%% internal consistency. We seek to define and explore the
+%% nature of this 
+%% engineering  notion in the current article
+%% (with the hope that it will help formalize how future
+%% 21st century computers can benefit from this engineering-style
+%% notion while, of course, respecting 
+%% %%% at the same time
+%% the strict prohibitions formalized by
+%% G\"{o}del's millennial result.)
+
+
+
+\section{Background Setting} 
+\label{pppp2}
+
+
+Let
+ $(  \alpha  , d  )$ 
+denote any axiom system
+and deduction method satisfying
+the
+simple {\bf ``Split Rule''}
+below$\,$\footnote{Our 
+ ``Split Rule'' 
+is  the trivial requirement
+                   that all the axiom sentences in
+$~\alpha~$ are 
+technically
+{\it proper axioms}, and
+  that 
+deduction method $~d~$ is 
+required
+to include 
+{\bf BOTH} a finite number of rules of inference
+and
+whatever ``logical axioms'' are needed
+{\it  (if any ? )} 
+by $\,d$'s methodology.
+(This
+trivial
+Split-Rule
+notation convention will 
+help  us to provide a 
+%%hhhh
+precisely formalized statement of our results.
+ .)}.
+This pair 
+will 
+be called  {\bf ``Self Justifying''} when:
+\begin{description}
+  \item[  i   ] one of $ \, \alpha \,$'s  theorems
+will
+state that the deduction method $ \, d, \, $ applied to the
+system $ \, \alpha, \, $ will 
+produce a consistent set of theorems, and
+\item[  ii   ]
+     the axiom system $ \, \alpha  \, $ is in fact consistent.
+\end{description}
+For any  $\,(\alpha,d) \,$, 
+it is 
+easy 
+to construct a 
+second
+ $ \, \alpha^d \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+requirement.
+For instance,  $ \, \alpha^d \, $  could
+consist of all of $~\alpha \,$'s axioms plus 
+an added {\bf $\,$``SelfRef$(\alpha,d)$''$\,$} sentence,
+defined as stating:
+\begin{quote} 
+$\bullet~$ 
+There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of the
+system $ \alpha  $
+with {\it this}
+sentence  ``SelfRef$(\alpha,d)$'' (looking at itself).
+\end{quote}
+Kleene 
+\cite{Kl38} 
+noted
+how
+to
+encode
+rough
+ analogs of 
+ ``SelfRef$(\alpha,d)$''.
+Each of
+Kleene, 
+Rogers and Jeroslow 
+ \cite{Kl38,Ro67,Je71}
+ noted
+$\alpha ^d$ 
+may,
+however,  be inconsistent
+(despite SelfRef$(\alpha,d)$'s assertion),
+thus causing 
+it
+to violate Part-ii's
+requirement.
+
+
+%% hhhh
+This problem arises in
+many
+contexts besides
+ G\"{o}del's
+paradigm,
+where $\alpha$  was an extension of Peano Arithmetic
+(see
+\cite{Ad2,AZ1,BS76,Bu86,BI95,Fe60,Fr79a,Go31,Ha7,Ha11,HP91,HB39,Ko6,KT74,Lo55,Pa71,Pa72,Pu85,Pu96,Ro67,Sa12,So94,Sv7,Vi5,WP87,ww2,wwlogos,ww7}).
+Such results formalize 
+paradigms where 
+self-justification is infeasible,
+due to  diagonalization issues.
+(It should,
+perhaps, 
+ be added that among this
+lengthy list of articles,
+it was especially 
+\cite{Ad2,Bu86,Go31,Lo55,Pu85,So94,WP87}'s
+incompleteness results that
+influenced our
+work in
+\cite{ww93}-\cite{ww14}.)
+% in \cite{ww93,sp0,ww1,ww2,tab2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.)
+In any case, the main point is that
+most
+logicians 
+have
+hesitated 
+to
+ employ 
+an
+analog of a
+ SelfRef$(\alpha,d)$
+ axiom
+because
+ $  \alpha^d  = \alpha+$SelfRef$(\alpha,d)  $
+is 
+typically 
+inconsistent. 
+
+
+
+
+
+
+
+
+
+Our research
+in \cite{ww93,ww1,ww5,ww6,wwapal}
+focused on
+paradigms
+where  
+self-justification is feasible.
+It
+involved weakening
+the properties 
+a 
+logic
+can prove 
+about
+addition and/or 
+multiplication
+(to avoid 
+potential
+difficulties).
+To be more precise, let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote 
+3-way predicates
+specifying
+$x+y=z$ and
+$x*y=z$.
+Then a
+logic
+will be said to
+{\bf recognize}
+successor, 
+ addition  and multiplication
+as {\bf Total Functions} iff it 
+includes
+sentences
+1-3 as axioms.
+
+\vspace*{- 0.4 em}
+{\small
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\vspace*{- 1.7 em}
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+
+\vspace*{- 1.2 em}
+
+A
+logic
+$\alpha$
+will be called 
+{\bf Type-M} iff it contains
+\ref{totdefxs}-\ref{totdefxm}
+as axioms,  
+{\bf $~$Type-A} iff it contains only
+\eq{totdefxs} and \eq{totdefxa} as axioms,
+{\bf $~$Type-S} iff it contains
+only \eq{totdefxs} as an
+ axiom, and 
+{\bf $\,$Type-NS$\,$} iff it  contains
+none of these axioms.
+The relationship of these constructs to 
+self-justification 
+is explained by
+items (a) and (b):
+\bed
+\item[  a.  ]
+The existence of
+Type-A systems that can  recognize 
+their own
+consistency under semantic tableaux deduction,
+while proving
+analogs of
+all 
+Peano Arithmetic's
+ $\Pi_1$ theorems (in a slightly different language),
+were
+%%hhhh
+demonstrated in
+\cite{tab2,ww5}.
+Also, \cite{ww1,wwapal} noted that 
+some 
+specialized
+forms 
+of
+Type-NS systems 
+can 
+likewise
+recognize their
+own Hilbert consistency.
+
+
+
+\item[   b.  ]
+The above 
+evasions of the Second Incompleteness
+Theorem are known to be near-maximal in a mathematical sense.
+This is because
+the
+combined work of Pudl\'{a}k, Solovay, Nelson and Wilkie-Paris
+\cite{Ne86,Pu85,So94,WP87} implied  no
+natural 
+Type-S system can recognize  its  Hilbert consistency,
+and   Willard
+subsequently
+ \cite{ww2,ww7,ww9} 
+hybridized their formalisms with some techniques of
+Adamowicz-Zbierski 
+\cite{Ad2,AZ1} 
+to establish that  most
+Type-M  systems cannot recognize their
+own semantic
+tableaux consistency.
+\ennd
+
+
+ 
+Other
+fascinating
+efforts to 
+evade the Second Incompleteness Theorem 
+have used
+the Kreisel-Takeuti    ``CFA''
+system \cite{KT74}
+or the 
+the {\it interpretational framework} of
+Friedman,
+Nelson, Pudl\'{a}k and Visser
+\cite{Fr79b,Ne86,Pu85,Vi5}.
+These systems are unrelated to our approach
+because they
+do not use
+Kleene-like {\it ``I am consistent''} axiom-sentences.
+Instead, CFA uses the 
+special
+properties of ``second order'' generalizations of Gentzen's
+{\it cut-free}
+Sequent Calculus, 
+and 
+the
+interpretational approach
+formalizes how some systems 
+recognize their
+ Herbrand consistency 
+on localized sets of integers,
+which 
+unbeknownst to 
+themselves,
+includes all
+integers.
+(These alternate results are interesting but
+unrelated to our approach.)
+
+ 
+
+
+
+
+
+\section{Defining Notation and Earlier Results}
+\label{pppp3}
+
+\label{sect3}
+
+
+
+A function $F $
+will be called {\bf Non-Growth}
+iff 
+$ F(a_1...a_j) 
+\leq   Maximum(a_1...a_j)$
+holds.
+Six  examples of  
+non-growth functions are
+\bee
+\small
+\parskip 0pt
+%hhhh
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$Maximum(x,y),$
+\item
+$ Logarithm(x)~=~\lfloor ~$Log$_2(x)~ \rfloor~$
+\item
+$\,Root(x,y) \, =  \, \lfloor  \, x^{1/y} \,  \rfloor~$. and
+\item$Count(x,j)$  designating the number of ``1'' bits
+among $  x$'s rightmost $  j  $ bits.
+\ene
+The term
+{\bf U-Grounding Function} 
+referred in \cite{ww5} to
+a set of 
+primitives,
+which included the
+preceding
+functions plus the {\it growth operations} of addition and
+{\it Double$(x)=x+x$}. 
+Our language $L^*$  was 
+built
+out of these 
+symbols, 
+plus the primitives of 
+``0'', ``1'',
+   ``$ \, = \, $''
+and ``$ \, \leq \, $''.
+
+
+
+
+In a context where  $~\, t \, ~$ is  any term in \cite{ww5}'s
+language $L^*$, 
+$\,$the quantifiers in
+%%  the wffs
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and $~ \exists ~ v \leq t~~ \Psi (v)$
+were called {\it bounded 
+quantifiers}.
+Any formula in 
+ $L^*$, 
+all of whose
+quantifiers are bounded, was called
+a $\Delta_0^*$ formula.
+The  $~\Pi_n^{* }~$ and  $~\Sigma_n^{* }~$ formulae 
+were
+then defined by the 
+usual
+rules that:
+\bee
+\item
+Every  
+$\Delta_0^*$ formula is considered to
+be 
+``$~\Pi_0^{* }~$''  and 
+also 
+ ``$~\Sigma_0^{* }~$''. 
+\item
+A
+wff
+is  called
+ $ \,\Pi_n^{* } \,$
+when it is encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  with
+$\Phi$ being  $\Sigma_{n-1}^{* }$
+\item
+Also,
+a wff
+is  called
+ $\Sigma_n^*$
+when it is encoded as 
+$\exists v_1 ..\exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{* }$.
+\ene
+
+%%bbb
+Our articles \cite{ww93,tab2,ww5} used the symbol $~D~$ to denote
+a deduction method. 
+They focused mostly around the 
+semantic tableaux deductive methodology,
+whose formal definition can be found in the textbooks
+by Fitting and Smullyan
+\cite{Fi90,Smul} and whose
+definition is also reviewed
+by Appendix A of the current article.
+
+%%bbb
+Our articles \cite{wwlogos,ww5}
+also considered an improved faster deductive technology,
+ called
+{\bf Tab-k
+ deduction}, that 
+consists of a 
+speeded-up version of a 
+tableaux,
+which 
+permits a
+{\it limited analog} of 
+Gentzen-style deductive
+cuts
+for  $\Pi_k^*$ and   $\Sigma_k^*$ formulae.
+Thus, if
+ $~H~$ 
+denotes a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Semantic Tableaux proof of the theorem $~t_i,~$
+then  $H$ 
+has been
+ called a
+{\bf ``Tab-k
+Proof''}
+of a theorem $~T~$ 
+from  $\alpha$'s axioms
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+have a complexity no greater than that of
+either a $\Pi_k^*$ or $\Sigma_k^*$ sentence.
+\item
+Each
+proper axiom in  $ p_i$'s
+proof 
+comes 
+either
+from $\alpha$ or is
+ one of $ t_1,t_2, \, ... \, , t_{i-1} $. 
+\end{enumerate}
+Thus, a 
+Tab-k proof is essentially a generalization of a classic
+semantic tableaux proof that essentially owns the equivalent of
+an
+extra specialized modus ponens rule for 
+ $\Pi_k^*$ and $\Sigma_k^*$ sentences.
+
+Let
+us say 
+an axiom system $\alpha$ 
+has a {\bf Level-J Understanding}
+of its own 
+consistency 
+under a deduction method $D$ 
+iff $\alpha$ can prove that there exists no proofs
+using
+its axioms and $D$'s deduction
+of both a
+$\Pi_J^*$ theorem and its negation.
+In this notation, items A and B summarize
+\cite{sp0,ww2,wwlogos,ww5,ww7}'s 
+main
+results:
+\bed
+\item[   A.   ] 
+ For 
+any
+axiom system $A$ using $L^*\,$'s
+ U-Grounding language, 
+\cite{ww5} 
+showed its
+IS$_D(A)$  formalism 
+could prove
+all $A$'s $\Pi_1^*$ theorems and simultaneously
+verify its 
+Level-1
+consistency under
+\txl{1} deduction.
+
+\smallskip
+
+\item[   B.   ] 
+Two negative results, tightly complementing
+item A's
+positive result,
+were exhibited
+in 
+\cite{sp0,ww2,wwlogos,ww7}. The first
+was that \cite{sp0,ww2,ww7} showed
+most 
+systems
+are 
+unable to verify their 
+Level-0 consistency under
+semantic tableaux 
+deduction, 
+ when they included 
+statement
+\eq{totdefxm}'s ``Type-M''
+axiom  that multiplication
+is a total function. Moreover, \cite{wwlogos}
+offered an alternate
+form
+of this
+ incompleteness
+result,
+showing statement
+\eq{totdefxa}'s
+{\it 
+far weaker} 
+Type-A 
+systems
+cannot
+verify
+their Level-0 consistency under
+\txl{2} deduction.
+\ennd
+
+
+
+
+The contrast between these
+positive and negative results
+has
+ led to our conjecture that
+automated
+theorem provers
+are likely
+ to 
+eventually
+achieve
+a fragmentary part of the ambitions 
+that were
+suggested by  Hilbert
+in 
+$**\,$.
+This is because
+the question of whether a
+formalism can support an 
+{\it idealized Utopian}
+conception of
+its own consistency is {\it 
+different} from 
+exploring the degrees to which 
+theorem-provers
+can possess 
+a {\it fragmentary
+knowledge} of 
+their own
+consistency. 
+The 
+Incompleteness Theorem 
+has demonstrated 
+an Utopian idealized form of self-justification
+is unobtainable,
+but our research has found some 
+diluted
+cousins
+of this construct  are 
+feasible
+%%% hhhh
+and warrant examination.
+
+
+%%%bbbbb
+In summary,
+%as a reader examines the remainder of this article,
+it should be kept in mind,
+during the remainder of this article,
+that the Hilbert-Bernays Derivability Conditions
+\cite{HP91,HB39,Mend}
+impose severe limits upon any  evasion of
+the Second Incompleteness Theorem.
+% that are inexorable. 
+On the other hand, 
+it appears that a
+ human's
+ faith in his own consistency
+is an essential
+prequisite to gain the needed
+ psychological
+motivation for 
+% cogitating.
+stimulating cogitation?
+% motivate  to cogitate. 
+%cogitation, is also a non-trivial agent.
+(This is why we suspect Hilbert was never willing
+to concede that all facets of his consistency program
+%would be 
+were
+hopeless.)  
+A broad theme of this paper will,
+% thus
+thus, 
+be that it
+is helpful to distinguish between the goals of
+a
+theoretical-oriented study of arithmetic from
+that of
+a more engineering-styled approach,
+since the
+Second Incompleteness Theorem is a perfect result
+from the first perspective while it permits
+for
+% some 
+well-defined
+limited-scale part-way exceptions from
+the second vantage point. 
+
+%% Above sentence replaces below
+
+
+%%  Our interest in
+%%  the contrast between Paradigms A and B, as well as
+%%  the  distinction between a
+%%  ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%%  was
+%%  % stimulated by such 
+%%  raised by these
+%%  considerations.
+
+
+%% It is for this reason that
+%% the contrast between Paradigms A and B, as well as
+%% the  distinction between a
+%% ``Fragmentary'' and ``Idealized-Utopian'' notion of consistency, 
+%% from the preceding two paragraphs,
+%% warrant investigation.
+%% 
+%are so important.
+
+
+%%  Their
+%% two subtle contrasts will be our
+%% main
+%% focus
+%% % of our attention 
+%% %in the remainder of this article.
+%% in the rest of this article.
+%% 
+
+
+\section{The IS$_D(A)$ Axiom System}
+\label{pppp4}
+
+
+\label{sect4}
+
+In a context where $~A~$ denotes any axiom
+system using
+ $L^*\,$'s 
+U-Grounding language, IS$_D(A)$ 
+was defined 
+in \cite{ww5}
+to be an axiomatic
+formalism  capable of recognizing all of 
+$A$'s $\Pi_1^*$ theorems and 
+corroborating 
+its own Level-1 consistency under $D$'s deductive method.
+It 
+consisted of the 
+following four
+groups of axioms:
+\begin{description}
+\item[Group-Zero:]
+Two of the Group-zero axioms 
+did
+% will 
+define
+the
+constant-symbols,
+$\bar{c}_0$
+and $\bar{c}_1$,
+designating the integers of 0 and 1.
+The
+Group-zero axioms
+will
+also define the
+growth functions of addition and 
+$~Double(x) \, = \, x+x.~$
+The
+net effect of these 
+axioms will be to set up a machinery to
+define
+any 
+integer
+$~n \geq 2~$ 
+using fewer than
+$3 \cdot \lceil \, $Log$~n~ \rceil \,$ 
+logic symbols.
+
+
+
+
+
+\item[Group-1:] 
+This axiom group 
+did
+% will 
+ consist of a
+finite set of $\Pi_1^{*} $ sentences, denoted as $~F~$, which
+can prove any $\Delta_0^*$ sentence that
+holds true under the standard model of the natural numbers.
+(Any finite set of 
+$\Pi_1^{*} $ sentences $~F~$ 
+with this property
+may be used to define Group-1,
+as    \cite{ww5}  noted.)
+
+
+
+\item[Group-2:]
+Let $\ulxyz \Phi \urxyz$ denote
+$\Phi$'s G\"{o}del Number, and
+HilbPrf$_A(\ulxyz \Phi \urxyz,p)$ denote a
+$\Delta_0^{*} $ formula indicating 
+$~p~$ is a
+Hilbert-styled proof of 
+theorem $~\Phi~$ from
+axiom system $A$.
+For each $\Pi_1^{*}  $ sentence  $\Phi$, the 
+Group-2 schema
+of \cite{ww5}
+did
+% will 
+ contain an axiom of 
+form \eq{group2}.
+(Thus IS$_D(A)$ can trivially prove
+ all $A$'s 
+$\Pi_1^{*}  $ theorems.) 
+\begin{equation}
+\forall ~p~~~\{~ \mbox{HilbPrf$_A(\ulxyz \Phi \urxyz,p)$}
+ ~~
+\Rightarrow ~~ \Phi~~\}
+\label{group2}
+\end{equation}
+\item[Group-3:]
+This final part of the IS$_D(\aaa)$ 
+essentially represented
+% will be 
+a
+self-referencing
+$\Pi_1^*$
+axiom,
+indicating 
+IS$_D(\aaa)$ meets 
+\textsection   3's criteria of being
+``Level-1 consistent'' 
+under deductive method $D$.
+It 
+amounts,
+%is, 
+thus, 
+to the following  declaration:
+\begin{quote} 
+\# $~${\it No two
+proofs exist
+for 
+a $\Pi_1^{*} $ sentence
+and its negation, when
+$D$'s deductive method is applied to an axiom system,
+consisting of  
+the {\it union}
+of 
+Groups 0, 1 and 2 with {\bf $\,$this sentence$\,$}
+(looking at itself).}
+\end{quote}
+
+One encoding of \#,
+$\,$as a self-referencing
+$\Pi_1^{*} $
+axiom,
+appears
+ in 
+\cite{ww5}.
+%% hhhh0000000000
+Thus, 
+the 
+below
+sentence
+\eq{group3} 
+represents 
+\cite{ww5}'s
+$\Pi_1^{*}\, $ styled
+encoding for$~$
+  \#
+in a context where:
+\bed
+\item[    i. ]
+$~~\mbox{Prf} \, _{\mbox{IS}_D(A)}(a,b) \, $ is
+a 
+ $\Delta_0^{*} $ formula 
+indicating 
+that 
+$ \, b \, $ is a proof
+ of a theorem $\, a\,$
+under
+  $\mbox{IS}_D(A)$'s axiom system
+and $D$'s deduction method, 
+$\,~$and 
+\item[   ii. ]
+$~~$Pair$(x,y)$ is a $\Delta_0^{*} $ formula
+indicating 
+that $ \, x \, $ is  a
+ $\Pi_1^{*} $ sentence 
+and 
+% that
+ $ \, y \, $ 
+represents 
+$ \, x \,$'s negation.
+\end{description}
+%% A summary of the formal techniques that
+%% \cite{ww5} used to encode
+%% sentence
+%% \eq{group3} is provided in Appendix B. 
+\end{description}
+\begin{equation}
+\forall  ~x~\forall  ~y~\forall  ~p~\forall  ~q~~~~ \neg ~~
+[~~ \mbox{Pair}(x,y)~ \wedge ~ 
+~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(x,p)~
+\wedge ~ ~\mbox{Prf}~_{\mbox{IS}_D(\aaa)}(y,q)~ ]
+\label{group3}
+\end{equation}
+ 
+\begin{remark} \label{remc}
+\rm
+A 
+fully formal
+summary of the techniques that
+\cite{ww5} used to encode
+%the
+sentence
+\eq{group3} is provided by
+the combination of  Appendices B and C. 
+The former appendix summarizes our
+methods for generating the G\"{o}del numbers
+of semantic tableaux and  \txl{k} proofs
+in an optimally compressed manner.
+The latter appendix explores how
+sentence
+\eq{group3}'s self-referencing statement is precisely encoded. 
+\end{remark}
+
+{\bf Notation.} An operation $~I(~\bullet~)~$ that maps
+an initial axiom system $\,\aaa \,$ onto an alternate
+system  $\,I(\aaa)\, $ will be called {\bf Consistency Preserving}
+iff  $\,I(\aaa)\, $ is consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers. In this
+context, 
+\cite{ww5} demonstrated:
+
+
+\begin{theorem}
+\label{ttt1}
+\label{thold}
+Suppose 
+the symbol $D$ denotes either semantic
+tableaux deduction or its \txl{1} generalization.
+Then the  IS$_D(~\bullet~)~$ mapping operation is consistency preserving
+(e.g. 
+IS$_D(\aaa) $ 
+will be consistent whenever all of $\aaa$'s axioms hold
+true under the standard model of the natural numbers).
+\end{theorem}
+
+We emphasize 
+the most difficult part of \cite{ww5}'s
+result was 
+neither the definition of its  
+IS$_D(\aaa) $'s axiom system nor the
+$\Pi_1^*$ fixed-point
+ encoding of \eq{group3}'s Group-3 axiom.
+Instead, 
+the key challenge
+ was the 
+confirming
+of \thx{thold}'s 
+``Consistency Preservation''
+property.
+
+
+The 
+confirming of
+this
+property
+is
+subtle
+because its invariant breaks down when 
+$~D~$ is a deduction method only slightly stronger than
+either semantic tableaux or  \txl{1} deduction.
+Thus,  Pudl\'{a}k's and Solovay's
+work \cite{Pu85,So94}
+implies  \thx{thold}'s analog fails when $D$ represents
+Hilbert deduction, and \cite{wwlogos} showed its generalization
+ fails
+even when  $D$ represents  \txl{2} deduction.
+
+
+
+
+
+
+
+
+\section{A Finitized Generalization of  \thx{thold}'s Methodology}
+\label{pppp5}
+
+
+\label{sect5}
+
+%%%mmmm
+One 
+difficulty with IS$_D(\aaa)$ 
+was
+is
+that it 
+employed
+an infinite number of different
+incarnations of
+sentence \eq{group2}
+in its Group-2 scheme (since it contained one incarnation
+of this sentence for each $\Pi_1^*$ sentence $\Phi$ in
+$L^*\,$'s language). Such a Group-2 schema is awkward because
+it simulates $A$'s 
+$\Pi_1^*$
+knowledge almost via a brute-force 
+enumeration.
+
+
+Our Definition \ref{dd-is2} and Theorems
+\ref{ttt2} and \ref{ttt3} will show how
+to 
+mostly
+overcome this problem by  
+compressing the infinite number
+of
+instances of sentence \eq{group2} in
+IS$_D(\aaa)$'s Group-2 schema into
+a purely finite structure.
+
+\smallskip
+
+\begin{definition}
+\label{dd-is2}
+\rm
+Let $~\beta~$ denote any
+finite set of
+axioms that have 
+ $\Pi_1^*$ encodings.
+Then 
+\I2 
+will denote an axiom system,
+similar to  IS$_D(\aaa)$, except 
+its Group-2
+scheme will employ $~\beta\,$'s set of axioms,
+instead of using an infinite number of applications
+of
+statement \eq{group2}'s scheme.
+(Thus,
+the 
+{\it ``I am consistent''} statement
+in \I2's Group-3
+axiom will be the same as before, except that
+the  {\it ``I am''} 
+fragment of its
+self-referencing
+statement
+will reflect 
+these
+ changes in Group-2 in the obvious manner.)
+\end{definition}
+ 
+
+
+\begin{theorem}
+\label{ttt2}
+Let 
+ $D$ again denote either
+semantic
+tableaux 
+or  \txl{1} deduction,
+and $\beta$ again denote a set of
+$\Pi_1^*$ axioms.
+Then
+\I2 
+will be consistent whenever all 
+$\beta$'s axioms hold
+true under the standard model.
+(In other words, 
+ \I2 
+will satisfy an analog of \thx{ttt1}'s
+consistency preservation property for IS$_D(\aaa) $.) 
+\end{theorem}
+
+%%bbbb
+\thx{ttt2}'s
+proof
+is almost identical to
+\cite{ww5}'s proof of \thx{ttt1}.
+Its proof is too lengthy to repeat here.
+Instead \textsection \ref{newppp9}
+will 
+briefly summarize its
+%%                                                        
+%%    provide
+%% a 
+%% brief
+%% %detailed
+%% % an intuitive 
+%% summary
+%% of the 
+%% formal
+%% % germane 
+%% 
+proof.
+This 
+abbreviated discussion
+%% discourse 
+should be sufficient to explain
+the gist behind the
+proof's core
+%needed 
+formalism,
+%proofs,
+without delving into
+\cite{ww5}'s
+full 
+%%%%% too many
+%full
+% formal 
+details.
+
+%%bbbb
+Our next definition will enable us to formalize
+the main application of 
+\thx{ttt2} that will be considered
+here.
+%during the present article.
+It will essentially explain how
+{\bf finite-sized}
+ self-justifying 
+ logics
+  can provide an
+  {\bf infinite amount }
+ of 
+ ``kernelized''
+   $\Pi_1^*$ 
+styled
+information.  
+
+
+
+%%% It will.
+%%% not be 
+%%% repeated in this extended abstract.
+%%% Instead, 
+%%% this section
+%%% will  apply
+%%% \thx{ttt2}
+%%% to 
+%%% show how
+%%% {\bf finite-sized}
+%%% self-justifying 
+%%% logics
+%%%  can provide an
+%%%  {\bf infinite amount }
+%%% of 
+%%% ``kernelized''
+%%%   $\Pi_1^*$ information.  
+%%% 
+
+\begin{definition}
+\label{dkern}
+\rm
+Let
+Test$_i(t,x)$ 
+denote any $\Delta_0^*$ formula,
+and $~\ulcorner \Psi \urcorner ~$ denote
+$\, \Psi\,$'s G\"{o}del number. Then
+Test$_i(t,x)$ will be called a {\bf Kernelized Formula}
+iff Peano Arithmetic can prove every $\Pi_1^*$ sentence
+$~\Psi~$ satisfies \eq{testker}'s
+identity:
+\beq
+\label{testker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+There are 
+infinitely
+many 
+ $\Delta_0^*$  predicates
+Test$_1(t,x)$, Test$_2(t,x)$, Test$_3(t,x)$ ...
+satisfying this kernelized condition
+(one of which is illustrated by Example \ref{eex1}).
+An enumerated list  of all 
+the available kernels
+is
+called a  {\bf Kernel-List}.
+\end{definition}
+
+\begin{example} \label{eex1} \rm
+The set of
+true $\Sigma_1^*$ sentences is
+r.e.
+This 
+implies
+there 
+exists a $\Delta_0^*$ formula,
+called say Probe$(g,x)$, 
+such
+that $~g~$ 
+is
+the G\"{o}del number of
+a $\Sigma_1^*$ statement that holds true in the Standard
+Model 
+if and only if
+%iff
+\eq{e-probe} is true: 
+\beq
+\label{e-probe}
+\exists ~x~~~ \mbox{Probe}(g,x)~\wedge~ x \geq g
+\enq
+Now, let Pair$(t,g)$ denote a $\Delta_0^*$ formula
+that specifies $~t~$ is the  G\"{o}del number of
+a $\Pi_1^*$ statement and
+ $~g~$ is
+the  $\Sigma_1^*$ formula which is its negation.
+Then our notation implies 
+that
+  $~t~$ 
+is
+a true 
+ $\Pi_1^*$ statement 
+if and only if \eq{e-2probe} holds true:
+\beq
+\label{e-2probe}
+\forall ~x~~~ 
+\neg~[~\exists ~g ~\leq~x~~~~~ \mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)~~]
+\enq
+Thus if
+Test$_0(t,x)$
+denotes the $\Delta_0^*$ formula of
+$~ \neg~[~\exists ~g \, \leq \, x~~ 
+\mbox{Pair}(t,g)~\wedge~\mbox{Probe}(g,x)]$,
+it
+is one example of what 
+Definition \ref{dkern}
+would
+call a
+``Kernelized Formula''.
+\end{example}
+
+\begin{definition}
+\label{def3}
+\rm
+Let us recall 
+Definition \ref{dkern}
+defined 
+{\bf Kernel-List} to be an enumeration of
+all the
+kernelized formulae 
+Test$_1(t,x)$,
+ Test$_2(t,x)$, Test$_3(t,x)...~$.
+Assuming
+Test$_i(t,x)$ is the $i-$th element in this
+list
+and 
+$\Psi$ is an arbitrary $\Pi_1^*$ sentence,
+the
+{\bf i-th Kernel Image}
+of $\, \Psi \,$
+ will be  
+defined as
+the 
+following $\Pi_1^*$
+sentence:
+\beq
+\label{imagker}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+\end{definition}
+
+\begin{example} \label{eex2}  \rm
+The Definitions 
+\ref{dkern}
+and \ref{def3} suggest that there is a
+ subtle relationship
+between a sentence $~\Psi~$ and its $i-$th kernel image.
+This is because 
+Definition \ref{dkern}
+indicates that Peano Arithmetic can prove the invariant
+\eq{testker},  indicating that 
+ $~\Psi~$ 
+is equivalent to
+ its $i-$th kernel image.
+However, a weak axiom system 
+can be plausibly uncertain about
+whether this 
+equivalence
+does formally hold.
+This invariant is duplicated below:
+\beq
+\label{againtestker}
+\Psi ~~~ \Longleftrightarrow~~~ \forall ~x~~
+\mbox{Test}_i  (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+
+% equivalence holds.
+
+%mm% 
+Thus if a weak axiom system proves statement 
+\eq{imagker} (rather than $~\Psi~$), 
+it 
+%% may
+will
+ not be able to equate these
+two
+results
+(unless it is able to verify
+\eq{againtestker}'s identity).
+This problem will apply to \thx{ttt3}'s
+formalism.
+However, \thx{ttt3} will
+%  be 
+still
+remain
+ of much interest
+because \textsection \ref{pppp6} will 
+illustrate a 
+methodology that
+can overcome
+many of \thx{ttt3}'s limitations.
+\end{example}
+
+
+
+
+
+
+
+\begin{theorem}
+\label{ttt3}
+Let $~A~$ denote any
+system, 
+whose
+ axioms hold 
+true 
+in arithmetic's standard model,
+and $~i~$ denote the index
+of any of
+Definition \ref{dkern}'s
+kernelized formulae
+ Test$_i(t,x)$.
+Then it is possible to construct a
+finite-sized 
+collection of $\Pi_1^*$ sentences, called say
+ $\beta_{A,i}$,
+where 
+\ik3
+satisfies the following invariant:
+\begin{quote}
+If $~\Psi~$ is one of the 
+$\Pi_1^*$ theorems of
+ $~A~$
+then  \ik3 can prove 
+\eq{imagker}'s
+statement 
+ (e.g. it will prove the
+``the $\, i-$th kernelized image'' 
+of 
+$~\Psi\,$).
+\end{quote}
+\end{theorem}
+
+\newpage
+
+\noindent
+{\bf Proof Sketch:}
+Our justification of 
+\thx{ttt3} will 
+use the following notation:
+\bee
+\item
+Check$(t)$ will denote a $\Delta_0^*$ formula
+that 
+produces a Boolean value of ``True'' when
+$t$ represents the G\"{o}del
+number of a $\Pi_1^*$ sentence.
+\item
+ $~\mbox{HilbPrf}_A \,(   t   ,   q   )~$ 
+will denote
+ a  $\Delta_0^*$ formula that indicates
+$~q~$ is a Hilbert-style proof of the theorem
+$~t~$ from axiom system   $~A~$.
+\item
+For any kernelized
+Test$_i(t,x)$ 
+formula, GlobSim$_i$ 
+will 
+denote \eq{globsim}'s $\Pi_1^*$ sentence.
+(It will be called $A$'s $i-$th
+{\bf ``Global Simulation Sentence''}.)
+\ene
+\beq
+\label{globsim}
+\forall ~t~~
+\forall ~q~~
+\forall ~x~~\{~~
+[~~\mbox{HilbPrf}_A \,(   t   ,   q   )~~ \wedge ~~
+\mbox{Check}(t)~~]~~~
+\Longrightarrow ~~~ 
+\mbox{Test}_i(t,x)~~~ \}
+\enq
+
+%%mm 
+In this notation, 
+%%%the requirements of 
+\thx{ttt3} 
+shall
+%will
+be satisfied by any
+version of the axiom system \I2, whose Group-2 schema $~\beta~$
+is a finite sized
+consistent set of $\Pi_1^*$ sentences
+that has 
+\eq{globsim}
+as an axiom.
+(This includes 
+the minimal sized such system,
+%  which we will 
+denoted as $~\beta_{A,i}~$, 
+that has only \eq{globsim} as an axiom.)
+This is because 
+%Thus,
+if 
+$\Psi$ is any 
+$\Pi_1^*$ theorem of $A$ whose proof
+is denoted as $~\bar{p}~$,  then both the
+$\Delta_0^*$ predicates of
+$\mbox{HilbPrf}_A \,( \ulxyz \Psi \urxyz , \bar{p}    )$ and
+$\mbox{Check}(  \ulxyz \Psi \urxyz )$ 
+will hold true.
+%are true. 
+Moreover,
+IS$^{\#}_D$'s
+%%%%%%%%%%%%%%  \I2's 
+Group-1 axiom subgroup was defined so that
+it can automatically prove all
+ $\Delta_0^*$ sentences that are true. 
+Hence,
+%Thus, 
+ \ik3 will
+ prove these two statements and 
+then automatically
+%hence 
+corroborate (via axiom
+\eq{globsim}) the further statement
+of:
+\beq
+\label{interm}
+\forall ~x~~
+\mbox{Test}_{\, i \,}(~  \ulxyz \Psi \urxyz ~,~x~ )
+\enq 
+%Hence 
+Thus
+for each of the infinite number of $\Pi_1^*$
+theorems that $~A~$ proves, the above defined
+formalism will prove a matching statement
+that corresponds to 
+its
+%% the 
+ $\, i-$th kernelized image.  $~~\Box$
+
+
+%% of 
+%% each
+%% such  proven theorem.
+%%  $~~\Box$
+
+\section{ L-Fold Generalizations of \thx{ttt3} } 
+\label{pppp6}
+
+
+
+
+\thx{ttt3} 
+is of
+interest
+because every axiom system $\,A\,$
+will have
+its formalism
+\ik3 
+prove the 
+ $\, i-$th kernelized image of every
+ $\Pi_1^*$  theorem that $A$ proves.
+This fact is helpful
+because 
+\eq{testker}'s invariance
+holds for all $\Pi_1^*$ sentences.
+Moreover, our  
+``U-Grounded''
+$\Pi_1^*$ sentences
+capture all 
+Conventional Arithmetic's
+{\it crucial}
+$\Pi_1$ 
+information
+because they can 
+view
+multiplication as a 3-way 
+ $\Delta_0^*$ 
+predicate
+Mult$(x,y,z)$
+via 
+\eq{neweq1}'s
+encoding of this predicate.
+\begin{equation}
+\label{neweq1}
+[~(x=0    \vee    y=0 ) \Rightarrow z=0~ ]~ ~\wedge ~~ 
+[~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~]
+\end{equation}
+
+One difficulty with 
+\I2
+and \ik3 
+was mentioned by 
+Example \ref{eex2}. 
+It was that 
+while
+ Peano 
+Arithmetic 
+can corroborate
+\eq{testker}'s invariance for every $\Pi_1^*$ sentence $\, \Psi \,$,
+these latter 
+systems 
+cannot  
+also do so.
+
+
+
+While there will
+probably
+never be a 
+perfect method for
+fully
+ resolving this 
+challenge,
+there is
+a  pragmatic engineering-style solution
+that is often available.
+This is essentially because
+ our proof of \thx{ttt3}
+employed
+a formalism $~\beta~$ that 
+used essentially
+only one axiom sentence (e.g. 
+\eq{globsim}'s $\Pi_1^*$ declaration ).
+
+
+Since  the  \I2 formalism was intended 
+for use
+by any finite-sized system $\beta$,
+it is clearly possible to
+include any
+finite number
+of formally true
+$\Pi_1^*$ sentences in $\beta$.
+Thus for some fixed constant $L$, 
+one can easily let
+ $\beta$ include
+$L$ copies of   \eq{globsim}'s axiom framework for a finite
+number of different
+Test$_1$, Test$_2$ ... Test$_L$ 
+predicates, each of which satisfy Definition \ref{dkern}'s
+criteria for being kernelized formulae. In this case, 
+\I2 
+will formally
+ map each  initial 
+$\Pi_1^*$ theorem 
+$\Psi$
+of some axiom system $~A~$ onto 
+$L$ 
+resulting
+different
+$\Pi_1^*$ 
+theorems
+of
+the form
+\eq{imagker}.
+
+
+
+\begin{remark} \label{remc2}
+\rm
+Our 
+basic
+conjecture is,
+essentially,
+that
+a goodly number of issues, concerning
+logic-based
+engineering applications
+called say $E$, 
+may
+have convenient solutions via
+self-justifying 
+logics,
+that follow the
+preceding 
+outlined 
+L-fold
+strategy. 
+Thus, we are suggesting that if 
+ $~\beta~$ is a large-but-finite set of axioms, that
+consists of 
+$L$ copies of   \eq{globsim}'s axiom framework for 
+different
+Test$_1$...Test$_L$ 
+predicates, then 
+some
+future
+engineering applications $~E~$
+may possibly
+ have their needs met by an
+\I2 formalisms, when a
+software
+ engineer meticulously  chooses
+an
+appropriately
+constructed
+finite-sized
+ $\beta$.
+\end{remark}
+
+
+
+
+\begin{remark} \label{rem2} \rm
+The preceding 
+was not meant to overlook 
+that the Second Incompleteness Theorem is a 
+robust  result,
+applying to
+all 
+logics of sufficient strength.
+Our suggestion, however,  is
+that computers
+are becoming 
+so 
+powerful, in both speed and memory size as
+the 21st century
+is progressing, that   there 
+will likely
+emerge engineering-style applications
+$~E~$
+ that will benefit from  \I2's 
+self-referencing
+formalisms when a {\it large-but-finite-sized}
+ $~\beta~$ is delicately chosen.
+Moreover, it is of interest to speculate
+whether such computers
+can partially
+imitate 
+a human being's 
+approximate instinctive 
+conjectures
+about
+his
+own consistency (that, as 
+common colloquially 
+held
+conjectures, 
+seem
+to serve as
+{\it essential prerequisites} for humans to 
+gain their 
+motivation
+to cogitate).
+\end{remark}
+
+Sections \ref{pppp7}-\ref{pppxppp10}
+ will examine the preceding 
+issues
+in
+further
+detail. 
+%% hhhh
+% 
+% Part of
+% their theme will be
+% that 
+% while boundary-case exceptions to the Second Incompleteness
+% Theorem are important and noteworthy in certain
+% specialized
+%  contexts,
+% they still
+% do not 
+% narrow the 
+% main intentions 
+% of
+%  G\"{o}del's
+% seminal  result.
+% 
+Also, 
+Section \ref{newppp9}
+will offer an intuitive
+summary
+of the techniques 
+that
+\cite{ww5} 
+used to prove 
+Theorem \ref{ttt1}, so that the reader
+can understand 
+%the intuition behind 
+\cite{ww5}'s gist without reading the full details of 
+\cite{ww5}'s
+formal
+proof.
+
+% its proof.
+
+
+\section{Comparing  Type-M and Type-A
+Formalisms} 
+
+\label{pppp7}
+
+Let us 
+recall 
+axioms
+\eq{totdefxs}-\eq{totdefxm} indicated
+Type-A 
+systems
+differ from Type$-$M formalisms by treating Multiplication
+as a 3-way relation (rather than as a total function).
+For the sake of accurately characterizing what our systems
+can and cannot do, we have described our results as
+being fringe-like exceptions to the Second Incompleteness
+Theorem, from the perspective of an Utopian view of Mathematics,
+while  perhaps 
+being
+more significant results from an
+engineering-style perspective of knowledge. Our goal in
+this section will be to 
+amplify
+upon
+this 
+perspective by taking a closer look at Type-A and Type-M
+formalisms.
+
+
+
+Let us assume that 
+$\,x_0 = \, 2 \, = \, y_0\,$ and that
+$      x_1,   x_2, x_3,     ...    $ 
+and  $      y_1,   y_2,  y_3,    ...  $
+are defined by the recurrence 
+rules of:
+\beq
+\label{smart-squeeze}
+x_{i+1}~=~x_{i}~+~x_{i}
+  ~~~~~~~ \mbox{AND} ~~~~~~~
+y_{i+1}~=~y_{i}~*~y_{i}
+\enq
+The sequences 
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+will 
+thus
+ represent the growth rates
+associated with the addition and multiplication primitives,
+lying in the 
+statements
+\eq{totdefxa} and \eq{totdefxm}'s
+{\bf ``Type-A''} and {\bf ``Type-M''} 
+axioms.
+
+Since $\,x_0 = \, 2 \, = \, y_0\,$,
+the rule
+\eq{smart-squeeze} implies 
+$ \, y_n \, = \, 2^{2^n} \, $ and
+ $ \, x_n \, = \, 2^{ n+1} \,  \, $. 
+The  $   y_0,   y_1,   y_2,     ...  $ sequence will,
+thus,
+grow 
+much more quickly than
+the
+$   x_0,   x_1,   x_2,     ...    $ 
+sequence (since
+ $ \, y_n \,$'s binary encoding will have an Log$(y_n)\, = \,2^n~$ 
+length
+while  $ \, x_n \,$'s binary encoding will have
+a shorter length 
+of size
+$~$Log$(x_n) \, = \, n+1~~$).
+
+
+
+Our prior papers noted that the 
+difference between these
+growth rates 
+was the 
+reason that \cite{sp0,ww2,ww7}
+showed 
+all natural 
+Type-M
+systems, recognizing 
+integer-multiplication as a total function, were unable
+to recognize their 
+tableaux-styled 
+consistency ---
+while \cite{ww93,ww1,ww5} showed 
+some Type-A 
+systems could 
+simultaneously prove all Peano Arithmetic $\Pi_1^*$ theorems and
+corroborate their own 
+tableaux consistency.
+Their gist
+was that a
+G\"{o}del-like diagonalization argument, which causes an
+axiom system to become inconsistent as soon as it proves
+a theorem affirming its own 
+tableaux consistency,
+stems,
+ultimately,
+ from
+the exponential growth in 
+the  series  $   y_0,   y_1,   y_2,     ...$ .
+
+This growth
+will, thus,
+facilitate
+% facilitates 
+an intense
+cascading
+amount of self-referencing,
+using 
+the identity
+Log$(y_n)\, \cong \,2^n~,~$
+that will,
+ultimately,
+ invoke the force of
+G\"{o}del's seminal
+diagonalization
+machinery.
+It 
+% thus raises
+will thus raise
+the following 
+enticing
+question: 
+\begin{quote} 
+$***~$ How natural are exponentially growing sequences,
+such as
+  $   y_0,   y_1,   y_2     ..  $, whose $n-$th member
+needs
+ $2^n$ bits 
+for its encoding, $~$when such lengths are
+greater than
+ the number of atoms in the universe
+when
+merely
+ $~n\,> \, 100~$?
+%hhhh
+Is the use of
+such a sequence
+%use, 
+for corroborating the Second Incompleteness
+Effect
+%  , thus essentially,
+%thereby 
+resting
+% , essentially,
+%, at least partially, 
+upon an
+% an inherently
+almost
+artificial construct 
+(with
+ an
+inherently 
+dizzying growth rate) ? 
+\end{quote}
+
+
+
+We will not attempt to derive a Yes-or-No answer to Question $***$
+because 
+we think that such a direct 
+response
+%%% answer 
+is too simplistic.
+Our point is that 
+both a positive and negative reply to
+ $***$
+are useful in different respects.
+%% 
+%% it
+%% is one of those epistemological questions that can be
+%%  debated
+%% endlessly.
+%% Our point is that  $***$
+%% probably does not require a definitive
+%% positive or negative answer because both perspectives
+%% are useful.
+%% 
+%% Thus,
+%% the theoretical existence of a sequence  
+This because 
+the theoretical existence of a sequence  
+integers 
+of $   y_0,   y_1,   y_2,     ...  $, whose binary
+encodings are doubling in length, is tempting
+from the perspective of 
+an Utopian view of mathematics, while 
+awkward from an engineering styled 
+perspective.
+We therefore ask: {\it ``Why not be tolerant
+of both perspectives? ''}
+
+One virtue of 
+this tolerance is 
+it 
+ushers in 
+a greater understanding
+for the statements $*$ and $**$ that G\"{o}del and
+Hilbert made during 
+1926 and 1931.
+This 
+is
+because the 
+Incompleteness Theorem
+demonstrates 
+no 
+formalism can display
+an understanding of its own consistency in an
+idealized
+ Utopian
+sense. On the other hand,  
+\textsection 6
+suggested
+these 
+two
+remarks by G\"{o}del and Hilbert 
+ might receive
+more sympathetic interpretations, 
+if one 
+sought to explore
+such questions from a less ambitious
+almost engineering-style perspective.
+
+
+
+
+Our
+main  thesis is 
+supported by a 
+theorem
+from \cite{ww6}.  It indicated that 
+tableaux
+variations of self-justifying systems have no difficulty
+in recognizing that an infinitized generalization of
+a computer's
+floating point multiplication (with rounding) is a total
+function. The latter 
+differs from integer-multiplication,
+by not having its output become double the length of
+its input when a number is multiplied by itself.
+Thus, the 
+intuitive
+reason 
+\cite{ww6}'s
+ multiplication-with-rounding operation
+is compatible with self-justification is
+because it
+ avoids the 
+inexorable
+exponential
+growth under
+rule \eq{smart-squeeze}'s sequence 
+ $   y_0,   y_1,   y_2     .. ~  $.
+
+\bigskip
+
+
+%\newpage
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+%% bbbbbbb
+Also,  \thx{ttt4} indicates 
+self-justifying logics
+can view
+double-precision
+integer multiplication
+similarly
+as
+ a  total function.
+In particular for 
+any arbitrary pair 
+of integers
+ $(a,b)$,
+let us employ a notation convention where:
+\bee
+\item 
+{\bf Size(a,b)} denotes the maximum of
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,a \, \rceil \,  $ 
+and 
+$ \, \lceil  \, 1 \, + \,$Log$_2 \,b \, \rceil \,  $. 
+% $\, 1 \, + \,$Log$_2 \,b \,$.
+\item The quantities 
+{\bf Left$(a,b)$}
+and {\bf Right$(a,b)$}
+represent the multiplicative product
+of 
+the integers
+$~a~$ and $~b~,~$ insofar as 
+Right$(a,b)$ 
+represents the rightmost bits of this product
+of length Size(a,b), and 
+Left$(a,b)$ encodes the remaining bits to the left
+of Right$(a,b)$ 
+(whose length will also be bounded by Size(a,b) ). 
+\ene
+Within this context,  
+\thx{ttt4} indicates 
+self-justifying logics
+self-justification 
+are able to view double-precision
+integer-multiplication as
+a total function.
+
+%% bbbbb
+\begin{theorem}
+\label{ttt4}
+Let us assume 
+the $ \,A \,$ in 
+IS$_D(\aaa)$ and 
+$\ \beta \,$ in 
+\I2
+are axiom systems all of whose $\Pi_1^*$ 
+theorems are true statements under the standard model
+of the natural numbers.
+Then 
+if $D$ corresponds to either semantic tableaux or
+\txl{1} deduction,
+it is possible to formalize 
+systems
+$~A^* \, \supseteq \, A~$
+and
+$~\beta^* \, \supseteq \, \beta~$
+such that \js and \ns are self-justifying
+extensions of respectively
+IS$_D(\aaa)$ and 
+\I2
+which can recognize 
+%that 
+each of
+the 
+double-multiplicative precision
+operations of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ 
+%(that define the double-precision multiplicative product
+%of $a$ and $b$) 
+as total functions.
+\end{theorem}
+
+%% bbbbb
+{\bf Proof Sketch;} The justification of \thx{ttt4}
+is
+% very 
+similar to 
+\cite{ww6}'s analysis of
+Floating Point Multiplication
+(with rounding). Our proof of   \thx{ttt4}
+will therefore be quite abbreviated.
+
+%% bbbbb
+The first point is that it is 
+% quite 
+straightforward
+to develop three $\Delta_0^*$ formulae,
+called $\theta_1(a,b,y)$,
+ $~\theta_2(a,b,y)$
+and
+ $\theta_3(a,b,y)$,
+that are the graphs of the functions
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+% Moreover, it 
+It
+is also easy to construct a
+finite set of $\Pi_1^*$ sentences,
+holding true in the Standard Model, 
+called $~\gamma~$,
+that  know how to correctly interpret these three
+ $\Delta_0^*$ formulae,
+insofar as $~\gamma~$ knows:
+\bee
+\item For each 
+%fixed 
+$a$ and $b$, there exists no more
+than one integer $~y~$ that satisfies each of our
+three  $\theta_j(a,b,y)$ formulae. 
+\item For each 
+%fixed 
+$a$ and $b$, 
+our three  $\theta_j(a,b,y)$ formulae 
+correctly simulate 
+the
+graphs of
+the respective
+functions of 
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$.
+\ene
+%Moreover since 
+Since 
+our U-Grounding language contains the built-in
+function primitives of ``Maximum'' and``Double$(x)$'',
+the Group-1 component of
+IS$_D$ 
+and IS$_D^{\#}$
+% formalisms 
+can 
+easily
+verify that
+the
+ operation
+$F(a,b)$, defined below is a total function:
+\beq
+\label{F-def}
+~F(a,b)~~=~~\mbox{ Double (Double (Double (Max}(a,b))))
+\enq
+This implies, in turn, that
+there exists a $\Pi_1^*$ sentence, called $\gamma^*$, that
+will enable our formalism to verify that each of
+ Size$(a,b)$, 
+Left$(a,b)$ and
+Right$(a,b)$ are total functions (simply because
+their output values are less than 
+$~F(a,b)$'s output).
+
+The main point is that the hypothesis of \thx{ttt4} 
+ indicated that
+all the axioms of
+ $ \,A \,$ and
+$\ \beta \,$ 
+did hold
+true under the Standard Model,
+and the preceding paragraph showed the same
+was
+ true for all the axioms in  
+ $~\gamma~$  and $~\gamma^*~$,
+Hence all the axioms in
+$~A^*~=~A~+~ \gamma~+~\gamma^*~$
+and 
+$~\beta^*~=~\beta~+~ \gamma~+~\gamma^*~$
+also
+hold true in the Standard Model.
+By Theorems \ref{ttt1} and \ref{ttt2},
+this implies that 
+IS$_D(\aaa)$ and 
+\I2 and are self-justifying formalism
+satisfying \thx{ttt4}'s claims. $~~\Box$
+
+
+
+%% \ik3  
+%% represents Peano Arithmetic. Then
+%% IS$_D(\aaa)$ and \ik3  
+%% can formalize
+%% two  total functions, called Left$(a,b)$
+%% and Right$(a,b)$, 
+%% where any  pair
+%% of integers
+%%  $(a,b)$
+%% is mapped onto
+%% the left and right halves of
+%%  $a$ and $b$'s multiplicative
+%% product.
+
+
+\begin{remark}
+\rm
+\label{rem-new}
+One 
+subtle
+%% slightly tricky 
+aspect is that our positive
+results,
+involving
+\cite{ww6}'s
+floating point multiplication
+primitive 
+and \thx{ttt4}'s
+analogous
+double precision multiplication
+operation, 
+{\it should
+not be confused} with a
+quite  different 
+exploration of integer multiplication
+in the context of our analysis of Herbrand
+consistency 
+in \cite{ww9}. 
+The latter took advantage
+of the fact that
+our deployed
+ Herbrand-styled proofs
+%%% in \cite{ww9}'s paradigm , are
+in \cite{ww9} were
+exponentially
+longer than their 
+tableaux
+counterparts
+(thus allowing \cite{ww9} 
+to formalize
+a limited use of multiplication).
+This was because 
+% its 
+\cite{ww9}'s
+deductive 
+methods 
+were
+%%%%% were, inherently,
+exponentially
+less efficient
+at an inherent
+level.
+Thus 
+  \cite{ww9}'s result,
+while 
+of
+%somewhat
+%%
+%%certainly
+%%perhaps
+%%
+theoretical 
+%theoretically 
+interest,
+is
+%essentially
+%%% hhhhh
+basically
+irrelevant to
+the core
+engineering environments,
+%e.g. 
+which 
+constitutes
+% are
+the
+ main 
+% central
+focus of
+ Theorems \ref{ttt1}--\ref{ttt4}.
+%% 
+%% (especially in regards to their
+%% particular interpretations
+%% given in
+%% Remark   \ref{rem2}).
+%% 
+\end{remark}
+
+
+%% In other words, Remark \ref{rem-new}'s
+%% observation is, once again, connected to
+%% the crucial distinction between
+%% % an 
+%% engineering
+%% and mathematical viewpoints 
+%% about
+%%  the 
+%% significance of theorem-proving.
+
+
+
+%%%bbbb
+Remark \ref{rem-new}'s
+contrast between 
+ \cite{ww9}'s results and \thx{ttt4}
+ is, once again, connected to
+the  distinction between
+the
+engineering
+and mathematical viewpoints 
+about
+ the main 
+intentions
+%importance
+%significance 
+of theorem-proving.
+% From an engineering perspective,
+\thx{ttt4}
+is helpful 
+from an engineering perspective
+because most 
+% of the 
+pragmatic
+%engineering 
+applications
+of integer multiplication 
+are analogous to either
+%% 
+%% correspond to 
+%% essentially
+%% % what correspond to be 
+%% the standard computerized word-oriented integer-multiplication
+%% primitive
+%% %operations 
+%% or 
+%% its
+%% %their 
+%% conventional
+%% 
+ computerized  double-precision 
+multiplication or its
+quadruple-precision or hexagonal
+% -precision 
+%  computerized
+generalizations.  
+
+\thx{ttt4} 
+(and its quadruple-precision 
+and 
+% hexagonal-precision generalizations)
+hexagonal generalizations)
+% helpfully 
+indicate 
+% such 
+these
+% pragmatic
+operations are
+%  fully
+compatible with a formalism recognizing its own
+semantic tableaux 
+%and \txl{1} 
+consistency.
+
+\section{A Different Type of Evidence Supporting 
+Our
+Thesis}
+
+\label{pppp8}
+
+
+Let us recall
+ Pudl\'{a}k and Solovay 
+\cite{Pu85,So94} 
+observed
+that 
+essentially all
+Type-S
+systems, 
+containing merely
+statement  \eq{totdefxs}'s
+axiom that successor is a total function,
+cannot verify their own consistency under 
+Hilbert deduction. 
+(See also related work by 
+Buss-Ignjatovic \cite{BI95},
+H\'{a}jek and
+ \v{S}vejdar \cite{Sv7},
+as well as  \cite{ww1}'s
+Appendix A.) 
+
+
+It turns out that
+\cite{wwlogos} generalized 
+these
+ results to
+show that 
+\ep{totdefxa}'s
+Type-A 
+systems are unable to verify their
+own consistency under the 
+\txl{2} deduction
+system 
+(defined 
+in
+\textsection  
+ \ref{pppp3}).
+At the same time,  
+the IS$_D$
+and IS$^{\#}_D$
+frameworks,
+from Sections \ref{pppp4}
+ and \ref{pppp5}, can verify
+their own consistency under
+\txl{1} deduction. Our goal in this section will be to
+illustrate how the
+tight
+ contrast between these positive and negative
+results 
+is
+analogous to the differing growth rates
+of
+the 
+sequences
+$   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $
+from
+ rule  \eq{smart-squeeze}. 
+
+
+
+
+During our discussion 
+$~G_i(v)~$ will denote 
+the scalar-multiplication
+operation  that maps
+an integer $~v~$ onto 
+$~ 2^{2^i}\cdot v~$. 
+Also, $~\Upsilon_i~$ will  denote 
+the statement, in the U-Grounding language, that 
+declares that 
+ $~G_i~$ is a total function.
+Our paper \cite{wwlogos}
+proved that  $~\Upsilon_i~$ has
+a $\Pi_2^*$ encoding. It also implied that $~G_i~$
+satisfied:
+\beq
+\label{e-Gi}
+G_{i+1}(v) ~~~ = ~~~ G_i(~ \, G_i(v)~ \, )
+\enq
+It was 
+noted in \cite{wwlogos} that 
+this identity
+implies  one
+can construct 
+an axiom system $  \beta  $, comprised of
+solely $\Pi_1^*$ sentences,
+where
+a semantic tableaux proof 
+can establish
+$  \Upsilon_{i+1}$  
+from
+$  \beta+\Upsilon_i$  
+in a constant number of steps. 
+This implies, in turn, that a \txl{2} proof from
+$  \beta  $ will require no more that O$(n)$ steps
+to prove $  \Upsilon_{n}$ (when it uses the obvious
+n-step process to
+confirm in chronological order 
+$~\Upsilon_1 \, , \,  \Upsilon_2 \, , \,  ... \Upsilon_n ~.~~)$
+
+
+\smallskip
+
+These observations are significant because 
+$G_n(1)=2^{2^n}$.
+Thus,
+\cite{wwlogos}
+% showed 
+established that
+a \txl{2} proof
+from $\beta$ can verify
+in 
+only
+ O$(n)$ steps   
+that this
+quite large
+ integer exists.
+
+
+\smallskip
+
+This example is  helpful because it illustrates
+the difference between the growth speeds
+under
+\txl{1} and \txl{2} deduction, is  analogous
+to the 
+differing
+growth 
+rates
+of
+the
+sequences $   x_0,   x_1,   x_2,     ...    $ 
+and  $   y_0,   y_1,   y_2,     ...  $ 
+from rule \eq{smart-squeeze}.
+Hence once again, a faster growth-rate 
+will usher in
+the Second Incompleteness Theorem's power
+(e.g. see \cite{wwlogos}).
+
+
+This analogy suggests
+that the 
+Second 
+Incompleteness
+Theorem has different implications from the perspectives
+of 
+Utopian and engineering
+theories about
+ the intended
+applications of mathematics. Thus, a Utopian
+may  possibly be 
+ comfortable
+with 
+a
+perspective, that contemplates sequences
+ $   y_0,   y_1,   y_2,     ...  $ 
+with
+elements growing in length
+at an exponential speed, but many engineers may be
+suspicious of such
+growths.
+
+
+
+
+
+
+A hard-core engineer,
+in contrast, might
+ surmise that the inability of self-justifying
+formalisms to be compatible with \txl{2} deduction is
+not 
+as disturbing
+ as it might 
+initially 
+appear to be.
+This is
+because \txl{2}
+differs from 
+ \txl{1} deduction
+by producing 
+exponential growths that are so sharp
+that their material realization has no analog
+in the everyday mechanical reality that is the
+focus of an engineer's 
+interest.
+
+Our personal preference is for
+a perspective lying
+half-way
+between 
+that of an Utopian mathematician and
+a hard-nosed engineer. 
+Its
+dualistic
+approach
+suggests
+some form of diluted
+partial agreement
+with  Hilbert's goals
+in  $**$ (in a context where the broad significance of
+the Second Incompleteness Theorem is obviously
+undeniable).
+
+
+
+
+
+
+
+
+\section{Outline of \thx{ttt2}'s Proof and 
+% Exploration of 
+% Further Discussion
+Its Implications}
+
+\label{new9}
+\label{newppp9}
+
+
+The prior two sections of this article 
+offered an intuitive explanation about why our
+self-justifying axiom systems needed omit the
+assumption that multiplication is a total function
+and
+could verify their  consistency 
+% verified their own consistency 
+only 
+ under
+% for 
+semantic tableaux and
+\txl{1} deduction.
+
+
+%%% \txl{1} deduction
+%%% (rather than a stronger \txl{2}  
+%%% rule of inference).
+
+
+We already noted 
+%that 
+\thx{ttt2}'s
+observation that
+ IS$_D^{\#}$ 
+%% proof 
+%% that 
+is  consistency-preserving
+%transformation
+has essentially an 
+analogous
+% hhhh
+%identical 
+proof as \cite{ww5}'s
+demonstration that 
+%\K1 
+ IS$_D$
+is  consistency-preserving.
+It is not our intention to repeat
+such a  proof  here.
+
+%%a
+%%virtual
+%% analog of
+%%\cite{ww5}'s proof  here.
+
+Instead, our goal will be to  provide a brief overview
+of the techniques 
+%appeared in \cite{ww5}'s proof. This
+that \cite{ww5} 
+had
+used. This
+overview
+will be
+% brief but
+%%% 
+%%% will not delve into all \cite{ww5}'s details.
+%%% It will,
+%%% however, be 
+%%% 
+sufficient
+for
+% so that 
+a reader
+to
+% can quickly
+appreciate
+the 
+% main 
+underlying
+intuition.
+
+%the underlying intuition.
+
+
+%%gain an intuition behind the 
+%%underlying nature
+%% of Theorems \ref{ttt1}
+%%and \ref{ttt2}. 
+
+\bigskip
+
+More precisely,
+two different types of proofs of \thx{ttt1}
+had appeared in our 2002 conference paper \cite{tab2}
+and subsequent journal paper \cite{ww5}. The
+latter 
+%result 
+was more appropriate for an archival 
+journal because its self-justification result
+applied to both semantic tableaux deduction and its
+\txl{1} generalization. 
+The more compressed conference paper 
+\cite{tab2} proved the analog of \thx{ttt1}
+only for tableaux deduction
+(using a technique
+% thus
+that was
+%pleasantly 
+somewhat
+shorter
+than \cite{ww5}'s more elaborate
+result). 
+Our
+% brief 
+summary of \thx{ttt1}'s
+proof,
+here,
+ will focus on the semantic tableaux deduction
+methodology so it can apply to either of
+\cite{tab2}
+or \cite{ww5}'s 
+methods.
+%results.
+
+%%
+%%Our discussion
+%%%in this section 
+%%will focus mostly on
+%%\cite{ww5}'s more 
+%%sophisticated
+%% result, but it should
+%%be also helpful to readers who 
+%%wish to
+%%examine only
+%%\cite{tab2}'s
+%%simpler
+%%but 
+%%%% 
+%%%% and slightly simpler
+%%%% presentation of a 
+%%%% 
+%%less ambitious result.
+
+Both of \cite{tab2,ww5}
+%% had
+% formalisms were
+justified \thx{ttt1} 
+by means of proofs by
+contradiction.
+Thus if \thx{ttt1} 
+was false,
+they 
+% both
+noted
+% then there would exist
+%two 
+a pair of
+proofs 
+%of 
+for
+a $\Pi_1^*$ sentence and its negation
+would exist
+from
+IS$_D(\aaa) $. 
+
+
+
+Let us call these two proofs $P$ and $Q$.
+Then \cite{tab2,ww5} both
+showed 
+(using different constructions) that
+one could construct from $(P,Q)$
+two other proofs  $(p,q)$ of another
+$\Pi_1^*$ sentence and its negation
+such that:
+\beq
+\label{catch}
+\mbox{Max}(p,q) ~~ < ~~ 
+\mbox{Max}(P,Q) 
+\enq
+The inequality in \eq{catch}
+is significant because it 
+will enable our proofs-by-contradiction to establish
+ the non-existence
+of an ordered pair
+  $(P,Q)$ violating \thx{ttt1}'s assumption.
+This is because 
+%otherwise 
+\eq{catch}
+would 
+otherwise 
+violate the Principle of Induction by showing
+there exists no such minimal ordered pair
+ $(P,Q)$
+eschewing \thx{ttt1}'s formalism. 
+
+The 
+exact
+details of these proofs by contradictions are too lengthy
+%for us 
+to fully summarize
+% them 
+here.
+For the case where $D$ in \thx{ttt1}
+is the semantic tableaux deduction method, they used the fact
+that  if $(P,Q)$ was the ordered pair with
+minimal $ \mbox{Max}(P,Q)$ value violating
+\thx{ttt1}'s hypothesis,
+then one could 
+isolate 
+two
+particular root-to-leaf paths in the tableaux
+proofs $P$ and $Q$ that would enable us to construct an
+additional pair $(p,q)$
+that violated \thx{ttt1} and satisfied
+\eq{catch}'s inequality.
+
+This  construction of 
+ $(p,q)$ from $(P,Q)$ 
+utilized the fact that
+ \thx{ttt1}'s 
+axiom system
+ IS$_D(\alpha) $ recognized addition but not multiplication
+as a total function.
+Otherwise,   \thx{ttt1}'s delicate
+proof-by-contradiction would collapse entirely
+(as a result of 
+the exponentially faster growth
+properties 
+of multiplication
+that was formalized by the
+series
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship).
+
+
+These observations reinforce the theme of
+\textsection \ref{pppp7}
+about the contrast between the slower growing series 
+ $      x_1,   x_2,  x_3,    ...  $
+and its exponentially faster counterpart
+ $      y_1,   y_2,  y_3,    ...  $
+under Line \eq{smart-squeeze}'s
+ recurrence 
+relationship.
+These two series defined the 
+% respective
+growth rates produced by the addition and
+multiplication function symbols
+% with  
+as, respectively,
+$ \, x_n \, = \, 2^{ n+1} \,  \, $ and
+$ \, y_n \, = \, 2^{2^n} \, $.
+They 
+thus illustrated
+% thus, once again, illustrate
+how multiplication's faster growth rate
+leads to such a
+%% 
+%% The themes of Sections \ref{ppp7} and
+%% \ref{ppp8} was that the latter growth rate
+%% represented a 
+%% 
+dizzying exponential speed-up,
+that 
+% will
+% would 
+makes
+one at least partially sympathetic to a
+hard-nosed engineer's skepticism about
+its 
+implications.
+
+%significance. 
+
+Thus if one were to
+preclude such a dizzying growth rate then
+a partial justification of a diluted version
+of Hilbert's consistency program would arise,
+in the context of systems possessing 
+{\it weak but well defined} knowledges of
+their own consistency.
+On the other hand, if the conventional assumption
+that multiplication is a total function is presumed,
+then the traditional interpretation of the
+Second Incompleteness Theorem will
+% , of course, fully 
+prevail.
+
+
+%% 
+%% 
+%% Hence some partial caveats can be attached to the
+%% Second Incompleteness Theorem that carry some
+%% credibility from an hard-nosed engineering
+%% perspective, while 
+%% simultaneously
+%% they 
+%%  fail to apply to a
+%% %at the same time not
+%% %be germane to a fully 
+%% pristine 
+%% mathematical
+%% perspective
+%% focused around the 
+%% Logical Platonism
+%% (that G\"{o}del
+%% had 
+%% explicitly explored).
+%% %wrote about).
+
+
+% \large
+
+% \baselineskip = 1.5 \normalbaselineskip 
+
+
+\section{Related Reflection Principles}
+
+
+\label{pppxppp10}
+
+An added point is that there are many 
+types of
+self-justifying systems  available, with some
+better suited for  engineering environments
+than others.
+
+
+% bbb 
+For instance, our initial 1993 paper \cite{ww93}
+employed a Group-3 {\it ``I am consistent''} axiom
+that was much weaker than 
+the current  specimen.
+The distinction was that
+\cite{ww93}'s self-consistency declaration 
+excluded 
+merely
+the existence of a semantic tableaux proof
+of $0=1$ from itself, while 
+the
+sentence \eq{group3} is
+more elaborate because
+it excludes the existence of simultaneous proofs
+of a $\Pi_1^*$ theorem and its negation.
+
+
+Ideally, one would  like to
+develop self-justifying
+systems $~S~$ that 
+% could 
+can
+corroborate the validity
+of \eq{brxefl}'s reflection principle for all sentences 
+$\Phi$.
+\beq
+\label{brxefl}
+\forall p ~~[~ Prf_S^D(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+L\"{o}b's Theorem 
+establishes,
+however,
+ that all
+ systems $S$,
+containing 
+Peano Arithmetic's
+strength, are able to prove
+\eq{brxefl}'s invariant 
+{\it only in the degenerate case} where they 
+do
+prove $\Phi$
+itself. Also, the Theorem 7.2 from \cite{ww1}
+showed 
+essentially all
+axiom systems,
+{\it weaker} than Peano Arithmetic, are unable to prove \eq{brxefl}
+for all $\Pi_1^*$ sentences $\Phi$
+simultaneously. Thus,
+\thx{ttt5}
+will be near optimal:
+
+%% xxxxx
+
+%%% bbbbb
+\begin{theorem}
+\label{ttt5}
+Let us recall that the difference between \thx{ttt1}'s
+axiom system 
+ IS$_D(A)$ 
+and \thx{ttt3}'s formalism
+\ik3 
+was that the latter replaced 
+ IS$_D(A)$'s infinite-sized Group-2 axiom schema
+with \ik3's compact 1-sentence axiom 
+\eq{globsim}, so that the latter system could at least verify 
+\eq{t5kern}'s kernelized statement
+for
+each $\Pi_1^*$ theorem that $A$ proved.
+\beq
+\label{t5kern}
+ \forall ~x~~
+\mbox{Test}_{\, i \,} (~\ulxyz~\Psi~\urxyz~,~x~)
+\enq
+Let likewise $IS^\lambda_\#( \, \beta_{A,i} \, )$
+denote the modification of \cite{ww1}'s  $IS^\lambda(A)$
+self-justifying
+system
+that replaces the latter's Group-2 schema with 
+\eq{globsim}'s more compact single-sentence axiom declaration
+(and 
+%  again
+%accordingly
+then
+has its Group-3 {\rm ``I am consistent''}
+axiom statement
+reflect this change, 
+once again).
+Then in a context where ``semtab'' is an abbreviation for
+semantic tableaux deduction, 
+the formalism $IS^\lambda_\#( \, \beta_{A,i} \, )$
+will be able to:
+\bee
+\item
+Verify that 
+semantic tableaux
+ deduction supports the
+following analog of 
+\eq{brxefl}'s 
+self-reflection principle
+under
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$
+%%%  $S$
+for any
+$\Delta_0^*$ and $\Sigma_1^*$ 
+sentences $\Phi~~$:
+\beq
+\label{nrxefl}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~ \Rightarrow  ~~ \Phi~~]
+\enq
+\item
+Verify 
+\eq{rdilute}'s more general
+{\bf ``root-diluted''} reflection principle
+for  $IS^\lambda_\#( \, \beta_{A,i} \, )$
+whenever
+$\theta$ is $\Sigma \, _{1}^*$ 
+and
+ $\Phi$ is a $\Pi_2^*$ sentence of the
+form ``$~\forall u_1  ... \forall u_n~~    
+  \theta(u_1... u_n  )~$''. 
+\beq
+\label{rdilute}
+\forall p ~[~ Prf_{IS^\lambda_\#( \, \beta_{A,i} \, )}^{\rm semtab}
+(\ulxyz \Phi \urxyz,p)
+  ~~  \Longrightarrow  ~   \forall x~
+ \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+  \theta(u_1... u_n  ) ~]
+\enq
+\ene
+\end{theorem}
+
+
+
+%% bbbb
+As is suggested by the similarity between the
+definitions of   $IS^\lambda(A)$ and
+ $IS^\lambda_\#( \, \beta_{A,i} \, )$,
+the proof of \thx{ttt5} is essentially
+identical to 
+\cite{ww1}'s
+analysis of  $IS^\lambda(A)$.
+For the sake of brevity, we will not repeat
+the relevant proof here.
+
+
+
+
+%%% 
+%%% \begin{theorem}
+%%% \label{tts5}
+%%% For any
+%%% input axiom system $A$,
+%%% it is possible to extend the self-justifying
+%%% IS$_D(\aaa)$ and \ik3
+%%% systems,
+%%% from  Theorems \ref{ttt1} and \ref{ttt3}, 
+%%% so
+%%% that the resulting 
+%%% self-justifying logics
+%%% $S$
+%%% can also:
+%%% \bee
+%%% \item
+%%% Verify that \txl{1} deduction supports the
+%%% following analog of 
+%%% \eq{brxefl}'s 
+%%% self-reflection principle
+%%% under $S$
+%%% for any
+%%% $\Delta_0^*$ and $\Sigma_1^*$ 
+%%% sentences $\Phi~~$:
+%%% \beq
+%%% \label{nrxefl}
+%%% \forall p ~~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~ \Rightarrow  ~~ \Phi~~]
+%%% \enq
+%%% \item
+%%% Verify 
+%%% \eq{rdilute}'s more general
+%%% {\bf ``root-diluted''} reflection principle
+%%% for $~S~$ 
+%%% whenever
+%%% $\theta$ is $\Sigma \, _{1}^*$ 
+%%% and
+%%%  $\Phi$ is a $\Pi_2^*$ sentence of the
+%%% form ``$~\forall u_1  ... \forall u_n~~    
+%%%   \theta(u_1... u_n  )~$''. 
+%%% \beq
+%%% \label{rdilute}
+%%% \forall p ~[~ Prf_S^{\rm Tab-1}
+%%% (\ulxyz \Phi \urxyz,p)
+%%%   ~~  \Longrightarrow  ~   \forall x~
+%%%  \forall u_1< \sqrt{x}~~     ... ~~ \forall u_n< \sqrt{x}~~      
+%%%   \theta(u_1... u_n  ) ~]
+%%% \enq
+%%% \ene
+%%% \end{theorem}
+%%% 
+
+
+%% \thx{ttt5}'s proof
+%% will
+%% rest 
+%% upon 
+%%  hybridizing
+%% the techniques from
+%% \cite{ww1}'s
+%% tangibility reflection principle 
+%%  with Theorem
+%%  \ref{ttt3}'s
+%% methodologies,
+%% in a 
+%% natural
+%% very
+%%  manner.
+%% %hhhh
+%% Its proof is summarized in Appendix D.
+
+
+
+% \baselineskip = 1.21 \normalbaselineskip 
+\parskip 4pt
+
+Analogous to our 
+other
+results,
+\thx{ttt5} 
+reinforces
+% the
+our
+ theme about how 
+exceptions 
+to
+the Second Incompleteness Theorem 
+may 
+appear to
+be 
+{\it quite
+minor} 
+from the perspective of
+an Utopian 
+view of mathematics,  
+while 
+being
+significant
+from an  engineering standpoint.
+In \thx{ttt5}'s
+particular case,
+this is 
+because:
+\bed
+\item[A. ]
+The ability of  \thx{ttt5}'s
+system
+%%% $S$ 
+to 
+support
+\eq{nrxefl}'s 
+self-reflection principle
+under
+tableaux
+%\txl{1} 
+proofs for
+any
+ $\Delta_0^*$ and $\Sigma_1^*$ sentence, 
+as well as
+to 
+support
+\eq{rdilute}'s
+root
+reflection principle 
+for  $\Pi_2^*$ sentences,
+is 
+clearly
+significant.
+\item[B. ] 
+The incompleteness result
+of  \cite{ww1}'s
+Theorem 7.2
+imposes,
+however, 
+sharp limitations upon Item A's
+generality
+(in that it  cannot be extended to
+fully all
+  $\Pi_1^*$ sentences, 
+{\it in an  undiluted sense).}
+\ennd
+%
+% \noindent
+Thus,
+the tight fit 
+between
+ A and B
+is
+reminiscent of
+other  
+slender 
+borderlines,
+that separated
+generalizations and 
+boundary-case exceptions
+for the 
+Incompleteness Theorem,
+explored
+earlier.
+Once again, 
+the Second Incompleteness
+Theorem 
+is
+seen
+ as robust,
+from an 
+idealized
+Utopian perspective on mathematics,
+while 
+permitting
+caveats
+from 
+engineering 
+styled 
+perspectives.
+
+This
+ dualistic 
+viewpoint
+allows one to
+nicely
+share 
+{\it   partial (and not full)} 
+agreement with 
+Hilbert's
+main aspirations in $**$, 
+$\,$while also 
+ appreciating
+the 
+ stunning
+achievement
+of
+the Second Incompleteness Theorem.
+
+
+
+
+
+
+
+
+\section{Concluding Remarks}
+
+\label{ppppp10}
+
+
+At a purely technical level,
+this  article has reached beyond 
+our prior papers in 
+several
+respects,
+including  
+\textsection \ref{pppp5}'s demonstration
+that any 
+initial 
+system $A$
+can have a kernelized image of its 
+ $\Pi_1^*$ knowledge duplicated by
+\ik3's {\bf strictly finite sized}
+self-justifying  
+system, 
+as well as
+%and also by
+ Section
+\ref{pppp6}'s
+and 
+Remark \ref{rem2}'s
+quite
+ pragmatic 
+ L-fold generalizations
+of 
+\thx{ttt3}.
+
+% this result. 
+
+
+
+ 
+These 
+perspectives
+%results 
+help resolve the mystery
+that has 
+enshrouded
+the Second Incompleteness Theorem and the statements
+$*$ and $**$ 
+of G\"{o}del and Hilbert.
+This is because 
+we have 
+{\it meticulously separated}
+the goals of a 
+pristine theoretical study of mathematical
+logic 
+from
+those of 
+a 
+ {\it 
+finite-sized}
+axiomatic
+subset of mathematics,
+intended
+ for modeling
+mostly  
+an engineering environment.
+
+
+
+
+
+
+
+
+
+There is no question that 
+G\"{o}del's Second 
+Theorem
+is  ideally robust,
+relative to a  
+purely pristine 
+approach to mathematics.
+On the other hand, we suspect
+Hilbert
+was
+{\it half-way
+correct} by
+ speculating 
+in 
+ $**$ 
+about humans
+possessing
+a knowledge 
+about
+ their own consistency,
+{\it in  at least some 
+% strikingly
+ weak
+and
+ tender sense,} as 
+essentially a
+% fundamental 
+prerequisite
+for
+{\it psychologically
+ motivating}
+their cogitations.
+%%%%  hhhhhh
+Thus in a context where the limitations of axiom systems, 
+that fail to recognize multiplication as a total function, 
+are manifestly
+obvious,
+%% 
+%% 
+%% 
+%% even when 
+%% such systems
+%% duplicate
+%% Peano Arithmetic's
+%% central
+%% $\Pi_1^*$ knowledge, 
+%% 
+it is legitimate to
+inquire
+ whether some 
+future 
+specialized
+21st century computers
+ might 
+find
+some 
+{\it partial-albeit-and-not-full} redeeming
+value
+in formalisms 
+having
+{\it weak-style}
+ knowledges
+of
+their 
+ \txl{1} consistency,
+as well as possessing a knowledge of 
+Peano Arithmetic's 
+$\Pi_1^*$ theorems.
+
+
+%%%% hhhh
+%%More precisely, 
+Sections 
+\ref{pppp5}-\ref{pppxppp10}
+were,
+thus,
+ intended 
+to provide
+a  
+unified 
+broad-scale
+interpretation of our
+diverse
+ earlier 
+results
+that had appeared
+%appearing
+in \cite{ww93}-\cite{ww9}.
+%from
+%\cite{ww93,sp0,ww1,ww2,wwlogos,ww5,wwapal,ww6,ww7,ww9}.
+In a
+context where
+the
+Incompleteness
+Theorem is 
+%% 
+%% firmly 
+%% understood 
+%% to be
+%%
+ sufficiently 
+ubiquitous
+ to preclude Hilbert's
+aspirations in $**$ 
+from 
+ever 
+being fully realized, 
+they show
+how
+some
+{\it fragmentary portion} of Hilbert's
+conjectures
+can
+be corroborated by
+{\it judiciously weakened} logics, 
+using a formalism, that is
+{\it much less} than ideally robust,
+{\it although
+not fully immaterial}.
+
+%\medskip
+
+\bigskip
+
+Such partial evasions of the Second Incompleteness Effect
+are certainly not broad-scale, but they
+do corroborate a fragment of what G\"{o}del and Hilbert
+%referred to
+had
+sought
+as 
+% ideal 
+their
+desired
+goals,
+expressed
+ in the statements $*$ and $**$.
+
+\newpage
+
+%\bigskip
+
+  {\bf Acknowledgments:} $~$I thank 
+  Bradley Armour-Garb  and Seth Chaiken  for 
+many
+ useful suggestions about how to
+improve the presentation of our results.
+%% I also thank the anonymous referees for their comments.
+This research was
+partially supported
+by  NSF Grant CCR  0956495.
+
+
+\small
+ \parskip 2 pt
+\baselineskip = 0.86 \normalbaselineskip 
+
+
+
+\bibliographystyle{abbrv}
+\bibliography{b15}
+
+
+
+
+% eeee end end
+% \newpage
+
+
+
+
+
+%\large
+% \baselineskip = 1.5 \normalbaselineskip 
+
+% \baselineskip = 1.2 \normalbaselineskip 
+
+ \parskip 4 pt
+
+\ssspace
+
+\section*{Appendix A: Definition of a
+Semantic Tableaux Proof }
+
+The 
+definition of a semantic tableaux proof,
+provided here,
+will be  similar to analogous definitions used in 
+say Fitting's or  Smullyan's textbooks
+ \cite{Fi90,Smul}.  
+
+%% For simplicity
+%% during our discourse, 
+%% a sentence $~\Psi~$
+%% will be called PRENEX$^*$ iff it is written in the
+%% form $Q_1 \, x_1~Q_2\, x_2...~Q_n \, x_n~~\theta(x_1,x_2...x_n)~$
+%% where $~\theta(x_1,x_2...x_n)~$ is a $\Sigma_0^-$ formula
+%% and $Q_i$ denotes either the symbol $\forall$ or $\exists$.
+
+During our 
+discussion, a
+% discourse, a
+{\bf $\Phi$-Based Candidate Tree} for
+an axiom system $\, \alpha \,$
+will be defined
+to be a  tree structure 
+whose root corresponds to
+the sentence $~\neg \, \Phi~,~$  rewritten in
+prenex normal form, and whose all other nodes are
+either axioms of $~\alpha~$ or deductions from higher
+nodes of the tree
+(using the Rules 1-6 defined below).
+More precisely, our six  rules
+(below)
+ have
+``$~ \cal{A} ~ \longmapsto ~ \cal{B} ~$''  denote
+that $~  \cal{B} ~$ 
+is a valid deduction
+from $~ \cal{A} ~$.
+They 
+% thus 
+specify when such a 
+descendant
+node $~  \cal{B} ~$  is allowed to
+appear below an ancestor $~  \cal{A} $ 
+%% 
+%%  is an ancestor of $~  \cal{B} ~$
+%% in the candidate tree $~T~$.  In this notation, the deduction
+%% rules allowed 
+%% 
+in a candidate tree:
+\begin{enumerate}
+ \parskip 1 pt
+\item $~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Upsilon
+~$ 
+and 
+$~ \Upsilon \wedge \Gamma \, ~ \longmapsto ~ \, \Gamma ~$ .
+\item $~ \neg  \,\neg \, \Upsilon ~ \longmapsto ~ \Upsilon~$.  
+Other 
+% valid Tableaux 
+rules for
+the ``$~ \neg ~$'' symbol  include: $~$
+$~\neg ( \Upsilon \vee \Gamma ) ~ \longmapsto ~ \neg \Upsilon
+\wedge \neg \Gamma~$,
+$ \, \neg ( \Upsilon \Rightarrow \Gamma )  \,  \longmapsto  \,   \Upsilon
+\wedge \neg \Gamma \, $,
+$ ~~~~\, \neg ( \Upsilon \wedge \Gamma )  \,  \longmapsto  \,  \neg
+\Upsilon \vee \neg \Gamma \, $,
+ $~ \,   \neg \, \exists v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\forall v   \neg \, \Upsilon  (v)  \, $ and
+ $ ~\,   \neg \, \forall v \, \Upsilon  (v)  \,  \longmapsto  \,  
+\exists v \,  \neg  \Upsilon  (v)$
+\item A pair of sibling nodes $~ \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+tree when their ancestor is
+$~\Upsilon \, \vee \, \Gamma~$.
+\item A pair of sibling nodes $~ \neg \Upsilon ~$ and $~ \Gamma ~$ is
+allowed in
+a 
+%candidate 
+proof
+ tree when their ancestor is
+$~\Upsilon \, \Rightarrow \, \Gamma~$.
+\item $~ \exists v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(u) ~$
+where $~u~$ denotes a newly introduced ``Parameter Symbol''. 
+\item $~ \forall v \, \Upsilon  (v) ~ \longmapsto ~ \, \Upsilon(t) ~$
+where $~t~$ denotes a ``Composite Term''.
+These terms  here are
+built out of
+combination of
+ the U-Grounding Function symbols,
+the constant symbols representing ``0'' and ``1''
+and  the parameter symbols $~u_1,u_2,..,u_n~$,
+where each 
+%symbol 
+$~u_i~$ {\bf was previously}
+introduced by 
+% instance of 
+applying 
+Rule 5 
+%applying 
+to
+an ancestor
+of the node storing 
+% the current new deduction
+ ``$ ~ \, \Upsilon(t) ~$''.
+\end{enumerate}
+Define a particular leaf-to-root branch in a candidate
+tree $~T~$ to be {\bf Closed} iff it contains both some sentence
+$~ \Upsilon ~$ and its negation $~ \neg \, \Upsilon ~$.  
+ A {\bf  Semantic
+Tableaux} proof of $~\Phi~$  will then be defined to be
+a candidate tree whose root stores the sentence
+$~ \neg \Phi~$ (written in prenex normal
+form) and all of whose  root-to-leaf branches are
+closed. 
+
+% All our theorems in the current article have,
+
+Our
+% discussion in the 
+current article has,
+% will, 
+for simplicity,
+used the preceding definition for a semantic tableaux proof.
+Some of our prior articles 
+%have 
+used a minor modification
+of this definition where there were two additional deduction
+rules for ``bounded quantifiers'' of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''.
+It is technically unnecessary to use special rules for
+such bounded quantifiers because these two expressions
+can be treated as being equivalent to 
+\eq{bex} and \eq{beu}, respectively.
+\beq
+\label{bex}
+\exists \, v ~~~~ v \leq t~\wedge~ \Upsilon  (v)
+\enq
+\beq
+\label{beu}
+\forall \, v ~~~~ v \leq t~\Rightarrow~ \Upsilon  (v)
+\enq
+Thus, we technically do not need special Elimination Rules
+for bounded quantifiers of the form
+``$~ \exists \, v \, \leq t~~ \Upsilon  (v)~$''
+and ``$~ \forall \, v \, \leq t~~ \Upsilon  (v)$''
+because statement
+\eq{bex} allows the 
+ former to be eliminated 
+by applying Rules 5 and 1, and  likewise
+\eq{beu} can 
+be processed via Rules 6 and 4.
+
+
+%% For simplicity, we will thus rely upon the above 6-part definition
+%% of semantic tableaux during the current article.
+%% 
+%% ???? Remove above sentence  ??? bbbbbbbbbbbbbbbbb
+
+\section*{Appendix B:  Summary of G\"{o}del Encoding Method}
+
+Every 
+%% formalization of either a 
+generalization and
+% a  
+boundary-case
+exception for
+ the Second Incompleteness
+Theorem 
+does
+require
+ deploying a 
+ G\"{o}del encoding methodology
+(to make it well defined).
+Such an encoding scheme will be 
+called
+{\bf Optimally Linearly Compressed} if  it requires: 
+\bed
+\item[   A.  ]
+Only
+$O(1)$ bits to store 
+each occurrence
+of  any
+logical symbol
+% any of  the logical symbols
+appearing in a tableaux proof 
+(except for the objects that 
+Items 5 and 6 of Appendix A called the $i-$th 
+``variable'' and ``parameter'' symbols). 
+\item[   B.  ]
+No more than
+$O(~1~+~$Log$(i) ~)$ bits to
+encode
+ a proof's
+$i-$th 
+``variable'' and ``parameter'' symbols. 
+(This  $O(~1~+~$Log$(i) ~)$ magnitude is unavoidable
+because  
+there is no finite limit to the number of different
+variable and parameter objects that may appear in
+one of Appendix A's
+semantic tableaux proofs.)
+\ennd
+All our published results about either 
+generalizations or 
+boundary-case
+exception 
+for the Second Incompleteness Theorem have used such optimally
+compressed encodings.  
+
+
+In particular,
+our  scheme for
+encoding
+a semantic tableaux proof
+ will use  
+the following
+24 language  symbols:
+\begin{enumerate}
+\small
+ \baselineskip = 1.1 \normalbaselineskip 
+\item The standard connective symbols of
+$\wedge ,~ \vee ,~ \neg ,~ \rightarrow ,~ \forall$
+and $~ \exists$.
+\item Two 
+left and two right parenthesis symbols
+denoted as: $~(~$ , $~)~$
+$~\underline{\, ( \,}~$ and $~\underline{\, ) \,}.~$
+\item 
+Two symbols to represent the special constants of ``0'' and ``1''.
+\item
+Eight function symbols for representing for representing
+the eight formal U-grounding functions of Addition, Doubling, Subtraction,
+Division, Logarithm, etc.
+\item 
+The relation symbols of
+``$~=~$'' and ``$~ \leq ~$''.
+\item The symbol $~ \hat{V} ~$  for designating
+the presence of a  basic variable $~v~$
+in a logical sentence.
+\item The symbol $~ \hat{U} ~$  for designating
+the presence of a parameter constant $~u~$
+in a logical sentence (which is produced by 
+Appendix A's
+deduction rule 5  for
+eliminating
+existential quantifiers).
+\end{enumerate}
+Define a byte to be an unit consisting of six bits. 
+We 
+may
+%will
+ think of a proof as
+comprising
+ either 
+ a sequence of
+bytes or  being an 
+equivalent
+integer
+written in base 64.
+Each of the 24 symbols (above) will be given
+some  unique 6-bit  code, ranging between  32 and
+55.
+Our method for representing the presence of
+the i-th variable $~v_i$
+will be to encode it is as
+a string 
+comprised
+of 
+$\, \lceil \, log_{\, 32 \,}(i+1) \, \rceil ~+~1~$ bytes, where the
+first byte is the ``$\, \hat{V} \,$'' symbol and the remaining bytes
+encode
+i as a base-32 number.
+% with the convention that the lead bit in each 
+%byte's 6-bit sequence  is ``0''.
+The same convention will be used to denote the presence of
+the i-th parameter  $~u_i~$
+except its first byte will be the ``$\, \hat{U} \,$'' symbol.
+
+
+
+Our notation has employed {\it two types} of
+parenthesis symbols because the first pair of
+parenthesis symbols will have their usual meaning in punctuating a 
+mathematical
+sentence, whereas the latter pair of symbols
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$
+will {\it separate} the individual sentences in
+a Semantic Tableaux proof tree.  For example,
+consider a tree which stores
+1) the sentence $~\psi_1~$ as its root, 2)
+the sentences $~\psi_2~$ and $~\psi_3~$ as the root's children, and 3)
+$~\psi_4~$ as the child of $~\psi_3.~$ There are several
+possible notation conventions for using the 
+ $~\underline{\, ( \,}~$ and  $~\underline{\, ) \,}~$ symbols
+to encode a Semantic Proof tree. 
+Our encoding
+convention will 
+presume
+%be that
+$~\psi_i~$
+is an ``ancestor'' of $~\psi_j~$ {\it if and only if} the range beginning
+with the
+parenthesis to $\psi_i$'s immediate left and continuing
+to the matching right parenthesis includes
+$~\psi_j.~$
+The example of our 4-node proof tree is thus 
+encoded as:  
+\begin{equation} 
+\label{paren}
+ ~~\underline{\, ( \,}~~ \psi_1
+ ~~\underline{\, ( \,}~~ \psi_2~
+ ~\underline{\, ) \,}~ 
+ ~~\underline{\, ( \,}~~ \psi_3
+ ~~\underline{\, ( \,}~~ \psi_4~
+ ~\underline{\, ) \,}~~ \underline{\, ) \,}~~ \underline{\, ) \,}~ 
+\end{equation}
+
+
+The preceding paragraph summarized our method for
+encoding semantic tableaux proofs. Its
+generalization 
+for 
+the
+encoding of \txl{1} proofs is
+straightforward. Thus if 
+ $~p_1,p_2,...p_n~$ 
+collectively constitute
+a list of  semantic tableaux proofs
+then the
+ natural  concatenation 
+of their byte strings will be the corresponding
+ \txl{1}
+proof.
+
+This ``Optimally Linearly Compressed'' encoding scheme
+is 
+%noteworthy 
+essential
+because all the core axiom systems, employed
+in this article, are Type-A formalisms, that recognize Addition
+but not Multiplication as a total function. If such formalisms
+were less than optimally compressed then our main theorems
+would lose relevance because the formalization 
+of
+unnecessarily expansive encodings would be awkward
+in the context of the slow growth properties of
+Type-A formalisms. Thus, 
+our results carry much greater significance when their
+% it is useful that our 
+encodings
+of a proof satisfy the maximal compression properties,
+% outlined in the first paragraph of 
+%that are 
+defined in
+this appendix. 
+
+
+%% 
+%% This byte-styled encoding method is approximately analogous
+%% to what Wilkie-Paris \cite{WP87} have called
+%% a {\it natural B-adic} encoding or a similar
+%% counterpart in the H\'{a}jek-Pudl\'{a}k textbook
+%% \cite{HP91}. Such
+%% compressed encodings are
+%%  considered  to be  more
+%% meaningful and efficient than an uncompressed encoding method,
+%% using say a Prime Number decomposition scheme \cite{Me97}
+%% (because the latter has an unnecessarily long bit-length). 
+%% All our theorems                 would also be
+%% valid for uncompressed
+%% encoding methods.
+%% However, they are more meaningful when one uses an
+%% efficiently compressed
+%% B-adic encoding method.
+%% 
+%% %\newpage
+%% 
+
+
+
+%% 
+%% \large
+%% \normalsize
+%%  \baselineskip = 2.0 \normalbaselineskip 
+
+
+
+\section*{Appendix C: Formal Encoding of 
+%Statmenent \eq{group3}'s 
+the
+Group-3 Axiom}
+
+Let us recall 
+%that 
+Appendix A 
+reviewed the definition of
+a
+semantic tableaux
+and \txl{1}
+ proof,
+ and Appendix B  formalized the 
+encodings
+of  such proofs. The goal of this appendix
+will be to summarize the methodology
+%% \cite{ww5} 
+%%  that was 
+used  to define
+Statmenent \eq{group3}'s Group-3 
+axiom
+in \cite{ww5} .
+
+%%% Passive Voice change in above sentence much
+%%% better because it understates my use of \cite{ww5} . 
+
+
+%% {\bf More Detailed Description of the Group-3 Axiom:} $~$ 
+%% A formal description of
+%%  IS$_D(A)$'s
+%% Group-3 axiom is more complicated than the abbreviated
+%% descriptions   given either by
+%% Sentence$~*~$ or by \ep{group3}'s analog.
+%% The
+%% main added complication is because
+%% the Group-3 axiom declares the consistency of
+%% a formal set of axioms that includes ``itself'' 
+%% (in the words of Sentence$~*~).~$ 
+%% As was noted in Section 1, the notion of an 
+%% axiom including
+%% ``itself'' when it refers to the consistency
+%% of an axiom schema dates back to Kleene's 1938 paper \cite{Kl38}.
+%% However, Kleene's abbreviated
+%% description is insufficient to establish that
+%% \ep{group3} can be encoded precisely as
+%% a 
+%% $\Pi_1^*$ sentence. The next two paragraphs will
+%% explain how this can be done.
+
+Let
+ UNION($A$)  denote the union of IS$_D(A)$'s Group-Zero,
+Group-1 and Group-2 axioms.    
+It will be useful to employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system  UNION($A)$ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+UNION(A) with the added axiom
+sentence specified by the integer
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+% the formally 
+an
+encoded term 
+% of 
+$~\underx{g}~$
+(that designates $g$'s G\"{o}del number.)
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{~UNION(A)}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+the conditions
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)$.
+
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^*$ formulae.
+Thus, Appendices C and D of  \cite{ww1}
+%% thus,
+ explained how 
+the first three of these predicates can receive
+ $\Delta_0^*$ encodings when one applies 
+the theory of LinH functions 
+\cite{HP91,Kr95,Wr78}.
+Hence, \eq{encode} illustrates
+one possible  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{~UNION(A)}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to
+the statement 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \eq{encode} is 
+ a $\Delta_0^*$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~UNION(A)}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~UNION(A)}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+Let us recall that
+$\mbox{Pair}(x,y)$ is a $\Delta_0^*$ sentence
+specifying  that
+ $~x~$ 
+and $~y~$
+are
+the encodings of 
+ a $\Pi_1^*$
+and $\Sigma_1^*$ sentence,
+that are logical negations of each other.
+Using
+ \eq{encode}'s
+  $\Delta_0^*$ encoding for 
+$\mbox{SubstPrf}_{UNION(A)}^D(   g   ,   t   ,   p  )$, 
+we can now explain 
+how
+statement 
+\eq{group3}'s Group-3 Axiom can
+be formally encoded.
+Let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, 
+% and let
+  $~n~$ denote $~\Gamma(g)$'s
+G\"{o}del number
+and  $\underx{n}$
+denote a term encoding $n$ in the U-Grounding language.
+$~\,$Then
+it will turn out that  $~$``$~\Gamma(~ \underx{n}~)~$''$~$ 
+will be a $\Pi_1^*$ sentence 
+that is equivalent to
+ this Group-3 axiom.
+\begin{equation}
+\label{encode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (    g    ,    y    ,    q   )  \,]
+\end{equation}
+More precisely, \eq{newencode2} formalizes the encoding
+of 
+ $~$``$~\Gamma(~ \underx{n}~)~$''.
+\begin{equation}
+\label{newencode2}
+\small
+\forall   \, x \, \forall   \, y \, \forall   \, p \, \forall   \, q \,  \,  \neg  \,  \, 
+[ \,\mbox{Pair}(x,y)  \wedge 
+\mbox{SubstPrf}_{UNION(A)}^D  (   \underx{n}   ,    x    ,    p   )
+\wedge  
+\mbox{SubstPrf}_{UNION(A)}^D  (      \underx{n}    ,    y    ,    q   )  \,]
+\end{equation}
+%In particular, 
+Thus,
+if we view 
+$~~$``$~\mbox{SubstPrf}_{~UNION(A)}^D~( \,
+ \underx{n} \, , \, t \, , \, p \,)~$''
+in \eq{newencode2} 
+as our formal method of
+encoding the concept that was previously informally
+called 
+``$~\mbox{Prf}~_{\mbox{IS}_D(A)}(t,p)~$''
+by Statement \eq{group3},
+then \eq{newencode2} amounts to
+the formal encoding of 
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration.
+
+\bigskip
+
+{\bf Reminder about
+the Significance of 
+ \eq{newencode2}'s Encoding :}
+The preceding construction 
+%shows 
+had showed
+merely that it is possible
+to encode
+Sentence
+\eq{group3}'s Group-3 
+{\it ``I am consistent''} axiom declaration
+in a well-defined manner as a $\Pi_1^*$
+sentence.
+It does not answer the more subtle question about whether or not
+its
+{\it ``I am consistent''} axiom declaration
+holds 
+true
+ under the Standard model.
+%of the natural numbers. 
+As we have noted before,
+most analogs of 
+%the above sentence
+\eq{newencode2}
+produce false statements
+%fail to hold True
+under the Standard Model 
+because a conventional G\"{o}del-like
+diagonalization argument will imply
+that 
+most deduction methods $D$ will produce
+%their resulting 
+axiom systems
+$\mbox{IS}_D(A)$
+that are
+ inconsistent. 
+
+\medskip
+
+The reason for our 
+particular
+interest in 
+\eq{newencode2}'s
+formal encoding is that 
+Theorems \ref{ttt1} and \ref{ttt2}
+indicate that $\mbox{IS}_D(A)$
+is 
+%indeed 
+consistent when $D$ denotes
+either the semantic tableaux or \txl{1}
+deduction methodologies. Thus
+\eq{newencode2}'s
+Fixed-Point construction should be seen as a
+methodology that has 
+%limited-but-subtle
+limited applications,
+but which is also
+quite helpful (when it is feasible).
+
+%quite significant.
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/2pm-jan11 b/nachlass/collected_dew_materials/Wheat-and-Chessboard/2pm-jan11
new file mode 100644
index 0000000..ce04c48
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/2pm-jan11
@@ -0,0 +1,50 @@
+
+Wheat and Chessboard Problem
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+or perhaps better, it can be called a
+   ``Compacting (J,K) Reduction Theorem''.
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex b/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex
new file mode 100644
index 0000000..ce04c48
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex
@@ -0,0 +1,50 @@
+
+Wheat and Chessboard Problem
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+or perhaps better, it can be called a
+   ``Compacting (J,K) Reduction Theorem''.
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex~ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex~
new file mode 100644
index 0000000..1caf914
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/n.tex~
@@ -0,0 +1,49 @@
+
+Wheat and Chessboard Problem
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/noon-30-bak b/nachlass/collected_dew_materials/Wheat-and-Chessboard/noon-30-bak
new file mode 100644
index 0000000..cd37386
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/noon-30-bak
@@ -0,0 +1,16 @@
+
+Wheat and Chessboard Probelm
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/v1 b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v1
new file mode 100644
index 0000000..7641bd8
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v1
@@ -0,0 +1,13 @@
+
+What and Chessboard Probelm
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 o third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/v4-bak b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v4-bak
new file mode 100644
index 0000000..0b7a18f
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v4-bak
@@ -0,0 +1,39 @@
+
+Wheat and Chessboard Probelm
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/v5-bak b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v5-bak
new file mode 100644
index 0000000..292f365
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v5-bak
@@ -0,0 +1,49 @@
+
+Wheat and Chessboard Probelm
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/v6 b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v6
new file mode 100644
index 0000000..1caf914
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v6
@@ -0,0 +1,49 @@
+
+Wheat and Chessboard Problem
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+
diff --git a/nachlass/collected_dew_materials/Wheat-and-Chessboard/v7 b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v7
new file mode 100644
index 0000000..ce04c48
--- /dev/null
+++ b/nachlass/collected_dew_materials/Wheat-and-Chessboard/v7
@@ -0,0 +1,50 @@
+
+Wheat and Chessboard Problem
+
+started 400-600 Ad not known when
+
+Sissa Ibn Dahir supposedly invented chess board.
+
+Asked by King how he should be rewarded for such an invention
+Reply was to put one grain of wheat on first square, 2 grains
+on second square 4 on third square etc. King laughed at his small request and
+replied he would be glad to do this as a reward to the inventor
+of the chess board.  But King  could not satisfy this request.
+
+This exemplifies difference between an arithmetic and exponial
+growth.
+
+I CHANGED MY MIND and term COMPACT REDUCER is better to use
+below than COMPACT INVERTER.
+
+In particular, my next paper will use the term  COMPACT REDUCER
+  `` COMPACT REDUCER for a base axiom system of \gamma 
+to describe and integer X that has the property that 
+ \gamma can prove:
+
+\forall Y  \exists Z such that X*Y= Z
+
+The basic point if that it is easy to construct
+systems \gamma that  can prove the identity that
+
+\forall X if X is a compact reducer (for \gamma) then
+so is X*X a compact reducer (for \gamma).
+
+I developped this theorem while teaching Robert. The point
+is that it leads to a simpler proof of a trivially
+modified version circle-dot (where Item I is changed to require
+that base axiom system A recognizes either addtion or
+doubling as total functions). Either of these two changes
+is good enough.  when formalism A can prove a fixed finite
+subset of PA's Pi_1* theorems.
+
+Possibly ``compact reducer'' to ``ironic reducer'' or
+''ironic compactifier'' or ``compact multiplier'' 
+but I like ``compact reducer''
+best.
+
+It also has advatage of introducing generalizations called
+Compact ``(J,K)'' Reducers.
+and similarily ``Compact (J,K) Reduction Theorems''.
+or perhaps better, it can be called a
+   ``Compacting (J,K) Reduction Theorem''.
diff --git a/nachlass/collected_dew_materials/asl.cls b/nachlass/collected_dew_materials/asl.cls
new file mode 100644
index 0000000..f1d56b6
--- /dev/null
+++ b/nachlass/collected_dew_materials/asl.cls
@@ -0,0 +1,6350 @@
+% asl.cls, Version 1.2, September 14, 2000 
+% Yiannis N. Moschovakis, ynm@math.ucla.edu
+% Copyright 2000 by the Association for Symbolic Logic
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% This file carries the usual TEX-file non-warranty for results
+% and it can be freely copied and re-distributed provided that: 
+%
+% IF IT IS CHANGED IN THE SLIGHTEST WAY, it must be RENAMED 
+%
+% With code cribbed from the 1996/11/05 Version 1.2q of amsart.cls
+% and ancillary AMSLATEX packages, in 1996 versions. 
+% An attempt to load one of these packages gives an error message.
+% For a list of the relevant packages and versions
+% search for \errmessage below or see asldoc.
+% 
+% The ASL-specific code has been adapted to latex2e by YNM from an
+% early version of AMSLATEX for the ASL prepared by the AMS
+% programming staff. This version includes the code from all
+% necessary stylefiles except for amsfonts, amssymb and latexsym
+% 
+% It is assumed that all LATEX2e files exist
+% in versions not earlier than 1994/12/01
+% including latexsym.sty, which is automatically loaded.
+%
+% The packages amsfonts.sty and amssymb.sty are loaded if they 
+% exist, otherwise a warning is given that ams fonts are not 
+% available and compilation does not stop until some amsfont 
+% symbol is required.  
+%
+%%%%%%%%%%%%%%%% Changes 
+% bibextra.sty minor (diagnostic) bug corrected, 4/6/00
+% Version 1.1, August 20, 2000
+% Introduced option bibother which works with 
+% non-asl bibliography styles and hand-set bibliographies
+%
+% Added an optional argument to \subjclass
+% which distinguishes between 1991 and 2000 classifications
+% to make it friendlier to new amslatex users
+%
+% 9/14/2000 Corrected numbering error in eqnarray environment
+%%%%%%%%%%%%%%%%%
+
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1994/12/01]% LaTeX date must December 1994 or later
+\ProvidesClass{asl}[2000/09/14 v1.2]
+\def\@tempa#1#2\@nil{\edef\@classname{#1}}
+\expandafter\@tempa\@currnamestack{}{}{}\@nil
+\ifx\@classname\@empty \edef\@classname{\@currname}\fi
+
+%%%%%%%%%%%%%%% Give error if called to load a known amslatex pkg
+\def\aslknown{0}
+\let\oldusepackage=\usepackage
+\renewcommand{\usepackage}[2][]{%
+\if\csname#2case\endcsname\aslknown\csname#2aslerr\endcsname
+\else \oldusepackage[#1]{#2}\fi}
+
+\def\amsmathcase{0}
+\def\amsmathaslerr{
+\errmessage{asl.cls already knows amsmath 
+version 1.2c [1996/11/01]^^J
+and cannot load amsmath.^^J
+Hit RETURN to continue!^^J
+}}
+
+\def\amsgencase{0}
+\def\amsgenaslerr{
+\errmessage{^^J asl.cls already knows amsgen 
+version 1.2b [1996/10/29]^^J
+and cannot load amsgen.^^J
+Hit RETURN to continue!^^J
+}}
+
+\def\amstextcase{0}
+\def\amstextaslerr{
+\errmessage{^^J asl.cls already knows amstext 
+version 1.2b [1996/10/28]^^J
+and cannot load amstext.^^J
+Hit RETURN to continue!^^J
+}}
+
+\def\amsbsycase{0}
+\def\amsbsyaslerr{
+\errmessage{^^J asl.cls already knows amsbsy 
+version 1.2b [1996/10/29]^^J
+and cannot load amsbsy.^^J
+Hit RETURN to continue!^^J
+}}
+
+\def\amsopncase{0}
+\def\amsopnaslerr{
+\errmessage{^^J asl.cls already knows amsopn 
+version 1.2b [1996/10/28]^^J
+and cannot load amsopn.^^J
+Hit RETURN to continue!^^J
+}}
+
+%%%%%%%%%%%%%%%%%%%%%%% Options from amsart
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+\DeclareOption{portrait}{}
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+\DeclareOption{e-only}{%
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+  \dateposted{Xxxx XX, XXXX}%
+}
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+\DeclareOption{12pt}{\def\@mainsize{12}\def\@ptsize{2}%
+  \def\@typesizes{%
+    \or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}\or{\@xipt}{13}%
+    \or{\@xiipt}{14}% normalsize
+    \or{\@xivpt}{17}\or{\@xviipt}{20}\or{\@xxpt}{24}%
+    \or{\@xxvpt}{30}\or{\@xxvpt}{30}}%
+  \normalsize \linespacing=\baselineskip
+}
+\DeclareOption{8pt}{\def\@mainsize{8}\def\@ptsize{8}%
+  \def\@typesizes{%
+    \or{5}{6}\or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}%
+    \or{8}{10}% normalsize
+    \or{9}{11}\or{10}{12}\or{\@xipt}{13}%
+    \or{\@xiipt}{14}\or{\@xivpt}{17}}%
+  \normalsize \linespacing=\baselineskip
+}
+\DeclareOption{9pt}{\def\@mainsize{9}\def\@ptsize{9}%
+  \def\@typesizes{%
+    \or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}%
+    \or{9}{11}% normalsize
+    \or{10}{12}\or{\@xipt}{13}\or{\@xiipt}{14}%
+    \or{\@xivpt}{17}\or{\@xviipt}{20}}%
+  \normalsize \linespacing=\baselineskip
+}
+\def\ps@empty{\let\@mkboth\@gobbletwo
+  \let\@oddhead\@empty \let\@evenhead\@empty
+  \let\@oddfoot\@empty \let\@evenfoot\@empty
+  \global\topskip\normaltopskip}
+\def\ps@plain{\ps@empty
+  \def\@oddfoot{\normalfont\scriptsize \hfil\thepage\hfil}%
+  \let\@evenfoot\@oddfoot}
+\let\sectionname\@empty
+\let\subsectionname\@empty
+\let\subsubsectionname\@empty
+\let\paragraphname\@empty
+\let\subparagraphname\@empty
+\def\leftmark{\expandafter\@firstoftwo\topmark{}{}}
+\def\rightmark{\expandafter\@secondoftwo\botmark{}{}}
+\long\def\@nilgobble#1\@nil{}
+\def\markboth#1#2{%
+  \begingroup
+    \@temptokena{{#1}{#2}}\xdef\@themark{\the\@temptokena}%
+    \mark{\the\@temptokena}%
+  \endgroup
+  \if@nobreak\ifvmode\nobreak\fi\fi}
+\def\ps@myheadings{\ps@headings \let\@mkboth\@gobbletwo}
+\newskip\normaltopskip
+\normaltopskip=10pt \relax
+\let\sectionmark\@gobble
+\let\subsectionmark\@gobble
+\let\subsubsectionmark\@gobble
+\let\paragraphmark\@gobble
+\DeclareOption{makeidx}{}
+%%%%%%%%%%%%%%%%%%%%%% End of amsart options
+
+%%%%%%%%%%%%%%%%%%%%%% amsgen.sty code
+\providecommand{\@saveprimitive}[2]{\begingroup\escapechar`\\\relax
+  \edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+    \edef\@tempb{\meaning#2}%
+    \ifx\@tempa\@tempb
+    \else
+      \@latex@error{Unable to properly define \string#2; primitive
+      \noexpand#1no longer primitive}\@eha
+    \fi
+  \fi
+  \endgroup}
+\let\@xp=\expandafter
+\let\@nx=\noexpand
+\newtoks\@emptytoks
+\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
+\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
+\long\def\@xifempty#1#2@#3#4#5\@nil{%
+  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
+\long\def\@ifnotempty#1{\@ifempty{#1}{}}
+\def\FN@{\futurelet\@let@token}
+\def\DN@{\def\next@}
+\def\RIfM@{\relax\ifmmode}
+\def\setboxz@h{\setbox\z@\hbox}
+\def\wdz@{\wd\z@}
+\def\boxz@{\box\z@}
+\def\relaxnext@{\let\@let@token\relax}
+\def\new@ifnextchar#1#2#3{%
+  \let\@tempe #1\def\@tempa{#2}\def\@tempb{#3}\futurelet
+    \@tempc\new@ifnch}
+\def\new@ifnch{\ifx\@tempc \@tempe \let\@tempd\@tempa
+             \else\let\@tempd\@tempb\fi\@tempd}
+\def\@ifstar#1#2{\new@ifnextchar *{\def\@tempa*{#1}\@tempa}{#2}}
+\@ifundefined{every@math@size}{%
+\let\every@math@size=\every@size
+\def\glb@settings{%
+     \expandafter\ifx\csname S@\f@size\endcsname\relax
+       \calculate@math@sizes
+     \fi
+     \csname S@\f@size\endcsname
+      \ifmath@fonts
+        \begingroup
+          \escapechar\m@ne
+          \csname mv@\math@version \endcsname
+          \globaldefs\@ne
+          \let \glb@currsize \f@size
+          \math@fonts
+        \endgroup
+        \the\every@math@size
+      \else
+      \fi
+}
+\def\set@fontsize#1#2#3{%
+    \@defaultunits\@tempdimb#2pt\relax\@nnil
+    \edef\f@size{\strip@pt\@tempdimb}%
+    \@defaultunits\@tempskipa#3pt\relax\@nnil
+    \edef\f@baselineskip{\the\@tempskipa}%
+    \edef\f@linespread{#1}%
+    \let\baselinestretch\f@linespread
+      \def\size@update{%
+        \baselineskip\f@baselineskip\relax
+        \baselineskip\f@linespread\baselineskip
+        \normalbaselineskip\baselineskip
+        \setbox\strutbox\hbox{%
+          \vrule\@height.7\baselineskip
+                \@depth.3\baselineskip
+                \@width\z@}%
+%%%     \the\every@size
+        \let\size@update\relax}%
+  }
+}{}% end \@ifundefined test
+\newdimen\ex@
+\addto@hook\every@math@size{\compute@ex@}
+\def\compute@ex@{%
+  \begingroup
+  \dimen@-\f@size\p@
+  \ifdim\dimen@<-20\p@
+    \global\ex@ 1.5\p@
+  \else
+    \advance\dimen@10\p@ \multiply\dimen@\tw@
+    \edef\@tempa{\ifdim\dimen@>\z@ -\fi}%
+    \dimen@ \ifdim\dimen@<\z@ -\fi \dimen@
+    \advance\dimen@-\@m sp % fudge factor
+    \vfuzz\p@
+    \def\do{\ifdim\dimen@>\z@
+      \vfuzz=.97\vfuzz
+      \advance\dimen@ -\p@
+      \@xp\do \fi}%
+    \do
+    \dimen@\p@ \advance\dimen@-\vfuzz
+    \global\ex@\p@
+    \global\advance\ex@ \@tempa\dimen@
+  \fi
+  \endgroup
+}
+\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
+\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
+  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
+\def\@mathmeasure#1#2#3{\setbox#1\hbox{\frozen@everymath\@emptytoks
+  \m@th$#2#3$}}
+\def\nomath@env{\@amsmath@err{%
+  \string\begin{\@currenvir} allowed only in paragraph mode%
+}\@ehb% "You've lost some text"
+}
+\def\Invalid@@{Invalid use of \string}
+%%%%%%%%%%%%%%%%%%% End amsgen.sty code
+
+\ExecuteOptions{centertags,letterpaper,portrait,%
+  10pt,twoside,onecolumn,final}
+
+% Do not allow compatibility mode --- enabling code deleted
+
+%%%%%%%%%%%%%%%%%%%%% amsmath.sty code
+\DeclareOption{intlimits}{\let\ilimits@\displaylimits}
+\DeclareOption{nointlimits}{\let\ilimits@\nolimits}
+\DeclareOption{sumlimits}{\let\slimits@\displaylimits}
+\DeclareOption{nosumlimits}{\let\slimits@\nolimits}
+\newif\ifctagsplit@
+\newif\iftagsleft@
+\DeclareOption{leqno}{\tagsleft@true}
+\DeclareOption{reqno}{\tagsleft@false}
+% add for eqnarray in asl
+\AtBeginDocument{\iftagsleft@%
+\renewcommand\@eqnnum{\hb@xt@.01\p@{}%   
+                      \rlap{\normalfont\normalcolor
+                        \hskip -\displaywidth(%
+%\theequation    ynm for tagging
+\ifasl@tag\tagsymbol\global\asl@tagfalse\else\theequation\fi%
+)}}\fi} 
+
+\DeclareOption{centertags}{\ctagsplit@true}
+\DeclareOption{tbtags}{\ctagsplit@false}
+\DeclareOption{cmex10}{%
+  \ifnum\cmex@opt=\@ne \def\cmex@opt{0}%
+  \else \def\cmex@opt{10}\fi
+}
+\@ifundefined{cmex@opt}{\def\cmex@opt{7}}{}
+%
+\newif\if@fleqn
+\newskip\@mathmargin
+\@mathmargin\@centering
+\DeclareOption{fleqn}{%
+    \@fleqntrue
+    \@mathmargin = -1sp
+    \AtBeginDocument{%
+        \ifdim\@mathmargin= -1sp
+            \@mathmargin\leftmargini
+        \fi
+    }%
+}
+\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
+\ifnum\cmex@opt=7 \relax
+  \catcode`\ =9
+  \DeclareFontShape{OMX}{cmex}{m}{n}{%
+    <-8> cmex7%
+    <8> cmex8%
+    <9> cmex9%
+    <10> <10.95> <12> <14.4> <17.28> <20.74> <24.88>cmex10%
+  }{}%
+  \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+  \catcode`\ =10
+\else
+  \ifnum\cmex@opt=\z@ % need to override cmex7 fontdef from amsfonts
+    \input{OMXcmex.fd}%
+    \expandafter\let\csname OMX/cmex/m/n/10\endcsname\relax
+    \def\cmex@opt{10}%
+  \fi
+\fi
+
+%%%%%%%%%%%%%%%%%%%%%% code from amstext.sty
+\DeclareRobustCommand{\text}{%
+  \ifmmode\expandafter\text@\else\expandafter\mbox\fi}
+\let\nfss@text\text
+\def\text@#1{\mathchoice
+  {\textdef@\displaystyle\f@size{#1}}%
+  {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
+  {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
+  \check@mathfonts
+}
+\def\textdef@#1#2#3{\hbox{{%
+                    \everymath{#1}%
+                    \let\f@size#2\selectfont
+                    #3}}}
+\newif\iffirstchoice@
+\firstchoice@true
+\def\stepcounter#1{%
+  \iffirstchoice@
+     \addtocounter{#1}\@ne
+     \begingroup \let\@elt\@stpelt \csname cl@#1\endcsname \endgroup
+  \fi
+}
+\def\addtocounter#1#2{%
+  \iffirstchoice@
+  \@ifundefined {c@#1}{\@nocounterr {#1}}%
+    {\global \advance \csname c@#1\endcsname #2\relax}%
+  \fi}
+\newif\ifmeasuring@
+\let\m@gobble\@empty
+\@xp\let\csname m@gobble4\endcsname\@gobblefour
+\long\@xp\def\csname m@gobble6\endcsname#1#2#3#4#5#6{}
+\toks@{%
+  \csname m@gobble\iffirstchoice@\ifmeasuring@ 4\fi\else 4\fi\endcsname
+  \protect}
+\edef\GenericInfo{\the\toks@
+  \@xp\@nx\csname GenericInfo \endcsname}
+\edef\GenericWarning{\the\toks@
+  \@xp\@nx\csname GenericWarning \endcsname}
+\toks@{%
+  \csname m@gobble\iffirstchoice@\ifmeasuring@ 6\fi\else 6\fi\endcsname
+  \protect}
+\edef\GenericError{\the\toks@
+  \@xp\@nx\csname GenericError \endcsname}
+\begingroup \catcode`\"=12
+\gdef\mathhexbox#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
+\endgroup
+%%%%%%%%%%%%%%%%%%%%%%% end of amstext.sty code
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%% code from amsbsy.sty
+\DeclareRobustCommand{\boldsymbol}[1]{%
+  \begingroup
+  \let\@nomath\@gobble \mathversion{bold}%
+  \math@atom{#1}{%
+  \mathchoice%
+    {\hbox{$\m@th\displaystyle#1$}}%
+    {\hbox{$\m@th\textstyle#1$}}%
+    {\hbox{$\m@th\scriptstyle#1$}}%
+    {\hbox{$\m@th\scriptscriptstyle#1$}}}%
+  \endgroup}
+\def\math@atom#1#2{%
+   \binrel@{#1}\binrel@@{#2}}
+\DeclareRobustCommand{\pmb}{%
+  \ifmmode\else \expandafter\pmb@@\fi\mathpalette\pmb@}
+\def\pmb@@#1#2#3{\leavevmode\setboxz@h{#3}%
+   \dimen@-\wdz@
+   \kern-.5\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
+   \kern\dimen@\kern.25\ex@\box\z@
+}
+\newdimen\pmbraise@
+\def\pmb@#1#2{\setbox8\hbox{$\m@th#1{#2}$}%
+  \setboxz@h{$\m@th#1\mkern.5mu$}\pmbraise@\wdz@
+  \binrel@{#2}%
+  \dimen@-\wd8 %
+  \binrel@@{%
+    \mkern-.8mu\copy8 %
+    \kern\dimen@\mkern.4mu\raise\pmbraise@\copy8 %
+    \kern\dimen@\mkern.4mu\box8 }%
+}
+\def\binrel@#1{\begingroup
+  \setboxz@h{\thinmuskip0mu
+    \medmuskip\m@ne mu\thickmuskip\@ne mu
+    \setbox\tw@\hbox{$#1\m@th$}\kern-\wd\tw@
+    ${}#1{}\m@th$}%
+  \edef\@tempa{\endgroup\let\noexpand\binrel@@
+    \ifdim\wdz@<\z@ \mathbin
+    \else\ifdim\wdz@>\z@ \mathrel
+    \else \relax\fi\fi}%
+  \@tempa
+}
+\let\binrel@@\relax
+%%%%%%%%%%%%%%%%%%%%%%%% end of amsbsy.sty code
+
+%%%%%%%%%%%%%%%%%%%%% code from amsopn.sty
+\@ifundefined{DN@}{%
+  \def\FN@{\futurelet\@let@token}%
+  \def\DN@{\def\next@}%
+}{}
+\def\nolimits@{%
+  \DN@{\nolimits\ifx\@let@token\limits\expandafter\@gobble\fi}%
+  \FN@\next@}
+\begingroup \catcode`\"=12
+\gdef\newmcodes@{\mathcode`\'39\mathcode`\*42\mathcode`\."613A%
+ \mathcode`\-45\mathcode`\/47\mathcode`\:"603A\relax}
+\endgroup
+\DeclareRobustCommand{\operatorname}{%
+  \@ifstar{\qopname\newmcodes@ m}%
+          {\qopname\newmcodes@ o}}%
+\DeclareRobustCommand{\qopname}[3]{%
+  \mathop{#1\kern\z@\operator@font#3}%
+  \csname n#2limits@\endcsname}
+\newcommand{\DeclareMathOperator}{%
+  \@ifstar{\@declmathop\@empty}{\@declmathop o}}
+\long\def\@declmathop#1#2#3{%
+  \@ifdefinable{#2}{%
+    \DeclareRobustCommand{#2}{\qopname\newmcodes@#1{#3}}}}
+\@onlypreamble\DeclareMathOperator
+\@onlypreamble\@declmathop
+\def\arccos{\qopname\relax o{arccos}}
+\def\arcsin{\qopname\relax o{arcsin}}
+\def\arctan{\qopname\relax o{arctan}}
+\def\arg{\qopname\relax o{arg}}
+\def\cos{\qopname\relax o{cos}}
+\def\cosh{\qopname\relax o{cosh}}
+\def\cot{\qopname\relax o{cot}}
+\def\coth{\qopname\relax o{coth}}
+\def\csc{\qopname\relax o{csc}}
+\def\deg{\qopname\relax o{deg}}
+\def\det{\qopname\relax\@empty{det}}
+\def\dim{\qopname\relax o{dim}}
+\def\exp{\qopname\relax o{exp}}
+\def\gcd{\qopname\relax\@empty{gcd}}
+\def\hom{\qopname\relax o{hom}}
+\def\inf{\qopname\relax\@empty{inf}}
+\def\injlim{\qopname\relax\@empty{inj\,lim}}
+\def\ker{\qopname\relax o{ker}}
+\def\lg{\qopname\relax o{lg}}
+\def\lim{\qopname\relax\@empty{lim}}
+\def\liminf{\qopname\relax\@empty{lim\,inf}}
+\def\limsup{\qopname\relax\@empty{lim\,sup}}
+\def\ln{\qopname\relax o{ln}}
+\def\log{\qopname\relax o{log}}
+\def\max{\qopname\relax\@empty{max}}
+\def\min{\qopname\relax\@empty{min}}
+\def\Pr{\qopname\relax\@empty{Pr}}
+\def\projlim{\qopname\relax\@empty{proj\,lim}}
+\def\sec{\qopname\relax o{sec}}
+\def\sin{\qopname\relax o{sin}}
+\def\sinh{\qopname\relax o{sinh}}
+\def\sup{\qopname\relax\@empty{sup}}
+\def\tan{\qopname\relax o{tan}}
+\def\tanh{\qopname\relax o{tanh}}
+\def\operator@font{\mathgroup\symoperators}
+\def\operatornamewithlimits{\operatorname*}
+\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
+ \hfil$#1\m@th\operator@font lim$\hfil\crcr
+ \noalign{\nointerlineskip\kern\ex@}#2#1\crcr
+ \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
+\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
+\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
+\def\varliminf{\mathpalette\varliminf@{}}
+\def\varliminf@#1{\mathop{\@@underline{\vrule\@depth.2\ex@\@width\z@
+   \hbox{$#1\m@th\operator@font lim$}}}}
+\def\varlimsup{\mathpalette\varlimsup@{}}
+\def\varlimsup@#1{\mathop{\@@overline
+  {\hbox{$#1\m@th\operator@font lim$}}}}
+\DeclareOption{namelimits}{\let\nmlimits@\displaylimits}
+\DeclareOption{nonamelimits}{\let\nmlimits@\nolimits}
+%%%%%%%%%%%%%%%%%%%%%%%%%% end of code from amsopn.sty
+
+\def\@amsmath@err{\PackageError{amsmath}}
+\def\AmS{{\protect\AmSfont
+  A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
+\def\AmSfont{%
+  \usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
+\def\pr@m@s{%
+  \ifx\@let@token'\DN@##1{\prim@s}\else\let\next@\egroup\fi\next@}
+\def\prim@s{\prime\futurelet\@let@token\pr@m@s}
+\let\@prime=\prime
+\renewcommand{\prime}{{\kern\z@\@prime}}
+\DeclareRobustCommand{\tmspace}[3]{%
+  \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+\renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+\let\thinspace\,
+\renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+\let\negthinspace\!
+\renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+\let\medspace\:
+\newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+\renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+\let\thickspace\;
+\newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+\newcommand{\mspace}[1]{\mskip#1\relax}
+\begingroup\catcode`\"=12
+\def\@tempa#1{\expandafter\@tempb\meaning#1%
+\relax\relax\relax\relax"0000\@nil#1}
+\def\@tempb#1"#2#3#4#5#6\@nil#7{%
+  \ifnum"#2=7 \count@"1#3#4#5\relax
+    \ifnum\count@<"1000 \else \global\mathchardef#7="0#3#4#5\relax \fi
+  \fi}
+\@tempa\Gamma \@tempa\Delta \@tempa\Theta \@tempa\Lambda \@tempa\Xi
+\@tempa\Pi \@tempa\Sigma \@tempa\Upsilon \@tempa\Phi \@tempa\Psi
+\@tempa\Omega
+\@ifundefined{varGamma}{%
+  \DeclareMathSymbol{\varGamma}{\mathord}{letters}{"00}
+  \DeclareMathSymbol{\varDelta}{\mathord}{letters}{"01}
+  \DeclareMathSymbol{\varTheta}{\mathord}{letters}{"02}
+  \DeclareMathSymbol{\varLambda}{\mathord}{letters}{"03}
+  \DeclareMathSymbol{\varXi}{\mathord}{letters}{"04}
+  \DeclareMathSymbol{\varPi}{\mathord}{letters}{"05}
+  \DeclareMathSymbol{\varSigma}{\mathord}{letters}{"06}
+  \DeclareMathSymbol{\varUpsilon}{\mathord}{letters}{"07}
+  \DeclareMathSymbol{\varPhi}{\mathord}{letters}{"08}
+  \DeclareMathSymbol{\varPsi}{\mathord}{letters}{"09}
+  \DeclareMathSymbol{\varOmega}{\mathord}{letters}{"0A}
+}{}
+\endgroup
+\begingroup \catcode`\"=12
+\gdef\@@sqrt#1{\radical"270370 {#1}}
+\endgroup
+\@saveprimitive\overline\@@overline
+\def\overline#1{\@@overline{#1}}
+\def\boxed#1{\fbox{\m@th$\displaystyle#1$}}
+\def\implies{\DOTSB\;\Longrightarrow\;}
+\def\impliedby{\DOTSB\;\Longleftarrow\;}
+\begingroup \catcode`\"=12 % in case activated by a preceding package
+\gdef\And{\DOTSB\;\mathchar"3026 \;}
+\@tempcnta=\@xp\@gobble\vert \advance\@tempcnta "4000000
+\xdef\lvert{\delimiter\number\@tempcnta\space }
+\advance\@tempcnta "1000000
+\xdef\rvert{\delimiter\number\@tempcnta\space }
+\@tempcnta=\@xp\@gobble\Vert \advance\@tempcnta "4000000
+\xdef\lVert{\delimiter\number\@tempcnta\space }
+\advance\@tempcnta "1000000
+\xdef\rVert{\delimiter\number\@tempcnta\space }
+\endgroup % restore "
+\@saveprimitive\over\@@over
+\@saveprimitive\atop\@@atop
+\@saveprimitive\above\@@above
+\@saveprimitive\overwithdelims\@@overwithdelims
+\@saveprimitive\atopwithdelims\@@atopwithdelims
+\@saveprimitive\abovewithdelims\@@abovewithdelims
+\DeclareRobustCommand{\primfrac}[1]{%
+  \PackageWarning{amsmath}{%
+Foreign command \@backslashchar#1; %
+\protect\frac\space or \protect\genfrac\space should be used instead%
+  }
+  \global\@xp\let\csname#1\@xp\endcsname\csname @@#1\endcsname
+  \csname#1\endcsname
+}
+\renewcommand{\over}{\primfrac{over}}
+\renewcommand{\atop}{\primfrac{atop}}
+\renewcommand{\above}{\primfrac{above}}
+\renewcommand{\overwithdelims}{\primfrac{overwithdelims}}
+\renewcommand{\atopwithdelims}{\primfrac{atopwithdelims}}
+\renewcommand{\abovewithdelims}{\primfrac{abovewithdelims}}
+\DeclareRobustCommand{\frac}[2]{{\begingroup#1\endgroup\@@over#2}}
+\newcommand{\dfrac}{\genfrac{}{}{}0}
+\newcommand{\tfrac}{\genfrac{}{}{}1}
+\DeclareRobustCommand{\binom}{\genfrac()\z@{}}
+\newcommand{\dbinom}{\genfrac(){0pt}0}
+\newcommand{\tbinom}{\genfrac(){0pt}1}
+\DeclareRobustCommand{\genfrac}[4]{%
+  \def\@tempa{#1#2}%
+  \edef\@tempb{\@nx\@genfrac\@mathstyle{#4}%
+    \csname @@\ifx @#3@over\else above\fi
+    \ifx\@tempa\@empty \else withdelims\fi\endcsname}
+  \@tempb{#1#2#3}}
+\def\@genfrac#1#2#3#4#5{{#1{\begingroup#4\endgroup#2#3\relax#5}}}
+\def\@mathstyle#1{%
+  \ifx\@empty#1\@empty\relax
+  \else\ifcase#1\displaystyle % case 0
+    \or\textstyle\or\scriptstyle\else\scriptscriptstyle\fi\fi}
+\def\colon{\nobreak\mskip2mu\mathpunct{}\nonscript
+  \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+\begingroup \catcode`\"=12
+\edef\@tempa{\string\mathchar"}
+\def\@tempb#1"#2\@nil{#1"}
+\edef\@tempc{\expandafter\@tempb\meaning\coprod "\@nil}
+\ifx\@tempa\@tempc
+  \global\let\coprod@\coprod
+  \gdef\coprod{\DOTSB\coprod@\slimits@}
+  \global\let\bigvee@\bigvee
+  \gdef\bigvee{\DOTSB\bigvee@\slimits@}
+  \global\let\bigwedge@\bigwedge
+  \gdef\bigwedge{\DOTSB\bigwedge@\slimits@}
+  \global\let\biguplus@\biguplus
+  \gdef\biguplus{\DOTSB\biguplus@\slimits@}
+  \global\let\bigcap@\bigcap
+  \gdef\bigcap{\DOTSB\bigcap@\slimits@}
+  \global\let\bigcup@\bigcup
+  \gdef\bigcup{\DOTSB\bigcup@\slimits@}
+  \global\let\prod@\prod
+  \gdef\prod{\DOTSB\prod@\slimits@}
+  \global\let\sum@\sum
+  \gdef\sum{\DOTSB\sum@\slimits@}
+  \global\let\bigotimes@\bigotimes
+  \gdef\bigotimes{\DOTSB\bigotimes@\slimits@}
+  \global\let\bigoplus@\bigoplus
+  \gdef\bigoplus{\DOTSB\bigoplus@\slimits@}
+  \global\let\bigodot@\bigodot
+  \gdef\bigodot{\DOTSB\bigodot@\slimits@}
+  \global\let\bigsqcup@\bigsqcup
+  \gdef\bigsqcup{\DOTSB\bigsqcup@\slimits@}
+\fi
+\endgroup
+\newcommand{\nobreakdash}{\leavevmode
+  \toks@\@emptytoks \def\@tempa##1{\toks@\@xp{\the\toks@-}\FN@\next@}%
+  \DN@{\ifx\@let@token-\@xp\@tempa
+       \else\setboxz@h{\the\toks@\nobreak}\unhbox\z@\fi}%
+  \FN@\next@
+}
+\def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha}
+\def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha}
+\newcount\uproot@
+\newcount\leftroot@
+\def\root{\relaxnext@
+  \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else
+   \ifx\@let@token\leftroot\let\next@\nextiii@\else
+   \let\next@\plainroot@\fi\fi\next@}%
+  \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
+  \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else
+   \DN@.{\FN@\nextv@}\fi\next@.}%
+  \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
+   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
+  \def\nextvii@{\ifx\@let@token\@sptoken
+   \DN@. {\FN@\nextviii@}\else
+   \DN@.{\FN@\nextviii@}\fi\next@.}%
+  \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else
+   \let\next@\plainroot@\fi\next@}%
+  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
+  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
+\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
+ $\m@th\scriptscriptstyle{#1}$}%
+ \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
+ {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
+\def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}%
+ \dimen@\ht\z@\advance\dimen@-\dp\z@
+ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
+ \advance\dimen@ by1.667\wd\@ne
+ \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
+ \mkern-10mu\mkern\leftroot@ mu\boxz@}
+\let\ifgtest@\iffalse                              % initial value
+\def\gtest@true{\global\let\ifgtest@\iftrue}
+\def\gtest@false{\global\let\ifgtest@\iffalse}
+\let\DOTSI\relax
+\let\DOTSB\relax
+\let\DOTSX\relax
+{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
+ \uppercase{%
+  \gdef\math@#1#2#3#4#5#6\math@{\gtest@false\ifx 7#1\ifx 8#2%
+  \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\gtest@true
+  \fi\fi\fi\fi\fi}}}
+{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
+ \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
+  \ifx 7#1\ifx 8#2\ifx 9#5\gtest@true\xdef\meaning@{9#6}\fi\fi\fi}}}
+\newcount\classnum@
+\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
+ \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
+ \gdef\thedots@{\dotsb@}\fi}
+{\uccode`4=`b \uccode`5=`i \uccode`6=`n
+ \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathbin@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \gtest@false\DN@##1\mathbin@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`4=`r \uccode`5=`e \uccode`6=`l
+ \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
+  \def\nextii@##1\mathrel@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\mathrel@{}%
+ \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
+{\uccode`5=`m \uccode`6=`a \uccode`7=`c
+ \uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
+  \ifx 5#1\ifx 6#2\ifx 7#3\gtest@true
+  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
+\def\macro@@#1->#2\macro@@{#2}
+\newcount\DOTSCASE@
+{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
+ \uppercase{\gdef\DOTS@#1#2#3#4#5{\gtest@false\DN@##1\DOTS@{}%
+  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@
+  \fi\fi\fi\fi\fi
+  \next@}}}
+{\uccode`3=`B \uccode`4=`I \uccode`5=`X
+ \uppercase{\gdef\DOTS@@#1{\relaxnext@
+  \def\nextii@##1\DOTS@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+  \DN@{\FN@\nextii@}%
+  \ifx 3#1\global\DOTSCASE@\z@\else
+  \ifx 4#1\global\DOTSCASE@\@ne\else
+  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
+  \fi\fi\fi\next@}}}
+{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
+ \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
+  \def\nextii@##1\not@{\ifx\@sptoken\@let@token\gtest@true\fi}%
+ \gtest@false\DN@##1\not@{}%
+ \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
+ \fi\next@}}}
+\def\keybin@{\gtest@true
+ \ifx\@let@token+\else\ifx\@let@token=\else
+ \ifx\@let@token<\else\ifx\@let@token>\else
+ \ifx\@let@token-\else\ifx\@let@token*\else\ifx\@let@token:\else
+   \gtest@false\fi\fi\fi\fi\fi\fi\fi}
+\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
+\DeclareRobustCommand{\dots}{\relax
+  \csname\ifmmode m\else t\fi dots@\endcsname}
+\def\tdots@{\leavevmode\unskip\relaxnext@
+ \DN@{$\m@th\@ldots\,
+   \ifx\@let@token,\,$\else\ifx\@let@token.\,$\else
+   \ifx\@let@token;\,$\else\ifx\@let@token:\,$\else
+   \ifx\@let@token?\,$\else\ifx\@let@token!\,$\else
+     $ \fi\fi\fi\fi\fi\fi}%
+  \ \FN@\next@}
+\def\mdots@{\FN@\mdots@@}
+\def\mdots@@{\gdef\thedots@{\dotso@}%
+ \ifx\@let@token\boldsymbol \gdef\thedots@\boldsymbol{\boldsymboldots@}%
+ \else\ifx,\@let@token \gdef\thedots@{\dotsc}%
+ \else\ifx\not\@let@token \gdef\thedots@{\dotsb@}%
+ \else\keybin@
+ \ifgtest@\gdef\thedots@{\dotsb@}%
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+   \xdef\meaning@@{\meaning@}%
+  \@xp\math@\meaning@\math@
+  \ifgtest@
+   \@xp\mathch@\meaning@\mathch@
+   \ifgtest@\@xp\getmathch@\meaning@\getmathch@\fi
+  \else\@xp\macro@\meaning@@\macro@
+  \ifgtest@
+   \@xp\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\DOTS@\meaning@\DOTS@
+  \ifgtest@
+   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
+    \or\gdef\thedots@{\dotsi}\else\fi
+  \else\@xp\math@\meaning@\math@
+  \ifgtest@\@xp\mathbin@\meaning@\mathbin@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \else\@xp\mathrel@\meaning@\mathrel@
+  \ifgtest@\gdef\thedots@{\dotsb@}%
+  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+ \thedots@}
+\def\boldsymboldots@#1{\bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
+ \boldsymbol#1\bold@false}
+\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
+\def\dotsi{\!\@cdots}
+\let\dotsb@\@cdots
+\def\rightdelim@{\gtest@true
+ \ifx\@let@token)\else
+ \ifx\@let@token]\else
+ \ifx\@let@token\rbrack\else
+ \ifx\@let@token\}\else
+ \ifx\@let@token\rbrace\else
+ \ifx\@let@token\rangle\else
+ \ifx\@let@token\rceil\else
+ \ifx\@let@token\rfloor\else
+ \ifx\@let@token\rgroup\else
+ \ifx\@let@token\rmoustache\else
+ \ifx\@let@token\right\else
+ \ifx\@let@token\bigr\else
+ \ifx\@let@token\biggr\else
+ \ifx\@let@token\Bigr\else
+ \ifx\@let@token\Biggr\else\gtest@false
+ \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+\def\extra@{%
+ \rightdelim@\ifgtest@
+ \else\ifx\@let@token$\gtest@true
+ \else\xdef\meaning@{\meaning\@let@token..........}%
+ \@xp\macro@\meaning@\macro@\ifgtest@
+ \@xp\DOTS@\meaning@\DOTS@
+ \ifgtest@
+ \ifnum\DOTSCASE@=\tw@\gtest@true\else\gtest@false
+ \fi\fi\fi\fi\fi}
+\newif\ifbold@
+\def\dotso@{\relaxnext@
+ \ifbold@
+  \let\@let@token\delayed@
+  \def\nextii@{\extra@\@ldots\ifgtest@\,\fi}%
+ \else
+  \def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
+ \fi
+ \nextii@}
+\def\extrap@#1{%
+ \DN@{#1\,}%
+ \ifx\@let@token,\else
+ \ifx\@let@token;\else
+ \ifx\@let@token.\else\extra@
+ \ifgtest@\else
+ \let\next@#1\fi\fi\fi\fi\next@}
+\DeclareRobustCommand{\ldots}{\relax
+  \ifmmode \DN@{\extrap@\@ldots}%
+  \else \let\next@\tdots@\fi
+  \FN@\next@}
+\DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@}
+\let\dotso\ldots
+\let\dotsb\cdots
+\let\dotsm\dotsb
+\DeclareRobustCommand{\dotsc}{%
+  \DN@{\ifx\@let@token;\@ldots\,%
+       \else \ifx\@let@token.\@ldots\,%
+       \else \extra@\@ldots \ifgtest@\,\fi
+       \fi\fi}%
+  \FN@\next@}
+\def\longrightarrow{\DOTSB\relbar\joinrel\rightarrow}
+\def\Longrightarrow{\DOTSB\Relbar\joinrel\Rightarrow}
+\def\longleftarrow{\DOTSB\leftarrow\joinrel\relbar}
+\def\Longleftarrow{\DOTSB\Leftarrow\joinrel\Relbar}
+\def\longleftrightarrow{\DOTSB\leftarrow\joinrel\rightarrow}
+\def\Longleftrightarrow{\DOTSB\Leftarrow\joinrel\Rightarrow}
+\def\mapsto{\DOTSB\mapstochar\rightarrow}
+\def\longmapsto{\DOTSB\mapstochar\longrightarrow}
+\def\hookrightarrow{\DOTSB\lhook\joinrel\rightarrow}
+\def\hookleftarrow{\DOTSB\leftarrow\joinrel\rhook}
+\def\iff{\DOTSB\;\Longleftrightarrow\;}
+\def\doteq{\DOTSB\mathrel{\mathop{\kern\z@ =}\limits^{\textstyle.}}}
+\newif\if@display
+\everydisplay\@xp{\the\everydisplay \@displaytrue}
+\def\int{\DOTSI\intop\ilimits@}
+\def\oint{\DOTSI\ointop\ilimits@}
+\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
+\def\intdots@{\mathchoice{\@cdots}%
+ {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
+ {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}
+\def\iint{\DOTSI\protect\ints@\tw@}
+\def\iiint{\DOTSI\protect\ints@\thr@@}
+\def\iiiint{\DOTSI\protect\ints@{4}}
+\def\idotsint{\DOTSI\protect\ints@\z@}
+\def\ints@#1{%
+  \mkern-7mu\mathchoice{\mkern-2mu}{}{}{}%
+  \mathop{\mkern7mu\mathchoice{\mkern2mu}{}{}{}%
+    \intop\ifnum#1=\z@\intdots@
+    \else\intkern@\fi
+    \ifnum#1>\tw@\intop\intkern@\fi
+    \ifnum#1>\thr@@\intop\intkern@\fi
+    \intop
+  }\ilimits@
+}
+\newbox\Mathstrutbox@
+\setbox\Mathstrutbox@=\hbox{}
+\def\Mathstrut@{\copy\Mathstrutbox@}
+\begingroup \catcode`\"=12
+\gdef\resetMathstrut@{%
+  \setbox\z@\hbox{%
+    \mathchardef\@tempa\mathcode`\(\relax
+    \def\@tempb##1"##2##3{\the\textfont"##3\char"}%
+    \expandafter\@tempb\meaning\@tempa \relax
+  }%
+  \ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@
+}
+\endgroup
+\addto@hook\every@math@size{\resetMathstrut@}
+\newbox\strutbox@
+\def\strut@{\copy\strutbox@}
+\addto@hook\every@math@size{%
+  \global\setbox\strutbox@\hbox{\lower.5\normallineskiplimit
+         \vbox{\kern-\normallineskiplimit\copy\strutbox}}}
+\def\big{\bBigg@\@ne}
+\def\Big{\bBigg@{1.5}}
+\def\bigg{\bBigg@\tw@}
+\def\Bigg{\bBigg@{2.5}}
+\def\bBigg@#1#2{%
+  {\@mathmeasure\z@{\nulldelimiterspace\z@}%
+     {\left#2\vcenter to#1\big@size{}\right.}%
+   \box\z@}}
+\addto@hook\every@math@size{%
+  \global\big@size 1.2\ht\Mathstrutbox@
+  \global\advance\big@size 1.2\dp\Mathstrutbox@ }
+\newdimen\big@size
+\def\accentclass@{7}
+\def\noaccents@{\def\accentclass@{0}}
+\DeclareFontEncoding{OML}{}{\noaccents@}
+\DeclareFontEncoding{OMS}{}{\noaccents@}
+\begingroup \catcode`\"=12
+\def\@tempa#1#2\@nil{\def\@tempc{#1}}\def\@tempb{\mathaccent}
+\expandafter\@tempa\hat \relax\relax\@nil
+\ifx\@tempb\@tempc
+  \def\@tempa#1\@nil{#1}%
+  \def\@tempb#1{\afterassignment\@tempa\mathchardef\@tempc=}%
+  \def\@tempe#1"{}
+  \def\do#1"#2{}
+  \def\@tempd#1#2{\expandafter\@tempb#1\@nil
+    \ifnum\@tempc<"1000
+      \edef\@tempc{"\@nx\accentclass@
+        \ifnum\@tempc<"100 0\fi
+        \@xp\@tempe\meaning\@tempc\space}%
+    \else
+      \edef\@tempc{"\@nx\@nx\@nx\accentclass@
+        \@xp\do\meaning\@tempc\space}%
+    \fi
+    \xdef#1{\mathaccent\@tempc}%
+    \toks@{%
+      \relax\ifmmode \else\DN@##1##2{\nonmatherr@{#2}}\@xp\next@\fi
+      \mathaccent@}%
+    \xdef#2{\the\toks@{\@tempc}}%
+  }
+  \@tempd\hat\Hat \@tempd\check\Check \@tempd\tilde\Tilde
+  \@tempd\acute\Acute \@tempd\grave\Grave \@tempd\dot\Dot
+  \@tempd\ddot\Ddot \@tempd\breve\Breve \@tempd\bar\Bar
+\fi
+\endgroup
+\newcount\skewcharcount@
+\newcount\familycount@
+\def\theskewchar@{\familycount@\@ne
+ \global\skewcharcount@\the\skewchar\textfont\@ne
+ \ifnum\mathgroup>\m@ne\ifnum\mathgroup<16
+  \global\familycount@\the\mathgroup\relax
+  \global\skewcharcount@\the\skewchar\textfont\the\mathgroup\relax\fi\fi
+ \ifnum\skewcharcount@>\m@ne
+  \ifnum\skewcharcount@<128
+  \multiply\familycount@256
+  \global\advance\skewcharcount@\familycount@
+  \global\advance\skewcharcount@28672
+  \mathchar\skewcharcount@\else
+  \global\skewcharcount@\m@ne\fi\else
+ \global\skewcharcount@\m@ne\fi}
+\newcount\pointcount@
+\def\getpoints@#1.#2\getpoints@{\pointcount@#1 }
+\newdimen\accentdimen@
+\newcount\accentmu@
+\def\dimentomu@{\multiply\accentdimen@ 100
+ \@xp\getpoints@\the\accentdimen@\getpoints@
+ \multiply\pointcount@18
+ \divide\pointcount@\@m
+ \global\accentmu@\pointcount@}
+\def\mathaccent@#1#2{\ifnum\mathgroup=\m@ne\xdef\thefam@{1}\else
+ \xdef\thefam@{\the\mathgroup}\fi
+ \accentdimen@\z@
+ \setboxz@h{\unbracefonts@$\m@th\mathgroup\thefam@\relax#2$}%
+ \ifdim\accentdimen@=\z@\DN@{\mathaccent#1{#2}}%
+  \setbox\@ne\hbox{\unbracefonts@
+    $\m@th\mathgroup\thefam@\relax#2\theskewchar@$}
+  \setbox\tw@\hbox{$\m@th\ifnum\skewcharcount@=\m@ne\else
+   \mathchar\skewcharcount@\fi$}%
+  \global\accentdimen@\wd\@ne\global\advance\accentdimen@-\wdz@
+  \global\advance\accentdimen@-\wd\tw@
+  \global\multiply\accentdimen@\tw@
+  \dimentomu@\global\advance\accentmu@\@ne
+ \else\DN@{{\mathaccent#1{#2\mkern\accentmu@ mu}%
+    \mkern-\accentmu@ mu}{}}\fi
+ \next@}
+\def\unbracefonts@{\let\math@bgroup\@empty\let\math@egroup\@empty}
+\def\nonmatherr@#1{\@amsmath@err{\protect
+  #1 allowed only in math mode}\@ehd}
+\begingroup \catcode`\"=12
+\def\@tempa#1#2{\gdef#1{\RIfM@\DN@{\mathaccent@{"\accentclass@#2 }}%
+  \else\DN@{\nonmatherr@{#1}}\fi\next@}}
+\@tempa\Hat{05E}\@tempa\Check{014}\@tempa\Tilde{07E}\@tempa\Acute{013}
+\@tempa\Grave{012}\@tempa\Dot{05F}\@tempa\Ddot{07F}\@tempa\Breve{015}
+\@tempa\Bar{016}
+\gdef\Vec{\RIfM@\DN@{\mathaccent@{"017E }}\else
+ \DN@{\nonmatherr@\Vec}\fi\next@}
+\endgroup
+\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+ \hbox{\normalfont ...}\vss}}}}
+\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
+ \hbox{\normalfont....}\vss}}}}
+\def\bmod{\mskip-\medmuskip\mkern5mu\mathbin
+  {\operator@font mod}\penalty900
+  \mkern5mu\mskip-\medmuskip}
+\def\pod#1{\allowbreak\if@display\mkern18mu\else\mkern8mu\fi(#1)}
+\def\pmod#1{\pod{{\operator@font mod}\mkern6mu#1}}
+\def\mod#1{\allowbreak\if@display\mkern18mu
+  \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+\newcommand{\cfrac}[3][c]{{\displaystyle\frac{%
+  \strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}%
+  \kern-\nulldelimiterspace}
+\def\overset#1#2{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
+\def\underset#1#2{\binrel@{#2}%
+  \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
+\def\sideset#1#2#3{%
+  \@mathmeasure\z@\displaystyle{#3}%
+  \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@
+  \setbox\tw@\box\@ne
+  \@mathmeasure4\displaystyle{\copy\tw@#1}%
+  \@mathmeasure6\displaystyle{#3\nolimits#2}%
+  \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@
+  \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}%
+}
+\renewcommand{\smash}[2][tb]{%
+  \def\smash@{#1}%
+  \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else
+        \@xp\makesm@sh\fi{#2}}
+\def\finsm@sh{\def\mb@t{\ht\z@\z@}\def\mb@b{\dp\z@\z@}%
+  \def\mb@tb{\mb@t\mb@b}%
+  {\csname mb@\smash@\endcsname}%
+  \leavevmode\boxz@}
+\def\rightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\copy\z@\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\leftarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\mathord\leftarrow\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
+  \mkern-6mu\box\z@$}
+\def\leftrightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
+  $#1\mathord\leftarrow\mkern-6mu\cleaders
+  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
+  \mkern-6mu\mathord\rightarrow$}
+\def\overarrow@#1#2#3{\vbox{\ialign{##\crcr#1#2\crcr
+ \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#2#3\hfil$\crcr}}}
+\def\overrightarrow{\mathpalette{\overarrow@\rightarrowfill@}}
+\def\overleftarrow{\mathpalette{\overarrow@\leftarrowfill@}}
+\def\overleftrightarrow{\mathpalette{\overarrow@\leftrightarrowfill@}}
+\def\underarrow@#1#2#3{%
+ \vtop{\ialign{##\crcr$\m@th\hfil#2#3\hfil$\crcr
+ \noalign{\nointerlineskip\kern-.5\ex@}#1#2\crcr}}}
+\def\underrightarrow{\mathpalette{\underarrow@\rightarrowfill@}}
+\def\underleftarrow{\mathpalette{\underarrow@\leftarrowfill@}}
+\def\underleftrightarrow{\mathpalette{\underarrow@\leftrightarrowfill@}}
+\newcommand{\xrightarrow}[2][]{%
+  \mathrel{\mathop{%
+    \setbox\z@\vbox{\m@th
+      \hbox{$\scriptstyle\;{#1}\;\;$}%
+      \hbox{$\m@th\scriptstyle\;{#2}\;\;$}%
+    }%
+    \hbox to\ifdim\wd\z@>\minaw@\wd\z@\else\minaw@\fi{%
+      \rightarrowfill@\displaystyle}}%
+  \limits^{#2}\@ifnotempty{#1}{_{#1}}}%
+}
+\newcommand{\xleftarrow}[2][]{%
+  \mathrel{\mathop{%
+    \setbox\z@\vbox{\m@th
+      \hbox{$\scriptstyle\;\;{#1}\;$}%
+      \hbox{$\m@th\scriptstyle\;\;{#2}\;\;$}%
+    }%
+    \hbox to\ifdim\wd\z@>\minaw@\wd\z@\else\minaw@\fi{%
+      \leftarrowfill@\displaystyle}}%
+  \limits^{#2}\@ifnotempty{#1}{_{#1}}}%
+}
+\@ifundefined{minaw@}{\newdimen\minaw@\minaw@11pt}{}
+\newcommand{\Sb}{\PackageError{amsmath}%
+  {Environment `Sb' is obsolete; use `substack' instead}%
+  {The \protect\\protect\ used to separate lines in a `Sb' environment can
+   cause problems if `Sb' is embedded in some aligning environments.}}
+\newcommand{\Sp}{\PackageError{amsmath}%
+  {Environment `Sp' is obsolete; use `substack' instead}%
+  {The \protect\\protect\ used to separate lines in a `Sp' environment can
+   cause problems if `Sp' is embedded in some aligning environments.}}
+\newenvironment{subarray}[1]{%
+  \vcenter\bgroup
+  \Let@ \restore@math@cr \default@tag
+  \baselineskip\fontdimen10 \scriptfont\tw@
+  \advance\baselineskip\fontdimen12 \scriptfont\tw@
+  \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
+  \lineskiplimit\lineskip
+  \ialign\bgroup\ifx c#1\hfil\fi
+    $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup
+}
+\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+\newenvironment{smallmatrix}{\null\,\vcenter\bgroup
+  \Let@\restore@math@cr\default@tag
+  \baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
+  \ialign\bgroup\hfil$\m@th\scriptstyle##$\hfil&&\thickspace\hfil
+  $\m@th\scriptstyle##$\hfil\crcr
+}{%
+  \crcr\egroup\egroup\,%
+}
+\newcount\c@MaxMatrixCols \c@MaxMatrixCols=10
+\renewenvironment{matrix}{%
+  \hskip -\arraycolsep\array{*\c@MaxMatrixCols c}%
+}{%
+  \endarray \hskip -\arraycolsep
+}
+\renewenvironment{pmatrix}{\left(\matrix}{\endmatrix\right)}
+\newenvironment{bmatrix}{\left[\matrix}{\endmatrix\right]}
+\newenvironment{Bmatrix}{\left\lbrace\matrix}{\endmatrix\right\rbrace}
+\newenvironment{vmatrix}{\left\lvert\matrix}{\endmatrix\right\rvert}
+\newenvironment{Vmatrix}{\left\lVert\matrix}{\endmatrix\right\rVert}
+\let\hdots\@ldots
+\newcommand{\hdotsfor}[1]{%
+  \ifx[#1\@xp\shdots@for\else\hdots@for\@ne{#1}\fi}
+\newmuskip\dotsspace@
+\def\shdots@for#1]{\hdots@for{#1}}
+\def\hdots@for#1#2{\multicolumn{#2}c%
+  {\m@th\dotsspace@1.5mu\mkern-#1\dotsspace@
+   \xleaders\hbox{$\m@th\mkern#1\dotsspace@.\mkern#1\dotsspace@$}%
+           \hfill
+   \mkern-#1\dotsspace@}%
+   }
+\renewenvironment{cases}{%
+  \left\{\def\arraystretch{1.2}%
+  \array{@{}l@{\quad}l@{}}%
+}{%
+  \endarray\right.%
+}
+\newcounter{parentequation}% Counter for ``parent equation''.
+\newenvironment{subequations}{%
+  \refstepcounter{equation}%
+  \begingroup % conservative approach
+  \let\protect\@nx
+  \edef\@tempa{\def\@nx\theparentequation{\theequation}}%
+  \@xp\endgroup\@tempa
+  \setcounter{parentequation}{\value{equation}}%
+  \setcounter{equation}{0}%
+  \def\theequation{\theparentequation\alph{equation}}%
+  \ignorespaces
+}{%
+  \setcounter{equation}{\value{parentequation}}%
+  \global\@ignoretrue
+}
+\def\numberwithin#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}{%
+  \@ifundefined{c@#2}{\@nocnterr{#2}}{%
+  \@addtoreset{#1}{#2}%
+  \toks@\@xp\@xp\@xp{\csname the#1\endcsname}%
+  \@xp\xdef\csname the#1\endcsname
+    {\@xp\@nx\csname the#2\endcsname
+     .\the\toks@}}}}
+\def\eqref#1{\textup{\tagform@{\ref{#1}}}}
+\newcount\dspbrk@lvl
+\dspbrk@lvl=-1
+\interdisplaylinepenalty\@M
+\def\allowdisplaybreaks{%
+  \new@ifnextchar[\allowdspbrks@{\allowdspbrks@[4]}}
+\def\allowdspbrks@[#1]{%
+  \interdisplaylinepenalty\getdsp@pen{#1}}
+\def\getdsp@pen#1{%
+  \ifcase #1\relax \@M
+    \or 9999
+    \or 6999
+    \or 2999
+    \or \z@\fi}
+\def\displaybreak{\@amsmath@err{\Invalid@@\displaybreak}\@eha}
+
+\def\displaybreak@{%
+  \def\displaybreak{\new@ifnextchar[\dspbrk@{\dspbrk@[4]}}}
+
+\def\dspbrk@[#1]{\global\dspbrk@lvl #1\relax}
+\def\math@cr{{\ifnum0=`}\fi
+  \@ifstar{\global\@eqpen\@M\math@cr@}%
+          {\global\@eqpen
+             \ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
+              \else -\@getpen\dspbrk@lvl \fi
+           \math@cr@}}
+\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
+\def\math@cr@@[#1]{\ifnum0=`{\fi}\math@cr@@@
+  \noalign{\vskip#1\relax}}
+\def\Let@{\let\\\math@cr}
+\def\restore@math@cr{\def\math@cr@@@{\cr}}
+\restore@math@cr
+\def\intertext{\@amsmath@err{\Invalid@@\intertext}\@eha}
+\def\intertext@{%
+  \def\intertext##1{%
+    \ifvmode\else\\\@empty\fi
+    \noalign{%
+      \penalty\postdisplaypenalty\vskip\belowdisplayskip
+      \vbox{\normalbaselines
+        \ifdim\linewidth=\columnwidth
+        \else \parshape\@ne \@totalleftmargin \linewidth
+        \fi
+        \noindent##1\par}%
+      \penalty\predisplaypenalty\vskip\abovedisplayskip%
+    }%
+}}
+\newhelp\tag@help
+  {tag cannot be used at this point.\space
+   If you don't understand why^^Jyou should consult
+   the documentation.^^JBut don't worry: just continue, and I'll
+   forget what happened.}
+\def\gobble@tag{\@ifstar\@gobble\@gobble}
+\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag}
+\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
+\def\default@tag{\let\tag\dft@tag}
+\default@tag
+\def\maketag@@{\@ifstar\maketag@@@\tagform@}
+\def\maketag@@@#1{\hbox{\m@th\normalfont#1}}
+\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip\@@italiccorr)}}
+\iftagsleft@
+  \def\@eqnnum{\hbox to1sp{}\rlap{\normalfont\normalcolor
+    \hskip -\displaywidth\tagform@\theequation}}
+\else
+  \def\@eqnnum{{\normalfont\normalcolor \tagform@\theequation}}
+\fi
+\def\thetag{\leavevmode\tagform@}
+\def\make@df@tag{\@ifstar\make@df@tag@@\make@df@tag@@@}
+\def\make@df@tag@@#1{%
+  \gdef\df@tag{\maketag@@@{#1}\def\@currentlabel{#1}}}
+\def\make@df@tag@@@#1{\gdef\df@tag{\tagform@{#1}%
+  \toks@\@xp{\p@equation{#1}}\edef\@currentlabel{\the\toks@}}}
+\let\ltx@label\label
+\def\label@in@display{%
+    \ifx\df@label\@empty\else
+        \@amsmath@err{Multiple \string\label's:
+            label '\df@label' will be lost}\@eha
+    \fi
+    \gdef\df@label
+}
+
+\let\df@label\@empty
+\def\make@display@tag{%
+    \if@eqnsw
+        \refstepcounter{equation}%
+        \tagform@\theequation
+    \else
+        \iftag@
+            \df@tag
+            \global\let\df@tag\@empty
+        \fi
+    \fi
+    \ifx\df@label\@empty\else
+        \ltx@label{\df@label}%
+        \global\let\df@label\@empty
+    \fi
+}
+\def\tag@in@align{%
+    \relax
+    \iftag@
+        \DN@{\invalid@tag{Multiple \string\tag}}%
+    \else
+    \global\tag@true
+    \nonumber
+        \let\next@\make@df@tag
+    \fi
+    \next@
+}
+\def\raisetag#1{\skip@#1\relax
+  \xdef\raise@tag{\vskip\iftagsleft@\else-\fi\the\skip@\relax}%
+}
+\let\raise@tag\@empty
+\def\notag{\nonumber}
+\newif\ifinany@
+\newif\ifinalign@
+\newif\ifingather@
+\newif\iftag@
+\newif\ifst@rred
+\newif\ifmeasuring@
+\newif\ifshifttag@
+\newcount\row@
+\newcount\column@
+\def\column@plus{%
+    \global\advance\column@\@ne
+}
+\newcount\maxfields@
+\def\add@amps#1{%
+    \begingroup
+        \count@#1
+        \DN@{}%
+        \loop
+            \ifnum\count@>\column@
+                \edef\next@{&\next@}%
+                \advance\count@\m@ne
+        \repeat
+    \@xp\endgroup
+    \next@
+}
+\newhelp\andhelp@
+{An extra & here is so disastrous that you should probably exit^^J
+and fix things up.}
+\newdimen\eqnshift@
+\newdimen\alignsep@
+\newdimen\tagshift@
+\def\mintagsep{.5\fontdimen6\textfont2}
+\def\minalignsep{10pt}
+\newdimen\tagwidth@
+\newdimen\totwidth@
+\newdimen\lineht@
+\def\tag@width#1{%
+    \ifcase\@xp#1\tag@lengths\fi
+}
+
+\def\savetaglength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@lengths{\tag@lengths\or \the\wdz@}%
+    \endgroup
+}
+
+\def\shift@tag#1{%
+    \ifcase\@xp#1\tag@shifts\fi\relax
+}
+
+\let\tag@shifts\@empty
+\def\saveshift@#1{%
+    \begingroup
+        \let\or\relax
+        \xdef\tag@shifts{\or#1\tag@shifts}%
+    \endgroup
+}
+\def\displ@y{\@display@init{}}
+\def\@display@init#1{%
+    \global\dt@ptrue
+    \openup\jot\m@th
+    \everycr{%
+        \noalign{%
+            #1%
+            \ifdt@p
+                \global\dt@pfalse
+                \vskip-\lineskiplimit
+                \vskip\normallineskiplimit
+            \else
+                \penalty\@eqpen
+            \fi
+        }%
+    }%
+}
+\def\displ@y@{\@display@init{%
+  \global\column@\z@ \global\dspbrk@lvl\m@ne
+  \global\tag@false \global\let\raise@tag\@empty
+}}
+\def\black@#1{%
+    \noalign{%
+        \ifdim#1>\displaywidth
+            \dimen@\prevdepth
+            \nointerlineskip
+            \vskip-\ht\strutbox@
+            \vskip-\dp\strutbox@
+            \vbox{\noindent\hbox to#1{\strut@\hfill}}%
+            \prevdepth\dimen@
+        \fi
+    }%
+}
+\def\savecounters@{%
+    \begingroup
+        \def\@elt##1{%
+          \global\csname c@##1\endcsname\the\csname c@##1\endcsname}%
+        \xdef\@tempa{%
+            \cl@@ckpt
+            \let\@nx\restorecounters@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorecounters@\@tempa
+}
+\let\restorecounters@\@empty
+\def\savealignstate@{%
+    \begingroup
+        \let\or\relax
+        \xdef\@tempa{%
+            \global\totwidth@\the\totwidth@
+            \global\row@\the\row@
+            \gdef\@nx\tag@lengths{\tag@lengths}%
+            \let\@nx\restorealignstate@\@nx\@empty
+        }%
+    \endgroup
+    \let\restorealignstate@\@tempa
+}
+
+\let\restorealignstate@\@empty
+\newtoks\@envbody
+\def\addto@envbody#1{\@envbody\@xp{\the\@envbody#1}}
+\def\collect@body#1{%
+    \@envbody{}%
+    \def\process@envbody{%
+        \@xp#1\@xp{\the\@envbody}%
+    }%
+    \@xp\let\csname\@currenvir\endcsname\collect@@body
+    \csname\@currenvir\endcsname
+}
+\def\collect@@body#1\end#2{%
+    \def\@tempa{#2}%
+    \ifx\@tempa\@currenvir
+        \addto@envbody{#1}%
+        \@xp\edef\csname\@currenvir\endcsname{%
+            \@nx\process@envbody\@nx\end{\@tempa}%
+            }%
+    \else
+        \addto@envbody{#1\end{#2}}%
+    \fi
+    \csname\@currenvir\endcsname
+}
+\newcommand{\start@aligned}[2]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{\@currenvir}}%
+    \fi
+    \null\,%
+    \if #1t\vtop \else \if#1b \vbox \else \vcenter \fi \fi \bgroup
+        \maxfields@#2\relax
+        \ifnum\maxfields@>\m@ne
+            \multiply\maxfields@\tw@
+            \let\math@cr@@@\math@cr@@@alignedat
+        \else
+            \restore@math@cr
+        \fi
+        \Let@
+        \default@tag
+        \ifinany@\else\openup\jot\fi
+        \column@\z@
+        \ialign\bgroup
+           &\column@plus
+            \hfil
+            \strut@
+            $\m@th\displaystyle{##}$%
+           &\column@plus
+            $\m@th\displaystyle{{}##}$%
+            \hfil
+            \crcr
+}
+\def\math@cr@@@alignedat{%
+    \ifnum\column@>\maxfields@
+        \begingroup
+          \measuring@false
+          \@amsmath@err{Extra & on this line}%
+            {\the\andhelp@}% "An extra & here is disastrous"
+        \endgroup
+    \fi
+    \column@\z@
+    \cr
+}
+\newenvironment{aligned}[1][c]{%
+    \start@aligned{#1}\m@ne
+}{%
+    \crcr\egroup\egroup
+}
+\newcommand{\alignedat}[2][c]{%
+        \start@aligned{#1}%
+}
+\let\endalignedat\endaligned
+\newcommand{\gathered}[1][c]{%
+    \RIfM@\else
+        \nonmatherr@{\begin{gathered}}%
+    \fi
+    \null\,%
+    \if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi \bgroup
+        \Let@
+        \restore@math@cr
+        \ifinany@\else\openup\jot\fi
+        \ialign\bgroup
+            \hfil\strut@$\m@th\displaystyle##$\hfil
+            \crcr
+}
+\let\endgathered\endaligned
+\def\start@gather#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred\else
+            \global\@eqnswtrue
+        \fi
+        \let\next@\gather@
+    \fi
+    \collect@body\next@
+}
+\def\gather{\start@gather\st@rredfalse}
+
+\@namedef{gather*}{\start@gather\st@rredtrue}
+\def\gather@#1{%
+    \ingather@true
+    \inany@true
+    \let\tag\tag@in@align
+    \let\label\label@in@display
+    \displaybreak@
+    \intertext@
+    \displ@y@
+    \Let@
+    \let\math@cr@@@\math@cr@@@gather
+    \gmeasure@{#1}%
+    \global\shifttag@false
+    \tabskip\z@skip
+    \global\row@\@ne
+    \halign to\displaywidth\bgroup
+        \strut@
+        \setboxz@h{$\m@th\displaystyle{##}$}%
+        \calc@shift@gather
+        \set@gather@field
+        \tabskip\@centering
+       &\setboxz@h{\strut@{##}}%
+        \place@tag@gather
+        \tabskip \iftagsleft@ \gdisplaywidth@ \else \z@skip \span\fi
+        \crcr
+        #1%
+}
+\def\endgather{%
+        \math@cr
+        \black@\totwidth@
+    \egroup
+    $$%
+    \global\@ignoretrue
+}
+
+\@xp\let\csname endgather*\endcsname\endgather
+\def\gmeasure@#1{%
+    \begingroup
+        \measuring@true
+        \totwidth@\z@
+        \global\let\tag@lengths\@empty
+        \savecounters@
+        \setbox\@ne\vbox{%
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \halign{%
+                \setboxz@h{$\m@th\displaystyle{##}$}%
+                \ifdim\wdz@>\totwidth@
+                    \global\totwidth@\wdz@
+                \fi
+               &\setboxz@h{\strut@{##}}%
+                \savetaglength@
+                \crcr
+                #1%
+                \math@cr@@@
+            }%
+        }%
+        \restorecounters@
+        \if@fleqn
+            \global\advance\totwidth@\@mathmargin
+        \fi
+        \iftagsleft@
+            \ifdim\totwidth@>\displaywidth
+                \global\let\gdisplaywidth@\totwidth@
+            \else
+                \global\let\gdisplaywidth@\displaywidth
+            \fi
+        \fi
+    \endgroup
+}
+\def\math@cr@@@gather{%
+    \ifst@rred\nonumber\fi
+   &\relax
+    \make@display@tag
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \global\advance\row@\@ne
+    \cr
+}
+\def\calc@shift@gather{%
+    \dimen@\mintagsep\relax
+    \tagwidth@\tag@width\row@\relax
+    \if@fleqn
+        \global\eqnshift@\@mathmargin
+        \ifdim\tagwidth@>\z@
+            \advance\dimen@\tagwidth@
+            \iftagsleft@
+                \ifdim\dimen@>\@mathmargin
+                    \global\shifttag@true
+                \fi
+            \else
+                \advance\dimen@\@mathmargin
+                \advance\dimen@\wdz@
+                \ifdim\dimen@>\displaywidth
+                   \global\shifttag@true
+                \fi
+            \fi
+        \fi
+    \else
+        \global\eqnshift@\displaywidth
+        \global\advance\eqnshift@-\wdz@
+        \ifdim\tagwidth@>\z@
+            \multiply\dimen@\tw@
+            \advance\dimen@\wdz@
+            \advance\dimen@\tagwidth@
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \ifdim\eqnshift@<4\tagwidth@
+                    \global\advance\eqnshift@-\tagwidth@
+                \fi
+            \fi
+        \fi
+        \global\divide\eqnshift@\tw@
+        \iftagsleft@
+            \global\eqnshift@-\eqnshift@
+            \global\advance\eqnshift@\displaywidth
+            \global\advance\eqnshift@-\wdz@
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+    \fi
+}
+\def\place@tag@gather{%
+    \iftagsleft@
+        \kern-\gdisplaywidth@
+        \ifshifttag@
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+            \global\shifttag@false
+        \else
+            \rlap{\boxz@}%
+        \fi
+    \else
+        \ifdim\totwidth@>\displaywidth
+            \dimen@\totwidth@
+            \advance\dimen@-\displaywidth
+            \kern-\dimen@
+        \fi
+        \ifshifttag@
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+            \global\shifttag@false
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+\def\set@gather@field{%
+    \iftagsleft@
+        \global\lineht@\ht\z@
+    \else
+        \global\lineht@\dp\z@
+    \fi
+    \kern\eqnshift@
+    \boxz@
+    \hfil
+}
+\newif\ifxxat@
+
+\newif\ifcheckat@
+
+\let\xatlevel@\@empty
+\def\start@align#1#2#3{%
+    \let\xatlevel@#1% always \z@, \@ne, or \tw@
+    \maxfields@#3\relax
+    \ifnum\maxfields@>\m@ne
+        \checkat@true
+        \ifnum\xatlevel@=\tw@
+            \xxat@true
+        \fi
+        \multiply\maxfields@\tw@
+    \else
+        \checkat@false
+    \fi
+    \ifingather@
+        {\ifnum0=`}\fi
+        \DN@{\vcenter\bgroup\savealignstate@\align@#2}%
+    \else
+        \ifmmode
+            \nomath@env
+            \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+        \else
+            $$%
+            \DN@{\align@#2}%
+        \fi
+    \fi
+    \collect@body\next@
+}
+\def\alignat{\start@align\z@\st@rredfalse}
+\@namedef{alignat*}{\start@align\z@\st@rredtrue}
+\def\xalignat{\start@align\@ne\st@rredfalse}
+\@namedef{xalignat*}{\start@align\@ne\st@rredtrue}
+\def\xxalignat{\start@align\tw@\st@rredtrue}
+\def\align{\start@align\@ne\st@rredfalse\m@ne}
+\@namedef{align*}{\start@align\@ne\st@rredtrue\m@ne}
+\def\flalign{\start@align\tw@\st@rredfalse\m@ne}
+\@namedef{flalign*}{\start@align\tw@\st@rredtrue\m@ne}
+\def\align@#1#2{%
+    \inany@true
+    \inalign@true
+    \displaybreak@
+    \intertext@
+    \ifingather@\else\displ@y@\fi
+    \Let@
+    \let\math@cr@@@\math@cr@@@align
+    \ifxxat@\else
+        \let\tag\tag@in@align
+    \fi
+    \let\label\label@in@display
+    #1% set st@r
+    \ifst@rred\else
+        \global\@eqnswtrue
+    \fi
+    \measure@{#2}%
+    \global\row@\z@
+    \tabskip\eqnshift@
+    \halign\bgroup
+        \span\align@preamble\crcr
+        #2%
+}
+\def\endalign{%
+        \math@cr
+        \black@\totwidth@
+    \egroup
+    \ifingather@
+        \restorealignstate@
+        \egroup
+        \nonumber
+        \ifnum0=`{\fi}%
+    \else
+        $$%
+    \fi
+    \global\@ignoretrue
+}
+\@xp\let\csname endalign*\endcsname\endalign
+\let\endxalignat\endalign
+\@xp\let\csname endxalignat*\endcsname\endalign
+\let\endxxalignat\endalign
+\let\endalignat\endalign
+\@xp\let\csname endalignat*\endcsname\endalign
+\let\endflalign\endalign
+\@xp\let\csname endflalign*\endcsname\endalign
+\def\math@cr@@@align{%
+    \kern-\alignsep@
+    \ifst@rred\nonumber\fi
+    \if@eqnsw \global\tag@true \fi
+    \global\advance\row@\@ne
+    \iftag@
+        \add@amps\maxfields@
+        \omit
+        \setboxz@h{\@lign\strut@{\make@display@tag}}%
+        \place@tag
+    \fi
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \global\lineht@\z@
+    \cr
+}
+\def\math@cr@@@align@measure{%
+   &\omit
+    \global\advance\row@\@ne
+    \ifst@rred\nonumber\fi
+    \if@eqnsw \global\tag@true \fi
+    \ifnum\column@>\maxfields@
+        \ifcheckat@
+            \begingroup
+              \measuring@false
+              \@amsmath@err{Extra & on this line}%
+                {\the\andhelp@}% "An extra & here is disastrous"
+            \endgroup
+        \else
+            \global\maxfields@\column@
+        \fi
+    \fi
+    \setboxz@h{\@lign\strut@{%
+        \if@eqnsw
+            \stepcounter{equation}%
+            \tagform@\theequation
+        \else
+            \iftag@\df@tag\fi
+        \fi
+    }}%
+    \savetaglength@
+    \ifst@rred\else\global\@eqnswtrue\fi
+    \cr
+}
+\let\field@lengths\@empty
+
+\def\savefieldlength@{%
+    \begingroup
+        \let\or\relax
+        \xdef\field@lengths{%
+            \field@lengths
+            \ifnum\column@=0
+                \or
+            \else
+                ,%
+            \fi
+            \the\wdz@
+        }%
+    \endgroup
+}
+
+\def\fieldlengths@#1{%
+    \ifcase\@xp#1\field@lengths\fi
+}
+\let\maxcolumn@widths\@empty
+\def\maxcol@width#1{%
+    \ifcase\@xp#1\maxcolumn@widths\fi\relax
+}
+\def\measure@#1{%
+    \begingroup
+        \measuring@true
+        \eqnshift@\z@
+        \alignsep@\z@
+        \global\let\tag@lengths\@empty
+        \global\let\field@lengths\@empty
+        \savecounters@
+        \global\setbox0\vbox{%
+            \let\math@cr@@@\math@cr@@@align@measure
+            \everycr{\noalign{\global\tag@false
+              \global\let\raise@tag\@empty \global\column@\z@}}%
+            \let\label\@gobble
+            \global\row@\z@
+            \tabskip\z@
+            \halign{\span\align@preamble\crcr
+                #1%
+                \math@cr@@@
+                \column@\z@
+                \add@amps\maxfields@\cr
+            }%
+        }%
+        \restorecounters@
+        \ifodd\maxfields@
+            \global\advance\maxfields@\@ne
+        \fi
+        \ifnum\xatlevel@=\tw@
+            \ifnum\maxfields@<\thr@@
+                \let\xatlevel@\z@
+            \fi
+        \fi
+        \setbox0\vbox{%
+            \unvbox0
+            \unpenalty
+            \global\setbox1\lastbox
+        }%
+        \global\totwidth@\wd1
+        \if@fleqn
+           \global\advance\totwidth@\@mathmargin
+        \fi
+        \global\let\maxcolumn@widths\@empty
+        \begingroup
+            \let\or\relax
+            \loop
+               \setbox1\hbox{%
+                   \unhbox1
+                   \unskip
+                   \global\setbox0\lastbox
+               }%
+                \ifhbox0
+                    \xdef\maxcolumn@widths{ \or \the\wd0 \maxcolumn@widths}%
+            \repeat
+        \endgroup
+        \dimen@\displaywidth
+        \advance\dimen@-\totwidth@
+        \ifcase\xatlevel@
+            \global\alignsep@\z@
+            \let\minalignsep\z@
+            \@tempcntb\z@
+            \if@fleqn
+                \@tempcnta\@ne
+                \global\eqnshift@\@mathmargin
+            \else
+                \@tempcnta\tw@
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+            \fi
+        \or
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \@tempcnta\@tempcntb
+            \advance\@tempcntb\m@ne
+            \if@fleqn
+                \global\eqnshift@\@mathmargin
+                \alignsep@\dimen@
+                \global\divide\alignsep@\@tempcnta
+            \else
+                \global\advance\@tempcnta\@ne
+                \global\eqnshift@\dimen@
+                \global\divide\eqnshift@\@tempcnta
+                \global\alignsep@\eqnshift@
+            \fi
+        \or
+            \@tempcntb\maxfields@
+            \divide\@tempcntb\tw@
+            \global\advance\@tempcntb\m@ne
+            \global\@tempcnta\@tempcntb
+            \global\eqnshift@\z@
+            \global\alignsep@\dimen@
+            \if@fleqn
+                \advance\alignsep@\@mathmargin\relax
+            \fi
+            \global\divide\alignsep@\@tempcntb
+        \fi
+        \ifdim\alignsep@<\minalignsep\relax
+            \global\alignsep@\minalignsep\relax
+            \ifdim\eqnshift@>\z@
+                \if@fleqn\else
+                    \eqnshift@\displaywidth
+                    \advance\eqnshift@-\totwidth@
+                    \advance\eqnshift@-\@tempcntb\alignsep@
+                    \global\divide\eqnshift@\tw@
+                \fi
+            \fi
+        \fi
+        \ifdim\eqnshift@<\z@
+            \global\eqnshift@\z@
+        \fi
+        \calc@shift@align
+        \tagshift@\totwidth@
+        \advance\tagshift@\@tempcntb\alignsep@
+        \if@fleqn
+            \ifnum\xatlevel@=\tw@
+                \global\advance\tagshift@-\@mathmargin\relax
+            \fi
+        \else
+            \global\advance\tagshift@\eqnshift@
+        \fi
+        \iftagsleft@ \else
+            \global\advance\tagshift@-\displaywidth
+        \fi
+        \dimen@\minalignsep\relax
+        \advance\totwidth@\@tempcntb\dimen@
+        \ifdim\totwidth@>\displaywidth
+            \global\let\displaywidth@\totwidth@
+        \else
+            \global\let\displaywidth@\displaywidth
+        \fi
+    \endgroup
+}
+\iftagsleft@\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \@tempdima\@mathmargin\relax
+            \advance\@tempdima-\mintagsep\relax
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@lf{%
+        \ifdim\eqnshift@=\z@
+            \global\eqnshift@\@mathmargin\relax
+            \alignsep@\displaywidth
+            \advance\alignsep@-\totwidth@
+            \global\divide\alignsep@\@tempcntb
+            \ifdim\alignsep@<\minalignsep\relax
+                \global\alignsep@\minalignsep\relax
+            \fi
+        \fi
+        \ifdim\tag@width\row@>\@tempdima
+            \saveshift@1%
+        \else
+            \saveshift@0%
+        \fi
+    }
+\fi\fi
+\iftagsleft@\else\if@fleqn
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rf
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@rf{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+            \advance\column@\m@ne
+            \divide\column@\tw@
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\eqnshift@
+            \advance\@tempdima\@tempdimc\relax
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\alignsep@\relax
+                \multiply\dimen@\@tempcntb
+                \advance\dimen@\@tempdima
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                    \else
+                        \global\alignsep@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+\iftagsleft@\else\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>0
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@rc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@rc{%
+        \column@\z@
+        \@tempdimb\z@
+        \@tempdimc\z@
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@rcalc@width
+        }%
+        \begingroup
+            \advance\column@\m@ne
+            \divide\column@\tw@
+            \ifnum\@tempcntb>\column@
+                \advance\@tempcnta-\@tempcntb
+                \advance\@tempcnta\column@
+                \@tempcntb\column@
+            \fi
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\@tempdimc
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\eqnshift@
+                \advance\dimen@\@tempdima
+                \advance\dimen@\@tempcntb\alignsep@
+                \advance\dimen@\tagwidth@
+                \ifdim\dimen@>\displaywidth
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \eqnshift@\displaywidth
+                        \advance\eqnshift@-\@tempdima
+                        \advance\eqnshift@-\@tempcntb\alignsep@
+                        \global\divide\eqnshift@\tw@
+                    \else
+                        \ifdim\dimen@<\eqnshift@
+                            \ifdim\dimen@<\z@
+                                \global\eqnshift@\z@
+                            \else
+                                \global\eqnshift@\dimen@
+                            \fi
+                        \fi
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+\fi\fi
+\iftagsleft@\else
+    \def\x@rcalc@width{%
+        \ifdim\@tempa > \z@
+            \advance\@tempdimc\@tempdimb
+            \ifodd\column@
+                \advance\@tempdimc\maxcol@width\column@
+                \@tempdimb\z@
+            \else
+                \advance\@tempdimc\@tempa\relax
+                \@tempdimb\maxcol@width\column@
+                \advance\@tempdimb-\@tempa\relax
+            \fi
+        \else
+            \advance\@tempdimb\maxcol@width\column@\relax
+        \fi
+    }
+\fi
+\iftagsleft@\if@fleqn\else
+    \def\calc@shift@align{%
+        \global\let\tag@shifts\@empty
+        \begingroup
+            \loop
+                \ifnum\row@>\z@
+                    \ifdim\tag@width\row@>\z@
+                        \x@calc@shift@lc
+                    \else
+                        \saveshift@0%
+                    \fi
+                    \advance\row@\m@ne
+            \repeat
+        \endgroup
+    }
+    \def\x@calc@shift@lc{%
+        \column@\z@
+        \@tempdima\z@ % ``width of equation''
+        \@tempdimb\z@ % ``indent of equation''
+        \edef\@tempb{\fieldlengths@\row@}%
+        \@for\@tempa:=\@tempb\do{%
+            \advance\column@\@ne
+            \x@lcalc@width
+        }%
+        \begingroup
+            \tagwidth@\tag@width\row@\relax
+            \@tempdima\totwidth@
+            \advance\@tempdima-\@tempdimb
+            \advance\@tempdima\tagwidth@
+            \dimen@\minalignsep\relax
+            \multiply\dimen@\@tempcntb
+            \advance\dimen@\mintagsep\relax
+            \ifnum\xatlevel@=\tw@ \else
+                \advance\dimen@\mintagsep\relax
+            \fi
+            \advance\dimen@\@tempdima
+            \ifdim\dimen@>\displaywidth
+                \saveshift@1%
+            \else
+                \saveshift@0%
+                \dimen@\alignsep@
+                \multiply\dimen@\count@
+                \advance\dimen@\eqnshift@
+                \advance\dimen@\@tempdimb
+                \ifdim\dimen@<2\tagwidth@
+                    \dimen@\displaywidth
+                    \advance\dimen@-\@tempdima
+                    \ifnum\xatlevel@=\tw@
+                        \advance\dimen@-\mintagsep\relax
+                    \fi
+                    \divide\dimen@\@tempcnta
+                    \ifdim\dimen@<\minalignsep\relax
+                        \global\alignsep@\minalignsep\relax
+                        \dimen@\displaywidth
+                        \advance\dimen@-\@tempdima
+                        \advance\dimen@-\@tempcntb\alignsep@
+                        \global\divide\dimen@\tw@
+                    \else
+                        \ifdim\dimen@<\alignsep@
+                            \global\alignsep@\dimen@
+                        \fi
+                    \fi
+                    \ifnum\xatlevel@=\tw@
+                        \dimen@\mintagsep\relax
+                    \fi
+                    \advance\dimen@\tagwidth@
+                    \advance\dimen@-\@tempdimb
+                    \advance\dimen@-\count@\alignsep@
+                    \ifdim\dimen@>\eqnshift@
+                        \global\eqnshift@\dimen@
+                    \fi
+                \fi
+            \fi
+        \endgroup
+    }
+    \def\x@lcalc@width{%
+        \ifdim\@tempdima = \z@
+            \ifdim\@tempa > \z@
+                \@tempdima\p@
+                \ifodd\column@
+                    \advance\@tempdimb \maxcol@width\column@
+                    \advance\@tempdimb-\@tempa
+                \fi
+                \count@\column@
+                \advance\count@\m@ne
+                \divide\count@\tw@
+                \advance\@tempcnta-\count@
+                \advance\@tempcntb-\count@
+            \else
+                \advance\@tempdimb \maxcol@width\column@\relax
+            \fi
+        \fi
+    }
+\fi\fi
+\def\place@tag{%
+    \iftagsleft@
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+            \rlap{\vbox{%
+                \normalbaselines
+                \boxz@
+                \vbox to\lineht@{}%
+                \raise@tag
+            }}%
+        \else
+            \rlap{\boxz@}%
+        \fi
+        \kern\displaywidth@
+    \else
+        \kern-\tagshift@
+        \if1\shift@tag\row@\relax
+            \llap{\vtop{%
+                \raise@tag
+                \normalbaselines
+                \setbox\@ne\null
+                \dp\@ne\lineht@
+                \box\@ne
+                \boxz@
+            }}%
+        \else
+            \llap{\boxz@}%
+        \fi
+    \fi
+}
+\def\align@preamble{%
+   &\hfil
+    \strut@
+    \setboxz@h{\@lign$\m@th\displaystyle{##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \tabskip\z@skip
+   &\setboxz@h{\@lign$\m@th\displaystyle{{}##}$}%
+    \ifmeasuring@\savefieldlength@\fi
+    \set@field
+    \hfil
+    \tabskip\alignsep@
+}
+\def\set@field{%
+    \column@plus
+    \iftagsleft@
+        \ifdim\ht\z@>\lineht@
+            \global\lineht@\ht\z@
+        \fi
+    \else
+        \ifdim\dp\z@>\lineht@
+            \global\lineht@\dp\z@
+        \fi
+    \fi
+    \boxz@
+}
+\def\split{%
+    \ifinany@
+        \@xp\insplit@
+    \else
+        \@xp\split@err
+    \fi
+}
+\edef\split@err{%
+    \@nx\@amsmath@err{%
+        \string\begin{split} won't work here%
+    }{%
+        \@xp\@nx\csname
+  Did you forget a preceding \string\begin{equation}?^^J%
+  If not, perhaps the `aligned' environment is what you want.\endcsname}%
+}
+\def\insplit@{%
+    \global\setbox\z@\vbox\bgroup
+        \Let@
+        \restore@math@cr
+        \default@tag % disallow use of \tag here
+        \ialign\bgroup
+            \hfil
+            \strut@
+            $\m@th\displaystyle{##}$%
+           &$\m@th\displaystyle{{}##}$%
+            \hfill % Why not \hfil?---dmj, 1994/12/28
+            \crcr
+}
+\def\endsplit{%
+            \crcr
+        \egroup
+    \egroup
+    \iftagsleft@
+        \@xp\lendsplit@
+    \else
+        \@xp\rendsplit@
+    \fi
+}
+\def\rendsplit@{%
+    \ifinalign@
+        \global\setbox9 \vtop{%
+            \unvcopy\z@
+            \global\setbox8 \lastbox
+            \unskip
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8
+            \unskip
+            \global\setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}%
+            }%
+        \else
+            \global\setbox7 \hbox{\unhbox\tw@\unskip}%
+            \gdef\split@{%
+                \global\@tempcnta\column@
+               &\setboxz@h{}%
+                \savetaglength@
+                \global\advance\row@\@ne
+                \vbox{\moveleft\wd\thr@@\box9}%
+                \crcr
+                \noalign{\global\lineht@\z@}%
+                \add@amps\@tempcnta
+                \box\thr@@
+               &\box7
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\boxz@}}%
+        \else
+            \gdef\split@{%
+                \boxz@
+            }%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+\def\lendsplit@{%
+    \global\setbox9\vtop{\unvcopy\z@}%
+    \ifinalign@
+        \setbox\@ne\vbox{%
+            \unvcopy\z@
+            \global\setbox8\lastbox
+        }%
+        \setbox\@ne\hbox{%
+            \unhcopy8%
+            \unskip
+            \setbox\tw@\lastbox
+            \unskip
+            \global\setbox\thr@@\lastbox
+        }%
+        \ifctagsplit@
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vcenter{\vbox{\moveleft\wd\thr@@\box9}}%
+            }%
+        \else
+            \gdef\split@{%
+                \hbox to\wd\thr@@{}%
+               &\vbox{\moveleft\wd\thr@@\box9}%
+            }%
+        \fi
+    \else
+        \ifctagsplit@
+            \gdef\split@{\vcenter{\box9}}%
+        \else
+            \gdef\split@{\box9}%
+        \fi
+    \fi
+    \aftergroup\split@
+}
+\newskip\multlinegap
+\multlinegap10pt
+\newskip\multlinetaggap
+\multlinetaggap10pt
+\def\start@multline#1{%
+    \RIfM@
+        \nomath@env
+        \DN@{\@namedef{end\@currenvir}{}\@gobble}%
+    \else
+        $$%
+        #1%
+        \ifst@rred
+            \nonumber
+        \else
+            \global\@eqnswtrue
+        \fi
+        \let\next@\multline@
+    \fi
+    \collect@body\next@
+}
+\def\multline{\start@multline\st@rredfalse}
+\@namedef{multline*}{\start@multline\st@rredtrue}
+\def\multline@#1{%
+    \inany@true
+    \Let@
+    \@display@init{\global\advance\row@\@ne \global\dspbrk@lvl\m@ne}%
+    \displaybreak@
+    \restore@math@cr
+    \let\tag\tag@in@align
+    \global\tag@false \global\let\raise@tag\@empty
+    \mmeasure@{#1}%
+    \let\tag\gobble@tag \let\label\@gobble
+    \tabskip \if@fleqn \@mathmargin \else \z@skip \fi
+    \totwidth@\displaywidth
+    \if@fleqn
+        \advance\totwidth@-\@mathmargin
+    \fi
+    \halign\bgroup
+        \hbox to\totwidth@{%
+            \if@fleqn
+                \hskip \@centering \relax
+            \else
+                \hfil
+            \fi
+            \strut@
+            $\m@th\displaystyle{}##$%
+            \hfil
+        }%
+        \crcr
+        \if@fleqn
+            \hskip-\@mathmargin
+        \else
+            \hfilneg
+        \fi
+        \iftagsleft@
+            \iftag@
+                \begingroup
+                    \ifshifttag@
+                        \rlap{\vbox{%
+                                \normalbaselines
+                                \hbox{%
+                                    \strut@
+                                    \make@display@tag
+                                }%
+                                \vbox to\lineht@{}%
+                                \raise@tag
+                        }}%
+                        \hskip\multlinegap
+                    \else
+                        \make@display@tag
+                        \hskip\multlinetaggap
+                    \fi
+                \endgroup
+            \else
+                \hskip\multlinegap
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+    #1%
+}
+\def\endmultline{%
+    \iftagsleft@
+        \@xp\lendmultline@
+    \else
+        \@xp\rendmultline@
+    \fi
+    \global\@ignoretrue
+}
+\@xp\let\csname endmultline*\endcsname=\endmultline
+\def\lendmultline@{%
+        \hfilneg
+        \hskip\multlinegap
+        \math@cr
+    \egroup
+    $$%
+}
+\def\rendmultline@{%
+    \iftag@
+        \begingroup
+            \ifshifttag@
+                \hskip\multlinegap
+                \llap{\vtop{%
+                    \raise@tag
+                    \normalbaselines
+                    \setbox\@ne\null
+                    \dp\@ne\lineht@
+                    \box\@ne
+                    \hbox{\strut@\make@display@tag}%
+                }}%
+            \else
+                \hskip\multlinetaggap
+                \make@display@tag
+            \fi
+        \endgroup
+    \else
+        \hskip\multlinegap
+    \fi
+    \hfilneg
+        \math@cr
+    \egroup$$%
+}
+\def\mmeasure@#1{%
+    \begingroup
+        \measuring@true
+        \def\label##1{%
+          \begingroup\measuring@false\label@in@display{##1}\endgroup}%
+        \def\math@cr@@@{\cr}%
+        \let\shoveleft\@iden \let\shoveright\@iden
+        \savecounters@
+        \global\row@\z@
+        \setbox\@ne\vbox{%
+            \global\let\df@tag\@empty
+            \halign{%
+                \setboxz@h{\@lign$\m@th\displaystyle{}##$}%
+                \iftagsleft@
+                    \ifnum\row@=\@ne
+                        \global\totwidth@\wdz@
+                        \global\lineht@\ht\z@
+                    \fi
+                \else
+                    \global\totwidth@\wdz@
+                    \global\lineht@\dp\z@
+                \fi
+                \crcr
+                #1%
+                \crcr
+            }%
+        }%
+        \ifx\df@tag\@empty\else\global\tag@true\fi
+        \if@eqnsw\global\tag@true\fi
+        \iftag@
+            \setboxz@h{%
+                \if@eqnsw
+                    \stepcounter{equation}%
+                    \tagform@\theequation
+                \else
+                    \df@tag
+                \fi
+            }%
+            \global\tagwidth@\wdz@
+            \dimen@\totwidth@
+            \advance\dimen@\tagwidth@
+            \advance\dimen@\multlinetaggap
+            \iftagsleft@\else
+                \if@fleqn
+                    \advance\dimen@\@mathmargin
+                \fi
+            \fi
+            \ifdim\dimen@>\displaywidth
+                \global\shifttag@true
+            \else
+                \global\shifttag@false
+            \fi
+        \fi
+        \restorecounters@
+    \endgroup
+}
+\iftagsleft@
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \hskip\multlinegap
+    }
+\else
+    \def\shoveright#1{%
+        #1%
+        \hfilneg
+        \iftag@
+            \ifshifttag@
+                \hskip\multlinegap
+            \else
+                \hskip\tagwidth@
+                \hskip\multlinetaggap
+            \fi
+        \else
+            \hskip\multlinegap
+        \fi
+    }
+\fi
+
+\if@fleqn
+    \def\shoveleft#1{#1}%
+\else
+    \iftagsleft@
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \iftag@
+                \ifshifttag@
+                    \hskip\multlinegap
+                \else
+                    \hskip\tagwidth@
+                    \hskip\multlinetaggap
+                \fi
+            \else
+                \hskip\multlinegap
+            \fi
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \else
+        \def\shoveleft#1{%
+            \setboxz@h{$\m@th\displaystyle{}#1$}%
+            \setbox\@ne\hbox{$\m@th\displaystyle#1$}%
+            \hfilneg
+            \hskip\multlinegap
+            \hskip.5\wd\@ne
+            \hskip-.5\wdz@
+            #1%
+        }
+    \fi
+\fi
+\def\[{%
+    \RIfM@
+        \@badmath
+    \else
+        \DN@{%
+            $$%
+            \ingather@true
+            \inany@true
+            \def\\{\@amsmath@err{\Invalid@@\\}\@eha}%
+            \tabskip\@mathmargin
+            \halign to\displaywidth\bgroup
+                \if@fleqn\else\hfil\fi
+                \setboxz@h{$\m@th\displaystyle{####}$}%
+                \global\totwidth@\wdz@
+                \boxz@
+                \hfil
+                \tabskip\@centering
+                \cr
+        }%
+        \@xp\next@
+    \fi
+}
+
+\def\]{%
+    \RIfM@
+        \DN@{%
+                \crcr
+            \black@\totwidth@
+            \egroup
+            $$%
+        }%
+        \@xp\next@
+    \else
+        \@badmath
+    \fi
+}
+\@xp\def\@xp\@arrayparboxrestore\@xp{\@arrayparboxrestore
+  \inany@false\ingather@false\inalign@false \default@tag}
+\def\equation{\gather\def\\{\@amsmath@err{\Invalid@@\\}\@eha}}
+\def\endequation{\endgather}
+\newenvironment{equation*}{%
+  \equation
+}{%
+  \nonumber\endequation
+}
+%%%%%%%%%%%%%%%%% end of amsmath.sty code
+
+\newcommand{\uppercasenonmath}[1]{\toks@\@emptytoks
+  \@xp\@skipmath\@xp\@empty#1$$%
+  \edef#1{\@nx\@upprep\the\toks@}%
+}
+\newcommand{\@upprep}{%
+  \spaceskip1.3\fontdimen2\font plus1.3\fontdimen3\font
+  \upchars@}
+\newcommand{\upchars@}{%
+  \def\ss{SS}\def\i{I}\def\j{J}\def\ae{\AE}\def\oe{\OE}%
+  \def\o{\O}\def\aa{\AA}\def\l{\L}\def\Mc{M{\scshape c}}}
+\newcommand{\@skipmath}{}
+\long\def\@skipmath#1$#2${%
+  \@xskipmath#1\(\)%
+  \@ifnotempty{#2}{\toks@\@xp{\the\toks@$#2$}\@skipmath\@empty}}%
+\newcommand{\@xskipmath}{}
+\long\def\@xskipmath#1\(#2\){%
+  \uppercase{\toks@\@xp\@xp\@xp{\@xp\the\@xp\toks@#1}}%
+  \@ifnotempty{#2}{\toks@\@xp{\the\toks@\(#2\)}\@xskipmath\@empty}}%
+\newcommand{\today}{%
+  \relax\ifcase\month\or
+  January\or February\or March\or April\or May\or June\or
+  July\or August\or September\or October\or November\or December\fi
+  \space\number\day, \number\year}
+\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
+\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
+\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
+\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
+\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
+\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
+\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
+\renewcommand*{\title}[2][]{\gdef\shorttitle{#1}\gdef\@title{#2}}
+\edef\title{\@nx\@dblarg
+  \@xp\@nx\csname\string\title\endcsname}
+\renewcommand{\author}[2][]{%
+  \ifx\@empty\authors
+    \gdef\shortauthors{#1}\gdef\authors{#2}%
+  \else
+    \g@addto@macro\shortauthors{\and#1}%
+    \g@addto@macro\authors{\and#2}%
+    \g@addto@macro\addresses{\author{}}%
+  \fi
+}
+\edef\author{\@nx\@dblarg
+  \@xp\@nx\csname\string\author\endcsname}
+\let\shortauthors\@empty   \let\authors\@empty
+\let\addresses\@empty      \let\thankses\@empty
+\newcommand{\address}[2][]{\g@addto@macro\addresses{\address{#1}{#2}}}
+\newcommand{\curraddr}[2][]{\g@addto@macro\addresses{\curraddr{#1}{#2}}}
+\newcommand{\email}[2][]{\g@addto@macro\addresses{\email{#1}{#2}}}
+\newcommand{\urladdr}[2][]{\g@addto@macro\addresses{\urladdr{#1}{#2}}}
+\renewcommand{\thanks}[1]{\g@addto@macro\thankses{\thanks{#1}}}
+\def\enddoc@text{\ifx\@empty\@translators \else\@settranslators\fi
+  \ifx\@empty\addresses \else\@setaddresses\fi}
+\AtEndDocument{\enddoc@text}
+\let\@date\@empty
+\def\dedicatory#1{\def\@dedicatory{#1}}
+\let\@dedicatory=\@empty
+\def\keywords#1{\def\@keywords{#1}}
+\let\@keywords=\@empty
+\def\twokname{2000}
+\newcommand\subjclass[2][]{%                 8/2000 ynm
+\def\testname{#1}\ifx\testname\twokname%
+\def\subjclassname{\textup{2000} Mathematics Subject 
+Classification}\fi\def\@subjclass{#2}}  % subj
+\let\@subjclass=\@empty
+\def\commby#1{\def\@commby{(Communicated by #1)}}
+\let\@commby=\@empty
+\def\translator#1{%
+  \ifx\@empty\@translators \def\@translators{#1}%
+  \else\g@addto@macro\@translators{\and#1}\fi}
+\let\@translators=\@empty
+\def\@settranslators{\par\begingroup
+  \addvspace{6\p@\@plus9\p@}%
+  \hbox to\columnwidth{\hss\normalfont\normalsize
+    Translated by %
+    \andify\@translators \uppercasenonmath\@translators
+    \@translators}
+  \endgroup
+}
+\newcommand{\xandlist}[4]{\@andlista{{#1}{#2}{#3}}#4\and\and}
+\def\@andlista#1#2\and#3\and{\@andlistc{#2}\@ifnotempty{#3}{%
+  \@andlistb#1{#3}}}
+\def\@andlistb#1#2#3#4#5\and{%
+  \@ifempty{#5}{%
+    \@andlistc{#2#4}%
+  }{%
+    \@andlistc{#1#4}\@andlistb{#1}{#3}{#3}{#5}%
+  }}
+\let\@andlistc\@iden
+\newcommand{\nxandlist}[4]{%
+  \def\@andlistc##1{\toks@\@xp{\the\toks@##1}}%
+  \toks@{\toks@\@emptytoks \@andlista{{#1}{#2}{#3}}}%
+  \the\@xp\toks@#4\and\and
+  \edef#4{\the\toks@}%
+  \let\@andlistc\@iden}
+\newcommand{\andify}{%
+  \nxandlist{\unskip, }{\unskip{} and~}{\unskip, and~}}
+\def\and{\unskip{ }and \ignorespaces}
+\def\maketitle{\par
+  \@topnum\z@ % this prevents figures from falling at the top of page 1
+  \@setcopyright
+  \uppercasenonmath\shorttitle
+  \ifx\@empty\shortauthors \let\shortauthors\shorttitle
+  \else \andify\shortauthors \uppercasenonmath\shortauthors \fi
+  \@maketitle@hook
+  \begingroup
+  \@maketitle
+  \toks@\@xp{\shortauthors}\@temptokena\@xp{\shorttitle}%
+  \edef\@tempa{\@nx\markboth{\the\toks@}{\the\@temptokena}}\@tempa
+  \endgroup
+  \thispagestyle{firstpage}% this sets first page specifications
+  \c@footnote\z@
+  \def\do##1{\let##1\relax}%
+  \do\maketitle \do\@maketitle \do\title \do\@xtitle \do\@title
+  \do\author \do\@xauthor \do\address \do\@xaddress
+  \do\email \do\@xemail \do\curraddr \do\@xcurraddr
+  \do\commby \do\@commby
+  \do\dedicatory \do\@dedicatory \do\thanks \do\thankses
+  \do\keywords \do\@keywords \do\subjclass \do\@subjclass
+}
+\def\@maketitle@hook{\global\let\@maketitle@hook\@empty}
+\def\@maketitle{%
+  \normalfont\normalsize
+  \let\@makefnmark\relax  \let\@thefnmark\relax
+  \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi
+  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
+  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
+  \ifx\@empty\thankses\else \@footnotetext{%
+    \def\par{\let\par\@par}\@setthanks}\fi
+  \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}%
+  \global\topskip42\p@\relax % 5.5pc   "   "   "     "     "
+  \@settitle
+  \ifx\@empty\authors \else \@setauthors \fi
+  \ifx\@empty\@dedicatory
+  \else
+    \baselineskip18\p@
+    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
+      \global\dimen@i\prevdepth}\prevdepth\dimen@i
+  \fi
+  \@setabstract
+  \normalsize
+  \if@titlepage
+    \newpage
+  \else
+    \dimen@34\p@ \advance\dimen@-\baselineskip
+    \vskip\dimen@\relax
+  \fi
+} % end \@maketitle
+\AtBeginDocument{%
+  \@ifundefined{publname}{%
+    \let\publname\@empty
+    \let\@serieslogo\@empty
+  }{%
+    \def\@serieslogo{%
+      \vbox to\headheight{%
+        \parindent\z@ \fontsize{6}{7\p@}\selectfont
+        \noindent\publname\newline
+        \volinfo, \pageinfo \@dateposted\newline \@PII\endgraf
+        \vss
+      }%
+    }%
+  }%
+}
+\AtBeginDocument{%
+  \@ifundefined{volinfo}{%
+    \def\volinfo{%
+      Volume \currentvolume, Number \number0\currentissue,
+      \currentmonth\ \currentyear
+    }%
+  }{}%
+}
+\def\issueinfo#1#2#3#4{\def\currentvolume{#1}\def\currentissue{#2}%
+  \def\currentmonth{#3}\def\currentyear{#4}}
+\issueinfo{00}% volume number
+  {0}%        % issue number
+  {Xxxx}%     % month
+  {XXXX}%     % year
+\def\copyrightinfo#1#2{\def\copyrightyear{#1}\def\copyrightholder{#2}}
+\copyrightinfo{0000}{(copyright holder)}
+\def\pagespan#1#2{\setcounter{page}{#1}%
+  \ifnum\c@page<\z@ \pagenumbering{roman}\setcounter{page}{-#1}\fi
+  \def\start@page{#1}\def\end@page{#2}}
+\pagespan{000}{000}
+\@ifundefined{pageinfo}{%
+  \def\pageinfo{%
+    \ifnum\start@page=\z@
+      Pages 000--000
+    \else
+      \ifx\start@page\end@page
+        Page \start@page
+      \else
+        Pages \start@page--\end@page
+      \fi
+    \fi}%
+}{}
+\@ifundefined{ISSN}{\def\ISSN{0000-0000}}{}
+\newcommand\PII[1]{\def\@PII{#1}}
+\PII{S \ISSN(XX)0000-0}
+\newinsert\copyins
+\skip\copyins=1.5pc                         % redefined in asl
+\count\copyins=1000 % magnification factor, 1000 = 100%
+\dimen\copyins=.5\textheight % maximum allowed per page
+\def\@combinefloats{%
+  \ifx \@toplist\@empty \else \@cflt \fi
+  \ifx \@botlist\@empty \else \@cflb \fi
+  \ifvoid\copyins \else \@cflci \fi
+}
+\def\@cflci{%
+  \if\if@twocolumn \if@firstcolumn F\else T\fi\else T\fi T%
+      \setbox\@outputbox\vbox{%
+        \unvbox\@outputbox
+        \vskip\skip\copyins
+        \hbox to\columnwidth{%
+          \hss\vbox to\z@{\vss\unvbox\copyins}}}%
+  \fi
+}
+\newcommand{\abstractname}{Abstract}
+\newcommand{\keywordsname}{Key words and phrases}
+\newcommand{\subjclassname}{\textup{1991} Mathematics Subject
+     Classification}
+\def\@tempb{amsart}
+\ifx\@classname\@tempb
+  \newcommand{\datename}{\textit{Date}:}
+\else
+  \newcommand{\datename}{Received by the editors}
+\fi
+\def\@setdate{\datename\ \@date\@addpunct.}
+\def\@setsubjclass{%
+  {\itshape\subjclassname.}\enspace\@subjclass\@addpunct.}
+\def\@setkeywords{%
+  {\itshape \keywordsname.}\enspace \@keywords\@addpunct.}
+% \def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses}
+\newbox\abstractbox
+\def\@setabstract{\@setabstracta \global\let\@setabstract\relax}
+\def\titlepage{%
+  \clearpage
+  \thispagestyle{empty}\setcounter{page}{0}}
+\def\endtitlepage{\newpage}
+\def\labelenumi{\theenumi.}
+\def\theenumi{\@arabic\c@enumi}
+\def\labelenumii{(\theenumii)}
+\def\theenumii{\@alph\c@enumii}
+\def\p@enumii{\theenumi}
+\def\labelenumiii{(\theenumiii)}
+\def\theenumiii{\@roman\c@enumiii}
+\def\p@enumiii{\theenumi(\theenumii)}
+\def\labelenumiv{(\theenumiv)}
+\def\theenumiv{\@Alph\c@enumiv}
+\def\p@enumiv{\p@enumiii\theenumiii}
+\def\labelitemi{$\m@th\bullet$}
+\def\labelitemii{\bfseries --}% \upshape already done by \itemize
+\def\labelitemiii{$\m@th\ast$}
+\def\labelitemiv{$\m@th\cdot$}
+\newenvironment{verse}{\let\\\@centercr
+  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent
+  \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]%
+}{%
+  \endlist
+}
+\let\endverse=\endlist % for efficiency
+\newenvironment{quote}{%
+  \list{}{\rightmargin\leftmargin}\item[]%
+}{%
+  \endlist
+}
+\let\endquote=\endlist % for efficiency
+\def\trivlist{\parsep\parskip\@nmbrlistfalse
+  \@trivlist \labelwidth\z@ \leftmargin\z@
+  \itemindent\z@
+  \let\@itemlabel\@empty
+  \def\makelabel##1{\upshape##1}}
+\renewenvironment{enumerate}{%
+  \ifnum \@enumdepth >3 \@toodeep\else
+      \advance\@enumdepth \@ne
+      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
+      {\csname label\@enumctr\endcsname}{\usecounter
+        {\@enumctr}\def\makelabel##1{\hss\llap{\upshape##1}}}\fi
+}{%
+  \endlist
+}
+\let\endenumerate=\endlist % for efficiency
+\renewenvironment{itemize}{%
+  \ifnum\@itemdepth>3 \@toodeep
+  \else \advance\@itemdepth\@ne
+    \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+    \list{\csname\@itemitem\endcsname}%
+      {\def\makelabel##1{\hss\llap{\upshape##1}}}%
+  \fi
+}{%
+  \endlist
+}
+\let\enditemize=\endlist % for efficiency
+\newcommand{\descriptionlabel}[1]{\hspace\labelsep \upshape\bfseries #1:}
+\newenvironment{description}{\list{}{%
+  \advance\leftmargini6\p@ \itemindent-12\p@
+  \labelwidth\z@ \let\makelabel\descriptionlabel}%
+}{
+  \endlist
+}
+\let\enddescription=\endlist % for efficiency
+\let\upn=\textup
+\AtBeginDocument{%
+  \settowidth\leftmargini{\labelenumi\hskip\labelsep}%
+  \advance\leftmargini by \normalparindent
+  \settowidth\leftmarginii{\labelenumii\hskip\labelsep}%
+  \advance\leftmarginii by 6pt
+  \settowidth\leftmarginiii{\labelenumiii\hskip\labelsep}%
+  \advance\leftmarginiii by 6pt
+  \settowidth\leftmarginiv{\labelenumiv\hskip\labelsep}%
+  \advance\leftmarginiv by 10pt
+  \leftmarginv=10pt
+  \leftmarginvi=10pt
+  \leftmargin=\leftmargini
+  \labelsep=5pt
+  \labelwidth=\leftmargini \advance\labelwidth-\labelsep
+  \@listi}
+\newskip\listisep
+\listisep\smallskipamount
+\def\@listI{\leftmargin\leftmargini \parsep\z@skip
+  \topsep\listisep \itemsep\z@skip
+  \listparindent\normalparindent}
+\let\@listi\@listI
+\def\@listii{\leftmargin\leftmarginii
+  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
+  \topsep\z@skip \parsep\z@skip \partopsep\z@skip \itemsep\z@skip}
+\def\@listiii{\leftmargin\leftmarginiii
+  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep}
+\def\@listiv{\leftmargin\leftmarginiv
+  \labelwidth\leftmarginiv \advance\labelwidth-\labelsep}
+\def\@listv{\leftmargin\leftmarginv
+  \labelwidth\leftmarginv \advance\labelwidth-\labelsep}
+\def\@listvi{\leftmargin\leftmarginvi
+  \labelwidth\leftmarginvi \advance\labelwidth-\labelsep}
+\def\@startsection#1#2#3#4#5#6{%
+\if@noskipsec \leavevmode \fi
+\par \@tempskipa #4\relax
+\@afterindenttrue
+\ifdim\@tempskipa<\z@\@tempskipa-\@tempskipa\@afterindentfalse\fi
+\if@nobreak\everypar{}\else
+\addpenalty\@secpenalty\addvspace\@tempskipa\fi
+\@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}%
+{\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
+}
+% This is in asl code, mildly ammended ynm
+%\def\@sect#1#2#3#4#5#6[#7]#8{%
+%  \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}%
+%  \ifnum #2>\c@secnumdepth \let\@secnumber\@empty
+%  \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi
+% \ifnum #2>\c@secnumdepth
+%   \let\@svsec\@empty
+% \else
+%    \refstepcounter{#1}%
+%    \edef\@svsec{\ifnum#2<\@m
+%       \@ifundefined{#1name}{}{%
+%         \ignorespaces\csname #1name\endcsname\space}\fi
+%       \@nx\textup{%
+%      \@nx\@secnumfont
+%         \csname the#1\endcsname.}\enspace
+%    }%
+%  \fi
+%  \@tempskipa #5\relax
+%  \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
+%    \begingroup #6\relax
+%    \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
+%    \endgroup
+%    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
+%  \else
+%  \def\@svsechd{#6\hskip #3\@svsec
+%    \@ifnotempty{#8}{\ignorespaces#8\unskip
+%       \@addpunct.}%
+%    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
+%  }%
+%  \fi
+%  \global\@nobreaktrue
+%  \@xsect{#5}}
+\let\@ssect\relax
+\newcounter{part}
+\newcounter{section}
+\newcounter{subsection}[section]
+\newcounter{subsubsection}[subsection]
+\newcounter{paragraph}[subsubsection]
+\renewcommand\thepart          {\arabic{part}}
+\renewcommand\theparagraph     {\thesubsubsection.\arabic{paragraph}}
+\setcounter{secnumdepth}{3}
+\def\partname{Part}
+\def\part{\@startsection{part}{0}%
+  \z@{\linespacing\@plus\linespacing}{.5\linespacing}%
+  {\normalfont\bfseries\raggedright}}
+\def\specialsection{\@startsection{section}{1}%
+  \z@{\linespacing\@plus\linespacing}{.5\linespacing}%
+  {\normalfont\centering}}
+\def\paragraph{\@startsection{paragraph}{4}%
+  \z@\z@{-\fontdimen2\font}%
+  \normalfont}
+\def\subparagraph{\@startsection{subparagraph}{5}%
+  \z@\z@{-\fontdimen2\font}%
+  \normalfont}
+\def\appendix{\par\c@section\z@ \c@subsection\z@
+   \let\sectionname\appendixname
+   \def\thesection{\@Alph\c@section}}
+\def\appendixname{Appendix}
+\def\@Roman#1{\@xp\@slowromancap
+  \romannumeral#1@}%
+\def\@slowromancap#1{\ifx @#1% then terminate
+  \else
+    \if i#1I\else\if v#1V\else\if x#1X\else\if l#1L\else\if
+    c#1C\else\if m#1M\else#1\fi\fi\fi\fi\fi\fi
+    \@xp\@slowromancap
+  \fi
+}
+\newcommand{\@pnumwidth}{1.6em}
+\newcommand{\@tocrmarg}{2.6em}
+\setcounter{tocdepth}{2}
+\def\@starttoc#1#2{\begingroup
+  \par\removelastskip\vskip\z@skip
+  \@startsection{}\@M\z@{\linespacing\@plus\linespacing}%
+    {.5\linespacing}{\centering\scshape}{#2}%
+  \ifx\contentsname#2%
+  \else \addcontentsline{toc}{section}{#2}\fi
+  \makeatletter
+  \@input{\jobname.#1}%
+  \if@filesw
+    \@xp\newwrite\csname tf@#1\endcsname
+    \immediate\@xp\openout\csname tf@#1\endcsname \jobname.#1\relax
+  \fi
+  \global\@nobreakfalse \endgroup
+  \addvspace{32\p@\@plus14\p@}%
+  \let\tableofcontents\relax
+}
+\def\contentsname{Contents}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\tableofcontents{\@starttoc{toc}\contentsname}
+\def\listoffigures{\@starttoc{lof}\listfigurename}
+\def\listoftables{\@starttoc{lot}\listtablename}
+\AtBeginDocument{%
+  \@for\@tempa:=-1,0,1,2,3\do{%
+    \@ifundefined{r@tocindent\@tempa}{%
+      \@xp\gdef\csname r@tocindent\@tempa\endcsname{0pt}}{}%
+  }%
+}
+\def\@writetocindents{%
+  \begingroup
+  \@for\@tempa:=-1,0,1,2,3\do{%
+    \immediate\write\@auxout{%
+      \string\newlabel{tocindent\@tempa}{%
+        \csname r@tocindent\@tempa\endcsname}}%
+  }%
+  \endgroup}
+\AtEndDocument{\@writetocindents}
+
+\let\indentlabel\@empty
+\def\@tochangmeasure#1{\sbox\z@{#1}%
+  \ifdim\wd\z@>\csname r@tocindent\@toclevel\endcsname\relax
+    \@xp\xdef\csname r@tocindent\@toclevel\endcsname{\the\wd\z@}%
+  \fi
+}
+\def\@toclevel{0}
+\def\@tocline#1#2#3#4#5#6#7{\relax
+  \ifnum #1>\c@tocdepth % then omit
+  \else
+    \par \addpenalty\@secpenalty\addvspace{#2}%
+    \begingroup \hyphenpenalty\@M
+    \@ifempty{#4}{%
+      \@tempdima\csname r@tocindent\number#1\endcsname\relax
+    }{%
+      \@tempdima#4\relax
+    }%
+    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
+    \rightskip\@pnumwidth plus1em \parfillskip-\@pnumwidth
+    #5\leavevmode\hskip-\@tempdima #6\relax
+    \hfil\hbox to\@pnumwidth{\@tocpagenum{#7}}\par
+    \nobreak
+    \endgroup
+  \fi}
+\def\@tocpagenum#1{\hss{\mdseries #1}}
+\def\@tocwrite#1{\@xp\@tocwriteb\csname toc#1\endcsname{#1}}
+\def\@tocwriteb#1#2#3{%
+  \begingroup
+    \def\@tocline##1##2##3##4##5##6{%
+      \ifnum##1>\c@tocdepth
+      \else \sbox\z@{##5\let\indentlabel\@tochangmeasure##6}\fi}%
+    \csname l@#2\endcsname{#1{\csname#2name\endcsname}{\@secnumber}{}}%
+  \endgroup
+  \addcontentsline{toc}{#2}%
+    {\protect#1{\csname#2name\endcsname}{\@secnumber}{#3}}}
+\def\l@section{\@tocline{1}{0pt}{1pc}{}{}}
+\newcommand{\tocsection}[3]{%
+  \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3}
+\def\l@subsection{\@tocline{2}{0pt}{1pc}{5pc}{}}
+\let\tocsubsection\tocsection
+\def\l@subsubsection{\@tocline{3}{0pt}{1pc}{7pc}{}}
+\let\tocsubsubsection\tocsection
+\def\l@part{\@tocline{-1}{12pt plus2pt}{0pt}{}{\bfseries}}
+\let\tocpart\tocsection
+\def\l@chapter{\@tocline{0}{8pt plus1pt}{0pt}{}{}}
+\let\tocchapter\tocsection
+\let\tocappendix\tocchapter
+\def\l@figure{\@tocline{0}{3pt plus2pt}{0pt}{}{}}
+\let\l@table=\l@figure
+\def\refname{References}               % redefined in asl
+\def\bibname{Bibliography}
+\def\bibliographystyle#1{%
+   \if@filesw\immediate\write\@auxout
+    {\string\bibstyle{#1}}\fi
+        \def\@tempa{#1}%
+        \def\@tempb{amsplain}%
+        \def\@tempc{}%
+        \ifx\@tempa\@tempb
+                \def\@biblabel##1{##1.}%
+                \def\bibsetup{}%
+        \else
+                \def\bibsetup{\labelsep6\p@}%
+        \ifx\@tempa\@tempc
+                \def\@biblabel##1{}%
+                \def\bibsetup{\labelwidth\z@ \leftmargin24\p@
+                \itemindent-24\p@
+                          \labelsep\z@ }%
+        \fi
+\fi}
+%\newenvironment{thebibliography}[1]{%
+%  \@xp\section\@xp*\@xp{\refname}%
+%  \normalfont\footnotesize\labelsep .5em\relax
+%  \renewcommand\theenumiv{\arabic{enumiv}}\let\p@enumiv\@empty
+%  \list{\@biblabel{\theenumiv}}{\settowidth\labelwidth{\@biblabel{#1}}%
+%    \leftmargin\labelwidth \advance\leftmargin\labelsep
+%    \usecounter{enumiv}}%
+%  \sloppy \clubpenalty\@M \widowpenalty\clubpenalty
+%  \sfcode`\.=\@m
+%}{%
+%  \def\@noitemerr{\@latex@warning{Empty `thebibliography' environment}}%
+%  \endlist
+%}
+\def\newblock{}
+\newcommand\MR[1]{\relax\ifhmode\unskip\spacefactor3000 \space\fi
+  \def\@tempa##1:##2:##3\@nil{%
+    \ifx @##2\@empty##1\else\textbf{##1:}##2\fi}%
+  \MRhref{#1}{MR \@tempa#1:@:\@nil}}
+\newcommand\URL{\begingroup
+  \def\@sverb##1{%
+    \def\@tempa####1##1{\@URL{####1}\egroup\endgroup}%
+    \@tempa}%
+  \verb}
+\let\URLhref\@gobble
+\def\@URL#1{\URLhref{#1}#1}
+\newif\if@restonecol
+\def\theindex{\@restonecoltrue\if@twocolumn\@restonecolfalse\fi
+  \columnseprule\z@ \columnsep 35\p@
+  \twocolumn[\@xp\section\@xp*\@xp{\indexname}]%
+  \thispagestyle{plain}%
+  \let\item\@idxitem
+  \parindent\z@  \parskip\z@\@plus.3\p@\relax
+  \footnotesize}
+\def\indexname{Index}
+\def\@idxitem{\par\hangindent 2em}
+\def\subitem{\par\hangindent 2em\hspace*{1em}}
+\def\subsubitem{\par\hangindent 3em\hspace*{2em}}
+\def\endtheindex{\if@restonecol\onecolumn\else\clearpage\fi}
+\def\indexspace{\par\bigskip}
+\def\footnoterule{\kern-.4\p@
+        \hrule\@width 5pc\kern11\p@\kern-\footnotesep}
+\def\@makefnmark{\hbox{$\m@th^{\@thefnmark}$}}
+\def\@makefntext{\indent\@makefnmark}
+\long\def\@footnotetext#1{\insert\footins{%
+  \normalfont\footnotesize
+  \interlinepenalty\interfootnotelinepenalty
+  \splittopskip\footnotesep \splitmaxdepth \dp\strutbox
+  \floatingpenalty\@MM \hsize\columnwidth
+  \@parboxrestore \parindent\normalparindent \sloppy
+  \edef\@currentlabel{\p@footnote\@thefnmark}%
+  \@makefntext{\rule\z@\footnotesep\ignorespaces#1\unskip\strut\par}}}
+\hfuzz=1pt \vfuzz=\hfuzz
+\def\sloppy{\tolerance9999 \emergencystretch 3em\relax}
+\setcounter{topnumber}{4}
+\setcounter{bottomnumber}{4}
+\setcounter{totalnumber}{4}
+\setcounter{dbltopnumber}{4}
+\renewcommand{\topfraction}{.97}
+\renewcommand{\bottomfraction}{.97}
+\renewcommand{\textfraction}{.03}
+\renewcommand{\floatpagefraction}{.9}
+\renewcommand{\dbltopfraction}{.97}
+\renewcommand{\dblfloatpagefraction}{.9}
+\setlength{\floatsep}{12pt plus 6pt minus 4pt}
+\setlength{\textfloatsep}{15pt plus 8pt minus 5pt}
+\setlength{\intextsep}{12pt plus 6pt minus 4pt}
+\setlength{\dblfloatsep}{12pt plus 6pt minus 4pt}
+\setlength{\dbltextfloatsep}{15pt plus 8pt minus 5pt}
+\setlength{\@fptop}{0pt}% removed ``plus 1fil''
+\setlength{\@fpsep}{8pt}% removed ``plus 2fil''
+\setlength{\@fpbot}{0pt plus 1fil}
+\setlength{\@dblfptop}{0pt}% removed ``plus 1fil''
+\setlength{\@dblfpsep}{8pt}% removed ``plus 2fil''
+\setlength{\@dblfpbot}{0pt plus 1fil}
+\newcommand{\fps@figure}{tbp}
+\newcommand{\fps@table}{tbp}
+\newcounter{figure}
+\def\@captionheadfont{\scshape}
+\def\@captionfont{\normalfont}
+\def\ftype@figure{1}
+\def\ext@figure{lof}
+\def\fnum@figure{\figurename\ \thefigure}
+\def\figurename{Figure}
+\newenvironment{figure}{%
+  \@float{figure}%
+}{%
+  \end@float
+}
+\newcounter{table}
+\def\ftype@table{2}
+\def\ext@table{lot}
+\def\fnum@table{\tablename\ \thetable}
+\def\tablename{Table}
+\newenvironment{table}{%
+  \@float{table}%
+}{%
+  \end@float
+}
+\def\@floatboxreset{\global\@minipagefalse \centering}
+\long\def\@makecaption#1#2{%
+  \setbox\@tempboxa\vbox{\color@setgroup
+    \advance\hsize-2\captionindent\noindent
+    \@captionfont\@captionheadfont#1\@xp\@ifnotempty\@xp
+        {\@cdr#2\@nil}{.\@captionfont\upshape\enspace#2}%
+    \unskip\kern-2\captionindent\par
+    \global\setbox\@ne\lastbox\color@endgroup}%
+  \ifhbox\@ne % the normal case
+    \setbox\@ne\hbox{\unhbox\@ne\unskip\unskip\unpenalty\unkern}%
+  \fi
+  \ifdim\wd\@tempboxa=\z@ % this means caption will fit on one line
+    \setbox\@ne\hbox to\columnwidth{\hss\kern-2\captionindent\box\@ne\hss}%
+  \else % tempboxa contained more than one line
+    \setbox\@ne\vbox{\unvbox\@tempboxa\parskip\z@skip
+        \noindent\unhbox\@ne\advance\hsize-2\captionindent\par}%
+\fi
+  \ifnum\@tempcnta<64 % if the float IS a figure...
+    \addvspace\abovecaptionskip
+    \moveright\captionindent\box\@ne
+  \else % if the float IS NOT a figure...
+    \moveright\captionindent\box\@ne
+    \nobreak
+    \vskip\belowcaptionskip
+  \fi
+\relax
+}
+\newskip\abovecaptionskip \abovecaptionskip=12pt \relax
+\newskip\belowcaptionskip \belowcaptionskip=12pt \relax
+\newdimen\captionindent \captionindent=3pc
+
+%%%%%%%%%%%%%%%%%%%%%%% code from amsthm.sty 
+\newcommand{\theoremstyle}[1]{%
+  \@ifundefined{th@#1}{%
+    \PackageWarning{amsthm}{Unknown theoremstyle `#1'}%
+    \thm@style{plain}%
+  }{%
+    \thm@style{#1}%
+  }%
+}
+\newtoks\thm@style
+\thm@style{plain}
+\newtoks\thm@bodyfont
+\thm@bodyfont{\itshape}
+\newtoks\thm@headfont
+\thm@headfont{\bfseries}
+\newtoks\thm@notefont
+\thm@notefont{}
+\newtoks\thm@headpunct
+\thm@headpunct{.}
+\newskip\thm@preskip \thm@preskip\topsep
+\newskip\thm@postskip \thm@postskip\topsep
+\renewcommand{\newtheorem}{\@ifstar{\@xnthm *}{\@xnthm \relax}}
+\def\@xnthm#1#2{%
+  \let\@tempa\relax
+  \@xp\@ifdefinable\csname #2\endcsname{%
+    \global\@xp\let\csname end#2\endcsname\@endtheorem
+    \ifx *#1% unnumbered, need to get one more mandatory arg
+      \edef\@tempa##1{%
+        \gdef\@xp\@nx\csname#2\endcsname{%
+          \@nx\@thm{\@xp\@nx\csname th@\the\thm@style\endcsname}%
+            {}{##1}}}%
+    \else % numbered theorem, need to check for optional arg
+      \def\@tempa{\@oparg{\@ynthm{#2}}[]}%
+    \fi
+  }%
+  \@tempa
+}
+\def\@ynthm#1[#2]#3{%
+  \ifx\relax#2\relax
+    \def\@tempa{\@oparg{\@xthm{#1}{#3}}[]}%
+  \else
+    \@ifundefined{c@#2}{%
+      \def\@tempa{\@nocounterr{#2}}%
+    }{%
+      \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#2\endcsname}%
+      \toks@{#3}%
+      \@xp\xdef\csname#1\endcsname{%
+        \@nx\@thm{%
+          \let\@nx\thm@swap
+            \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
+          \@xp\@nx\csname th@\the\thm@style\endcsname}%
+            {#2}{\the\toks@}}%
+      \let\@tempa\relax
+    }%
+  \fi
+  \@tempa
+}
+\def\@xthm#1#2[#3]{%
+  \ifx\relax#3\relax
+    \newcounter{#1}%
+  \else
+    \newcounter{#1}[#3]%
+    \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#3\endcsname
+      \@thmcountersep\@thmcounter{#1}}%
+  \fi
+  \toks@{#2}%
+  \@xp\xdef\csname#1\endcsname{%
+    \@nx\@thm{%
+      \let\@nx\thm@swap
+        \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
+      \@xp\@nx\csname th@\the\thm@style\endcsname}%
+      {#1}{\the\toks@}}%
+}
+\let\@ythm\relax
+\let\thmname\@iden \let\thmnote\@iden \let\thmnumber\@iden
+\providecommand\@upn{\textup}
+\def\thmhead@plain#1#2#3{%
+  \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }#2}%
+  \thmnote{ {\the\thm@notefont(#3)}}}
+\let\thmhead\thmhead@plain
+\def\swappedhead#1#2#3{%
+  \thmnumber{#2}\thmname{\@ifnotempty{#2}{. }#1}%
+  \thmnote{ {\the\thm@notefont(#3)}}}
+\let\thmheadnl\relax
+\def\@begintheorem#1#2[#3]{%
+  \item[\normalfont % reset in case body font is abnormal
+  \hskip\labelsep
+  \the\thm@headfont
+  \thm@indent
+  \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%
+  \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%
+  \@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%
+  \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%
+  \the\thm@headpunct]%
+  \@restorelabelsep
+  \thmheadnl % possibly a newline.
+  \ignorespaces}
+\def\nonslanted{\relax
+  \@xp\let\@xp\@tempa\csname\f@shape shape\endcsname
+  \ifx\@tempa\itshape\upshape
+  \else\ifx\@tempa\slshape\upshape\fi\fi}
+\def\swapnumbers{\edef\thm@swap{\if S\thm@swap N\else S\fi}}
+\def\thm@swap{N}%
+\let\@opargbegintheorem\relax
+\def\th@plain{%
+  \itshape % body font
+}
+\def\th@definition{%
+  \normalfont % body font
+}
+\def\th@remark{%
+  \thm@headfont{\itshape}%
+  \normalfont % body font
+  \thm@preskip\topsep
+  \divide\thm@preskip\tw@
+  \thm@postskip\thm@preskip
+}
+\def\@endtheorem{\endtrivlist\@endpefalse }
+\newcommand{\newtheoremstyle}[9]{%
+  \@ifempty{#5}{\dimen@\z@skip}{\dimen@#5\relax}%
+  \ifdim\dimen@=\z@
+    \toks@{#4\let\thm@indent\noindent}%
+  \else
+    \toks@{#4\def\thm@indent{\noindent\hbox to#5{}}}%
+  \fi
+  \def\@tempa{#8}\ifx\space\@tempa
+    \toks@\@xp{\the\toks@ \labelsep\fontdimen\tw@\font\relax}%
+  \else
+    \def\@tempb{\newline}%
+    \ifx\@tempb\@tempa
+      \toks@\@xp{\the\toks@ \labelsep\z@skip
+        \def\thmheadnl{%
+          \@noskipsectrue \global\@nobreaktrue
+          \everypar{\global\@minipagefalse \global\@newlistfalse
+            \global\@inlabelfalse \global\@nobreakfalse
+            {\setbox\z@\lastbox}\box\@labels\par
+            \nobreak\vskip-\parskip
+            \everypar{}\noindent}}%
+      }%
+    \else
+      \toks@\@xp{\the\toks@ \labelsep#8\relax}%
+    \fi
+  \fi
+  \begingroup \th@plain % to set \thm@preskip and postskip
+  \@defaultunits\@tempskipa#2\thm@preskip\relax\@nnil
+  \@defaultunits\@tempskipb#3\thm@postskip\relax\@nnil
+  \xdef\@gtempa{\thm@preskip\the\@tempskipa
+    \thm@postskip\the\@tempskipb\relax}%
+  \endgroup
+  \@temptokena\@xp{\@gtempa
+    \thm@headfont{#6}\thm@headpunct{#7}%
+  }%
+  \@ifempty{#9}{%
+    \let\thmhead\thmhead@plain
+  }{%
+    \@namedef{thmhead@#1}##1##2##3{#9}%
+    \@temptokena\@xp{\the\@temptokena
+      \@xp\let\@xp\thmhead\csname thmhead@#1\endcsname}%
+  }%
+  \@xp\xdef\csname th@#1\endcsname{\the\toks@ \the\@temptokena}%
+}
+\DeclareRobustCommand{\qed}{%
+  \ifmmode % if math mode, assume display: omit penalty etc.
+  \else \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
+  \fi
+  \quad\hbox{\qedsymbol}}
+\newcommand{\openbox}{\leavevmode
+  \hbox to.77778em{%
+  \hfil\vrule
+  \vbox to.675em{\hrule width.6em\vfil\hrule}%
+  \vrule\hfil}}
+%%%%%%%%%%%%%%%%%%%%%%% end of code from amsthm.sty
+
+\def\@swapped#1#2{#2%
+  \@ifnotempty{#1}{\@addpunct{.}\quad#1\unskip}}
+\def\thmhead@plain#1#2#3{%
+  \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
+  \thmnote{ \textmd{\upshape(#3)}}}
+\def\swappedhead@plain#1#2#3{%
+  \thmnumber{\@upn{#2}}\thmname{\@ifnotempty{#2}{. }#1}%
+  \thmnote{ \textmd{\upshape(#3)}}}
+\def\th@plain{%
+  \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain
+  \thm@preskip.5\baselineskip\@plus.2\baselineskip
+                                    \@minus.2\baselineskip
+  \thm@postskip\thm@preskip
+  \itshape
+}
+\def\th@definition{%
+  \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain
+  \thm@preskip.5\baselineskip\@plus.2\baselineskip
+                                    \@minus.2\baselineskip
+  \thm@postskip\thm@preskip
+  \upshape
+}
+\def\th@remark{%
+  \thm@headfont{\itshape}% heading font bold
+  \let\thmhead\thmhead@plain \let\swappedhead\swappedhead@plain
+  \thm@preskip.5\baselineskip\@plus.2\baselineskip
+                                    \@minus.2\baselineskip
+  \thm@postskip\thm@preskip
+  \upshape
+}
+\if@compatibility
+\let\@newpf\proof \let\proof\relax \let\endproof\relax
+\newenvironment{pf}{\@newpf[\proofname]}{\qed\endtrivlist}
+\newenvironment{pf*}[1]{\@newpf[#1]}{\qed\endtrivlist}
+\fi
+\def\nonbreakingspace{\unskip\nobreak\ \ignorespaces}
+\def~{\protect\nonbreakingspace}
+\def\@biblabel#1{\@ifnotempty{#1}{[#1]}}
+\def\@cite#1#2{{%
+  \m@th\upshape\mdseries[{#1\if@tempswa, #2\fi}]}}
+\@ifundefined{cite }{%
+  \expandafter\let\csname cite \endcsname\cite
+  \edef\cite{\@nx\protect\@xp\@nx\csname cite \endcsname}%
+}{}
+\def\fullwidthdisplay{\displayindent\z@ \displaywidth\columnwidth}
+\edef\@tempa{\noexpand\fullwidthdisplay\the\everydisplay}
+\everydisplay\expandafter{\@tempa}
+\newcommand\seename{see also}%
+\newcommand\see[2]{{\em \seename\/} #1}%
+\newcommand\printindex{\@input{\jobname.ind}}%
+\DeclareRobustCommand\textprime{\leavevmode
+  \raise.8ex\hbox{\check@mathfonts\the\scriptfont2 \char48 }}
+\hyphenation{acad-e-my acad-e-mies af-ter-thought anom-aly anom-alies
+an-ti-deriv-a-tive an-tin-o-my an-tin-o-mies apoth-e-o-ses
+apoth-e-o-sis ap-pen-dix ar-che-typ-al as-sign-a-ble as-sist-ant-ship
+as-ymp-tot-ic asyn-chro-nous at-trib-uted at-trib-ut-able bank-rupt
+bank-rupt-cy bi-dif-fer-en-tial blue-print busier busiest
+cat-a-stroph-ic cat-a-stroph-i-cally con-gress cross-hatched data-base
+de-fin-i-tive de-riv-a-tive dis-trib-ute dri-ver dri-vers eco-nom-ics
+econ-o-mist elit-ist equi-vari-ant ex-quis-ite ex-tra-or-di-nary
+flow-chart for-mi-da-ble forth-right friv-o-lous ge-o-des-ic
+ge-o-det-ic geo-met-ric griev-ance griev-ous griev-ous-ly
+hexa-dec-i-mal ho-lo-no-my ho-mo-thetic ideals idio-syn-crasy
+in-fin-ite-ly in-fin-i-tes-i-mal ir-rev-o-ca-ble key-stroke
+lam-en-ta-ble light-weight mal-a-prop-ism man-u-script mar-gin-al
+meta-bol-ic me-tab-o-lism meta-lan-guage me-trop-o-lis
+met-ro-pol-i-tan mi-nut-est mol-e-cule mono-chrome mono-pole
+mo-nop-oly mono-spline mo-not-o-nous mul-ti-fac-eted mul-ti-plic-able
+non-euclid-ean non-iso-mor-phic non-smooth par-a-digm par-a-bol-ic
+pa-rab-o-loid pa-ram-e-trize para-mount pen-ta-gon phe-nom-e-non
+post-script pre-am-ble pro-ce-dur-al pro-hib-i-tive pro-hib-i-tive-ly
+pseu-do-dif-fer-en-tial pseu-do-fi-nite pseu-do-nym qua-drat-ic
+quad-ra-ture qua-si-smooth qua-si-sta-tion-ary qua-si-tri-an-gu-lar
+quin-tes-sence quin-tes-sen-tial re-arrange-ment rec-tan-gle
+ret-ri-bu-tion retro-fit retro-fit-ted right-eous right-eous-ness
+ro-bot ro-bot-ics sched-ul-ing se-mes-ter semi-def-i-nite
+semi-ho-mo-thet-ic set-up se-vere-ly side-step sov-er-eign spe-cious
+spher-oid spher-oid-al star-tling star-tling-ly sta-tis-tics
+sto-chas-tic straight-est strange-ness strat-a-gem strong-hold
+sum-ma-ble symp-to-matic syn-chro-nous topo-graph-i-cal tra-vers-a-ble
+tra-ver-sal tra-ver-sals treach-ery turn-around un-at-tached
+un-err-ing-ly white-space wide-spread wing-spread wretch-ed
+wretch-ed-ly Eng-lish Euler-ian Feb-ru-ary Gauss-ian
+Hamil-ton-ian Her-mit-ian Jan-u-ary Japan-ese Kor-te-weg
+Le-gendre Mar-kov-ian Noe-ther-ian No-vem-ber Rie-mann-ian Sep-tem-ber}
+\def\calclayout{\advance\textheight -\headheight
+  \advance\textheight -\headsep
+  \oddsidemargin\paperwidth
+  \advance\oddsidemargin -\textwidth
+  \divide\oddsidemargin\tw@
+  \ifdim\oddsidemargin<.5truein \oddsidemargin.5truein \fi
+  \advance\oddsidemargin -1truein
+  \evensidemargin\oddsidemargin
+  \topmargin\paperheight \advance\topmargin -\textheight
+  \advance\topmargin -\headheight \advance\topmargin -\headsep
+  \divide\topmargin\tw@
+  \ifdim\topmargin<.5truein \topmargin.5truein \fi
+  \advance\topmargin -1truein\relax
+}
+\calclayout % initialize
+\pagenumbering{arabic}
+\thispagestyle{plain}
+
+%%%%%%%%%%%%%%%%%%%%%%% code from asl.sty
+
+\let\oldcal\mathcal              % so mathcal is always available
+
+% To make truescript give math cal with CMR fonts
+\def\truescript#1{\ifxfonts\truescriptalphabet{#1}%
+\else\oldcal{#1}\fi}
+
+%------------------------------- Basic dimension defaults
+\textheight=48pc          % = 19.395 cm
+\textwidth=29pc           % = 12.227 cm
+\headsep=12pt
+\headheight=6pt
+\topskip=12pt
+\footskip=24pt                    % It is 12pt in amsart.cls
+\parskip=0pt
+\parindent=1em \normalparindent\parindent
+\skip\copyins=2pc                 % .5pc larger than in amsart.cls
+
+\floatsep=\baselineskip           % Make these smaller, still not rigid
+\textfloatsep=\floatsep           % occassionally they need to be set rigid
+\intextsep=\floatsep
+
+% These have been tighter, but always needed fixing.
+% Here are the values in amsart:
+% \floatsep 24pt plus 2pt minus 2pt
+% \textfloatsep 24pt plus 2pt minus 2pt
+% \intextsep 24pt plus 2pt minus 2pt
+
+\calclayout % initialize again
+
+% Fudgefactor, from jsl-l
+\advance\textheight by.5\normalbaselineskip
+\divide\textheight by\normalbaselineskip
+\multiply\textheight by\normalbaselineskip
+
+%------------------------------- Logo and Copyright ifs
+
+\newif\ifLogoOn
+\LogoOnfalse
+\def\LogoOn{\global\LogoOntrue}
+
+% Set the fontsize of the logo larger for mainsize 11 or more
+\def\@logofontsize{\ifnum\@mainsize=10 \fontsize{6}{9\p@}
+   \else \fontsize{7}{10\p@}\fi}%
+
+\newif\ifCopyRight
+\CopyRightfalse
+\def\CopyRightOn{\global\CopyRighttrue}
+
+
+%------------------------------- The default is JSL with CMR 
+%                                fonts and no logo or copyright
+
+\def\ISSN{0022-4812}
+\def\journalname{The Journal of Symbolic Logic}
+
+% The following forms are used to compute the length of the logobox
+\def\shortjournalname{The Journal of Symbolic Logic}
+\def\longjournalname{The \hfil Journal \hfil of \hfil Symbolic \hfil Logic}
+\AtBeginDocument{\ifLogoOn\@setlogo\fi}
+
+%------------------------------- jsl option
+
+\DeclareOption{jsl}{%
+\LogoOn
+\CopyRightOn
+\def\jslname{this \textsc{Journal}} % for the bibliography
+} % End \DeclareOption{jsl}
+
+%------------------------------- xfonts option, uses the xgt fonts
+\newif\ifxfonts  
+\xfontsfalse
+
+\DeclareOption{xfonts}{%
+\xfontstrue
+\renewcommand{\sfdefault}{cmss}
+\renewcommand{\rmdefault}{xgt}
+\renewcommand{\ttdefault}{cmtt}
+
+\DeclareMathAlphabet{\mathrm}{OT1}{xgt}{m}{n}
+\DeclareMathAlphabet{\mathit}{OT1}{xgt}{m}{it}
+\DeclareMathAlphabet{\mathbf}{OT1}{xgt}{b}{it}
+
+\DeclareSymbolFont{letters}{OML}{xgt}{m}{it}
+\SetSymbolFont{letters}{bold}{OML}{xgt}{b}{it}
+\DeclareMathAlphabet{\truescriptalphabet}{OT1}{xgt}{m}{scr}
+
+\SetSymbolFont{operators}{normal}{OT1}{xgt}{m}{n}
+\SetSymbolFont{operators}{bold}{OT1}{xgt}{b}{n}
+
+% Must redefine \boldsymbol to correct an error in the .vf file
+\let\oldboldsymbol=\boldsymbol
+\def\boldsymbol#1{\ifx#1\Phi\oldboldsymbol{\Ypsilon}%
+\else\ifx#1\Psi\oldboldsymbol{\Phi}%
+\else\ifx#1\Ypsilon\oldboldsymbol{\Psi}\else\oldboldsymbol{#1}%
+\fi\fi\fi}
+
+% This allows the script font with all options
+\newcommand{\script}{\normalfont\fontshape{scr}\selectfont}
+
+% Use the fontshape th for theorems
+% This font uses italicizes all but punctuation and numerals
+\DeclareRobustCommand\xitshape{\fontshape{th}\selectfont}
+\def\th@plain{\xitshape}
+
+% For use in \textth{...} outside theorems (rare), Brian
+\DeclareTextFontCommand{\textth}{\xitshape}
+
+% Fix the lack of \l in small caps in the xgt fonts
+% by using the CMR small caps \l
+\def\@sc{sc}
+\let\@polishl=\l
+\def\l{\ifx\f@shape\@sc \raisebox{.06ex}{\fontfamily{cmr}\selectfont%
+\@polishl}\else\@polishl\fi}
+
+}% end \DeclareOption{xfonts}
+
+%------------------------------- xjsl option, jsl with Monotype Times fonts
+
+\DeclareOption{xjsl}{%
+\ExecuteOptions{jsl,xfonts}
+\let\mathcal=\truescriptalphabet    % \mathcal comes out script
+                                    % \oldcal gives cal
+}% end \DeclareOption{xjsl}
+
+%---------------------------------- bsl option, bsl with cmr fonts
+
+\DeclareOption{bsl}{%
+% We want Option 11pt to be executed.
+  \def\@mainsize{11}\def\@ptsize{1}%
+  \def\@typesizes{%
+    \or{6}{7}\or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}%
+    \or{\@xipt}{13}% normalsize
+    \or{\@xiipt}{14}\or{\@xivpt}{17}\or{\@xviipt}{20}%
+    \or{\@xxpt}{24}\or{\@xxvpt}{30}}%
+  \normalsize \linespacing=\baselineskip
+\def\ISSN{1079-8986}
+\def\journalname{The Bulletin of Symbolic Logic}
+\def\shortjournalname{The Bulletinof SymbolicLogic}
+\def\longjournalname{The \hfil Bulletin \hfil of \hfil Symbolic \hfil
+Logic}
+\def\bslname{this \textsc{Bulletin}}  % for the bibliography
+} % end \DeclareOption{bsl}
+
+%------------------------------- xbsl option, bsl with Monotype Times fonts
+
+\DeclareOption{xbsl}{%
+\ExecuteOptions{xfonts,bsl}
+\LogoOn
+\CopyRightOn
+} % end \DeclareOption{xbsl}
+
+%------------------------------- meeting option
+
+\DeclareOption{meeting}{%
+\ExecuteOptions{bsl}
+\LogoOn
+\CopyRightOn
+% Default sizes, initialization.
+% Assumes that we are setting at 11pt.
+\def\AbstractsOn{%
+        \normalfont
+        \let\large=\small           % This is 10/12%
+       \let\normalsize=\Small       % {9}{11\p@}
+      \let\small=\scriptsize        % This is OK, 8/10
+     \let\footnotesize=\scriptsize  % This is OK, 8/10
+    \let\scriptsize=\Tiny           % We want this to be 6/7
+    \fontsize{9}{11\p@}\selectfont  % the normalsize
+    \@adjustvertspacing
+}
+\AtBeginDocument{\AbstractsOn}
+
+\widowpenalty=0             % This is needed, to help with pagination
+\clubpenalty=0
+
+% The subtitle is, typically, "cosponsored by ..."
+% but it may be different, and it may have math in it
+% We put it where the Authors go
+% but allowing math and not affecting the headings
+
+\def\subtitletext{}
+\def\subtitle#1{\gdef\subtitletext{\uppercase{#1}}}
+
+% Here there are differences in vertical space from amsart
+\def\@setsubtitle{%
+  \begingroup
+  \trivlist
+\centering\fontsize{8}{9\p@}\selectfont       % Fontsize from jsl-l
+\@topsep36\p@\relax
+  \advance\@topsep by -2.5\baselineskip
+  \item\relax
+\subtitletext
+  \endtrivlist
+  \endgroup
+\vskip -0.5\baselineskip
+}
+
+\def\location#1{\vskip 14pt
+       {\gdef\@location{, #1}           % used by aslprod.sty
+\normalfont\large\bfseries\centering #1\endgraf}\nobreak
+                \vskip 14\p@}
+
+\def\conferencehead#1{\vskip 14pt
+             {\normalfont\large\bfseries\centering #1\endgraf}\nobreak
+                \vskip 5\p@}
+
+% Repeat \@maketitle from amsart.cls to change one skip.
+% and to replace setauthors by setsubtitle
+\def\@maketitle{%
+  \normalfont\normalsize
+  \let\@makefnmark\relax  \let\@thefnmark\relax
+  \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi
+  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
+  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
+  \ifx\@empty\thankses\else \@footnotetext{%
+    \def\par{\let\par\@par}\@setthanks}\fi
+  \@mkboth{\@nx\shorttitle}{\@nx\shorttitle}%
+  \global\topskip42\p@\relax % 5.5pc   "   "   "     "     "
+  \@settitle
+\ifx\@empty\subtitletext
+  \else
+  \@setsubtitle
+\fi
+  \ifx\@empty\@dedicatory
+  \else
+    \baselineskip18\p@
+    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
+      \global\dimen@i\prevdepth}\prevdepth\dimen@i
+  \fi
+  \@setabstract
+  \normalsize
+  \if@titlepage
+    \newpage
+  \else
+%    \dimen@34\p@ \advance\dimen@-\baselineskip   % (in amsart.csl)
+    \dimen@14\p@ \advance\dimen@-\baselineskip    % (from jsl-l)
+    \vskip\dimen@\relax
+  \fi
+} % end \@maketitle
+
+\def\blurbsig#1#2{\vspace{1pc}{\noindent\hfill #1\endgraf
+        \noindent\hfill {\sc #2}}\vskip\baselineskip}
+
+%  Resetting counters within each abstract
+\def\absauth#1{%
+        \setcounter{equation}{0}\setcounter{abscounter}{0}
+        \vspace{8pt}\noindent
+        {\llap{\raisebox{.3ex}{\small$\blacktriangleright$\hskip\labelsep}}%
+         \uppercase{#1}}\unskip,}
+
+% For proofs, redefine \absauth
+\let\standardabsauth=\absauth
+\def\proofcopy{\def\absauth{\newpage\standardabsauth}}
+
+\def\meetau#1{\endgraf#1\unskip,}
+\def\meettitle#1{{\it #1}\unskip.}
+% For the cases where the title ends with punctuation:
+\def\noperiodmeettitle#1{{\it #1}\unskip}
+\def\meetemail#1{\endgraf\noindent{\it E-mail}: {\tt #1}.}
+\def\affil#1{\endgraf\noindent{#1.}}
+\def\msort#1{\endgraf\noindent{\it Sorting}: #1.}
+
+\def\article#1{{\it #1}\unskip.\hspace{1em}}
+
+% Rarely, article names end with their own punctuation:
+\def\noperiodarticle#1{{\it #1}\unskip\hspace{1em}}
+
+% Even more rarely, the period at the end of the article following the
+% \hspace jumps to another line.
+\def\nospacearticle#1{{\it #1.}}
+
+% No title or spacing for REFERENCES
+% and use hand-set bibliography
+\oldbib
+%\let\cite\oldcite
+%\let\@citex\old@citex
+
+\renewenvironment{thebibliography}[1]{%
+  \vskip 2pt
+  \def\and{{\normalfont \lowercase{and}\ }}%
+  \list{[\@arabic\c@enumiv]\ }{%
+        \leftmargin\z@ \labelwidth\z@ \itemindent12\p@
+  \labelsep\z@ \usecounter{enumiv}}%
+  \sloppy \clubpenalty4000\relax \widowpenalty\clubpenalty
+  \sfcode`\.\@m}
+  {\endlist}
+
+% Fake theorem, to fix the theorem numbering,
+% so that it starts anew with each abstract.
+
+\newtheorem{abscounter}{abscoutner}
+
+\newtheorem{theorem}[abscounter]{Theorem}
+\newtheorem{lemma}[abscounter]{Lemma}
+\newtheorem{corollary}[abscounter]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{definition}[abscounter]{Definition}
+\newtheorem{example}[abscounter]{Example}
+
+\theoremstyle{plain} % 1/25/00, in case more theorems are introduced
+} % end \DeclareOption{meeting}
+
+%-------------------------------------- xmeeting option, with xgt fonts
+
+\DeclareOption{xmeeting}{%
+\ExecuteOptions{xbsl,meeting}
+} % end DeclareOption
+
+%------------------------------- notices option
+
+\DeclareOption{notices}{%
+\ExecuteOptions{bsl}
+\LogoOn
+\CopyRightfalse
+
+\AtBeginDocument{\Notices}
+\papertype{end notices}
+% \Notices is defined globally, as it is used in the covers
+
+% Title is not set, may be set to anything.
+
+\def\marginmark#1{%
+\global\let\email=\nemail               % sets this after \maketitle
+\global\let\urladdr=\nemail             % add so this makes sense
+\vskip 4pt\noindent\llap{\raisebox{.3ex}%
+{$\bullet$\hskip\labelsep}}{\sc #1}}
+
+\def\meetingdates#1{{\em #1,}}
+
+%% This command prints location information (if there is any) in italic
+%% and adds a period at the end:
+\def\meetingplace#1{{\em #1.}}
+
+% Repeat \@maketitle from amsart.cls to change one skip.
+\def\@maketitle{%
+  \normalfont\normalsize
+  \let\@makefnmark\relax  \let\@thefnmark\relax
+  \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi
+  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
+  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
+  \ifx\@empty\thankses\else \@footnotetext{%
+    \def\par{\let\par\@par}\@setthanks}\fi
+  \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}%
+  \global\topskip42\p@\relax % 5.5pc   "   "   "     "     "
+  \@settitle
+  \ifx\@empty\@dedicatory
+  \else
+    \baselineskip18\p@
+    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
+      \global\dimen@i\prevdepth}\prevdepth\dimen@i
+  \fi
+  \@setabstract
+  \normalsize
+  \if@titlepage
+    \newpage
+  \else
+%    \dimen@34\p@ \advance\dimen@-\baselineskip   % (in amsart.csl)
+    \dimen@24\p@ \advance\dimen@-\baselineskip    % (from jsl-l, adjusted)
+    \vskip\dimen@\relax
+  \fi
+} % end \@maketitle
+
+% The macros for \nemail are in the general macros below,
+% because they involve changing catcodes and this does not work
+% with the \ProcessOptions command.
+% Here we have only the re-definition for the notices option
+% \email=\nemail is set in the \marginmark command
+% 
+
+}% end \DeclareOption{notices}
+
+%------------------------------- xnotices option, with the xgt fonts
+
+\DeclareOption{xnotices}{%
+\ExecuteOptions{notices}
+\ExecuteOptions{xfonts}
+} % end \DeclareOption
+
+%------------------------------- reviews option, with the cmr fonts
+
+\DeclareOption{reviews}{%
+\ExecuteOptions{bsl}
+\LogoOn
+\CopyRightOn
+
+\title{Reviews}
+\papertype{reviews}
+\typesetter{H. Enderton}
+
+% Setting the sizes
+
+\def\BookRev{%
+        \normalfont
+        \let\large=\small           % 10/12
+       \let\normalsize=\Small       % 9/11
+      \let\small=\SMALL             % 8/10
+     \let\footnotesize=\SMALL       % 8/10
+    \let\scriptsize=\Tiny           % 6/7
+    \fontsize{9}{11\p@}\selectfont  % the normalsize
+    \@adjustvertspacing
+}
+\AtBeginDocument{\BookRev}
+
+\newcommand{\revaddress}[1]{\g@addto@macro\localaddresses{#1. }}%
+
+\newcommand{\revemail}[1]{\if#1\@empty \else%
+\g@addto@macro\localaddresses{%
+\unskip\nobreak\hfil\penalty50%
+\hskip .5em\nobreak
+\hbox{#1. }\hspace*{\fill}}\fi%
+\g@addto@macro\localaddresses{\par}}
+
+\newenvironment{review}{\vskip 15pt plus 1pt minus 1pt%
+\let\localaddresses\@empty%
+\let\reviewers\@empty%
+\let\revreviewers\@empty\let\@toctitle\@empty%
+\let\earlysignature\@empty%
+\let\@titleindex=0%
+\let\@reviewtitle\@empty}%
+{\if\earlysignature X\else\signature\fi%
+\if\@toctitle\@empty\tocerror\fi}
+
+% Macros to place the signature correctly
+
+\def\signature{\gdef\earlysignature{X}\@doreviewers\par%
+\localaddresses}
+% \signature places the signature where it occurs
+% (rather than at the end of the review)
+
+%\def\@doreviewers{%
+%\andify\reviewers%
+%\unskip\nobreak\hfil\penalty50%
+%\hskip2em\hbox{}\nobreak\hfill%
+%\hbox{\sc \reviewers}\hspace*{1em}}
+
+\def\@doreviewers{%
+\andify\reviewers%
+\par\rightline{\textsc{\reviewers}\hspace*{1em}}}
+
+% error message - redo to make better
+\def\tocerror{\typeout{^^J%
+No toctitle!^^J\stop}}
+
+\newcommand{\reviewer}[3][]{%
+  \ifx\@empty\reviewers
+    \gdef\reviewers{#2}%
+  \else
+    \g@addto@macro\reviewers{\protect\and#2}%
+  \fi
+\ifx\@empty\revreviewers
+\gdef\revreviewers{#3}%
+\gdef\firstreviewer{#3}
+\else
+\g@addto@macro\revreviewers{\and#3}%
+\fi
+}
+
+\def\toctitle#1{\gdef\@toctitle{#1}}
+\def\reviewtitle#1{\gdef\@titleindex{1}\gdef\@reviewtitle{#1}}
+
+\newcommand{\makereviewheading}[1][]%
+{% \ydowrite in reprod.sty
+{\noindent\large\if\@titleindex0%
+{{\bfseries \@toctitle}\hskip 1em{#1}\vskip 3pt}%
+\else\if\@reviewtitle\empty %
+\else{{\bfseries \@reviewtitle}}\fi\fi}}
+
+% To put some extra space between the preamble and the reviews
+% we introduce a command for the preamble
+
+\long\def\revinstructions#1{{#1}\vskip 5pt plus 1pt minus 1pt}
+
+
+\def\horspace{\hspace{.3em}} % to test and control the spaces
+
+\def\revau#1{%                 % same as anau, for compatibility
+    \def\and{{\normalfont and\ }}%
+        \vskip 0pt plus 1pt minus 1pt
+    {\sc #1}\unskip.\horspace}
+
+\def\anau#1{%
+    \def\and{{\normalfont and\ }}%
+        \vskip 0pt plus 1pt minus 1pt
+    {\sc #1}\unskip.\horspace}
+
+\renewcommand{\andify}{%
+  \nxandlist{\unskip, }{\unskip{} {\normalfont\lowercase{and}}~}%
+{\unskip, {\normalfont\lowercase{and}}~}}
+
+\def\and{\unskip{ }\mbox{\normalfont \lowercase{and}} \ignorespaces}
+
+
+\def\book#1{{\bfseries\itshape #1}\unskip{\bfseries\itshape.}\quad}
+
+\def\article#1{{\it #1}\unskip.\horspace}
+
+\def\journal#1{{\bfseries\itshape #1}\unskip, }
+
+% Repeat \@maketitle from amsart.cls to change one skip.
+\def\@maketitle{%
+  \normalfont\normalsize
+  \let\@makefnmark\relax  \let\@thefnmark\relax
+  \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi
+  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
+  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
+  \ifx\@empty\thankses\else \@footnotetext{%
+    \def\par{\let\par\@par}\@setthanks}\fi
+  \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}%
+  \global\topskip42\p@\relax % 5.5pc   "   "   "     "     "
+  \@settitle
+  \ifx\@empty\@dedicatory
+  \else
+    \baselineskip18\p@
+    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
+      \global\dimen@i\prevdepth}\prevdepth\dimen@i
+  \fi
+  \normalsize
+  \if@titlepage
+    \newpage
+  \else
+%    \dimen@34\p@ \advance\dimen@-\baselineskip   % (in amsart.csl)
+    \dimen@24\p@ \advance\dimen@-\baselineskip    % (from jsl-l, adjusted)
+    \vskip\dimen@\relax
+  \fi
+} % end \@maketitle
+
+% For the production version
+\InputIfFileExists{revprod.sty}{}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Follow Herb's macros
+
+%-------- Font changes
+
+\def\bi{\bfseries\itshape}
+\def\lss{\sf}
+\def\bss{\sf\bfseries}
+
+%--------
+
+\hyphenation{Eng-lish}
+\abovedisplayskip=2pt plus2pt
+\belowdisplayskip=2pt plus2pt
+
+%%%%%%%%%%%%%%%%%% revlocal.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Note: \normalfont does what \normalshape did before
+\def\Box{\square} %\square is defined in JSL stylefile; it does not
+\def\Boxs{\square\hskip 0.1em}     %reduce properly in scriptstyle.
+\def\Diamond{\lozenge}
+\def\lor{\mathbin{\mbox{\boldmath$\vee$}}}
+\def\ifthen{\mathbin{\mbox{\boldmath$\supset$}}} % horseshoe
+\def\land{\mathbin{\mbox{\boldmath$\wedge$}}}
+\def\lnot{\mbox{\boldmath$\sim\mskip -2.0mu$}} %9/98
+\def\amp{\mathbin{\mbox{\&}}}
+\def\turnStile{\mathbin{{|\hskip -0.47em\sim}}}  % symbol |~ "snake"
+\def\simr{\raise 0.5ex\hbox{$\sim$}}
+\def\preorder{\underset\simr\to\sqsubset}
+\def\sqr#1#2{{\vbox % was \vcenter   %%from Texbook p. 320
+        {\hrule height.#2pt
+    \hbox{\vrule width.#2pt height#1pt \kern#1pt
+        \vrule width.#2pt}
+    \hrule height.#2pt}}}
+%first number is size in pts, second is thickness in tenths of pts
+\def\soft{$\,^\prime\!$} %for use in italics
+\def\lag{\vbox{\offinterlineskip
+   \hbox{$\scriptscriptstyle\langle\!\langle$}\kern-0.1pt}
+   \hskip -0.6pt plus.3pt minus.2pt~ \hskip -2.7pt}
+\def\rag{~\hskip -2.7pt plus.3pt minus.2pt\vbox{\offinterlineskip
+   \hbox{$\scriptscriptstyle\rangle\!\rangle$}\kern-0.1pt}}
+\def\smirk{\mathrel
+    {\,\mid\hskip -2.1ex\lower 1ex\hbox{$\smile$}\hskip 0.1ex }}
+\def\notsmirk{\mathrel{\smirk\hskip -2.5ex / \hskip 0.5ex}}
+\def\ltx{La{\hskip -0.9pt}T{\hskip -0.8pt}e{\hskip -0.6pt}X} %LaTeX
+\def\sqdot{\mathbin{\vrule width0.3ex height0.3ex depth0ex}} %square dot
+\def\monus{-\kern -1.6ex{\raise1.0ex\hbox{.}}\kern 1.3ex}
+\def\supmonus{-\kern -1.1ex{\raise0.7ex\hbox{.}}\kern 0.5ex}
+\def\polishaccent{{\family{cmr}\shape{n}\selectfont\char32}}
+\def\smcl{\rlap{\polishaccent}l} %small cap Polish L
+
+\def\fishhook{\mathbin{{\vrule width0.4ex height0.80ex  %%for eight-
+    depth-0.62ex}\!{\text{\sf 3}}}}   %see 60:3 p 1009  %%point
+\def\boldsquare{\hspace{0.1em}
+     \mathchoice\sqr{4.8}4\sqr{4.8}4\sqr{3.6}3\sqr{2.7}2\hspace{0.1em}}
+
+%%end of revlocal.tex%%%%%%%%%%
+
+}% end \DeclareOption
+
+%--------------------------------------------- xreviews option, with xfonts
+
+\DeclareOption{xreviews}{%
+\ExecuteOptions{xfonts,reviews}
+}% end \DeclareOption
+
+%--------------------------------------------- obituary option
+% Typeset obits exactly as if they were articles
+% except for the following:
+% (1) Start with \documentclass[obituary]{amsart}
+%     or \documentclass[xobituary]{amsart}
+% (2) Enter the command
+% \subject{Smith, George W.}
+% of the person honored, in this "reversed" way.
+% (4) The author(s) is typeset at the end of the paper (\rightline)
+%     If you want it at some place earlier, put the command
+%     \signature at the place where you want it
+%     (This is sometimes needed if there are references,
+%      and the signature comes before the references.)
+%
+% The format ends up exactly as in articles, except that
+% addresses and email addresses will not print,
+% and so they need not be set. (They do not hurt.)
+
+\DeclareOption{obituary}{%
+\ExecuteOptions{bsl}
+
+% Make sure that address, email do not print
+\AtBeginDocument{\let\addresses\@empty \recdate{}}
+
+% Redefine \author so that the names come at the end
+\let\obitauthors\@empty
+\renewcommand{\author}[2][]{%
+  \ifx\@empty\obitauthors
+    \gdef\shortobitauthors{#1}\gdef\obitauthors{#2}%
+  \else
+    \g@addto@macro\shortobitauthors{\and#1}%
+    \g@addto@macro\obitauthors{\and#2}%
+  \fi
+}
+\renewcommand{\andify}{%
+  \nxandlist{\unskip, }{\unskip{} {\normalfont\lowercase{and}}~}%
+{\unskip, {\normalfont\lowercase{and}}~}}
+
+\def\and{\unskip{ }\mbox{\normalfont \lowercase{and}} \ignorespaces}
+
+% Macros to place the signature correctly
+\let\earlysignature\@empty
+\def\signature{\gdef\earlysignature{@ok}\@doobitauthors}
+\AtEndDocument{\ifx\@empty\earlysignature\@doobitauthors\fi}
+\def\@doobitauthors{\par\bigskip\noindent%
+\andify\obitauthors
+\rightline{\sc \obitauthors}}
+
+} % end \DeclareOption{obituary}
+
+%------------------------------- xobituary option
+\DeclareOption{xobituary}{%
+\ExecuteOptions{xfonts,obituary}
+\LogoOn
+\CopyRightOn
+} % end \DeclareOption{xobituary}
+
+%--------------------------------------------- errata option
+% Follow notices style
+\DeclareOption{errata}{%
+\LogoOnfalse
+\CopyRightfalse
+\AtBeginDocument{\Notices}
+% must specify xjsl or xbsl along with this, to make the difference
+% After the \maketitle use
+% \centerline{\uppercase{volume 3}}
+% \bigskip
+% Page 230.  ...
+% and then a \medskip separating the different errata
+} % end option errata
+
+%------------------------------------------- conference option
+
+\DeclareOption{conference}{%
+% Use the size of LC '98
+% and otherwise the JSL defaults  
+\textwidth=11.9cm
+\textheight=18.6cm
+%\let\ps@firstpage=\ps@empty
+% Redefine \ps@plain, used on the first page
+% to give identification and copyright
+
+% These must be defined for each conference
+\def\copyrightyear{1000}
+\def\confname{Meeting}
+
+% We set the Conference name and copyright ionfo
+% using \maketitle
+
+\let\oldmaketitle\maketitle
+\def\maketitle{\oldmaketitle\confinfo}
+
+\long\def\confinfo{%
+\begin{table}[b]%
+\vbox to -\footskip{%
+\begin{minipage}[t]{\textwidth}
+\fontsize{7}{8pt}\selectfont%
+\vskip -\baselineskip
+\textbf{\confname}\\
+\copyright\ \copyrightyear, 
+\textsc{Association for Symbolic Logic}
+\end{minipage}}
+\end{table}
+}
+
+} % End DeclareOption conference
+
+%----------------------------------------- option xconference
+
+\DeclareOption{xconference}{%
+\ExecuteOptions{conference,xfonts}
+}
+
+%----------------------------------------- option endnotes
+
+
+%*********************** Changes from amsart ***********************%
+
+% The changes from amsart are noted and commented out
+
+%------------------------------------ email for notices
+
+% These commands are also used in the covers,
+% so they are defined for all options.
+
+\def\Notices{%
+        \normalfont
+        \let\normalsize=\Small     % 9/11, per jsl-l
+        \let\small=\scriptsize     % 8/10, per jsl-l
+        \let\scriptsize=\Tiny      % 6/7,per jsl-l
+        \fontsize{9}{11\p@}\selectfont % the normalsize
+        \@adjustvertspacing}
+
+\def\weakdot{.\linebreak[0]}
+\def\weakat{@\linebreak[0]}
+\def\weakcolon{:\linebreak[0]}
+\def\weakslash{/\linebreak[0]}
+
+\newcommand{\emailcatcodes}{%
+  \catcode`\:=\active
+  \catcode`\.=\active
+  \catcode`\@=\active
+  \catcode`\/=\active
+}
+
+{\emailcatcodes
+\immediate \global\let:=\weakcolon
+\immediate \global\let.=\weakdot
+\immediate \global\let@=\weakat
+\immediate \global\let/=\weakslash
+}
+
+%% The catcodes are decided when the argument is read.  Thus we need
+%% to set the catcodes before reading the argument to \nemail.
+
+%\def\nemail{\begingroup\emailcatcodes\def~{$\sim$}\xemail}
+\def\nemail{\begingroup\emailcatcodes\def~{\urltilde}\xemail}
+%% The group ended here was started by the \nemail command:
+\def\xemail#1{\tt#1\endgroup}
+
+% \email=\nemail is implemented by the \marginmark command,
+% after \maketitle
+
+%-------------------------------- Thanks
+
+% Skip the . in \thanks, to allow for other punctuation (rare).
+%(amsart) \def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses}
+\def\@setthanks{\def\thanks##1{\par##1}\thankses}
+
+%-------------------------------- Maketitle
+
+% Title not  in boldface
+\def\@settitle{\begin{center}%
+  \baselineskip14\p@\relax
+%(amsart)    \bfseries
+\fontsize{10}{13\p@}\selectfont     % At 11pt, it is still 10pt, by jsl-l
+\uppercasenonmath\@title
+  \@title
+  \end{center}%
+}
+
+% Vertical space adjustment, per jsl-l
+\def\@setabstracta{%
+  \ifvoid\abstractbox
+  \else
+%(amsart)    \skip@20\p@ \advance\skip@-\lastskip
+    \skip@36\p@ \advance\skip@-\lastskip           % to match the top
+    \advance\skip@-.5\baselineskip                 % adjust, to match
+    \advance\skip@-\baselineskip \vskip\skip@
+    \box\abstractbox
+    \prevdepth\z@ % because \abstractbox is a vtop
+  \fi
+}
+
+% Vertical space adjustment and font fixing
+\def\@setauthors{%
+  \begingroup
+  \trivlist
+%(amsart)  \centering\footnotesize                % Fix, per jsl-l
+% 1998: follow amsart with footnotesize
+% 1998: which is 8/10 at 10pt and 9/11 at 11pt
+\centering\fontsize{7}{8\p@}\selectfont
+%(amsart) \@topsep30\p@\relax
+\@topsep36\p@\relax                             % increase, per jsl-l
+  \advance\@topsep by -\baselineskip
+  \item\relax
+  \andify\authors
+\uppercasenonmath\authors
+  \authors
+  \endtrivlist
+  \endgroup
+}
+
+%-------------------------------- Topmatter
+
+\def\recdate#1{\def\@date{#1}\def\@setdate{#1}}
+
+% Default commands for the topmatter
+% These are reset automatically in production
+
+% They need not be entered
+\def\copyrightyear{0000}
+\def\copyrightyearmodC{00}
+\def\paperID{0000-0000}
+\def\copyrightprice{\$00.00}
+\def\currentvolume{00}
+\def\currentnumber{0}
+\def\currentyear{0000}
+\def\currentmonth{XXX}
+
+%-------------------------------- Address, email, url
+
+% We enrich amsart with the \twoaddreess and the \genaddr
+% macros.
+
+% For authors with two affiliations:
+% \twoaddress{firstaddress}{secondaddress}{c}
+% where c is the number of lines in the first address
+\newcommand\addrand{{\normalfont and}}   % Same as \biband
+
+\newcommand{\twoaddress}[4][]{\g@addto@macro\addresses{%
+\address{#1}{#2 \protect\advance\parindent by -#4em\protect\\\addrand %
+\protect\\ #3}}}
+
+% For other stuff in the addreses place,
+% for example "Correspondence address", or "additional address"
+% The optional #1 gives the appellation
+% Enter as \genaddr{Correspondence address}{....}
+
+\newcommand{\genaddr}[2]{\g@addto@macro\addresses{\genaddr{#1}{#2}}}
+
+% Modify amsart to allow returns in \address, and indent on them
+% \curraddr gives current address as in amsart
+
+\def\@setaddresses{\par
+  \nobreak \begingroup
+\footnotesize
+  \def\author##1{\nobreak\addvspace\bigskipamount}%
+%(amsart)  \def\\{\unskip, \ignorespaces}%
+  \def\\{\unskip \newline\advance\parindent by 1em\indent}%  (and indent)
+  \interlinepenalty\@M
+
+   \def\address##1##2{\begingroup
+    \par\addvspace\bigskipamount\indent
+    \@ifnotempty{##1}{(\ignorespaces##1\unskip) }%
+%(amsart)    {\scshape\ignorespaces##2}\par\endgroup}%
+    {\ignorespaces\scriptsize\uppercase{##2}}\par\endgroup}%
+
+  \def\curraddr##1##2{\begingroup
+    \def\\{\unskip, \ignorespaces}   % use amsart style for curraddr
+    \@ifnotempty{##2}{\nobreak\indent{\itshape Current address}%
+      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
+      ##2\par\endgroup}}%
+
+  \def\genaddr##1##2{\begingroup
+    \def\\{\unskip, \ignorespaces}   % use amsart style for curraddr
+     \nobreak\indent{\textit{##1}\unskip}\/:\space
+      ##2\par\endgroup}%
+
+  \def\email##1##2{\begingroup
+%(amsart)    \@ifnotempty{##2}{\nobreak\indent{\itshape E-mail address}%
+    \@ifnotempty{##2}{\nobreak\indent{\itshape E-mail}%
+      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
+%(amsart)      \ttfamily##2\par\endgroup}}%
+      \normalfont ##2\par\endgroup}}%
+
+\def\urladdr##1##2{\begingroup
+    \@ifnotempty{##2}{\nobreak\indent{\itshape URL}%
+      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
+%(amsart)      \ttfamily##2\par\endgroup}}%
+      \normalfont ##2\par\endgroup}}%
+  \addresses
+  \endgroup
+}
+
+\def\urltilde{$^\sim$}            % To use in URL addresses
+
+%------------------------------- Abstract
+
+% The Abstract is wider, with a smaller fontsize and
+% a larger baselineskip than the one in amsart.
+% 1998: keep the wider width but use the larger typefont
+% which is just set to \Small.
+
+%\renewenvironment{abstract}{%
+\newenvironment{abstract}{%
+\newcommand{\abstractfontsize}{\ifnum\@mainsize=10 \fontsize{7}{10\p@}
+   \else \fontsize{8}{11\p@}\fi}
+\ifx\maketitle\relax
+    \ClassWarning{\@classname}{Abstract should precede
+      \protect\maketitle\space in AMS documentclasses; reported}%
+  \fi
+  \global\setbox\abstractbox=\vtop \bgroup
+ \abstractfontsize\selectfont
+  \list{}{\labelwidth\z@
+    \leftmargin1.5pc \rightmargin\leftmargin
+    \listparindent\normalparindent \itemindent\parindent
+    \parsep\z@ \@plus\p@
+    \let\fullwidthdisplay\relax
+    }%
+%(amsart)  \item[\hskip\labelsep\scshape\abstractname.]%
+  \item[\hskip\labelsep\bfseries\abstractname.]%
+}{%
+  \endlist\egroup
+  \ifx\@setabstract\relax \@setabstracta \fi
+}
+
+%------------------------------- Logo setting
+
+\def\@setlogo{%
+% Compute the width of the logobox
+\newcommand{\datenametext}{Volumei\currentvolume iNumber
+\currentnumber i\currentmonth i\currentyear}
+\newcommand{\longdatenametext}%
+{Volume \currentvolume,\hfill Number \currentnumber,\hfill
+        \currentmonth\hfill \currentyear}
+\newlength{\@datename}
+\settowidth{\@datename}{\normalfont\@logofontsize\selectfont \datenametext}
+\newlength{\@logoname}
+\settowidth{\@logoname}{%
+     \normalfont\@logofontsize\scshape\selectfont%
+\journalname}
+\newlength{\@logowidth}
+\setlength{\@logowidth}{\ifdim\@logoname>\@datename \@logoname%
+\else \@datename\fi}
+% Pass the logobox to the amsart routine
+    \def\@serieslogo{%
+\vbox to\headheight{%
+  \parindent\z@ \@logofontsize\selectfont%
+  \hsize\@logowidth{\scshape\selectfont \longjournalname}\newline
+\longdatenametext\newline\endgraf\vss}}
+}
+
+%------------------------------- Copyright setting
+
+\def\@copyrightfontsize{\ifnum\@mainsize=10 \fontsize{6}{8\p@}
+   \else \fontsize{7}{9\p@}\fi}%
+
+\def\@setcopyright{%
+\ifCopyRight{%
+   \insert\copyins{\par\hsize\textwidth
+  \parfillskip\z@ \leftskip\z@\@plus.9\textwidth
+\@copyrightfontsize\normalfont\upshape
+      \everypar{}%
+      \vskip-\skip\copyins \nointerlineskip
+      \noindent\vrule\@width\z@\@height\skip\copyins
+  \copyright\ \copyrightyear, Association for Symbolic Logic\break
+  \ISSN/\copyrightyearmodC/\paperID/\copyrightprice\endgraf
+      \par
+      \kern\z@}%
+}
+\else\relax
+\fi
+}
+
+%------------------------------- Sectioning commands
+
+
+% For conditionals 
+% \iete{#1}{#2}{#3} =
+%    if #1 is empty then #2 else #3.
+% Uses settowidth to avoid expanding #1
+
+\newlength\ietetest
+\def\iete#1#2#3{\settowidth\ietetest{#1}%
+\ifdim\ietetest=0pt{#2}\else{#3}\fi}
+
+\def\@secnumfont{\bfseries}
+
+% Section, has the \S symbol in \thesection
+% while the deeper environments do not
+
+\renewcommand\thesection       {\arabic{section}}
+\renewcommand\thesubsection    {\thesection.\arabic{subsection}}
+\renewcommand\thesubsubsection {\thesubsection.\arabic{subsubsection}}
+
+\def\section{\@secstartsection{section}{1}%   % to treat \S in \section*
+{\parindent\bfseries}{12\p@\@plus3\p@}{-.5em}
+{\normalfont\bfseries}}
+
+% The adjustment gives the right spacing for \section{}
+
+\def\@secstartsection#1#2#3#4#5#6{%
+ \if@noskipsec \leavevmode \fi
+ \par \@tempskipa #4\relax
+ \@afterindenttrue
+ \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi
+ \if@nobreak \everypar{}\else
+     \addpenalty\@secpenalty\addvspace\@tempskipa\fi
+ \@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}%
+         {\@dblarg{\@sect{#1}{#2}{#3\S}{#4}{#5}{#6}}}%    % \S here
+}
+
+\def\subsection{\@startsection{subsection}{2}%
+ {\parindent}{2\p@}{-.5em}%      (some vertical space above)
+{\normalfont\bfseries}}
+
+\def\subsubsection{\@startsection{subsubsection}{3}%
+{\parindent}{1\p@}{-.5em}%      (same as for paragraphs in bsl)
+{\normalfont\itshape}}
+
+\def\@sect#1#2#3#4#5#6[#7]#8{%
+  \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}%
+  \ifnum #2>\c@secnumdepth \let\@secnumber\@empty
+  \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi
+ \ifnum #2>\c@secnumdepth
+   \let\@svsec\@empty
+ \else
+    \refstepcounter{#1}%
+    \edef\@svsec{\ifnum#2<\@m
+       \@ifundefined{#1name}{}{%
+         \ignorespaces\csname #1name\endcsname\space}\fi 
+       \@nx\textup{%
+      \@nx\@secnumfont
+          \csname the#1\endcsname.}\enspace
+    }%
+  \fi
+  \@tempskipa #5\relax
+  \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
+    \begingroup #6\relax
+    \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
+    \endgroup
+    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
+  \else
+   \def\@svsechd{#6\hskip #3\@svsec 
+% Change the next two lines to correct horizontal spacing
+% in the case of \section{}
+% Use \iete rather than \@ifempty to avoid expansion
+%    \@ifnotempty{#8}{\ignorespaces#8\unskip 
+%    \@addpunct.}%                    
+%     \iete{#8}{\hskip -.4em}{\ignorespaces#8\unskip\@addpunct.}%
+    \iete{#8}{\hskip -.4em}{\ignorespaces#8\unskip\@addpunct.}%
+    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
+  }%
+  \fi
+  \global\@nobreaktrue
+  \@xsect{#5}}
+
+%-------------------------------------- Quotation
+
+\newenvironment{quotation}{\list{}{%
+%    \leftmargin3pc \listparindent\normalparindent
+    \leftmargin1.5pc \listparindent\normalparindent
+    \itemindent\z@
+    \rightmargin\leftmargin \parsep\z@ \@plus\p@}%
+  \item[]%
+}{%
+  \endlist
+}
+\let\endquotation=\endlist % for efficiency
+
+%-------------------------------- Theorems
+
+% We follow amsthm.sty for the most part.
+
+\def\@thm#1#2#3{\normalfont
+  \trivlist
+  \edef\@restorelabelsep{\labelsep\the\labelsep}%
+  \labelsep.5em\relax \let\thmheadnl\relax
+%  \let\thm@indent\noindent                      % no indent
+  \let\thm@indent\indent                         % we indent
+  \let\thm@swap\@gobble
+%  \thm@headfont{\bfseries}                      % heading font bold
+  \thm@headfont{\scshape}                        % heading font smallcaps
+  \thm@headpunct{.}% add period after heading
+  \thm@notefont{\normalfont}                     % optional note font, added
+  \thm@preskip\topsep
+  \thm@postskip\thm@preskip
+  #1% style overrides
+  \@topsep \thm@preskip               % used by first \item
+  \@topsepadd \thm@postskip           % used by \@endparenv
+  \def\@tempa{#2}\ifx\@empty\@tempa
+    \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
+  \else
+    \refstepcounter{#2}%
+    \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
+  \fi
+  \@tempa
+}
+%------------------------------------------ Proofs
+
+\newcommand{\qedsymbol}{\mbox{$\dashv$}}
+\newcommand{\noqed}{\def\qedsymbol{}}
+
+\newcommand{\proofname}{Proof}
+
+% \begin{proof}[Proof of Claim]
+% gives {\scshape Proof of Claim}. ...
+% \begin{proof}[\it Proof]
+% gives a proof with the proofname in italics
+
+\newif\ifqedneeded  % to deal with the \proofend command
+\qedneededtrue
+\newlength{\@back}
+\setlength{\@back}{\baselineskip}
+\addtolength{\@back}{\jot}
+
+\newenvironment{proof}[1][\proofname]{\par
+\let\oldqedsymbol\qedsymbol             % Store \qedsymbol
+\global\qedneededtrue      % just in case \proofend was invoked
+  \normalfont
+%  \topsep6\p@\@plus6\p@ \trivlist      % ynm debug
+\trivlist                               % debug
+  \item[\hskip\labelsep\hskip\parindent\scshape
+    #1\@addpunct{.}]\ignorespaces
+}{%
+\ifqedneeded\qed\else\global\qedneededtrue%
+\vspace*{-\baselineskip}\fi\endtrivlist
+\global\let\qedsymbol\oldqedsymbol      % Restore it, if it had been changed
+}
+
+% Add the environment with no period after Proof
+% For construction of the form
+% Proof is by induction ...
+
+\newenvironment{proofplain}[1][\proofname]{\par
+\let\oldqedsymbol\qedsymbol             % Store \qedsymbol
+\global\qedneededtrue      % just in case \proofend was invoked
+  \normalfont
+  \topsep6\p@\@plus6\p@ \trivlist
+  \item[\hskip\labelsep\hskip\parindent\scshape
+%    #1\@addpunct{.}]\ignorespaces
+     #1]
+}{%
+\ifqedneeded\qed\else\global\qedneededtrue%
+\vspace*{-\baselineskip}\fi\endtrivlist
+\global\let\qedsymbol\oldqedsymbol      % Restore it, if it had been changed
+}
+
+%-------------------------------------- Bibliography
+
+\def\refname{REFERENCES}
+\newcommand{\bysame}{\leavevmode\hbox{%
+    \vsize.7ex \advance\vsize-.4pt
+    \vrule height.7ex depth-\vsize width3em}%
+  \thinspace}
+
+\newcommand{\weaktie}{\penalty9999\spacefactor1000 \space}
+
+\newcommand\biband{{\normalfont and}}
+\newcommand\bibetal{{\normalfont et\weaktie al.}}
+
+% Define \jslname and \bslname which are produced by asl.bst
+% The options jsl and bsl redefine these
+\def\jslname{{\bfit The Journal of Symbolic Logic}}
+\def\bslname{{\bfit The Bulletin of Symbolic Logic}}
+
+% The pointsizes here depend on the overall pointsize
+% and we deal with this by simply taking cases.
+% 1998: The results for 12pt will not be very good.
+% 1998: and it may be useful to add a third case, if we want a 12pt
+
+\newenvironment{thebibliography}[1]{%
+\ifnum\@mainsize=10                 % (at 10pt, the normal for the jsl)
+   \par
+        \vspace{18pt}%
+        \centerline{\fontsize{7}{7\p@}\selectfont \refname}%
+        \nobreak\vspace*{5pt}\nobreak%
+  \fontsize{8}{10\p@}\selectfont\relax
+  \def\and{{\normalfont \lowercase{and}\ }}%
+  \list{[\@arabic\c@enumi]\ }{%
+        \leftmargin\z@ \labelwidth\z@ \itemindent12\p@
+  \labelsep\z@ \usecounter{enumi}}%
+  \sloppy \clubpenalty4000\relax \widowpenalty\clubpenalty
+  \sfcode`\.\@m
+\else                              % (at 11pt or bigger)
+  \par
+        \vspace{20pt}%
+        \centerline{\fontsize{8}{8\p@}\selectfont \refname}%
+        \nobreak\vspace*{6pt}\nobreak%
+  \fontsize{9}{11\p@}\selectfont\relax
+  \def\and{{\normalfont \lowercase{and}\ }}%
+  \list{[\@arabic\c@enumi]\ }{%
+        \leftmargin\z@ \labelwidth\z@ \itemindent12\p@
+  \labelsep\z@ \usecounter{enumi}}%
+  \sloppy \clubpenalty4000\relax \widowpenalty\clubpenalty
+  \sfcode`\.\@m%
+\fi} % end of the pointsize choice
+{\endlist}
+
+%-------------------------------------- bibliography options
+% \input bibextra.sty
+% This (until "endbibextra") is Kapoutsis' bibextra.sty
+% implementing the commands required by asl.bst
+% It defines the options bibalpha, bibay1 and bibay2
+%%%====================================================================
+%%%==0== General description ==========================================
+%%%====================================================================
+
+% WHAT \bib?item AND \bibxcite DO
+%
+% Commands \bibfitem, \bibritem replace \bibitem, providing more
+% flexibility. Their exact syntax is (see also file asl.bst)
+%   \bib?item 1. {citation key}
+%             2. {guys-list}
+%             3. {year}
+%             4. {tag}
+%             5. {`number' of editors}
+%             6. {hardcite}
+%             7. {hardlist}
+% Like \bibitem, each one of \bib?item lies in a `thebibliography'
+% environment (which is just a special `list' environment) and is
+% followed by some text that describes a bibliography entry. The
+% only difference between the two commands is that: in a group of
+% successive entries that have the same author(s)/editor(s), BibTeX
+% will introduce the f-irst entry with \bibfitem and the r-est with
+% \bibritem. So, differences in the definitions of the commands may
+% produce nice effects. (E.g., see \intergroupsep below.)
+%
+% As an example, here is a possible case (with any of \bibfitem,
+% \bibritem in place of \bib?item):
+%   \begin{thebibliography}
+%   ...
+%   \bib?item{easy}
+%   {\guy{C.}{Christos}{}{Kapoutsis}{}}
+%   {2020}{a}{0}
+%   {}{}
+%   \guysmagic{Christos Kapoutsis} {\itshape $\P=\NP$},
+%   {\bfseries\itshape Ciao}, vol.\weaktie 30\yearmagic{}{(2020)},
+%   no.\weaktie 50, pp.\weaktie 1--2.
+%   ...
+%   \end{thebibliography}
+%
+% There are two things the command should do:
+%    (i) provide an \item[label] of this `list'-like `thebibliography'
+%        environment. The label should be relevant to the corresponding
+%        bibliography entry. Obviously, it should be constructed out of
+%        the arguments of \bib?item.
+%   (ii) write a \bibxcite command to the .aux file. At the beginning
+%        of its next pass, LaTeX will execute this command and thus
+%        define the citation label for the specific entry (that is,
+%        the label that will appear wherever in the text there is a
+%        reference to this entry).
+%
+% [ Note: Below, both commands use \@bib@item as a `submacro' (\bibritem
+%   is exactly that; \bibfitem also inserts a vertical space). So, we
+%   refer to `the definition of \bib?item', meaning `the definition
+%   of \@bib@item'. ]
+%
+% The first three lines in the definition of \bib?item handle (i):
+%    \item[\@wrap@list{%
+%       \@ifisempty{#7}{\composelist{#1}{#2}{#3}{#4}{#5}}{#7}%
+%    }]%
+% This is a typical \item[..] command. (Note that, because of the
+% brackets, this does not advance \@listctr, the counter of the list).
+% \@wrap@list encloses the result of the second line into brackets (its
+% definition is simply that: \def\@wrap@list#1{[#1]}; but, of course,
+% it may change; it does, in option bibay2). And the result of the
+% second line is
+%    -- the hardlist field of the entry (argument #7), if there was
+%       such a field in the entry.
+%    -- the result of the \composelist macro, otherwise.
+%
+% It is evident that the syntax of \composelist is
+%         \composelist 1. {citation key}
+%                      2. {guys-list}
+%                      3. {year}
+%                      4. {tag}
+%                      5. {`number' of editors}
+% and that its purpose is to construct a label for the item out of its
+% arguments. It is important to note that, being within the environment
+% `thebibliography', \composelist has access to the \@listctr counter,
+% the counter of the list. Therefore, interesting effects are possible.
+% For example, a definition of the form
+%    \def\composelist#1#2#3#4#5{%
+%        \addtocounter{\@listctr}{1}\the\value{\@listctr}}
+% would result in the list label being (almost) the order of the item
+% in the bibliography list. (This is the actual definition of the
+% command when none of the bibalpha, bibay1, bibay2 options is set.)
+%
+% [ Note: (To explain `almost'.) Even if \composelist remembers to
+%   advance \@listctr, this advance will not be performed at entries
+%   that had their hardlist field defined. Indeed, \bib?item will
+%   select the value of that field as the label of the entry in the
+%   bibliography listing and will not call \composelist at all. ]
+%
+% The rest of the definition of \bib?item handles (ii). Lines
+%    \@auxstring={\bibxcite{#1}{#2}{#3}{#4}{#5}{#6}}
+%    \immediate\write\@auxout{\the\@auxstring{\the\value{\@listctr}}}}
+% result in printing into the .aux file the command
+%    \bibxcite{citation key}{guys-list}{year}{tag}{noe}{hardcite}{n}
+% where n = the order of the item in the list (almost). For example,
+% assuming the case cited above, the next pass of LaTeX will print
+% into the .aux file the command
+% \bibxcite{easy}{\guy{C.}{Christos}{}{Kapoutsis}{}}{2020}{a}{0}{}{7}
+% if this was (almost) the seventh item in the list...
+
+
+% CITATION
+%
+% ...In its next pass, LaTeX will start (as always) by reading the
+% .aux file. So, it will execute this \bibxcite command. The result
+% will be (see the definition of \bibxcite below) the definition of
+% the following macros:
+%  \gu@easy -> \guy{C.}{Christos}{}{Kapoutsis}{}
+%  \yr@easy -> 2020
+%  \yt@easy -> a
+%  \cp@easy -> \composecitepre{easy}
+%              {\guy{C.}{Christos}{}{Kapoutsis}{}}{2020}{a}{0}{7}
+%  \ci@easy -> \composeciteid{easy}
+%              {\guy{C.}{Christos}{}{Kapoutsis}{}}{2020}{a}{0}{7}
+% (but if there was a nonempty hardcite argument, the \ci@easy macro
+% would have been set to that).
+%    In the sequel, LaTeX will read the .tex file and (see the
+% definition for \citeauth, \citeyear, \citeytag) use \gu@easy to
+% replace all \citeauth{easy} references, \yr@easy,\yt@easy for all
+% \citeyear{easy} references, etc.
+%
+% As for the classical \cite command, this is redefined in order
+% to serve the following intention: Each citation label consists of
+% two parts: `pre' and `id'; `pre' may be empty, but `id' shouldn't.
+% For the entry named `easy', the `pre' part is whatever \cp@easy
+% has been defined to be and the `id' part is whatever \ci@easy has
+% been defined to be. Typically, `pre' will be empty or the list of
+% authors' names and `id' will be the year followed by the year tag.
+%   [ One can alter these two definitions by properly redefining the
+%     \composecitepre and \composeciteid commands. But redefinition
+%     of \composeciteid will have no effect if the entry has a
+%     hardcite field. See the \bibxcite command below.
+%
+%     Specifically, the syntax of \composeciteXXX (XXX=pre,id) is
+%       \composeciteXXX  1. {citation key}
+%                        2. {guys-list}
+%                        3. {year}
+%                        4. {tag}
+%                        5. {`number' of editors}
+%                        6. {n}
+%     where n is as in the syntax for \bibxcite. The intension is that
+%     the command should construct a cite label out of its arguments.
+%     E.g.,
+%           \def\composecitepre#1#2#3#4#5#6{%
+%               \let\oldguy=\guy%
+%               \def\guy##1##2##3##4##5{%
+%                   ##1\@ifnonempty{##3}{ ##3}~##4}
+%               #2%                                                (<-)
+%               \global\let\guy=\oldguy}
+%     defines the `pre' part to be the list of author(s)/editor(s),
+%     in line (<-). The other lines redefine \guy so that a
+%     `first-name-initials von last' format is used for the names.
+%     So, applied to the running example, this definition would
+%     result in this `pre' part of the citation label: `C.~Kapoutsis'.
+%   ]
+% The `id' part will be always present in a citation, while the `pre'
+% part may be gobbled up.
+%
+% The code `\cite[opt]{easy}' in the text is equivalent to:
+%   \@wrap@cite{\@wrap@alone@label{\@make@alone@label{opt}{easy}}}
+% Here, \@wrap@cite may provide a wrap for the whole citation, while
+% \@wrap@alone@label may provide a wrap for the label. The command
+% \@make@alone@label should construct the actual label out of the
+% two arguments of \cite and using the \cp@easy, \ci@easy macros
+% defined by \bibxcite.
+%    The code `\cite{easy}' is equivalent to `\cite[]{easy}'.
+%    Infix `alone' implies that the label that is being constructed
+% is the only label in the citation.
+%
+% There may be more than one label, e.g.,
+%   \cite{easy,hard}
+% where `hard' is the name of some other entry. This code is equivalent
+% to this one:
+%   \@wrap@cite{%
+%       \@wrap@first@label{\@make@first@label{opt}{easy}}%
+%       \@wrap@final@label{\@make@final@label{opt}{hard}}%
+%   }
+% Here, `first' and `final' imply that the corresponding labels appear
+% first (and, at the same time, nonlast) and last (and, at the same
+% time, nonfirst) in the list, respectively.
+%
+% There may also be more than two labels. E.g., the command
+%   \cite{easy,typical1,..,typicalN,hard}             (for N>0)
+% is equivalent to this:
+%   \@wrap@cite{%
+%       \@wrap@first@label{\@make@first@label{opt}{easy}}%
+%       \@wrap@inner@label{\@make@inner@label{opt}{typical1}}%
+%       ...
+%       \@wrap@inner@label{\@make@inner@label{opt}{typicalN}}%
+%       \@wrap@final@label{\@make@final@label{opt}{hard}}%
+%   }
+% where `inner' implies the construction of a both nonfirst and nonlast
+% label.
+%
+% Properly defining the \@wrap@xxxxx@label commands we can get some
+% nice effects. But wrapping also occurs deeper in the construction
+% of the label: \@make@xxxxx@label{opt}{easy} is equivalent to
+%   \@wrap@xxxxx@label@pre{\@make@label@pre{opt}{easy}}%
+%   \@wrap@xxxxx@label@id{\@make@xxxxx@label@id{opt}{easy}}
+% which constructs and wraps the `pre' and `id' parts of the label.
+%
+% The two \@make commands in this level construct the two parts of the
+% label in the expected way. The only important thing here is that:
+%   in a maximal list of successive labels that have the same `pre'
+%   part, \@make@label@pre will return the `pre' part of the first
+%   of these labels but just an empty string for every other label.
+% This is how, e.g., a list like
+%    \cite{Church1949a,Church1949b,Church1950}
+% ends up to be
+%    Church [1949a], [1949b], [1950]
+% and not
+%    Church [1949a], Church [1949b], Church [1950]
+% in style `bibay2'.
+%
+% Also note that, in all options, the optional argument `opt' of the
+%   \cite[opt]{name1,name2,..,nameN}
+% command is appended as a comment to the last entry, that is to the
+% antry named `nameN'. But this may also change, by properly redefining
+% the \@make@xxxxx@label commands.
+
+
+
+
+
+
+
+%%%====================================================================
+%%%==1== General purpose ==============================================
+%%%====================================================================
+
+% Some numbers.
+\mathchardef\@oh=100
+\mathchardef\@fh=500
+
+
+% To select on the emptiness of an argument. \@ifisempty checks whether
+% its first argument is empty. If so, the second argument is selected.
+% Otherwise, the third argument is selected.
+%
+% The trick is to construct a new temporary macro \@iietest as
+%    {},  if #1 is empty       or       {blabla},  if #1 is blabla.
+% Then, \ifx compares the `top level' expansion of \@iietest and of
+% \empty  (whose definition is: \def\empty{}).
+\def\@ifisempty#1#2#3{%
+    \def\@iietest{#1}%
+    \ifx\@iietest\empty #2\else #3\fi}
+
+% To do something only when an argument is nonempty & not `\empty'.
+% If #1 is empty, \ifx compares \empty with \else, finds them
+%   different and just gobbles up everything up to \fi, doing nothing.
+% Otherwise: if #1 is the \empty command, \ifx compares it with
+%   \empty, finds them identical, executes everything (nothing) upto
+%   the \else command and ignores everything else upto \fi. So, in
+%   total, it does nothing at all.
+% Otherwise: if #1 is not the \empty command, \ifx compares \empty
+%   with (the starting of) #1, finds them different, gobbles up
+%   everything up to \else, and executes #2.
+% Apparently, #1 shouldn't contain an unmatched \else.
+\def\@ifnonempty#1#2{\ifx\empty#1\else#2\fi}
+
+% Redefinition of the \@for#1:=#2\do macro. This macro allows iteration
+% over a comma separated list of strings. For example, this code:
+%   \@for\temp:= one, two ,   three\do{<\temp>}
+% is equivalent to this code:
+%   <one><two ><three>
+% (Notice that leading spaces are trimmed off, while trailing blanks
+% contract to a single space.)
+%
+% We first copy the set of relevant commands from latex.ltx and then
+% we add the lines with the comments on the right. This way, we have
+% two new if's:
+%  -- \if@sol, which is true iff the current item is the first item
+%              of the list, and
+%  -- \if@eol, which is true iff the current item is the last item
+%              of the list.
+\newif\if@sol                             %  !insertion: \@sol created.
+\newif\if@eol                             %  !insertion: \@eol created.
+
+\def\@predictor{?}                        % !insertion: updating \@eol.
+\def\@end@of@list{\@nil,\@nil}            %
+\def\@update@eol#1{%                      %
+    \def\@predictor{#1}%                  %
+    \ifx\@predictor\@end@of@list%         %
+        \@eoltrue%                        %
+    \else%                                %
+        \@eolfalse%                       %
+    \fi%                                  %
+}                                         %
+
+\long\def\@for#1:=#2\do#3{%
+    \@soltrue%                            %   !insertion: \@sol inited.
+    \@eolfalse%                           %   !insertion: \@eol inited.
+    \expandafter\def\expandafter\@fortmp\expandafter{#2}%
+    \ifx\@fortmp\@empty \else
+        \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}\fi}
+\long\def\@forloop#1,#2,#3\@@#4#5{%
+    \@update@eol{#2,#3}%                  %  !insertion: \@eol updated.
+    \def#4{#1}%
+    \ifx #4\@nnil \else%
+        #5%
+        \@solfalse%                       %  !insertion: \@sol updated.
+        \@update@eol{#3}%                 %  !insertion: \@eol updated.
+        \def#4{#2}%
+        \ifx #4\@nnil \else%
+            #5%
+            \@iforloop #3\@@#4{#5}%
+        \fi%
+    \fi%
+}
+\long\def\@iforloop#1,#2\@@#3#4{%
+    \@update@eol{#2}%                     %  !insertion: \@eol updated.
+    \def#3{#1}%
+    \ifx #3\@nnil%
+        \expandafter\@fornoop%
+    \else%
+        #4\relax%
+        \expandafter\@iforloop%
+    \fi%
+    #2\@@#3{#4}%
+}
+
+
+
+
+
+
+
+
+
+%%%====================================================================
+%%%==2== Material used by all options =================================
+%%%====================================================================
+
+
+
+%%%==2A. What MUST be defined =========================================
+% See asl.bst, Section `COMMANDS THAT THE TEX USER SHOULD DEFINE'.
+%     (\bysame, \weaktie, \jslname, \bslname are defined elsewhere.)
+% See the discussion above.
+
+\def\yearappear{\def\yearmagic##1##2{##1 ##2}}
+\def\yeardisappear{\def\yearmagic##1##2{}}
+
+\def\guysappear{\def\guysmagic##1{##1,}}
+\def\guysdisappear{\def\guysmagic##1{}}
+
+\def\guy#1#2#3#4#5{#4}
+
+\def\TheSortKeyIs#1{}
+
+\newtoks\@auxstring
+\newlength{\intergroupsep}
+\def\@bib@item#1#2#3#4#5#6#7{%
+\item[\@wrap@list{%
+  \@ifisempty{#7}{\composelist{#1}{#2}{#3}{#4}{#5}}{#7}%
+}]%
+\if@filesw{%
+  \def\protect##1{\string ##1\space}%
+  \@auxstring={\bibxcite{#1}{#2}{#3}{#4}{#5}{#6}}
+  \immediate\write\@auxout{\the\@auxstring{\the\value{\@listctr}}}}
+\fi
+\mbox{ }\ignorespaces
+}
+\def\bibfitem{\vspace{\intergroupsep}\@bib@item}
+\let\bibritem=\@bib@item
+
+
+%%%==2B. Citation enhancements ========================================
+
+% Define all necessary entry data as macros.
+\def\bibxcite#1#2#3#4#5#6#7{
+  \global\@namedef{gu@#1}{#2}                      % guys list
+  \global\@namedef{yr@#1}{#3}                      % year
+  \global\@namedef{yt@#1}{#4}                      % year tag
+  \global\@namedef{cp@#1}{\composecitepre{#1}{#2}{#3}{#4}{#5}{#7}}  %%%
+  \@ifisempty{#6}                                  % cite  pre  &  id %
+    {\global\@namedef{ci@#1}{\composeciteid{#1}{#2}{#3}{#4}{#5}{#7}}} %
+    {\global\@namedef{ci@#1}{#6}}                                   %%%
+}
+
+% Action to be taken when a citation key is used without having been
+% defined yet: A bold question mark is left and a warning is printed.
+\def\cite@nondef@type{{\reset@font\bf ?}}
+\def\cite@nondef@warn#1{%
+    \@warning{Citation `#1' on page \thepage\space undefined}}
+\def\cite@nondef#1{\cite@nondef@type\cite@nondef@warn{#1}}
+
+
+% CITATION SPACING
+%
+% For the construction of the citations we define two spaces: a hard
+% one (\@hard@cispace) which does not break at line end and a soft one
+% (\@soft@cispace) which breaks. \@cispace is always one of these two
+% spaces. We use \@cispace each time we want a space which we do not
+% know for sure if it should be hard or soft.
+%    When preceding a \cite or \fullcite command, the \XXXXcispacehere
+% command (XXXX=hard,soft) makes \@cispace be XXXX for this citation.
+%    Command \XXXXcispace makes \@cispace be XXXX for all citations
+% after it and before the next \XXXXcispace command.
+%    This behavior is achieved through the use of \@global@cispace and
+% the reassignment of \@cispace to that, at the end of the \cite
+% command definition.
+\def\@hard@cispace{\penalty\@m\space}
+\def\@soft@cispace{\space}
+\def\hardcispacehere{\let\@cispace=\@hard@cispace}
+\def\softcispacehere{\let\@cispace=\@soft@cispace}
+\def\hardcispace{
+    \let\@global@cispace=\@hard@cispace
+    \let\@cispace=\@hard@cispace}
+\def\softcispace{
+    \let\@global@cispace=\@soft@cispace
+    \let\@cispace=\@soft@cispace}
+
+
+% REDEFINITION OF THE \cite MACRO    (See a discussion above.)
+%
+% Store the old \cite and \@citex 
+% and put this in a command
+\let\oldcite\cite
+\let\old@citex\@citex
+\let\oldbibitem\bibitem
+
+% To turn the old style on
+\def\oldbib{\let\cite\oldcite\let\@citex\old@citex\bibitemfalse}
+\newif\ifbibitem
+\bibitemtrue
+\def\bibitem{\ifbibitem\errbibitem\bibitemfalse\fi\oldbibitem}
+
+\def\errbibitem{%
+\errmessage{**************************************************^^J
+There are (ugh!) bibitems in your .bbl file!^^J^^J
+Use the option bibother to get the correct citations.^^J^^J
+And think about using bibliographystyle{asl}^^J
+which is highly recommended for ASL publications!^^J
+**************************************************^^J}}
+
+
+
+\DeclareRobustCommand\cite{%
+    \@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}}
+
+\def\@citex[#1]#2{%
+%properly wrap the whole list of citations:
+    \@wrap@cite{%
+%now traverse the list, \@citeb always being the current item.
+        \@for\@citeb:=#2\do{%
+%gobble one initial space, if present:
+            \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
+%output sth like `\citation{Church36a}' to the auxiliary file:
+            \if@filesw%
+                \immediate\write\@auxout{\string\citation{\@citeb}}%
+            \fi%
+%select the proper \@wrap@xxxxx@label, \@make@xxxxx@label commands:
+            \if@sol%
+                \if@eol%
+                    \def\@wrap@label{\@wrap@alone@label}%
+                    \def\@make@label{\@make@alone@label}%
+                \else%
+                    \def\@wrap@label{\@wrap@first@label}%
+                    \def\@make@label{\@make@first@label}%
+                \fi%
+            \else%
+                \if@eol%
+                    \def\@wrap@label{\@wrap@final@label}%
+                    \def\@make@label{\@make@final@label}%
+                \else%
+                    \def\@wrap@label{\@wrap@inner@label}%
+                    \def\@make@label{\@make@inner@label}%
+                \fi%
+            \fi%
+%make this item's label, and print it properly wrapped.
+            \@wrap@label{\@make@label{#1}{\@citeb}}%
+        }%
+    }%
+%reset spacing to be as in global definition.
+    \let\@cispace=\@global@cispace%
+}
+
+% making label parts:
+\def\@make@alone@label#1#2{%
+    \@wrap@alone@label@pre{\@make@label@pre{#1}{#2}}%
+    \@wrap@alone@label@id{\@make@alone@label@id{#1}{#2}}%
+}
+\def\@make@first@label#1#2{%
+    \@wrap@first@label@pre{\@make@label@pre{#1}{#2}}%
+    \@wrap@first@label@id{\@make@first@label@id{#1}{#2}}%
+}
+\def\@make@final@label#1#2{%
+    \@wrap@final@label@pre{\@make@label@pre{#1}{#2}}%
+    \@wrap@final@label@id{\@make@final@label@id{#1}{#2}}%
+}
+\def\@make@inner@label#1#2{%
+    \@wrap@inner@label@pre{\@make@label@pre{#1}{#2}}%
+    \@wrap@inner@label@id{\@make@inner@label@id{#1}{#2}}%
+}
+\def\@prev@label@pre{}              % \@prev@label@pre:    *init
+\def\@make@label@pre#1#2{%
+    \@ifundefined{cp@#2}
+        {\cite@nondef@type}
+        {\edef\@curr@label@pre{\csname cp@#2\endcsname}%
+         \ifx\@curr@label@pre\@prev@label@pre\else%
+            \@wrap@label@pre{\@curr@label@pre}%
+         \fi%
+         \global\edef\@prev@label@pre{\@curr@label@pre}%   *update
+        }%
+}
+\def\@make@alone@label@id#1#2{%
+    \@ifundefined{ci@#2}
+        {\cite@nondef{#2}%
+         \@ifnonempty{#1}{,\@cispace#1}%
+        }
+        {\csname ci@#2\endcsname%
+         \@ifnonempty{#1}{,\@cispace#1}%
+         \global\def\@prev@label@pre{}%                    *reset
+        }%
+}
+\def\@make@first@label@id#1#2{%
+    \@ifundefined{ci@#2}
+        {\cite@nondef{#2}}
+        {\csname ci@#2\endcsname}%
+}
+\def\@make@final@label@id#1#2{%
+    \@ifundefined{ci@#2}
+        {\cite@nondef{#2}%
+         \@ifnonempty{#1}{,\@cispace#1}%
+        }
+        {\csname ci@#2\endcsname%
+         \@ifnonempty{#1}{,\@cispace#1}%
+         \global\def\@prev@label@pre{}%                    *reset
+        }%
+}
+\def\@make@inner@label@id#1#2{%
+    \@ifundefined{ci@#2}
+        {\cite@nondef{#2}}
+        {\csname ci@#2\endcsname}%
+}
+
+
+% VARIANTS OF THE \cite MACRO
+%
+% For each variant, the (unique) argument should contain only one
+% entry name (that is, a comma-separated list of entry names is not
+% supported). Moreover, there is no optional argument.
+\def\citeauth#1{%     For the list of author(s).
+\@ifundefined{gu@#1}
+   {\cite@nondef{#1}}
+   {\csname gu@#1\endcsname}}
+
+\def\citeyear#1{%     For the year + year tag label.
+\@ifundefined{yr@#1}
+   {\cite@nondef{#1}}
+   {\csname yr@#1\endcsname\csname yt@#1\endcsname}}
+
+\def\citeytag#1{%     For the year tag.
+\@ifundefined{yt@#1}
+   {\cite@nondef{#1}}
+   {\csname yt@#1\endcsname}}
+
+\DeclareRobustCommand\fullcite{%      For a full-name citation.
+    \@ifnextchar [{\@tempswatrue\@fcitex}{\@tempswafalse\@fcitex[]}}
+\def\@fcitex[#1]#2{%
+    \let\oldguy=\guy%
+    \def\guy##1##2##3##4##5{%
+        ##2 \@ifnonempty{##3}{##3}~##4\@ifnonempty{##5}{, ##5}}%
+    \@citex[#1]{#2}%
+    \global\let\guy=\oldguy%
+}
+
+
+% REDEFINITION OF THE \nocite MACRO    (See a discussion above.)
+%
+% This is almost identical to the definition of the command in
+% latex.ltx. The only difference is the first argument of the
+% \@ifundefined command, which is `gu@\@citeb' instead of
+% `b@\@citeb'.
+\def\nocite#1{%
+%some initial hacking I don't understand:
+    \@bsphack%
+%traverse the comma-separated list, \@citeb being the current item:
+    \@for\@citeb:=#1\do{%
+%gobble one initial space, if present:
+        \edef\@citeb{\expandafter\@firstofone\@citeb}%
+%output sth like `\citation{Church36a}' to the auxiliary file:
+        \if@filesw%
+            \immediate\write\@auxout{\string\citation{\@citeb}}%
+        \fi%
+%check if the citation has been defined previously:
+        \@ifundefined{gu@\@citeb}%
+            {\G@refundefinedtrue%
+             \@latex@warning{Citation `\@citeb' undefined}}%
+            {}%
+    }%
+%some final hacking I don't understand:
+  \@esphack%
+}
+% Also define the control sequence \gu@*, to prevent a warning when
+% the \nocite{*} command is used.
+\expandafter\let\csname gu@*\endcsname\@empty
+
+
+
+
+
+
+
+
+%%%====================================================================
+%%%==3== The Options ==================================================
+%%%====================================================================
+
+
+
+%%%==3A. Option None (default style) ==================================
+\guysappear
+\yearappear
+\setlength{\intergroupsep}{0ex}
+\def\composelist#1#2#3#4#5{%
+    \addtocounter{\@listctr}{1}\the\value{\@listctr}}
+\def\@wrap@list#1{[#1]}
+
+\def\composecitepre#1#2#3#4#5#6{}
+\def\composeciteid#1#2#3#4#5#6{#6}
+\def\@wrap@cite#1{[#1]}
+\def\@wrap@alone@label#1{#1}
+\def\@wrap@first@label#1{#1,\@cispace}
+\def\@wrap@final@label#1{#1}
+\def\@wrap@inner@label#1{#1,\@cispace}
+\def\@wrap@alone@label@pre#1{#1}
+\def\@wrap@first@label@pre#1{#1}
+\def\@wrap@final@label@pre#1{#1}
+\def\@wrap@inner@label@pre#1{#1}
+\def\@wrap@alone@label@id#1{#1}
+\def\@wrap@first@label@id#1{#1}
+\def\@wrap@final@label@id#1{#1}
+\def\@wrap@inner@label@id#1{#1}
+\def\@wrap@label@pre#1{#1}
+\hardcispace
+
+
+%%%==3B. Option bibalpha ==============================================
+\DeclareOption{bibalpha}{
+\guysappear
+\yearappear
+\def\composelist#1#2#3#4#5{#1}
+
+\def\composecitepre#1#2#3#4#5#6{}
+\def\composeciteid#1#2#3#4#5#6{#1}
+\def\@wrap@first@label#1{#1,\@cispace}
+\def\@wrap@inner@label#1{#1,\@cispace}
+}
+
+
+%%%==3C. Option bibay1 (author-year style) ============================
+\DeclareOption{bibay1}{
+\guysappear
+\yearappear
+\def\composelist#1#2#3#4#5{#3#4}
+\def\composecitepre#1#2#3#4#5#6{#2}
+\def\composeciteid#1#2#3#4#5#6{#3#4}
+\def\@wrap@cite#1{#1}
+\def\@wrap@first@label#1{#1}
+\def\@wrap@inner@label#1{#1}
+\def\@wrap@alone@label@id#1{,\@cispace#1]}
+\def\@wrap@first@label@id#1{,\@cispace#1}
+\def\@wrap@final@label@id#1{,\@cispace#1]}
+\def\@wrap@inner@label@id#1{,\@cispace#1}
+\def\@wrap@label@pre#1{\if@sol\else],\space\fi[#1}
+}
+
+
+%%%==3D. Option bibay2 (author-year style) ============================
+\DeclareOption{bibay2}{
+\guysdisappear
+\yeardisappear
+\def\edlabel{\space(editor)\space}
+\def\edslabel{\space(editors)\space}
+\def\composelist#1#2#3#4#5{%
+    \let\oldguy=\guy%
+    \def\guy##1##2##3##4##5{%
+        ##2 \@ifnonempty{##3}{##3~}##4\@ifnonempty{##5}{, ##5}}%
+    \ifcase#5\textsc{#2}\space\or#2\edlabel\or#2\edslabel\fi[#3#4],%
+    \global\let\guy=\oldguy}
+\def\@wrap@list#1{#1}
+
+\def\composecitepre#1#2#3#4#5#6{#2}
+\def\composeciteid#1#2#3#4#5#6{#3#4}
+\def\@wrap@cite#1{#1}
+\def\@wrap@first@label#1{#1}
+\def\@wrap@inner@label#1{#1}
+\def\@wrap@alone@label@id#1{[#1]}
+\def\@wrap@first@label@id#1{[#1],\space}
+\def\@wrap@final@label@id#1{[#1]}
+\def\@wrap@inner@label@id#1{[#1],\space}
+\def\@wrap@label@pre#1{#1\@cispace}
+}
+
+%%%%%%%%%%% Bib styles other than asl.bst
+
+\DeclareOption{bibother}{\oldbib}
+
+%%%%%%%%%%% endbibextra code
+
+%-------------------------------------- endnotes options
+% \input ntsextra.sty
+% Until "endntsextra" this is the Kapoutsis customization of
+% endnotes.sty
+% It defines options endnotes and mixednotes
+
+%%%====================================================================
+%%%==1== The options ==================================================
+%%%====================================================================
+
+% The standard option is just the file aslnotes.sty. Both options that
+% follow (endnotes, mixednotes) first execute the standard one.
+
+%%%==1A. The standard option ==========================================
+\DeclareOption{stdnotes}{
+% This is file aslnotes.sty, version 1.0, inserted.
+% A version of John Lavagnino's endnotes.sty customized for asl.cls
+% by Christos Kapoutsis. For the customization, see sections marked
+% ckap at end. For instructions search for instructions (at the end).
+
+\message{^^J
+Using a version of John Lavagnino's endnotes.sty [1991/09/24]^^J
+customized for asl.cls by Christos Kapoutsis^^J^^J}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%       ****************************************
+%       *              ENDNOTES                *
+%       ****************************************
+%
+%  Date of this version: 24 September 1991.
+%
+%  Based on the FOOTNOTES section of
+%  LATEX.TEX (VERSION 2.09 - RELEASE OF 19 April 1986), with
+%  "footnote" changed to "endnote" and "fn" changed to "en" (where
+%  appropriate), with all the minipage stuff pulled out, and with
+%  some small changes for the different operation of endnotes.
+%
+%  Uses an extra external file, with .ENT extension, to hold the
+%  text of the endnotes.  This may be deleted after the run; a new
+%  version is generated each time.
+%
+%  This code does not obey \nofiles.  Perhaps it should.
+%
+%   John Lavagnino (lav@brandeis.bitnet), 9/23/88
+%   Department of English and American Literature,
+%      Brandeis University
+%
+%  To turn all the footnotes in your documents into endnotes, say
+%
+%     \let\footnote=\endnote
+%
+%  in your preamble, and then add something like
+%
+%     \newpage
+%     \begingroup
+%     \parindent 0pt
+%     \parskip 2ex
+%     \def\enotesize{\normalsize}
+%     \theendnotes
+%     \endgroup
+%
+% as the last thing in your document.
+%
+%       ****************************************
+%       *            CHANGE LOG                *
+%       ****************************************
+%
+% JL  Modified to include \addtoendnotes.  JL, 10/22/89.
+%
+% JK  Modification by J"org Knappen 25. 2. 1991:
+% JK
+% JK  Introduced \notesname in the spirit of international \LaTeX.
+% JK  \notesname is set per default to be {Notes}, but can easily
+% JK  be redifined, e.g. for german language
+% JK  \renewcommand{\notesname}{Anmerkungen}
+%
+% DW Modification by Dominik Wujastyk, London, 19 September 1991:
+% DW
+% DW Moved the line
+% DW         \edef\@currentlabel{\csname p@endnote\endcsname\@theenmark}
+% DW out of the definition of \@endnotetext and into the definition
+% DW of \@doanenote so that \label and \ref commands work correctly in
+% DW endnotes.  Otherwise, the \label just pointed to the last section
+% DW heading (or whatever) preceding the \theendnotes command.
+%
+% JL Revised documentation and macros.  24 Sept 1991.
+%
+%       ****************************************
+%       *        ENDNOTE COMMANDS              *
+%       ****************************************
+%
+%
+%   \endnote{NOTE}       : User command to insert a endnote.
+%
+%   \endnote[NUM]{NOTE}  : User command to insert a endnote numbered
+%                           NUM, where NUM is a number -- 1, 2,
+%                           etc.  For example, if endnotes are numbered
+%                           *, **, etc. within pages, then \endnote[2]{...}
+%                           produces endnote '**'.  This command does not
+%                           step the endnote counter.
+%
+%   \endnotemark[NUM]    : Command to produce just the endnote mark in
+%                           the text, but no endnote.  With no argument,
+%                           it steps the endnote counter before generating
+%                           the mark.
+%
+%   \endnotetext[NUM]{TEXT} : Command to produce the endnote but no
+%                              mark.  \endnote is equivalent to
+%                              \endnotemark \endnotetext .
+%
+%   \addtoendnotes{TEXT} : Command to add text or commands to current
+%                              endnotes file: for inserting headings,
+%                              pagebreaks, and the like into endnotes
+%                              sections.  TEXT a moving argument:
+%                              \protect required for fragile commands.
+%
+%       ****************************************
+%       *        ENDNOTE USER COMMANDS         *
+%       ****************************************
+%
+%   Endnotes use the following parameters, similar to those relating
+%   to footnotes:
+%
+%   \enotesize   : Size-changing command for endnotes.
+%
+%   \theendnote : In usual LaTeX style, produces the endnote number.
+%
+%   \@theenmark : Holds the current endnote's mark--e.g., \dag or '1' or 'a'.
+%
+%   \@makeenmark : A macro to generate the endnote marker from \@theenmark
+%                  The default definition is \hbox{$^\@theenmark$}.
+%
+%   \@makeentext{NOTE} :
+%        Must produce the actual endnote, using \@theenmark as the mark
+%        of the endnote and NOTE as the text.  It is called when effectively
+%        inside a \parbox, with \hsize = \columnwidth.  For example, it might
+%        be as simple as
+%               $^{\@theenmark}$ NOTE
+%
+%
+%       ****************************************
+%       *        ENDNOTE PSEUDOCODE            *
+%       ****************************************
+%
+% \endnote{NOTE}  ==
+%  BEGIN
+%    \stepcounter{endnote}
+%    \@theenmark :=G eval (\theendnote)
+%    \@endnotemark
+%    \@endnotetext{NOTE}
+%  END
+%
+% \endnote[NUM]{NOTE} ==
+%  BEGIN
+%    begingroup
+%       counter endnote :=L NUM
+%       \@theenmark :=G eval (\theendnote)
+%    endgroup
+%    \@endnotemark
+%    \@endnotetext{NOTE}
+%  END
+%
+% \@endnotetext{NOTE} ==
+%  BEGIN
+%    write to \@enotes file: "\@doanenote{ENDNOTE MARK}"
+%    begingroup
+%       \next := NOTE
+%       set \newlinechar for \write to \space
+%       write to \@enotes file: \meaning\next
+%               (that is, "macro:->NOTE)
+%    endgroup
+%  END
+%
+% \addtoendnotes{TEXT} ==
+%  BEGIN
+%    open endnotes file if not already open
+%    begingroup
+%       let \protect to \string
+%       set \newlinechar for \write to \space
+%       write TEXT to \@enotes file
+%    endgroup
+%  END
+%
+% \endnotemark      ==
+%  BEGIN \stepcounter{endnote}
+%        \@theenmark :=G eval(\theendnote)
+%        \@endnotemark
+%  END
+%
+% \endnotemark[NUM] ==
+%   BEGIN
+%       begingroup
+%         endnote counter :=L NUM
+%        \@theenmark :=G eval(\theendnote)
+%       endgroup
+%       \@endnotemark
+%   END
+%
+% \@endnotemark ==
+%   BEGIN
+%    \leavevmode
+%    IF hmode THEN \@x@sf := \the\spacefactor FI
+%    \@makeenmark          % put number in main text
+%    IF hmode THEN \spacefactor := \@x@sf FI
+%   END
+%
+% \endnotetext      ==
+%    BEGIN \@theenmark :=G eval (\theendnote)
+%          \@endnotetext
+%    END
+%
+% \endnotetext[NUM] ==
+%    BEGIN begingroup  counter endnote :=L NUM
+%                      \@theenmark :=G eval (\theendnote)
+%          endgroup
+%          \@endnotetext
+%    END
+%
+%       ****************************************
+%       *           ENDNOTE MACROS             *
+%       ****************************************
+%
+\@definecounter{endnote}
+\def\theendnote{\arabic{endnote}}
+
+% Default definition
+\def\@makeenmark{\hbox{$^{\@theenmark}$}}
+
+\newdimen\endnotesep
+
+\def\endnote{\@ifnextchar[{\@xendnote}{\stepcounter
+   {endnote}\xdef\@theenmark{\theendnote}\@endnotemark\@endnotetext}}
+
+\def\@xendnote[#1]{\begingroup \c@endnote=#1\relax
+   \xdef\@theenmark{\theendnote}\endgroup
+   \@endnotemark\@endnotetext}
+
+%  Here begins endnote code that's really different from the footnote
+% code of LaTeX.
+
+\let\@doanenote=0
+\let\@endanenote=0
+
+\newwrite\@enotes
+\newif\if@enotesopen \global\@enotesopenfalse
+
+\def\@openenotes{\immediate\openout\@enotes=\jobname.ent\relax
+      \global\@enotesopentrue}
+
+%  The stuff with \next and \meaning is a trick from the TeXbook, 382,
+% there intended for setting verbatim text, but here used to avoid
+% macro expansion when the footnote text is written.  \next will have
+% the entire text of the footnote as one long line, which might well
+% overflow limits on output line length; the business with \newlinechar
+% makes every space become a newline in the \@enotes file, so that all
+% of the lines wind up being quite short.
+
+\long\def\@endnotetext#1{%
+     \if@enotesopen \else \@openenotes \fi
+     \immediate\write\@enotes{\@doanenote{\@theenmark}}%
+     \begingroup
+        \def\next{#1}%
+        \newlinechar='40
+        \immediate\write\@enotes{\meaning\next}%
+     \endgroup
+     \immediate\write\@enotes{\@endanenote}}
+
+% \addtoendnotes works the way the other endnote macros probably should
+% have, requiring the use of \protect for fragile commands.
+
+\long\def\addtoendnotes#1{%
+     \if@enotesopen \else \@openenotes \fi
+     \begingroup
+        \newlinechar='40
+        \let\protect\string
+        \immediate\write\@enotes{#1}%
+     \endgroup}
+
+%  End of unique endnote code
+
+\def\endnotemark{\@ifnextchar[{\@xendnotemark
+    }{\stepcounter{endnote}\xdef\@theenmark{\theendnote}\@endnotemark}}
+
+\def\@xendnotemark[#1]{\begingroup \c@endnote #1\relax
+   \xdef\@theenmark{\theendnote}\endgroup \@endnotemark}
+
+\def\@endnotemark{\leavevmode\ifhmode
+  \edef\@x@sf{\the\spacefactor}\fi \@makeenmark
+   \ifhmode\spacefactor\@x@sf\fi\relax}
+
+\def\endnotetext{\@ifnextchar
+    [{\@xendnotenext}{\xdef\@theenmark{\theendnote}\@endnotetext}}
+
+\def\@xendnotenext[#1]{\begingroup \c@endnote=#1\relax
+   \xdef\@theenmark{\theendnote}\endgroup \@endnotetext}
+
+
+%  \theendnotes actually prints out the endnotes.
+
+%  The user may want separate endnotes for each chapter, or a big
+% block of them at the end of the whole document.  As it stands,
+% either will work; you just say \theendnotes wherever you want the
+% endnotes so far to be inserted.  However, you must add
+% \setcounter{endnote}{0} after that if you want subsequent endnotes
+% to start numbering at 1 again.
+
+%  \enoteformat is provided so user can specify some special formatting
+% for the endnotes.  It needs to set up the paragraph parameters, start
+% the paragraph, and print the label.  The \leavemode stuff is to make
+% and undo a dummy paragraph, to get around the games \section*
+% plays with paragraph indenting.
+
+%%%%%%changes-----------------------------------------------------ckap-
+%%% So that the style is compatible with that of ASL.
+%%%
+\def\notesname{NOTES}
+
+\def\enoteheading%
+{\ifnum\@mainsize=10               % (at 10pt, the normal for the jsl)
+ \par
+ \vspace{18pt}%
+ \centerline{\fontsize{7}{7\p@}\selectfont \notesname}%
+ \nobreak\vspace*{5pt}\nobreak%
+ \fontsize{8}{10\p@}\selectfont\relax
+\else                              % (at 11pt or bigger)
+  \par
+  \vspace{20pt}%
+  \centerline{\fontsize{8}{8\p@}\selectfont \notesname}%
+  \nobreak\vspace*{6pt}\nobreak%
+  \fontsize{9}{11\p@}\selectfont\relax
+\fi
+}
+
+\def\enoteformat{\rightskip\z@ \leftskip\z@ %
+   \leavevmode\llap{\hbox{$^{\@theenmark}$}}}
+
+\def\enotesize{\footnotesize}
+
+% Restarts counting the endnotes as 1,2,...
+\def\startoverendnotes{\setcounter{endnote}{0}}
+
+% The definition of \ETC. is needed only for versions of TeX prior
+% to 2.992.  Those versions limited \meaning expansions to 1000
+% characters; in 2.992 and beyond there is no limit.  At Brandeis the
+% BIGLATEX program changed the code in the token_show procedure of
+% TeX to eliminate this problem, but most ``big'' versions of TeX
+% will not solve this problem.
+
+\def\theendnotes%
+{\immediate\closeout\@enotes \global\@enotesopenfalse
+  \begingroup
+    \makeatletter
+    \def\@doanenote##1##2>{\def\@theenmark{##1}\par\begingroup
+        \edef\@currentlabel{\csname p@endnote\endcsname\@theenmark} %DW
+        \enoteformat}
+    \def\@endanenote{\par\endgroup}%
+    \def\ETC.{\errmessage{Some long endnotes will be truncated; %
+                            use BIGLATEX to avoid this}%
+    \def\ETC.{\relax}}
+    \enoteheading
+    \enotesize
+    \@input{\jobname.ent}%
+  \endgroup%
+  }
+%%%
+%%%%%%end changes-------------------------------------------------ckap-
+
+% End of file aslnotes.sty, version 1.0, inserted.
+}
+
+
+
+%%%==1B. The default (no option) ======================================
+\let\endnote=\footnote
+\let\endnotemark=\footnotemark
+\let\endnotetext=\footnotetext
+\let\addtoendnotes=\@gobble
+\let\theendnotes=\empty
+\let\startoverendnotes=\empty
+
+
+%%%==1C. Option endnotes ==============================================
+\DeclareOption{endnotes}{
+\ExecuteOptions{stdnotes}
+\let\footnote=\endnote
+\setcounter{endnote}{0}
+}
+
+
+%%%==1D. Option mixednotes ============================================
+% The footnotes are numbered as 1,2,..,9,1,2,..,9,1,2,.. etc.
+% Attention: This way, for all k=1,2,.. : after the 9k'th footnote and
+% before the (9k+1)'th footnote the footnote counter is 0.
+\DeclareOption{mixednotes}{
+\ExecuteOptions{stdnotes}
+\newcounter{prevfootnotepage}
+\def\thefootnote{%
+    \fnsymbol{footnote}%
+    \ifnum\value{footnote}>8\setcounter{footnote}{0}\fi}
+}
+
+% endntsextra code
+
+%-------------------------------------- Additional commands (mod)
+
+% Add the Greek cap Ypsilon
+\def\Ypsilon{\char7}
+
+% In xfonts, \boldsymbol is redefined because the font-coding
+% has an error.
+
+% For the game quantifier, requires rotate.sty
+\def\game{\rotate[r]{\rotate[r]{$\mathsf{G}$}}}
+
+% For arithmetic subtraction
+\newcommand\dotminus{\mathop{\mbox{$-^{\hspace{-.5em}\cdot}\,\,$}}}
+
+% For the boldface \Delta, \Pi,. in descriptive set theory, to
+% make them more noticeably different from the lightface
+
+% For the boldface symbols \Pi, \Sigma, \Delta, etc. in DST
+\newcommand\tboldsymbol[1]{\underset{\widetilde{}}{\boldsymbol{#1}}}
+
+% To get McCusker to print correctly in the title, headings,
+% and Tables of contents, use \author{Guy \Mc Cusker}
+\def\Mc{{\sc Mc}}
+\def\Mac{{\sc Mac}}
+% This preserves the small cap font when the \uppercase command is applied,
+% and the c turns out to be a \sc C
+
+\def\cmr#1{{\fontfamily{cmr}\selectfont #1}}   % For single symbols in CMR
+% Use \cmr{\pounds} for \pounds
+
+% The \implies symbol comes out wrong with Times, redefine with cmr:
+
+\def\implies{\,\text{\fontfamily{cmr}\fontshape{n}\selectfont\char61%
+$\joinrel\Rightarrow\,$}}
+
+% Boldface italics are used often, so we have a command for them:
+\def\bfit{\bfseries\itshape}
+
+% LaTeX symbols are available, because latexsym.sty has been loaded.
+% These are the following:
+% \mho  \Join  \Box  \Diamond  \leadsto  \sqsubset  \sqsupset
+% \lhd  \unlhd  \rhd  \unrhd
+% Additional symbols from amsmath
+
+% Prints an OK? sign on the margin
+\def\ok{\marginpar{\text{OK?}}}
+
+% Use to leave space so lines line up
+% This should have the sanme effect as \phantom
+% but is left here for compatibility
+\newlength{\ypushlen}
+\def\ypush#1{\settowidth{\ypushlen}{#1}\hspace*{\ypushlen}}
+
+% \ytext{text1}{text2} puts text2 at the right end of a box
+%                      of size text1, in normalshape, roman.
+% For handsetting lists, when necessary.
+\newlength{\yboxlen}
+\def\ytext#1#2{\settowidth{\yboxlen}{#1}%
+\makebox[\yboxlen][r]{\normalfont #2}}
+
+% Eqnarray modification, tighter and with a third column
+%%% labels work fine, but the \label command must be given
+%%% after the tag command
+%%% fleqn does not work in align
+
+% to detect \label - \tag error in eqnarray
+\newif\iflabelgiven
+
+% Fix the standard label definition
+\let\standardlabel\label
+
+\def\labeltagerr{%
+\errmessage{^^J
+The sequence label{..} 
+tag{..}^^J
+is not allowed in the eqnarray environment^^J
+give tag{..} 
+label{..} instead to get right numbering^^J
+Hit X to stop and correct the error^^J
+}}
+
+% to detect if \tag has been given
+% the command \asl@tagfalse is never needed, because of the locality
+% of the \@@eqnrc redefinition, which works only for the line 
+% in which the tag occurs
+\newif\ifasl@tag
+
+
+% the redone eqnarray environment
+\def\eqnarray{%
+% Adjust label to record \labelgiven
+\def\label##1{\labelgiventrue\standardlabel{##1}}
+% keep the old definition of \@currentlabel
+% \let\old@currentlabel=\@currentlabel   % 9/14/2000, ynm do not
+\let\notag\nonumber      % so notag can be used
+%% make the tag definition local to the eqnarray environment
+\def\tag##1{%
+% \label -\tag error handling
+\iflabelgiven\immediate\labeltagerr \fi
+\global\asl@tagtrue\gdef\tagsymbol{##1}%
+\xdef\@currentlabel{\tagsymbol}
+\iftagsleft@\addtocounter{equation}{-1}
+\else%
+\def\@@eqncr{
+\let\reserved@a\relax
+    \ifcase\@eqcnt \def\reserved@a{& & &}\or \def\reserved@a{& &}%
+     \or \def\reserved@a{&}\else
+       \let\reserved@a\@empty
+       \@latex@error{Too many columns in eqnarray environment}\@ehc\fi
+     \reserved@a \ifasl@tag(\tagsymbol)\else\fi
+     \global\@eqnswtrue\global\@eqcnt\z@\cr}\fi}
+%%% end of the local tag
+   \stepcounter{equation}%
+   \def\@currentlabel{\p@equation\theequation}%
+  \global\@eqnswtrue
+   \m@th
+   \global\@eqcnt\z@
+   \if@fleqn\else\tabskip\@centering\fi
+   \let\\\@eqncr
+   $$\everycr{}\halign to\displaywidth\bgroup%
+\hskip\@centering$\displaystyle\tabskip\z@skip%
+\hskip \leftmargini{##}$\@eqnsel&%
+ \global\@eqcnt\@ne\hskip.5\arraycolsep\hfil$\displaystyle{##}$\hfil%
+&\hskip .5\arraycolsep$\displaystyle{##}\hskip \leftmargini$\hfil%
+&\global\@eqcnt\tw@%
+$\displaystyle{##}$\hfil\tabskip\@centering%
+&\global\@eqcnt\thr@@\hb@xt@\z@\bgroup\hss##\egroup% 
+\gdef\@currentlabel{\p@equation\theequation}%
+\labelgivenfalse
+\tabskip\z@skip%
+\cr%
+%% restore the standard def. of \@currentsymbol
+% \gdef\@currentlabel{\old@currentlabel} 9/14/00, ynm do not
+}
+
+% To place a qedsymbol in the last equation of a proof
+
+\def\proofend{\global\qedneededfalse
+\\[-\@back] &&                 % This is the fake entry
+\def\@@eqncr{\let\@tempa\relax
+    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}
+      \else \def\@tempa{&}\fi
+     \@tempa \if@eqnsw{\qedsymbol}\fi
+     \global\@eqnswtrue\global\@eqcnt\z@\cr}}
+
+% endproofeqnarray environment, for compatibility
+\@namedef{endproofeqnarray}{\def\@eqncr{\@seqncr}\eqnarray}
+\@namedef{endendproofeqnarray}{%
+ \proofend\endeqnarray}
+
+\@namedef{endproofeqnarray*}{\def\@eqncr{\nonumber\@seqncr}\eqnarray}
+\@namedef{endendproofeqnarray*}{%
+\global\proofend\endeqnarray}
+
+% The next command can be used to make the entry not count 
+% in the centering, for example
+% \begin{eqnarray}
+% a &=& b    &\text{comment}\\
+% c &=& \forget{very long formula}\\
+% e &=& f(y) &\text{comment}
+% \end{eqnarray}
+% which lines up the comments as they appear to be.
+\def\forget#1{\protect\makebox[0pt]{$#1$}}
+
+%-------------------------------------- Heads
+
+% First page has only the page number, but normalsize
+\def\ps@firstpage{\ps@plain
+\def\@oddfoot{%                                             % jsl-l
+\normalfont\fontsize{8}{10\p@}\selectfont \hfil\thepage\hfil
+     \global\topskip\normaltopskip}%
+  \let\@evenfoot\@oddfoot
+  \def\@oddhead{\@serieslogo\hss}%
+  \let\@evenhead\@oddhead % in case an article starts on a left-hand page
+}
+
+\def\ps@headings{\ps@empty
+ \def\@evenhead{\normalfont\fontsize{10}{13\p@}\selectfont
+    \rlap{\thepage}\hfil
+\fontsize{7}{8\p@}\selectfont\leftmark{}{} \hfil}%
+ \def\@oddhead{\normalfont\fontsize{7}{8\p@}\selectfont
+    \hfil\rightmark{}{} \hfil
+    \fontsize{10}{13\p@}\selectfont \llap{\thepage}}%
+  \let\@mkboth\markboth
+}
+
+\pagestyle{headings}    % Initialize again
+
+%%%%%%%%%%%%%%%% For the production version, aslprod.sty
+
+% Definition of commands redefined in aslprod.sty
+
+\def\typesetter#1{}
+\def\papertype#1{}
+\def\revauthor#1{}
+\def\revdate#1{}
+
+\def\@location{}   % There is also a change in the definition of \location
+
+
+\InputIfFileExists{aslprod.sty}{}
+
+%------------------------------- End of asl.sty code
+
+%******************** Font Definitions for Monotype Times *************%
+
+%-------------------------------- OT1 Definitions
+\DeclareFontFamily{OT1}{xgt}{}
+
+\DeclareFontShape{OT1}{xgt}{b}{n}{
+   <-> gamatb
+}{}
+
+\DeclareFontShape{OT1}{xgt}{b}{sc}{
+   <-> sub * xgt/b/n                    % corr. 1/16/2000, ynm
+}{}
+
+\DeclareFontShape{OT1}{xgt}{b}{sl}{<->ssub * xgt/b/it}{}
+
+\DeclareFontShape{OT1}{xgt}{b}{it}{
+   <-> gamatbti
+}{}
+
+\DeclareFontShape{OT1}{xgt}{m}{n}{
+   <-> gamatr
+}{}
+
+\DeclareFontShape{OT1}{xgt}{m}{sc}{
+   <-> gamatcsc
+}{}
+
+\DeclareFontShape{OT1}{xgt}{m}{sl}{<->ssub * xgt/m/it}{}
+
+\DeclareFontShape{OT1}{xgt}{m}{it}{
+   <-> gamatti
+}{}
+
+%% Fontshape scr gives true script.
+
+\DeclareFontShape{OT1}{xgt}{m}{scr}{
+   <-> gamascr
+}{}
+
+%% Fontshape th sets in italics but with Roman punctuation.
+
+\DeclareFontShape{OT1}{xgt}{m}{th}{
+   <-> gamatth
+}{}
+
+\DeclareFontShape{OT1}{xgt}{bx}{n}{<->ssub * xgt/b/n}{}
+\DeclareFontShape{OT1}{xgt}{bx}{sc}{<->ssub * xgt/b/sc}{}
+\DeclareFontShape{OT1}{xgt}{bx}{sl}{<->ssub * xgt/b/it}{}
+\DeclareFontShape{OT1}{xgt}{bx}{it}{<->ssub * xgt/b/it}{}
+\DeclareFontShape{OT1}{xgt}{l}{n}{<->ssub * xgt/m/n}{}
+\DeclareFontShape{OT1}{xgt}{l}{sc}{<->ssub * xgt/m/sc}{}
+\DeclareFontShape{OT1}{xgt}{l}{sl}{<->ssub * xgt/m/sl}{}
+\DeclareFontShape{OT1}{xgt}{l}{it}{<->ssub * xgt/m/it}{}
+\DeclareFontShape{OT1}{xgt}{bx}{th}{<->ssub * xgt/b/it}{} % for bold in thms.
+
+
+%------------------------------------- OMS Definitions
+
+\DeclareFontFamily{OMS}{xgt}{\skewchar\font48 }
+\DeclareFontShape{OMS}{xgt}{m}{n}
+   {<-> ssub * cmsy/m/n}{}
+\DeclareFontShape{OMS}{xgt}{m}{it}
+   {<-> ssub * cmsy/m/n}{}
+\DeclareFontShape{OMS}{xgt}{m}{sl}
+   {<-> ssub * cmsy/m/n}{}
+\DeclareFontShape{OMS}{xgt}{m}{sc}
+   {<-> ssub * cmsy/m/n}{}
+\DeclareFontShape{OMS}{xgt}{bx}{n}
+   {<-> ssub * cmsy/b/n}{}
+\DeclareFontShape{OMS}{xgt}{bx}{it}
+   {<-> ssub * cmsy/b/n}{}
+\DeclareFontShape{OMS}{xgt}{bx}{sl}
+   {<-> ssub * cmsy/b/n}{}
+\DeclareFontShape{OMS}{xgt}{bx}{sc}
+   {<-> ssub * cmsy/b/n}{}
+
+%------------------------------------------ OML Definitions
+
+\DeclareFontFamily{OML}{xgt}{\skewchar\font127 }
+
+\DeclareFontShape{OML}{xgt}{m}{it}{
+      <-> gamatmi
+      }{}
+\DeclareFontShape{OML}{xgt}{b}{it}{%
+      <-> gamatmib
+      }{}
+
+% This is the true script
+
+\DeclareFontShape{OML}{xgt}{m}{scr}{
+   <-> gamascr
+}{}
+
+\DeclareFontShape{OML}{xgt}{bx}{it}
+   {<-> ssub * xgt/b/it}{}
+
+%% Includes psamsfonts and all corrections needed for the psfonts
+
+\InputIfFileExists{amsbug.sty}{}
+
+
+%--------------------------------------------
+\ExecuteOptions{leqno}
+
+\ProcessOptions\relax
+\RequirePackage{latexsym}
+
+% Load amssymb.sty (which loads amsfonts.sty) if they exist.
+\def\amssymbtest{1}
+\IfFileExists{amssymb.sty}{\gdef\amssymbtest{0}}%
+{\GenericWarning{}{WARNING:^^JPackage amssymb.sty not found
+^^Jso AMS fonts not loaded.}}
+
+\if\amssymbtest 0\let\mathfrak\relax\RequirePackage{amssymb}\fi
+
+% Do not allow compatibility mode
+%\if@compatibility \else\endinput\fi
+%\def\tiny{\Tiny}
+%\def\defaultfont{\normalfont}
+%\def\rom{\textup}
+\endinput
+%%
+%% End of file `asl.cls'.
diff --git a/nachlass/collected_dew_materials/aug25,2020/#n.tex# b/nachlass/collected_dew_materials/aug25,2020/#n.tex#
new file mode 100644
index 0000000..c0699c7
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/#n.tex#
@@ -0,0 +1,136 @@
+% 2020 Aug 25 3.01     pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+On Aug 20, I reported that after I gave
+Robert a class about logic (over internet),
+ I developped a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Opps''
+ Here False, True, Zippy and Opps carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developped after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Compleness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
\ No newline at end of file
diff --git a/nachlass/collected_dew_materials/aug25,2020/#silly# b/nachlass/collected_dew_materials/aug25,2020/#silly#
new file mode 100644
index 0000000..661d8ed
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/#silly#
@@ -0,0 +1,140 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a
+%%% xxx
+glitch
+ in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/done b/nachlass/collected_dew_materials/aug25,2020/done
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/done
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/n.aux b/nachlass/collected_dew_materials/aug25,2020/n.aux
new file mode 100644
index 0000000..f23e546
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/n.aux
@@ -0,0 +1 @@
+\relax 
diff --git a/nachlass/collected_dew_materials/aug25,2020/n.log b/nachlass/collected_dew_materials/aug25,2020/n.log
new file mode 100644
index 0000000..9566035
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/n.log
@@ -0,0 +1,157 @@
+This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex 2020.4.7)  25 AUG 2020 15:11
+entering extended mode
+ restricted \write18 enabled.
+ %&-line parsing enabled.
+**n
+(./n.tex
+LaTeX2e <2020-02-02> patch level 5
+L3 programming layer <2020-03-06>
+(/usr/local/texlive/2020/texmf-dist/tex/latex/base/latex209.def
+File: latex209.def 2018/08/11 v0.54 Standard LaTeX file
+
+
+          Entering LaTeX 2.09 COMPATIBILITY MODE
+ *************************************************************
+    !!WARNING!!    !!WARNING!!    !!WARNING!!    !!WARNING!!   
+ 
+ This mode attempts to provide an emulation of the LaTeX 2.09
+ author environment so that OLD documents can be successfully
+ processed. It should NOT be used for NEW documents!
+ 
+ New documents should use Standard LaTeX conventions and start
+ with the \documentclass command.
+ 
+ Compatibility mode is UNLIKELY TO WORK with LaTeX 2.09 style
+ files that change any internal macros, especially not with
+ those that change the FONT SELECTION or OUTPUT ROUTINES.
+ 
+ Therefore such style files MUST BE UPDATED to use
+          Current Standard LaTeX: LaTeX2e.
+ If you suspect that you may be using such a style file, which
+ is probably very, very old by now, then you should attempt to
+ get it updated by sending a copy of this error message to the
+ author of that file.
+ *************************************************************
+
+\footheight=\dimen134
+\@maxsep=\dimen135
+\@dblmaxsep=\dimen136
+\@cla=\count167
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diff --git a/nachlass/collected_dew_materials/aug25,2020/n.pdf b/nachlass/collected_dew_materials/aug25,2020/n.pdf
new file mode 100644
index 0000000..eed1ba7
Binary files /dev/null and b/nachlass/collected_dew_materials/aug25,2020/n.pdf differ
diff --git a/nachlass/collected_dew_materials/aug25,2020/n.tex b/nachlass/collected_dew_materials/aug25,2020/n.tex
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/n.tex
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/n.tex~ b/nachlass/collected_dew_materials/aug25,2020/n.tex~
new file mode 100644
index 0000000..c981657
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/n.tex~
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.1     pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/o.tex b/nachlass/collected_dew_materials/aug25,2020/o.tex
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/o.tex
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/r.tex b/nachlass/collected_dew_materials/aug25,2020/r.tex
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/r.tex
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/restart b/nachlass/collected_dew_materials/aug25,2020/restart
new file mode 100644
index 0000000..b4997d0
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/restart
@@ -0,0 +1,143 @@
+% 2020 Aug 25 1.30   pm
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  20,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+ \parskip 4 pt
+
+ After giving Robert some classes about logic (over internet),
+ I developped a much refined undersanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook).  I   NOW FULLY understand how it can be used to prove the
+ conjecture about the $\Theta$
+ primitive in my arXiv paper
+
+ The old probability argument would also work, but a cleaner
+ proof follows when using Compactness.  Both show that
+ $\Theta$
+ arithmetic is a self justifying formalism
+ when Enderton's deductive calculus for First Order logic is
+ used.  (Both employ ZF set theory to prove  the consistency preservation
+ property holds for my self justifying  $\Theta$ arithetic axiom system.) 
+
+ The result holds for other Hilbert-style deductive methods,
+ but Enderton's formalism simplifies the 
+ proof because it uses only modus ponens as a rule of inference
+ (thereby making propositional calculus easier to apply
+ to what Enderton has called ``prime formulae'' ).
+
+ This proof was designed by me without the help of Nate and Cameron.
+ I am tentatively planning to allow them co-authorship if they write
+ up the paper.  (I will not object to listing names in alphabetical
+ order if they do the write-up AND THE PAPER EXPLAINS MY ROLE IN CONCEIVING
+ the result.)
+
+ The latter is needed because I want to get a new job
+ (and this will be challenging when I am 72 years old and
+ suffering from macular eye degeration)
+ I am actually being quite generous in this offer because they
+ can use the tex files from my arXiv paper and Florida articles to
+ AS ROUGH DRAFTS to write some sections.
+
+
+
+
+
+ 
+ \end{document}
+ 
+
+
diff --git a/nachlass/collected_dew_materials/aug25,2020/silly b/nachlass/collected_dew_materials/aug25,2020/silly
new file mode 100644
index 0000000..661d8ed
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/silly
@@ -0,0 +1,140 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a
+%%% xxx
+glitch
+ in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/silly-bak b/nachlass/collected_dew_materials/aug25,2020/silly-bak
new file mode 100644
index 0000000..661d8ed
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/silly-bak
@@ -0,0 +1,140 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a
+%%% xxx
+glitch
+ in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/silly~ b/nachlass/collected_dew_materials/aug25,2020/silly~
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/silly~
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/start b/nachlass/collected_dew_materials/aug25,2020/start
new file mode 100644
index 0000000..5797d51
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/start
@@ -0,0 +1,141 @@
+ % 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  20,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+ \parskip 4 pt
+
+ After giving Robert some classes about logic (over internet),
+ I developped a much refined undersanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook).  I   NOW FULLY understand how it can be used to prove the
+ conjecture about the $\Theta$
+ primitive in my arXiv paper
+
+ The old probability argument would also work, but a cleaner
+ proof follows when using Compactness.  Both show that
+ $\Theta$
+ arithmetic is a self justifying formalism
+ when Enderton's deductive calculus for First Order logic is
+ used.  (Both employ ZF set theory to prove  the consistency preservation
+ property holds for my self justifying  $\Theta$ arithetic axiom system.) 
+
+ The result holds for other Hilbert-style deductive methods,
+ but Enderton's formalism simplifies the 
+ proof because it uses only modus ponens as a rule of inference
+ (thereby making propositional calculus easier to apply
+ to what Enderton has called ``prime formulae'' ).
+
+ This proof was designed by me without the help of Nate and Cameron.
+ I am tentatively planning to allow them co-authorship if they write
+ up the paper.  (I will not object to listing names in alphabetical
+ order if they do the write-up AND THE PAPER EXPLAINS MY ROLE IN CONCEIVING
+ the result.)
+
+ The latter is needed because I want to get a new job
+ (and this will be challenging when I am 72 years old and
+ suffering from macular eye degeration)
+ I am actually being quite generous in this offer because they
+ can use the tex files from my arXiv paper and Florida articles to
+ AS ROUGH DRAFTS to write some sections.
+
+
+
+
+
+ 
+ \end{document}
+ 
+
+
diff --git a/nachlass/collected_dew_materials/aug25,2020/temp.tex b/nachlass/collected_dew_materials/aug25,2020/temp.tex
new file mode 100644
index 0000000..da5dc9e
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/temp.tex
@@ -0,0 +1,34 @@
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/temp.tex.bak b/nachlass/collected_dew_materials/aug25,2020/temp.tex.bak
new file mode 100644
index 0000000..abcde28
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/temp.tex.bak
@@ -0,0 +1,34 @@
+
+On Aug 20, I reported that after I gave
+Robert a class about logic (over internet),
+ I developped a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Opps''
+ Here False, True, Zippy and Opps carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developped after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Compleness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/temp.tex~ b/nachlass/collected_dew_materials/aug25,2020/temp.tex~
new file mode 100644
index 0000000..07a00b9
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/temp.tex~
@@ -0,0 +1,136 @@
+% 2020 Aug 25 2.55     pm
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+On Aug 20, I reported that after I gave
+Robert a class about logic (over internet),
+ I developped a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Opps''
+ Here False, True, Zippy and Opps carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developped after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Compleness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/v1 b/nachlass/collected_dew_materials/aug25,2020/v1
new file mode 100644
index 0000000..9b7cc49
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v1
@@ -0,0 +1,170 @@
+% 2020 Aug 25 1.30   pm
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  20,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+On Aug 20, I reported
+ After giving Robert some classes about logic (over internet),
+ I developped a much refined undersanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding that uses a logic with 2.2 logic symbols, where the
+ symbol submols is an ``unknown'' symbol when a Boolean value is unknown.
+ called ``Zippy'' and the fourth symbol is a never-employed
+ symbol ``Opps'' Here False, True, Zippy and Opps caried the numeral values of 0, 1, 2, and 3 in a ``quadruple'' analog of a decimal encoding.
+
+ Also, I discovered yesterday. I don.t have time to go into details here
+ of either of the above because I need to do my taxes today.  I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workrs in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ pbvious security breach is possible.
+
+ \end{document}
+ \newpage
+
+ \section{Most of Old Notes with frist pargarph Deleted}
+
+
+
+ I   NOW FULLY understand how it can be used to prove the
+ conjecture about the $\Theta$
+ primitive in my arXiv paper
+
+ The ol
+ probability argument would also work, but a cleaner
+ proof follows when using Compactness.  Both show that
+ $\Theta$
+ arithmetic is a self justifying formalism
+ when Enderton's deductive calculus for First Order logic is
+ used.  (Both employ ZF set theory to prove  the consistency preservation
+ property holds for my self justifying  $\Theta$ arithetic axiom system.) 
+
+ The result holds for other Hilbert-style deductive methods,
+ but Enderton's formalism simplifies the 
+ proof because it uses only modus ponens as a rule of inference
+ (thereby making propositional calculus easier to apply
+ to what Enderton has called ``prime formulae'' ).
+
+ This proof was designed by me without the help of Nate and Cameron.
+ I am tentatively planning to allow them co-authorship if they write
+ up the paper.  (I will not object to listing names in alphabetical
+ order if they do the write-up AND THE PAPER EXPLAINS MY ROLE IN CONCEIVING
+ the result.)
+
+ The latter is needed because I want to get a new job
+ (and this will be challenging when I am 72 years old and
+ suffering from macular eye degeration)
+ I am actually being quite generous in this offer because they
+ can use the tex files from my arXiv paper and Florida articles to
+ AS ROUGH DRAFTS to write some sections.
+
+
+
+
+
+ 
+ \end{document}
+ 
+
+
diff --git a/nachlass/collected_dew_materials/aug25,2020/v2-before-lunch b/nachlass/collected_dew_materials/aug25,2020/v2-before-lunch
new file mode 100644
index 0000000..193cbbc
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v2-before-lunch
@@ -0,0 +1,134 @@
+% 2020 Aug 25 2.15   pm
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+On Aug 20, I reported that after I gave
+Robert a class about logic (over internet),
+ I developped a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   Called ``Opps''
+ Here False, True, Zippy and Opps carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also developped after talking to robert an improved
+ proof of  Enderton's
+ A-version of the Compleness Theorem
+ I don't have time to go into details here
+ of either of the above because I need to do my taxes today.
+
+ I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/v3-also-before-lunch b/nachlass/collected_dew_materials/aug25,2020/v3-also-before-lunch
new file mode 100644
index 0000000..211d1d7
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v3-also-before-lunch
@@ -0,0 +1,136 @@
+% 2020 Aug 25 2.35    pm
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+On Aug 20, I reported that after I gave
+Robert a class about logic (over internet),
+ I developped a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an Aug 24 class, I developped a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   Called ``Opps''
+ Here False, True, Zippy and Opps carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tommorow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developped after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Compleness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/v4 b/nachlass/collected_dew_materials/aug25,2020/v4
new file mode 100644
index 0000000..c981657
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v4
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.1     pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+\Large
+\baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/v5 b/nachlass/collected_dew_materials/aug25,2020/v5
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v5
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/aug25,2020/v5-done b/nachlass/collected_dew_materials/aug25,2020/v5-done
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/aug25,2020/v5-done
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/baltimore.tex b/nachlass/collected_dew_materials/baltimore.tex
new file mode 100644
index 0000000..3735acc
--- /dev/null
+++ b/nachlass/collected_dew_materials/baltimore.tex
@@ -0,0 +1,139 @@
+%%%%  septem10 correction  10.2am  MISSING WORD FIX UP 
+
+
+
+\documentclass[bsl,meeting]{asl}
+
+\pagestyle{plain}
+
+\def\urladdr#1{\endgraf\noindent{\it URL Address}: {\tt #1}.}
+
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\NP}{}
+
+\begin{document}
+\thispagestyle{empty}
+
+
+\NP  
+\absauth{Dan E. Willard}
+\meettitle{On 
+the Utility of 
+Partial Evasions
+of the Second Incompleteness Theorem in the Modern
+Digital Era}
+\affil{Computer Science \& Mathematics Departments, 
+University at Albany, NY 12222}
+\meetemail{dwillard@albany.edu}
+
+ \normalsize
+ % \Large
+% \baselineskip = 2.0 \normalbaselineskip 
+ \baselineskip = 1.0 \normalbaselineskip 
+
+We have published several articles
+about 
+generalizations of the Second Incompleteness  
+ Theorem
+and partial
+evasions of it under
+formalisms that own a partial
+knowledge about their own self-consistency.  
+This research has included six articles
+in the JSL and APAL plus several additional papers,
+all of which are 
+cited by us in
+paper \cite{ww18}.
+In a context where $\alpha$ is a set of proper axioms and
+$d$ is one of several possible deduction methods, the ordered pair
+$(\alpha,d)$ will be called {\bf Self Justifying} iff:
+\begin{description}
+  \item[  i   ] one of  $~(  \alpha  , d  )$'s  theorems
+(or possibly one of $\alpha$'s axioms)
+will
+state that the deduction method $ \, d, \, $ applied to the
+axiom
+system $ \, \alpha, \, $ 
+is formally consistent.
+\item[  ii   ]
+   and also  the axiom system $ \, \alpha  \, $ is actually consistent.
+\end{description}
+We noted  the 
+Fixed Point Theorem enables one to map 
+any ordered pair $(\alpha,d)$ onto an axiom system $\alpha^d$
+that includes all $\alpha$'s axioms plus an added axiom
+statement  declaring:
+\begin{quote} 
+$\oplus~~~$ {\it
+``There is no proof 
+(using 
+$d$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+statement $~\oplus ~$ (looking at itself)''.}
+\end{quote}
+The difficulty with statement 
+$\oplus$ is that it is typically false, {\bf  although it can be quite easily
+encoded.} This is because if the ordered pair $(\alpha,d)$ is too
+strong then the system   $\alpha^d$ will be rendered
+inconsistent via a classic
+G\"{o}del
+ Diagonalization construction (e.g.
+causing $(\alpha^d,d)$ to violate Part-ii of
+Self-Justification's
+ definition).
+
+These issues  raise  {\bf legitimate concerns}
+about whether  boundary-case
+exceptions to the Second Incompleteness Theorem are 
+an awkward  wrinkle
+from a theoretical  standpoint.
+ Aside from generalizing our earlier
+results, our new paper suggests its proposals can be
+ significant  from a {\it pragmatic perspective.}
+For instance,
+ the late physicist
+Stephen Hawking
+has predicted that global warming
+and other dangers will likely be so severe
+that civilization in our Solar System
+will find it difficult to persist without
+employing
+ artificially
+intelligent
+computers, 
+{\it in at least some respect.}
+
+One would
+ideally
+ prefer computers to imitate a human's approximate
+instinctive appreciation 
+of  his own consistency.  
+Our paper \cite{ww18} observes  self-justifying
+computers can do this in  a pragmatic manner,
+when $(\alpha,d)$ is sufficiently weak.
+Moreover
+{\it with quite comparable efficiency}, our formalisms can
+nicely
+ reconstruct  isomorphic 
+analogs of
+all  the  $\Pi_1$ theorems of Peano Arithmetic  under a 
+slightly revised language.
+\begin{thebibliography}{99}
+\bibitem{ww18}
+Dan E. Willard,
+``About   the 
+Chasm Separating the Goals of
+Hilbert's Consistency Program From the
+Second Incompleteness Theorem''
+http://arXiv.org/abs/1807.04717
+
+% (in the Cornell Archives).
+
+\end{thebibliography}
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/changes.tex b/nachlass/collected_dew_materials/changes.tex
new file mode 100644
index 0000000..81bfe11
--- /dev/null
+++ b/nachlass/collected_dew_materials/changes.tex
@@ -0,0 +1,57 @@
+
+Suggested Changes for Dan Willard's Wiki Page
+
+Thanks for your help with Dan Willard's wiki page, and below
+is a list of four minor new updates (one of which I suspect you will
+find to be quite interesting and surprisingly intriguing):
+
+1) Please change the first clause in the ``Contributions'' Section
+to read as given below. Thus, the initial words of this passage will be
+the exact same as your words.  The only change is the new suggested words on the
+second line below. This added new short passage starts with phrase of ``as well as''
+
+``Although trained as a mathematician and employed as a computer scientist
+     as well as an expert in the proof-theoretic aspects of Symbolic Logic,...''
+
+2) Just before the ``SELECTED PUBLICATIONS'' section, please add the following tiny
+new section entitled ``MORE DETAILS about WILLARD'S LOGIC RESEARCH''. (You may
+either copy the below passage verbatim or edit it, if you so wish. I suspect you
+will find the below passage to be surprisingly interesting and informative):
+
+Willard's most recent published paper during his logic investigations appeared
+during 2021 in Oxford University's Journal of Logic and Computation (publication 8
+whose exact citation you can find in the Item A-1 of my vitae). This article
+both extends Willard's prior research, and it also summarizes the implications
+of his prior and on-going research. It explains more clearly than before the
+motivation for Willard's unorthodox exploration
+of boundary-case exceptions to G\"{o}del's Second Incompleteness Theorem.
+It thereby contains a pointer to an 80-minute year-2007 You-Tube lecture,
+delivered by Prof. Gerald Sacks (who had visited Kurt G\"{o}del for two extended 
+periods at the Institute for Advanced Studies and also did hold two simultaneous
+joint appointments with Harvard and MIT). Sacks explains that
+the reticent Kurt G\"{o}del had expressed startling
+private opinions ``that were almost the  opposite of what everyone
+else would have expected''. Sacks thus indicated that  G\"{o}del believed some type
+of expanded Gentzen-like exception to  G\"{o}del's G-2 result would eventually
+accompany  G\"{o}del's G-2 formalism.  In essence, Willard's logic research, which
+started in 1993, was based on intuitions that were approximately similar
+to the comments that Gerald Sacks had much later attributed to Kurt G\"{o}del during
+Sacks's year-2007 You-Tube lecture. More details about this 80-minute You-Tube lecture
+and its exact relationship to Willard' research can be found in Willard's recent
+year-2021 JLC article.
+
+3. Please replace my old vitae with the  attached  updated year-2021 version
+of this vitae.
+
+4. Please insert an eighth article into my list of selected publications on the
+wiki page.  It correspond to Item A-1 in my vitae's publication list.
+
+Again, I thank you very warmly for setting up my wiki page, and I have
+carefully described these changes so that you can easily undertake them.
+It is obviously significant that a Harvard-MIT professor, who was a close
+friend of Kurt G\"{o}del, recorded a year-2007 You-Tube lecture, that was consistent
+with a theory that I started advocating in 1993. To save you work, my Item 2
+has outlined a new paragraph that you can add to my wiki-page. Thus, you can either
+employ Item 2 verbatim or revise it, if you so wish. You can telephone me at
+518-475-1622 or email me at dan.willard.albany@gmail.com  ---- Very Best Regards, Dan
+
diff --git a/nachlass/collected_dew_materials/dwillard-lfcs.tex b/nachlass/collected_dew_materials/dwillard-lfcs.tex
new file mode 100644
index 0000000..82a72ab
--- /dev/null
+++ b/nachlass/collected_dew_materials/dwillard-lfcs.tex
@@ -0,0 +1,3520 @@
+%% 9sept 2.1 pm       Final Copy Ready Submit AFTER SPELL DONE
+%%% AND dwillard-lfcs will NOW be READY
+%%   Artemov "recently" added to page 1
+
+\documentclass[12pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+%\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-0.8in}
+\setlength{\textheight}{9.2 in}
+%\setlength{\textwidth}{6.0 in}
+ \setlength{\textwidth}{6.7 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}
+%%% \newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+\newtheorem{comm}[theorem]{$~~~~$Comment}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\aaa{\beta}
+\def\ccc{Class}
+
+
+\def\ulxyz{\lceil}
+\def\urxyz{\rceil}
+
+\def\ulxyz{\ulcorner}
+\def\urxyz{\urcorner}
+
+
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{SFact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+
+\title{On a 3-Part ``Tripod'' Styled Reply
+  to Hilbert's Mysterious Second Problem}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%\thanks{This research 
+%was partially supported
+%by the NSF Grant CCR  0956495.}}
+
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+
+ 
+
+
+
+
+\begin{abstract}
+\baselineskip = 4.2 \normalbaselineskip  
+\Large
+Hilbert's mysterious year-1900 Second Problem asked
+mathematicians to devise a methodology whereby
+Peano Arithmetic  can confirm its own consistency.
+G\"{o}del's famous 1931 paper showed that a fully
+positive reply can never be made to Hilbert's question.
+This article will explain how Hilbert's question is
+such a complicated issue that it can be better receive
+a 3-way styled  ``Tripod'' reply.
+
+\bigskip
+
+We  also provide substantial evidence  that G\"{o}del would likely
+agree with  the main opinions   expressed in this article.
+
+\end{abstract}
+
+\bigskip
+
+\bigskip
+
+\bigskip
+
+\bigskip
+
+ 
+ \normalsize
+ {\bf Keywords and Phrases:}
+ G\"{o}del's Second Incompleteness Theorem,  Hilbert's Second
+ Problem, Consistency, Smullyan-Fitting Semantic Tableau Deduction,
+ Hilbert-Frege Deduction.
+
+
+\bigskip
+
+\bigskip
+
+%% 
+%% {\bf Mathematics Subject Classification:}
+%% 03B52; 03F25; 03F45; 03H13 
+%% 
+%% \bigskip
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.10 \normalbaselineskip  \parskip 2pt }
+\def\ttt2c{ \baselineskip = 0.98 \normalbaselineskip  \parskip 0pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+% \def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \large \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+\def\gvx{ \large \baselineskip = 1.6 \normalbaselineskip  \parskip    3pt}
+
+%fffff
+\def\gvs{ \Large \baselineskip = 1.8 \normalbaselineskip  \parskip   5pt}
+
+\def\gvs{ \LARGE \baselineskip = 2.0 \normalbaselineskip  \parskip   5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.0 \normalbaselineskip  \parskip   5pt}
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip   5pt}
+
+ \def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip   5pt}
+ \def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip   5pt}
+\def\gvs{ \normalsize \baselineskip = 2.0 \normalbaselineskip  \parskip   5pt}
+\def\gvs{ \normalsize \baselineskip = 2.5 \normalbaselineskip  \parskip   5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip   5pt}
+\def\gvs{ \large      \baselineskip = 1.7 \normalbaselineskip  \parskip   5pt}
+\def\gvs{ \normalsize \baselineskip = 1.7 \normalbaselineskip  \parskip   5pt}
+
+
+% \def\gvs{ \normalsize \baselineskip = 1.07 \normalbaselineskip  \parskip =  5pt}
+
+%\def\gvs{ \large \baselineskip = 2.4 \normalbaselineskip}
+%\def\gvs{ \normalsize \baselineskip = 3.1 \normalbaselineskip}
+
+%\def\gvs{ \normalsize \baselineskip = 3.4 \normalbaselineskip}
+
+%\def\gvs{ \normalsize \baselineskip = 1.07 \normalbaselineskip  \parskip  5pt}
+
+
+%\def\gvs{ \normalsize \baselineskip = 2.07 \normalbaselineskip}
+
+\def\gvs{ \large \baselineskip = 1.6 \normalbaselineskip}
+
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip}
+
+\def\gvs{ \Large \baselineskip = 1.8 \normalbaselineskip}
+
+\def\gvs{ \normalsize \baselineskip = 2.2 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 2.7 \normalbaselineskip}
+%\def\gvs{ \normalsize \baselineskip = 2.0 \normalbaselineskip}
+
+\def\gvs{ \normalsize \baselineskip = 2.9 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 3.0 \normalbaselineskip}
+
+\def\gvs{ \normalsize \baselineskip = 2.0 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 1.7 \normalbaselineskip}
+
+\def\gvs{ \normalsize \baselineskip = 1.07 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 1.02 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 1.0 \normalbaselineskip}
+\def\gvs{ \normalsize \baselineskip = 1.01 \normalbaselineskip}
+
+\def\nvs{ \Large \baselineskip = 2.5 \normalbaselineskip}
+
+\def\nvs{ \normalsize \baselineskip = 1.5   \normalbaselineskip}
+
+%
+
+% \def\nvs{ \large \baselineskip = 1.8   \normalbaselineskip}
+
+%%%\def\nvs{ \normalsize \baselineskip = 1.10   \normalbaselineskip}
+
+\def\nvs{ \normalsize \baselineskip = 1.08   \normalbaselineskip}
+
+% \def\nvs{ \normalsize \baselineskip = 2.2   \normalbaselineskip}
+\def\nvs{ \normalsize \baselineskip = 0.96   \normalbaselineskip}
+
+ % fffffffff
+
+
+
+\label{ss11}
+\gvs
+\parskip 3pt
+\tttc
+
+\def\dvs{ \normalsize \baselineskip = 2.5   \normalbaselineskip}
+\def\dvs{ \normalsize \baselineskip = 3.0   \normalbaselineskip}
+ \def\dvs{ }
+\newpage
+
+\section{Introduction}
+%  11111111111111111 }
+
+\label{aaa1}
+\nvs
+\baselineskip = 1.02   \normalbaselineskip
+\parskip 4pt
+\tttc
+\dvs
+
+This article is a continuation of a series of papers
+that began
+with the 1993 article \cite{ww93} 
+and continued until and through the year-2021 article \cite{ww21}.
+This series,
+which included six papers appearing in the JSL and APAL,
+had focused on discussing
+generalizations and boundary case exceptions for the
+Second Incompleteness Theorem.
+The two goals of this paper will be to explore the underlying
+philosophy that motivated this series and to explain how
+it is related to several generalizations and
+boundary-case exceptions to the
+Second Incompleteness Theorem , including some important new
+results formalized by Artemov  \cite{Ar19,Ar21}.
+
+
+Our general theme will be that Hilbert's year-1900 Second Problem
+ is too complex an issue to receive a 1-directional or
+ even 2-directional
+ answer. Instead, it will require a
+ 3-directional answer, called the ``Tripod Reply'' to
+ Hilbert's question. 
+
+
+ 
+ The  first leg of this 3-part response to
+ Hilbert's Second Problem
+will rest on the combination
+ of G\"{o}del's initial version of his Second Incompleteness
+ Theorem and its numerous generalizations.
+ % cite????.
+ They
+collectively
+establish that 
+many logical formalisms, besides Peano Arithmetic (PA),
+are unable to corroborate their own consistency
+in a fully extensive respect.
+While these generalizations of G\"{o}del's Second 
+Incompleteness paradigm are indisputably important,
+the second section of this article will explain
+%% how
+Hilbert and G\"{o}del
+strongly doubted they constituted a full answer to  
+  Hilbert's year-1900 problem.
+
+%%% citations in above paragraph MAYBE NOT? PROBABLY NOT NOT NOT NOT
+  
+ A second leg of a 3-part reply to Hilbert's Second
+Problem
+ was formalized by  
+ Artemov
+            recently
+ in \cite{Ar19,Ar21}.
+ He noted Peano Arithmetic (PA) can prove
+ an infinite
+%  -sized
+ schema of
+ theorems, whose collective
+ union
+ confirms PA's own consistency.
+Let us call Artemov's method a
+Step-By-Step Infinite-Schema  approach  
+ {\bf ($~$SBSIS$~$)}.
+This technique is an extension of the 
+ Justification Logics explored by Artemov and Beklemishev 
+ \cite{Ar1,AB5,Be5}. It was  motivated by
+Artemov's
+ observation
+ that
+the year-1900 logic community (including Hilbert)
+were not aware  that PA would require an
+ infinite number of proper axioms.
+$~$Thus, an  SBSIS schema-driven logic approach
+is a valid reply to Hilbert's year-1900 second problem,
+although its potential significance
+was not
+well appreciated during the
+era when Hilbert  posed his Second  Problem.
+
+Artemov's analysis
+\cite{Ar19,Ar21} 
+uses Tarski's partial definitions
+of truth for sentences with a bounded number of quantifiers
+as an intermediate step. It essentially
+constructs a sequence of finite subsets of
+Peano Arithmetic $~S_1~\subset ~S_2~\subset ~S_3~\subset~ ...~$
+where
+\bee
+\item
+$~~~$PA$~=~~S_1~\cup ~S_2~\cup ~S_3~\cup~ ...~$
+\item
+\label{step2}
+  Each $~S_{j+1}~$ can prove
+a $\Pi_1$ theorem asserting
+  there exists no proof from
+  $~S_j~$
+  of 0=1, in a context where all  logical axioms
+  in the concerned proof from $~S_j~$
+are no
+  more complex than $\Pi_j$
+  or $\Sigma_j$ statements. (We will henceforth
+  call this theorem $T_{j+1}$. )
+  \ene
+While  
+Artemov's SBSIS-style response to Hilbert's
+year-1900 second question
+is intriguing,
+one would ideally still like access to 
+a system that does not rely upon his infinite series
+$S_1 \, S_2 \, S_3  ...~$.
+
+\nvs
+
+\dvs
+
+
+\parskip 9 pt
+
+A third possible leg  of a proposed Tripod
+reply to Hilbert's second problem  
+involves formal systems using Fixed Point axiomatic
+sentences that confirm their own consistency.
+For example,
+% the articles
+\cite{ww93,ww1,ww5,wwapal,ww21}
+examined systems
+strictly weaker than PA,
+that 
+ verified their own self-consistency,
+under mostly semantic tableau deduction
+\cite{Fi96,Sm95}.
+
+
+
+This third leg,
+called the {\bf Declarative Approach},
+rests on using a self-referencing 
+{\it ``I am consistent''} axiomatic declaration,
+so that a formalism can confirm its own consistency.
+This approach, studied in
+\cite{ww93,ww1,ww5,wwapal,ww21},
+will  essentially be defined by
+the current paper's statement $~\oplus~$.
+Its advantage is
+that its declaration of self-consistency is compressed into
+one single sentence,
+while its {\bf non-trivial drawback} is that its particular  statement $~\oplus~$
+(defined in \textsection\ref{aaa3})
+can be successfully used  only when the 
+surrounding formalism is sufficiently weak.
+
+In other words, each of the three legs of a unified ``Tripod''
+response to Hilbert's Second Problem
+% do
+own drawbacks, as well as significant
+virtues. A 
+multi-facet
+Tripod reply is attractive
+because it 
+straddles nicely between
+%these
+all  three
+legs, simultaneously.
+
+
+\section{The Main Weakness of the First Leg}
+
+%%%% 2222222222222 
+
+\label{aaa2}
+
+It is  known  Hilbert always suspected  the
+Second Incompleteness Theorem, while correct, would
+ultimately  display established exceptions.
+For instance, Hilbert \cite{Hil26} wrote:
+\begin{quote}
+\small
+\baselineskip = 1.0 \normalbaselineskip
+\ttt2c 
+$*~$
+{\it ``
+Let us admit that the situation in which we presently
+find ourselves with respect to paradoxes is in the long
+run intolerable. Just think: in mathematics, this paragon of
+reliability and truth, the very notions and inferences,
+as everyone learns, teaches, and uses them, lead to absurdities.
+And 
+where 
+else 
+would 
+reliability and truth be found 
+if  even mathematical thinking fails?''}
+\end{quote}
+Also, it is known 
+G\"{o}del's seminal 1931 paper appeared to
+agree with
+the goals of Hilbert's
+Consistency Program
+in one of its closing paragraphs:
+\begin{quote}
+\small
+\ttt2c 
+$~**~~$ 
+{\it ``It must be expressly noted that
+Theorem XI''}
+(i.e. the Second Incompleteness Theorem) 
+{\it ``represents no contradiction of the formalistic
+standpoint of Hilbert. For this standpoint
+presupposes only the existence of a consistency
+proof by finite means, and {\it there might
+conceivably be finite proofs} which cannot
+be stated in P or in ... ''}
+\end{quote}
+Several 
+biographies
+of 
+ G\"{o}del
+\cite{Da97,Go5,Yo5}
+have observed 
+ G\"{o}del's 
+ intention (prior to 1930)
+was to
+establish
+Hilbert's proposed objectives, before
+he developed
+his famous
+nearly
+opposing theorem. 
+Indeed,
+ Yourgrau's
+biography
+ \cite{Yo5}
+ of G\"{o}del 
+had traversed beyond this point.
+It
+recorded
+ how
+von Neumann 
+found it necessary during the early 1930's to
+{\it ``argue 
+against G\"{o}del 
+himself''}
+ about the definitive 
+ termination of Hilbert's
+consistency program,  
+which
+{\it ``for several years''} after \cite{Go31}'s publication,
+G\"{o}del 
+{\it ``was cautious not to prejudge''}.
+
+In a context where
+G\"{o}del
+published fewer than 85 pages during his
+career,
+historians 
+will
+probably
+never
+% be able to
+fully characterize
+the nature of
+G\"{o}del's
+finished  philosophical position about his Second Incompleteness
+Theorem. From informal notes taken during a 1933 
+ Vienna
+ lecture  \cite{Go33}, it is known
+ % that 
+ G\"{o}del was more positive about its
+% long-range
+ implications
+in 1933 than he was in 1931.
+
+On the other hand, Gerald Sacks   recorded
+a
+% quite
+stunning
+80-minute YouTube recollection 
+about his interactions with
+G\"{o}del in the year 2007.
+These recordings
+explicitly  
+state that
+G\"{o}del
+held  {\it ``contrarious''} opinions about some of
+his 
+most
+famous results.
+Also Sacks explicitly recalled (see footnote
+\footnote{These two quotations
+can be found in
+  the 7-th and and 9-th minutes of the YouTube recording
+  \cite{YouSa14}
+  that Gerald Sacks had made about Kurt
+  G\"{o}del.
+  } )
+G\"{o}del  
+  communicating
+private
+opinions
+% to Sacks
+that were
+{\it ``almost the opposite of what every one
+else would have expected''.}
+
+These remarks by Sacks deserve to be taken
+seriously,
+given that Sacks 
+interacted  twice
+with  G\"{o}del
+at the Institute of Advanced Studies
+(once in 1959-1960 and a second time
+during the 1970s). Moreover, Sacks went beyond the preceding quoted
+remarks.
+Other
+memorable comments  from
+ \cite{YouSa14}'s
+  YouTube lecture
+are that:
+\bed
+\item{   a)  } 
+ G\"{o}del
+ {\it ``did not  think''}
+the objectives of Hilbert's Consistency Program 
+{\it ``were erased''} 
+by
+the Incompleteness Theorem,
+\item{   b)  } 
+G\"{o}del believed (according to Sacks) 
+ it left
+  Hilbert's program 
+{\it ``very much alive and
+even more interesting than it initially was''}. 
+\ennd
+Also,  Nerode \cite{Ne20}    
+indicated in private communications that
+Stanley Tennenbaum
+shared similar conversations 
+with G\"{o}del, as those remembered by
+Sacks.
+These
+conversations
+%%%  with  Tennenbaum also, apparently, 
+%% expressed
+likewise noted
+the
+need for some type of revival of
+Hilbert's
+Consistency Program.
+
+Thus,
+ the last two paragraphs,
+have
+summarized the quandary
+that Symbolic
+Logic has faced since 1931.
+They suggest some type of
+ Tripod-like response
+to Hilbert's  Second  Problem
+may become necessary to formulate
+a more comprehensive
+response to Hilbert's question.
+
+\section{Starting Notation with the WCB \& USEGR Paradigms}
+
+%%  3333333
+  \label{aaa3}
+
+  We will summarize the contents of the prior articles
+  \cite{wwapal,ww21} so it will be
+ unnecessary to read these papers.
+The particular annotated paragraphs in
+\cite{ww21} had used
+decimal reference  numbers
+  such as ``Example 3.1'' or ``Definition 3.2''. A non-decimal paragraph
+  notation will be used in the current paper,
+thus having its
+  first  annotated paragraph instead
+called ``Definition 1''. This type of
+adjusted notation
+should
+ cross-reference the prior paper \cite{ww21}
+in a manner that avoids
+any ambiguity and/or
+  confusion.
+
+  
+\begin{deff}
+  \label{defx1}
+\rm  
+An
+ordered pair $(\alpha,D)$ is called a
+{\it Generalized Arithmetic Configuration}
+(abbreviated as a {\bf ``GenAC'' })
+when  its 
+first and second 
+components 
+are 
+defined 
+as 
+follows:
+\bee
+\item
+The {\bf Axiom Basis} ``$~\alpha~$'' 
+for a 
+ GenAC
+is defined as
+its set of
+ proper axioms. 
+\item
+The second component 
+ ``$\, D \,$''
+  of a 
+ GenAC,  called its 
+ {\bf Deductive Apparatus},
+is
+defined as
+the union of its
+ logical axioms ``$\,L_D$'' with its
+        rules for obtaining inferences.
+\ene
+\end{deff}    
+
+The Example 3.1 from \cite{ww21} provided several examples of
+GenAC formalisms. Its Definition 3.2 defined a GenAC $(\alpha,D)$
+to be {\bf ``Self-Justifying''} when:
+\begin{description}
+  \item[  i.   ] one of  $~(  \alpha  , D  )$'s  theorems
+(or possibly one of $\alpha$'s axioms)
+states that the deduction method $ \, D, \, $ applied to the
+basis
+system $ \, \alpha, \, $ 
+produces a consistent set of theorems, and
+\item[  ii.   ]
+     the GenAC formalism $ \,( \alpha,D)  \, $ is
+actually, in fact,
+ consistent.
+\end{description}
+
+\begin{exx}
+  \label{ex3}
+\rm  
+For any  $\,(\alpha,D) \,$, 
+it is
+easy
+to construct a 
+system $ \, \alpha^D \, \supseteq  \,  \alpha  \, $
+ that  satisfies
+the
+Part-i 
+condition
+in an isolated context  where the Part-ii condition is
+ not also
+satisfied.
+Thus,
+  $ \, \alpha^D \, $  can
+consist of all of $~\alpha \,$'s axioms plus 
+the added {\bf $\,$``SelfRef$(\alpha,D)$''$\,$} sentence,
+defined below: 
+\begin{quote} 
+$\oplus~~~$ 
+There is no proof 
+(using 
+$D$'s deduction method)
+of  $0=1$
+from the  {\it union}
+ of
+the
+ axiom system $\, \alpha \, $
+with {\it this}
+sentence  ``SelfRef$(\alpha,D) \,$'' (looking at itself).
+\end{quote}
+Each of
+Kleene and Rogers
+\cite{Kl38,Ro67}
+noticed
+how
+to
+encode
+ analogs of 
+SelfRef$(\alpha,D) \,$'s  above statement,
+which we will often 
+ call an 
+ {\bf $\,$``I AM CONSISTENT'' 
+ axiom.}
+ The catch is
+%%%  that
+$\alpha ^D$ 
+may
+be inconsistent
+(e.g. 
+violate
+ Part-ii of   self-justification's
+definition
+ despite
+the assertion in
+ SelfRef$(\alpha,D)$'s 
+particular
+statement).
+This is because if the 
+ pair $(\alpha,D)$ is too strong
+then a
+quite conventional
+G\"{o}del-style diagonalization argument can
+be applied to the axiom basis of
+$~\alpha^D~=~ \alpha \, + \, $ SelfRef$(\alpha,D), ~$
+where the added presence of the statement 
+SelfRef$(\alpha,D)$ 
+will cause this extended version of 
+$\, \alpha\,$, ironically,
+ to
+ become automatically inconsistent.
+ Thus, while an
+ encoding for statement $~\oplus \,$'s {\it ``I am consistent''} declaration 
+ is a  routine consequence of the Fixed Point Theorem,
+its computational implementation is
+ potentially
+devastating.
+
+% consequences.
+ 
+\end{exx}
+
+\begin{definition}
+\label{defy3}
+\rm
+Let
+ $Add(x,y,z)$ and    $Mult(x,y,z)$ 
+denote two 3-way predicates  specifying 
+ $x+y=z$ and $x*y=z$.
+Let us say
+ a formalized
+system of
+$~\alpha~$
+recognizes successor,  addition  and multiplication
+as {\bf Total Functions} iff 
+it can  prove all of
+\eq{totdefxs} - \eq{totdefxm}
+as theorems:
+\end{definition}
+{ \small
+\baselineskip =  .9 \normalbaselineskip 
+\beq 
+\label{totdefxs}
+\forall x ~ \exists z ~~~Add(x,1,z)~~
+\enq
+\beq 
+\label{totdefxa}
+\forall x ~\forall y~ \exists z ~~~Add(x,y,z)~~
+\enq
+\beq 
+\label{totdefxm}
+\forall x ~\forall y ~\exists z ~~~Mult(x,y,z)~
+\enq }
+\noindent
+Also a GenAC system $(\alpha,D)$ will be called
+{\bf Type-M} 
+formalism
+iff it proves
+\eq{totdefxs} - \eq{totdefxm}
+as theorems, {\bf Type-A} if it proves
+only \eq{totdefxs} and \eq{totdefxa},
+and
+it will be called
+ {\bf Type-S} if it proves
+only \eq{totdefxs} as a
+ theorem. 
+Furthermore,
+ $(\alpha,D)$ 
+ will be 
+called 
+{\bf Type-NS} iff it proves
+none of \eq{totdefxs} - \eq{totdefxm}. 
+
+
+
+
+%% bbbbbbbbbbbbbbb  DONE AND DELETE
+
+\bigskip
+{\bf BROADER OUTLOOK:$~$}
+The remainder of this chapter will step backwards 
+roughly 1,500 years in time, and explore the relationship
+between Definitions 1 and \ref{defy3}  with
+India's famous 
+``Wheat-and-Chess-Board''  paradigm (often denoted 
+as the
+{\bf ``WCB''} phenomenon.)
+
+This paradigm will
+ help clarify the persistent confusion that has
+ surrounded G\"{o}del's
+   Second  Incompleteness
+   Theorem. It will also help us 
+formulate a new interpretation for the
+   Remarks $*$ and $**$ that
+Hilbert and  G\"{o}del had made.
+
+
+ According to Wikipedia,
+ scholars are
+ %actually
+ uncertain about exactly when the WCB fable had
+ emerged in ancient India. (It may possibly
+ have originated as early as 400-600 AD.)
+
+ % but  scholars have debated the accuracy of such a date.)
+
+
+
+ Under one variant of the WCB fable, a Brahmin named
+ Sissa ibn Dahir invented
+a type of
+ Indian predecessor
+to the game of Chess. At some
+subsequent
+ juncture, the king
+ of Sissa's province
+ offered to award him
+a prize 
+for the  game
+that
+ Sissa had invented. Sissa
+ replied that it would be sufficient to place one
+ grain of wheat on the first square of a Chess Board,
+ two grains
+ on its second square, four grains on its third square, etc.
+
+ There are several versions of the Sissa fable, and no one knows
+ which (if any ?) is accurate. One variant  involves
+ the king executing Sissa after  discovering the last
+chess-board
+ square would
+ require  $~2^{63}~$ grains of wheat. Another
+ variant concludes with
+the
+ king smiling upon realizing the nature
+of Sissa's 
+ puzzle and declaring that this
+puzzle
+is  more fascinating
+and stimulating
+than the game of chess (itself).
+
+ While many  precise details have been lost over time,
+ the underlying 
+message of the WCB  
+ fable has survived for  roughly
+ 1,500 years.
+This is because it is thought
+to
+ convey an instructive lesson about
+the unsustainable
+nature of an
+ exponential growth.
+
+ For instance,
+ the 64 squares
+of a Chess Board
+has been found to 
+require  an amount of wheat that is
+ a factor of 1,000 greater than the world's full
+ %  annual
+ year-2019 production.
+ Moreover, a direct analog of the WCB paradigm, involving
+a few hundred doubling operations,  will require more  wheat
+than there are  atoms in the  Universe.
+ 
+Our point is that this fable would not have survived the
+millennial test of time if there were not many examples in human
+history where individuals
+have
+accidentally got involved in
+exponential growth processes that started at a gradual rate
+but then ran wild.  One of these examples seems to concern 
+Hilbert's statement $~*~$ (i.e. see Section \ref{aaa2} ).
+It appears to  ask   mathematicians to  perform
+an
+analog of Sissa's
+unsustainable  WCB task.
+
+This is
+because if one seeks to perform
+several hundred squarings of the number 2 than its
+binary encoding will own more zero-digits than there are
+atoms in the universe. This can be appreciated by comparing
+the sequences 
+$   x_0,~   x_1,~   x_2,~     ...    $ 
+and  $   y_0, ~  y_1, ~  y_2,     ...  $
+     defined below:
+\vspace*{- 0.7 em}
+\beq
+\label{zs}
+x_0~~~=~~~2~~~=~~~y_0
+\enq
+\beq
+x_{i}~~~=~~~x_{i-1}~+~x_{i-1}
+\label{as}
+\enq
+\beq
+y_{i}~~~=~~~y_{i-1}~*~y_{i-1}
+\label{bs}
+\enq
+For $\, i \, > \,0 \,$, 
+$\,$let $ \, \phi_{i} \, $ 
+and $ \, \psi_{i} \, $ 
+denote the  
+sentences in 
+\eq{as} and \eq{bs}
+respectively.
+Then if
+  $ \, \phi_{0} \, $ and
+$ \, \psi_{0} \, $ 
+denote \eq{zs}'s
+sentence, it is apparent that
+ $ \, \phi_0, \, \phi_1, \, ... \, \phi_n \, $
+imply
+ $ \, x_n \, = \, 2^{ n+1} \, , \, $ while 
+ $ \, \psi_0, \, \psi_1, \, ... \, \psi_n \, $
+ imply $ \, y_n \, = \, 2^{2^n} \, $.
+Thus, the  latter sequence
+will grow
+at an
+exponentially 
+faster
+rate than 
+the former.
+(E.g. the respective  quantities of
+Log$_{ \, 2 \,}(\, y_n \,) \, = \, 2^{n} \, $ 
+and
+Log$_{ \, 2  \,}(\, x_n \,) \, = \, {n+1} \, $ 
+ represent
+the  lengths for the binary codings 
+for
+$ \, y_n \, $
+ and 
+$ \, x_n \, $.)
+
+In other words,
+ $ \, y_n\,$'s 
+     binary     
+encoding 
+will have a length
+$\,  2^{n} \,  $,
+roughly analogous to the n-th square belonging to
+the WCB's chess board paradigm.
+Leaving aside technical details that were largely
+explored in \cite{ww21} and shall be briefly
+reviewed later, we are suggesting that
+Hilbert's statement $*$ is partially analogous to
+ Sissa's WCB paradigm.
+
+The latter is not meant to deny that there are some intriguing
+interpretations of the Hilbert and G\"{o}del statements $*$ and $**$
+deserving, certainly, partial sympathy.
+This is because there are other types of 
+amended axiomatizations of arithmetic that avoid 
+\el{bs}'s malignant assumption that multiplication is
+a total function.
+
+The theme  of this article is, thus,  largely that one needs to be
+more flexible when approaching the 
+ Hilbert and G\"{o}del statements $*$ and $**$.
+Our proposed ``Tripod Reply'' to
+ Hilbert's Second Problem 
+ will thus be
+ a gentle      hybrid-styled reply, that
+accepts   $*$ and $**$ half-way and
+ views Hilbert's Second Problem
+from all three Tripod  perspectives
+summarized in Section \ref{aaa1}.
+
+
+  
+\begin{deff}
+  \label{defx4}
+ \rm 
+During the remainder of this article, the acronym
+{\bf ``USEGR''} will refer to an {\it Unsustainable Exponential Growth Rate},
+similar
+to the dizzying number of zero digits
+that was produced by \el{bs}'s sequence of
+  $   y_0, ~  y_1, ~  y_2,   ~  ... ~ $.
+We will argue that such sequences resemble 
+Sissa's WCB paradigm,
+and a mathematical realist should avoid them as much as
+feasible.
+\end{deff}
+
+
+
+
+Our theme will be that a
+1-way response to Hilbert's Second Open Problem
+was observed  by  Sacks to be 
+questionable
+when his remarks 
+(a) and (b)  
+(from Section \ref{aaa2})
+noted G\"{o}del viewed  the
+ Second Incompleteness Theorem
+as essentially 
+a too
+pessimistic reply
+ to Hilbert's second problem.
+Also,
+ a broader
+ 2-way response is inadequate  because the various
+self-justifying formalisms that Willard developed
+between the times of \cite{ww93}'s  initial 1993
+%%%  announcement
+presentation
+and that of  \cite{ww21}'s
+recent  LFCS-2020 paper
+are weaker 
+than would be  preferred.
+
+Artemov's third type of response
+uses
+his elegant 
+infinite ranged SBSIS-styled
+exceptions \cite{Ar19,Ar21}
+to the Second Incompleteness Theorem. It is 
+almost
+%% very close to
+ideal.
+Yet, it also  has drawbacks
+(see footnote\footnote{Section 
+   \ref{aaa1}    explained how
+   \cite{Ar19,Ar21} also had
+some
+   hidden difficulties. They are
+  that Peano Arithmetic (PA)
+  can  verify only the consistency of finite subsets
+  of itself. Thus,
+\cite{Ar19,Ar21}'s 
+  perspective 
+  is
+  useful, but it does not
+fully
+  explain how PA
+  can formalize the consistency of its
+  infinitely drawn
+  expanse.
+  In this context, we shall argue 
+  a hybridized
+  Tripod-style response is a
+more
+ comprehensive
+reply to
+  Hilbert's Second  Problem.} )
+because its  SBSIS approach
+is a mixture of tempting positive results and compromises.
+This
+is the  reason the current article advocates using
+several perspectives simultaneously,
+under
+our  hybrid-styled 3-part
+``Tripod''
+reply to Hilbert's
+120 year-old problem. 
+
+\section{Characterizations of Several Base Languages}
+
+    %%% 4444444444444444
+
+  \label{aaa4}
+% \label{sect4}
+
+  Sissa's 1,500-year old ``WCB'' paradigm,
+as well as
+  Definition \ref{defx4}'s   
+  ``USEGR'' effect
+have  similar themes.
+This is because both involve
+unsustainable exponential growth processes,
+having   often
+unwelcome and
+unanticipated effects.
+In particular, their circumstances can be made applicable
+to essentially all the
+known variants of the Second Incompleteness Theorem
+\cite{AZ1,Ar1,Be14,BS76,Bu86,BI95,Fe60,Fr79a,Ha11,HP91,KT74,Pa71,PD83,Pa72,Pu85,Pu96,So88,So94,Sv7,Vi5,WP87,ww1,ww2,ww7,wwapal,ww16}.  
+% cite?????????.
+One  
+of its
+most
+interesting variants
+arose during a  1994
+private telephone conversation
+with Robert Solovay
+\cite{So94}.
+% that he chose never to publish.
+Its formalism  (due to Solovay)
+can be thought of as a generalization
+  of a methodology that was initially developed by Pudl\'{a}k in
+  \cite{Pu85} and which was further explored
+by  Nelson and Wilkie-Paris in \cite{Ne86,WP87}.
+%%% Essentially,  it is
+It essentially amounts to
+the following observation:
+\begin{description}
+\item{\bf Theorem ++ $~$ : }
+{\it 
+$~~~$
+(Solovay's  
+modification
+\cite{So94}
+of Pudl\'{a}k \cite{Pu85}'s formalism 
+using some of 
+Nelson and Wilkie-Paris \cite{Ne86,WP87}'s
+methods)} :
+Let 
+$ \, (\alpha,D) \, $ 
+denote 
+a 
+Type-S
+GenAC system
+which assures
+the successor operation
+will
+% provably
+satisfy
+both 
+ $  \,   x'     \neq 0     $ and
+$     x'     =     y' \Leftrightarrow x=y $.
+$~$Then
+$ \, (\alpha,D  ) \, $  
+cannot verify its own
+consistency
+whenever
+simultaneously
+ $D$ is some type of
+a Hilbert-Frege
+deductive
+apparatus and
+$~\alpha~$
+ treats addition and multiplication
+as 3-way relations, 
+satisfying 
+their usual % identity,
+associative, commutative, 
+ distributive 
+and identity 
+axioms.
+\end{description}
+%\end{quote}
+\medskip
+Essentially, Solovay \cite{So94}
+had
+privately communicated to Willard  
+an approximate analog of Theorem $++$.
+(This communication was similar to
+several
+other often-privately-communicated comments 
+that the
+% \newpage
+% \noindent
+literature  \cite{BI95,HP91,Ne86,PD83,Pu85,WP87} 
+has often attributed to 
+Robert Solovay's unpublished observations.)
+It also should be mentioned that
+partial
+analogs of 
+ $++$'s statement
+ were  explored 
+subsequently
+ by  Buss-Ignjatovi\'{c},
+H\'{a}jek 
+and
+\v{S}vejdar in \cite{BI95,Ha11,Sv7},
+as well as in Appendix A of 
+the paper
+\cite{ww1} 
+and in \cite{wwapal}.
+
+Furthermore, we stress that
+Pudl\'{a}k's initial 1985 article  \cite{Pu85} 
+had captured
+the majority 
+of $++$'s 
+implications, chronologically before Solovay's observations.
+Also,
+Friedman did nicely
+ related
+work as early as 1979
+ in
+\cite{Fr79a}.
+
+\bigskip
+
+In order to explain how
+Sissa's 1,500-year old ``WCB'' paradigm
+and Definition \ref{defx4}'s similar  ``USEGR'' effect
+are related to  this material, it will be useful to
+employ
+Definition \ref{defx5}'s notation. (Its predecessor
+can be found 
+in \cite{wwapal}, but the latter did not discuss the
+ ``USEGR'' effect.)
+
+\begin{deff}
+\label{defx5}
+\rm
+A  function $ F(a_1, \ldots , \, a_j)$
+is called
+a {\bf Non-Growth} operation
+when 
+$ F(a_1, \ldots , \, a_j) 
+\leq  Maximum(a_1 , \ldots ,  \, a_j)$
+holds.
+Seven  examples of  
+non-growth functions are:
+\bee
+ \parskip 0pt
+\small 
+ \baselineskip = 0.70 \normalbaselineskip
+\item
+{\it Integer Subtraction} 
+(where $~x-y~$ is defined to equal zero when
+ $~x \leq y~),~~$
+\item
+{\it Integer 
+Division}
+(where $x \div y$ 
+equals
+$~x~$ when $y=0$, and
+it equals $~\lfloor ~x/y ~\rfloor~$ otherwise),
+\item
+$~Maximum(x,y),$
+\item
+$~ Log_{ \, \spadesuit \, }(x)~$ 
+ which
+is an abbreviation for
+$~\lceil~$Log$_2(~x+1~)~\rceil~~$ under 
+the conventional
+notation.
+%%% 
+%%% 
+%%% (The  footnote 
+%%% \footnote{
+%%% The H\'{a}jek-Pudl\'{a}k textbook \cite{HP91} uses the 
+%%% notation ``$~\mid  x  \mid~$'' 
+%%%  to 
+%%% designate
+%%%  what we shall call ``$~ Log_{ \, \spadesuit \, }(x)~$''
+%%% Thus for $~x \geq 1 \,$, 
+%%%  $~ Log_{ \, \spadesuit \, }(x)~$
+%%% denotes
+%%% the number of symbols that will 
+%%% encode the number $~x\,$, when it is written 
+%%% in  a binary format. }
+%%% explains the 
+%%% significance
+%%% of this
+%%% concept.)
+%%% 
+%%%
+\item
+$\,~Root(x,y) \, =  \, \lceil  \, x^{1/y} \,  \rceil$ 
+\item $~Count(x,j)$   designating the number of
+physical
+ ``1'' bits
+stored among $    \,  x$'s rightmost $    \, j    \,  $ bits.
+\item 
+  $~Bit(x,i)~$
+  designating  the $i-$th rightmost bit
+  of the string $x$ (as explained by footnote \footnote{ The 
+$~Bit(x,i)~$ operator is technically unnecessary
+because it can be encoded as:
+$\mbox{Bit}(x,i)~~=~~   \mbox{Count}(x,i) ~-~  \mbox{Count}(x,i-1)$.}
+ $~$.)
+\ene  
+These function were called 
+{\bf Grounding Functions} in \cite{wwapal}.
+Also, $L^0  \,$ will denote a quite weak
+language 
+built from the Grounding functions, together with
+three constant symbols $c_0$,  $c_1$ and   $c_2$
+(representing  0, 1 and 2) and 
+the   ``$ \, = \, $''
+and ``$ \, \leq \, $'' primitives.
+\end{deff}
+
+Since  $L^0$ contains absolutely no growth
+functions, it is so weak that it cannot even
+formalize the integer 3 as an isolated term.
+Thus,
+\cite{wwapal} found it necessary
+to strengthen  $L^0$'s language
+with the two additional Type-NS 3-way predicates
+$Add(x,y,z)$ and $Mult(x,y,z)$ (that formalize the concepts
+of   ``$~x+y=z~$'' and ``$~x*y=z~$''). Their two formal
+definitions are given below:
+\beq
+\small
+  \label{newadd}
+z ~ - ~ x ~~ =~~y ~~~~\wedge~~~~ z~~ \geq~~ x
+\enq 
+\begin{equation}
+\small
+  \label{newmult}
+ \{ ~(x=0    \vee    y=0 ) \Rightarrow z=0~ \} ~ ~\wedge ~~ 
+  \{ ~(x \neq 0 \wedge y \neq 0~) ~ \Rightarrow ~
+(~ \frac{z}{x}=y  ~\wedge \, ~  \frac{z-1}{x}<y~~)~ \}
+\end{equation}
+More precisely to help Type-NS self-justifying axiom systems encode
+integers distinctly larger than 2,
+%%% the year-2006 paper
+\cite{wwapal}
+introduced some additional constant
+symbols,
+of $~a_2,~a_3,~a_4~....~$
+and $~b_2,~b_3,~b_4~....~$  
+into  $L^0~$'s language,  satisfying  the following
+three constraints:
+\newpage
+\beq
+\label{new1}
+a_2~~=~~b_2~~=~~c_2~~=~~2
+\enq
+\beq
+\label{new2}
+a_{j+1}~~=~~a_{j}~+~a_{j} ~~~~\mbox{ for } j \geq 2
+\enq
+\beq
+\label{new3}
+b_{j+1}~~=~~b_{j}~*~b_{j} ~~~~\mbox{ for } j \geq 2
+\enq
+In particular, 
+\el{new2} was called
+ an
+ {\bf ``Additive Naming Convention''  (ANC)}
+ in \cite{wwapal}, 
+ and
+\el{new3} was called an {\bf ``Multiplicative Naming Convention'' (MNC)}.
+
+The precise encoding for Lines
+\eq{new2} and \eq{new3}  in \cite{wwapal}
+is delicate 
+because the earlier Theorem  ++
+(of Pudl\'{a}k and Solovay)  implied
+a Self-Justifying axiom system cannot treat either addition
+or multiplications as total functions. 
+This difficulty was resolved by applying
+the  $Add(x,y,z)$ and  $Mult(x,y,z)$ 3-way predicates
+(from Lines \eq{newadd} and \eq{newmult}$~).$
+Thus,   reworded forms of Lines \eq{new2}
+and
+\eq{new3}, evading  $++$'s
+generalization of the Second Incompleteness Theorem, 
+appear below:
+\beq
+\label{new4}
+\label{addcov}
+ADD(a_j,a_j,a_{j+1}) ~~~~\mbox{ for any } j \geq 2
+\enq
+  \beq
+  \label{new5}
+  \label{multcov}
+Mult(b_j,b_j,b_{j+1}) ~~~~\mbox{ for any } j \geq 2
+\enq
+
+\begin{deff}
+\label{defx6}
+\rm
+The term {\bf Additive Naming Language} refers to the
+extension of the  
+$L^0~$ language
+that defines the constant symbols
+$~a_2,~a_3,~a_4~....~$
+with the formalisms from
+Lines \eq{new1}
+and \eq{new4}. This revised version
+of $L^0~$'s language will be denoted as $L^{\rm ANC}.$
+Likewise
+the phrase {\bf Multiply-Additive Naming Language}  refers to the
+revision of the  
+$L^0~$'s language
+that defines all the constant symbols of
+$~b_2,~b_3,~b_4~....~$
+and $~a_2,~a_3,~a_4~....~,~$  
+using the formalisms of
+%via
+Lines \eq{new1}, \eq{new4} and \eq{new5}. 
+Its language
+% will be
+is
+called  $L^{\rm MNC}.$
+\end{deff}
+
+An examination of these two naming conventions was
+%% essentially
+the core topic in \cite{wwapal}.
+Roughly summarized, the Theorem 3 from \cite{wwapal} 
+showed that  the language $L^{\rm ANC}$
+can house
+robust forms of self-justifying systems, and its Theorem 4
+showed that the similar evasions of the Second Incompleteness
+Theorem are impossible under  $L^{\rm MNC}$. 
+
+
+What was missing from \cite{wwapal}
+was an intuitive explanation about why  $L^{\rm MNC}~$'s failure
+did not out-shine the partial (but limited)
+success that $L^{\rm ANC}$
+achieved. In other  words, one needs to ask:
+{\it Which of these two languages 
+provide a better standpoint
+for
+  approximating conventional
+ theorem-proving?}
+
+Our reply to the preceding question is that the language 
+$L^{\rm ANC}$ is preferable because
+Definition \ref{defx4}'s
+ awkwardly
+exponential
+``USEGR''
+growth suggests one must 
+look merely at 
+the  600-th item in the sequence 
+$~b_2,~b_3,~b_4~....~$  to find an
+element
+whose
+binary encoding-length exceeds the number of atoms lying inside
+the universe.
+
+\begin{remm}
+  \label{remx}
+  \rm
+  \dvs
+More specifically, we are suggesting the reader keep in mind
+this dichotomy
+when the next section
+compares the Parts (a) and (b) of its Propositions
+\ref{prop11} and  \ref{prop12}.
+This is because the negativistic  Propositions
+\ref{prop11}.b and  \ref{prop12}.b will
+% shall
+appear to be
+% even more
+divorced from reality, when their 
+excessive
+USEGR  growth rates
+are noticed to
+resemble
+Sissa's  quite
+outlandish exponential
+ rates of growth. 
+%% $\, 2^{64}  \,- \, 1 \,$ grains of
+%% wheat.
+\end{remm}
+
+\section{More Details}
+
+    %%% 555555555555555555555555
+
+  \label{aaa5}
+
+  
+The primary goal 
+in \cite{wwapal} 
+was to determine when
+%it was possible to formulate 
+self-justification
+was possible
+under 
+either
+\eq{addcov}'s
+ ``Additive'' 
+or  \eq{multcov}'s 
+ ``Multiplicative'' 
+ naming
+schema.
+Some added notation is needed to briefly review
+\cite{wwapal}'s
+main
+results.
+In a context where
+ $\, t \,$ denotes   any term in $\, L^{ANC} \,$'s   
+language that does not contain the variable symbol of ``$~v~$'',
+the quantifiers in
+%%% the  wffs of
+$~ \forall ~ v \leq t~~ \Psi (v)~$ and
+$\exists ~ v \leq t~~ \Psi (v)$
+will be  called $\, L^{ANC} \,$'s   
+{\bf ``$~$v-Restricting 
+  Quantifiers$~$''}. Then Definition \ref{def-3.8}
+formalizes the 
+analogs of
+%% a
+conventional arithmetic's
+$\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+formulae
+in  $L^{ANC}\,$'s 
+language:
+
+\begin{deff}
+\label{def-3.8}
+\rm
+Any formula in
+$L^{ANC}\,$'s 
+language will be called  $\Delta^{ANC}_0$
+iff all its quantifiers variables $v$ meet the preceding
+v-Restricting  constraint.  The  $\Pi^{ANC}_n$
+ and $\Sigma^{ANC}_n$.
+sentences of  $L^{ANC}$
+are
+then
+defined by
+%%% Items
+1-3.
+(They are analogous to
+conventional arithmetic's
+%%% $\Delta_0$, $\Pi_n$ and $\Sigma_n$ 
+counterparts):
+\bee
+%\small
+%\parskip -2 pt
+%\baselineskip = 0.8 \normalbaselineskip 
+\item
+Every  
+$\Delta_0^{ANC}$ formula is
+%considered to be 
+also 
+a
+$\Pi_0^{ANC}$  and 
+an
+$\Sigma_0^{ANC}  $ expression.
+%% 
+%% ``$~\Pi_0^{ANC}~ \,$''  and 
+%% %  also 
+%%  ``$~\Sigma_0^{ANC}~ \, $''. 
+%% 
+\item
+A
+formula
+is
+called
+% defined to be
+ $ \,\Pi_n^{ANC} \,$
+when it
+% is 
+can be
+encoded as 
+$\forall v_1 ~ ...~ \forall v_k ~ \Phi$  
+where
+%with
+$\Phi$ is  $\Sigma_{n-1}^{ANC}$.
+\item
+A formula
+is
+called
+% defined to be
+ $\Sigma_n^{ANC}$
+when it can be encoded as 
+$\exists v_1~ ...~ \exists v_k ~ \Phi,$  where
+$\Phi$ is  $\Pi_{n-1}^{ANC}$.
+\ene
+\end{deff}
+
+
+
+%%\begin{deff}
+%%\label{def3.9}
+%%\rm
+
+\bigskip
+ 
+
+
+Given an initial axiom system $\beta,$
+the Theorem 3 of \cite{wwapal}
+% defined 
+%formalized
+did formalize
+a 
+self-justifying logic, called
+{\bf ISCE$(\beta)$}, 
+that could prove all 
+$~\beta\,$'s $\Pi_1^{ANC}$ theorems and 
+which could also
+verify its own consistency under any Hilbert-Frege
+style deductive
+apparatus.
+The axiom basis for ISCE$(\beta)$ 
+was comprised, formally,  of the following four
+distinct
+groups of axioms:
+%
+%  \newpage
+\begin{description}
+ \parskip 0pt
+\item
+{\bf GROUP-ZERO:} 
+This 
+schema
+%  axiom group 
+will formally represent the integers of 0,1 and 2
+with the defined constant-symbols of
+$\,c_0\,$, $\,c_1\,$ and $\,c_2\,$. 
+It will also apply
+\el{addcov}'s Additive Naming 
+schema
+to formally define 
+ the further constants of
+ $    a_2,~  a_3,~ a_4,~  ...   $  
+\item
+{\bf GROUP-1:} 
+This axiom group  will consist of a
+finite set of  
+ $\Pi_1^{ANC} $ sentences, whose union with Group-zero
+axioms is consistent and
+can prove every $\Delta_0^{ANC}$ sentence that
+holds true in the standard model.
+(It was explained in \cite{wwapal} that any
+set $~H~$, meeting these conditions,
+may formalize the  
+Group-1 axioms.)
+\item
+{\bf GROUP-2:}
+Let
+$\ulcorner \, \Phi \, \urcorner$ denote $\Phi$'s G\"{o}del number, and
+$\mbox{HilbPrf}_{ \beta  }(x,y)$
+denote a 
+$\Delta _0^{ANC}$ 
+formula indicating  $y$ is a
+Hilbert-styled
+proof
+from axiom system $\beta  $ of the theorem 
+$x$.
+For each 
+$\Pi_1^{ANC} $
+ sentence  $\Phi$,
+the Group-2 schema
+will 
+contain
+an 
+% approximate
+encoding for 
+\eq{group2old}'s
+$\Pi_1^{ANC} $
+axiom:
+\begin{equation}
+\label{group2old}
+\forall ~y~~~\{~ \mbox{HilbPrf}~_\beta
+~(~ \ulcorner \Phi \urcorner ~,~y~)~~
+\Rightarrow ~~ \Phi~~\}
+\end{equation}
+\item
+{\bf GROUP-3:}
+This last part of
+%%%%%%%%%%%%%%%  
+\cite{wwapal}'s
+ISCE$(\beta)$
+formalism
+is
+% was
+ a single 
+self-referencing
+$\Pi_1^{ANC}$
+sentence  
+stating:
+ %% essentially declaring:
+\begin{quote}
+% \small
+%%%%%%%%%%%%% $ \oplus ~ \oplus ~~~$
+$ \oplus  \oplus ~~~$
+ ``There 
+%is 
+exists
+no
+Hilbert-style proof of 0=1 from the union of the Group-0, 1 and 2
+axioms  with {\it THIS  SENTENCE} (referring to itself)''.
+\end{quote}
+\end{description}
+More details about  $ \oplus  \oplus$'s exact formal 
+encoding ``fixed point'' encoding will not be mentioned here
+because our earlier articles
+\cite{ww1,ww5,wwapal} discussed this topic quite thoroughly.
+
+  
+\begin{deff}
+  \label{defx8}
+  \rm
+  The symbols
+  $\Pi^{MNC}_n$,
+ $\Sigma^{MNC}_n$
+  and $\Delta_0^{MNC}$
+will denote the obvious analogs of  
+Definition \ref{def-3.8}'s
+  $\Pi^{ANC}_n$,
+ $\Sigma^{ANC}_n$
+and $\Delta_0^{ANC}$ sentences,
+when the broader language $L^{MNC}$
+of the Multiplicative Naming Convention
+replaces the initial language of   $L^{ANC}$.
+%
+% utilized by the Additive Naming Convention.
+%
+Also, the symbol
+ISCE$^{\rm MNC}(\beta)$
+will denote the  intended
+generalization of the preceding  
+ISCE$(\beta)$
+system where 
+all references to the language   $L^{ANC}$
+and its Additive Naming Convention
+are
+%% changed into
+replaced by
+statements about
+the language   $L^{MNC}$,
+along with
+its Multiplicative Naming Convention,
+ its
+associated
+ $\Pi^{MNC}_n$ 
+and $\Delta_0^{MNC}$ sentences
+and a natural $\Pi^{MNC}_1$ counterpart of
+ISCE$(\beta)$'s particular
+Group-3 {\it ``I am consistent''}
+axiom.
+(It  turns out that Proposition
+\ref{prop11} will formalize that 
+ISCE$^{\rm MNC}(\beta)$
+typically
+differs from 
+ISCE$(\beta)$ by being
+inconsistent.)
+\end{deff}
+
+\begin{deff}
+  \label{defx9}
+  \rm
+Let $~I(~\bullet~)~$ denote
+an operation that maps
+an initial axiom basis $\, \beta \,$ onto an alternate
+system  $\,I(\beta)\, $.
+(One example of
+such an operation is the
+  ISCE$( \, \bullet \, )$ 
+framework,
+that maps 
+an initial axiom basis of 
+  $~\beta~$ onto 
+the alternate formalism of
+ ISCE$(\beta).~)~$ 
+Such an operation  $~I(~\bullet~)~$
+is  called {\bf Consistency Preserving}
+iff  $\,I(\beta)\, $ is consistent whenever 
+the union of
+ $\beta$ with the Groups 0 and 1 axiom schemas is
+consistent.
+\end{deff}
+
+\begin{propp}
+  \label{prop11}
+\rm
+The   ISCE$(~\bullet~)~$
+and  ISCE$^{\rm MNC} (~\bullet~)~$ transformations
+have different properties, insofar as only the former is
+Consistency Preserving. In particular, if PA$^{Ground}$
+denotes the
+extension of Peano Arithmetic that includes
+both the conventional arithmetic operators and the
+seven Grounding functions, then the following two invariants
+do hold:
+\bed
+\item[  a.  ]
+  ISCE(PA$^{Ground~}$)
+  is a consistent system.
+\item[  b.  ]
+ISCE$^{\rm MNC}$(PA$^{Ground~}$) fails to be  consistent.
+\ennd  
+\end{propp}
+
+It is unnecessary to prove  Propositions \ref{prop11}.a
+and \ref{prop11}.b
+ because  they both are easy consequences
+of the Theorems 3 and 4 from \cite{wwapal}.
+Instead, we will discuss in this paper how 
+Proposition \ref{prop11} and some
+ of its
+generalizations
+are related
+to a needed Tripod-style response to Hilbert's
+% famous
+second problem.
+
+It firstly should be noted that
+Definition \ref{defx4}'s  very fast ``USEGR'' growth is
+essential for appreciating   the sharp contrast  between 
+Proposition \ref{prop11}.a
+and 
+Proposition \ref{prop11}.b.
+This is because
+the generalization of the Second Incompleteness Theorem, in 
+Proposition \ref{prop11}.b,
+arises because the MNC supports 
+an unfortunately fatal
+USEGR rate of
+exponential  growth among numbers.
+
+It next should be remarked that some half-analogs of 
+Proposition \ref{prop11}'s twin 2-part statement for semantic
+tableau deduction were established by \cite{ww1,ww2,ww5,ww9}.
+Moreover, the recent year-2021 article \cite{ww21}
+summarized and extended
+these  articles.
+Thus, let 
+ IS$_{\rm Tab}(~\beta~)$
+ be
+\cite{ww21}'s
+ 4-part axiom system,
+whose relationship to the ISCE$(\beta)$
+system
+is as follows:
+\bed
+\item[    i. ]
+  The Group-1 and Group-2
+  schemes
+  for   IS$_{\rm Tab}(~\beta~)$
+and  ISCE$(\beta)$
+are
+  essentially identical.
+\item[   ii. ]
+  The Group-Zero
+  scheme
+  for   IS$_{\rm Tab}(~\beta~)$
+is stronger than that of
+  ISCE$(\beta)$ because it replaces the 
+  Additive Naming Convention with
+a stronger 
+more compact ``Type-A''
+statement  declaring that the operations of
+  Addition and Double$(x)~=~x+x~$ are total functions.
+\item[   iii. ]
+  The Group-3
+  scheme
+  for   IS$_{\rm Tab}(~\beta~)$
+is, however, weaker that its analog under
+ISCE$(\beta)$ because  it only recognizes its
+self-consistency under a semantic tableau form of deduction.
+\ennd
+
+%%%% ggggggggggggggggggggggg
+
+ \baselineskip = 1.12 \normalbaselineskip 
+
+ \dvs
+
+
+
+ \parskip 8pt
+
+
+ 
+ \noindent
+ In essence,
+  IS$_{\rm Tab}(~\beta~)$
+  is a self-justifying formalism that avoids being inconsistent
+  by using a different type of trade-off
+  than 
+  ISCE$(\beta)$, where its Group-Zero schema  is stronger while
+its Group-3 statement uses a weaker deductive methodology.
+
+%%%%% \newpage
+%nnnnnnn
+
+\bigskip
+
+% \parskip 11pt
+
+Also, we will employ the Example 5.1 from \cite{ww21}, where
+IS$^M_{\rm Tab}(~\beta~)$ denotes the natural modification
+of  IS$_{\rm Tab}(~\beta~)$ wherein:
+\bee
+\item
+  The  Group-Zero axiom further recognizes integer-multiplication
+  as a total function.
+\item
+  The Group-3 axiom is the same as that described in (iii)
+  except  that its Group-3
+  {\it ``I am consistent''} statement has been  adjusted 
+  to recognize
+  that the Group-Zero formalism now treats
+  integer-multiplication as a total function.
+  \ene
+  Then  Proposition \ref{prop12}
+  is
+%  represents
+  \cite{ww21}'s  counterpart
+to
+%   for
+Proposition \ref{prop11}.
+(We again remind the reader about  Remark$~$\ref{remx}'s
+warnings about excessively
+demanding
+USEGR  growth rates.
+It is the
+ intuitive
+reason behind the two inconsistency
+results 
+appearing in both
+the Parts (b) of the
+Propositions \ref{prop11}
+and \ref{prop12}.)
+
+
+\begin{propp}
+  \label{prop12}
+  \rm
+  The
+ IS$_{\rm Tab}(~$PA$^{\rm Ground}~)$
+ and  IS$^M_{\rm Tab}(~$PA$^{\rm Ground}~)$
+ systems
+ differ
+  insofar as:
+\bed
+\item[  a.  ]
+   IS$_{\rm Tab}(~$PA$^{\rm Ground}~)$
+  is  consistent.
+\item[  b.  ]
+   IS$^M_{\rm Tab}(~$PA$^{\rm Ground}~)$
+  fails to be  consistent.
+  \ennd
+\end{propp}
+
+
+ Formal   analogs of Propositions \ref{prop12}.a and   \ref{prop12}.b
+ were established in \cite{ww21}, and we shall not repeat that
+ discussion here. Instead, our discussion will focus on the two consistency
+ results established by
+ Propositions \ref{prop11}.a and   \ref{prop12}.a.
+ The Remark \ref{rm13} will explain how there exists a quite natural
+ interface between these
+ two results
+with
+ the 
+SBSIS-styled approach that  Artemov had
+advocated in   \cite{Ar19,Ar21}.
+ 
+
+
+
+%% \def\fggg{ \normalsize  \baselineskip = 2.4 \normalbaselineskip } 
+
+%%% \fggg
+
+\begin{remm}
+  \label{rm13}
+ \baselineskip = 1.17 \normalbaselineskip 
+ \rm
+ \dvs
+     {\bf (about how the Second and Third Legs of a ``Tripod''
+      Reply to Hilbert's Second  Problem will interact
+Quite Nicely
+       with Each Other) :}
+%%% in a Beautiful Manner) :}        
+Let us recall that Step \ref{step2}
+of  Artemov's
+SBSIS-styled   \cite{Ar19,Ar21}
+constructions
+(summarized earlier
+in \textsection\ref{aaa1})
+%%% proved
+constructed
+a class of $\Pi_1-$like theorems, 
+$~T_1,~T_2,~T_3,~...~$,
+which confirmed 
+the consistency of  the respective
+ $S_0,~S_1,~S_2,~...~~$ subsets of PA.
+It turns out that the  validity of these
+ $~T_j~$ theorems
+is  known
+ to both
+ the preceding
+ISCE$(~$PA$^{\rm Ground}~)$
+and
+IS$_{\rm Tab}(~$PA$^{\rm Ground}~)$ formalisms from
+Propositions \ref{prop11}.a
+and \ref{prop12}.a
+(see Footnote
+  \footnote{This is because the Group-2 schemes of
+ISCE$(~$PA$^{\rm Ground}~)$
+and
+IS$_{\rm Tab}(~$PA$^{\rm Ground}~)$
+% do
+both
+formally
+recognize the validity of all of PA's $\Pi_1$ theorems.
+Thus using a direct analog of Artemov's
+proposed
+methodology in \cite{Ar19,Ar21},
+they can
+prove each of
+\cite{Ar19,Ar21}'s
+theorems of
+$~T_1,~T_2,~T_3,~...~$.  }
+$~$).
+  The remainder of this section will explain
+  how this subtle interconnection causes the Second and Third legs of
+a proposed Tripod Reply to Hilbert's Second Open Question   
+to interact
+quite eloquently
+%%% nicely
+with each other.
+\bigskip
+\end{remm}
+
+
+The central point is that both Artemov's formalisms in 
+\cite{Ar19,Ar21} and Willard's approach
+in \cite{ww93}-\cite{ww21}
+own distinct partial drawbacks. It turns out that one can
+essentially 
+overcome
+these deficiencies by using Remark \ref{rm13}'s
+theoretical hybrid methodology.
+In particular, the partial drawbacks which 
+ need to be addressed
+are listed below:
+\bed
+\item[   a.  ]
+  The partial drawback to Artemov's 
+SBSIS-styled method \cite{Ar19,Ar21}
+is that it does not produce one single theorem, asserting
+a formalism's overall consistency. (Instead, it produces a sequence of
+theorems  $~T_1,~T_2,~T_3,~...~$,
+whose approximate union
+comprises the desired {\it ``I am consistent''} statement.)
+\item[   b.  ]
+  The comparable drawback in Willard's
+self-justifying
+  formalisms in
+\cite{ww93}-\cite{ww21}'s
+  28-year long
+  series of 
+  articles 
+  is that none of these
+papers involve a system as powerful as Peano Arithmetic
+  declaring its own self-consistency.
+  \ennd
+  The nice aspect of
+Remark \ref{rm13}'s
+  hybridization of these two methods is   
+  that its
+  methodology
+  views the two formalisms from
+%%  in  
+Propositions \ref{prop11}.a
+and \ref{prop12}.a as declaring their self-consistencies under certain
+%%% particular
+unifying perspectives,
+while owning a formalized awareness about
+Artemov's
+broader expanding  series of
+theorems   $~T_1,~T_2,~T_3,~...~$.
+This paradigm overcomes the main challenges
+that were
+posed by Items (a) and (b).
+
+{\it For the sake of clarity, 
+Remark \ref{rm13}'s observations
+certainly should not be viewed
+as being a full panacea}. Thus,
+the 
+Second Incompleteness Theorem certainly imposes
+%%% some very
+severe
+restrictions
+upon any attempts to
+%%% fully
+evade it.
+Thus
+while not fully embracing the  philosophical positions
+of Hilbert's and G\"{o}del's
+statements of
+$*$ and $**$, our perspective in Remark \ref{rm13}
+certainly conveys definite strong partial support for 
+Hilbert's and G\"{o}del's general approach.
+
+In other words, a 3-way ``Tripod'' reply for Hilbert's Second
+Open Problem appears to be attractive
+because a
+more
+compressed
+% abbreviated
+one-or-two leg reply to Hilbert's
+foundational
+Second Problem
+appears to be essentially
+%%%much
+over-simplistic.
+
+\section{Additional Curious Aspects about Self-Justification}
+
+    %%% 66666666666666666666666 
+
+  \label{aaa6}
+
+
+Contrary to the impression that may have been conveyed by
+the 28-year-long literature
+about self-justification
+\cite{ww93}-\cite{ww21},
+these papers {\it were
+ not actually}
+ intended to completely 
+ reject the assumption
+that integer-multiplication is a total function.
+This is because several {\it near-cousins} of  integer-multiplication
+% were
+are
+allowed
+within the languages of  \cite{ww93}-\cite{ww21}. 
+
+This point was first made in \cite{ww93}'s  initial 1993 paper. 
+Its Item (viii) on page 326
+suggested using a function called ``AndMultiply$(x,y,z)$''
+The latter function first
+computes an integer $~v~$, derived by
+taking the
+multiplicative product of $~x~$ and  $~y~$, and it then
+computes the Bit-Wise-AND of
+$~v~$ and  $~z~$ so that its final
+output
+is produced.
+
+% of  ``$~w~$'' can be produced.
+
+This ``AndMultiply$(x,y,z)$'' operation can be viewed as
+a non-growth operator
+because AndMultiply$(x,y,z)~\leq~$Max$(x,y,z)$. Thus, it can
+be easily incorporated into the Type-A
+%%% ``U-Grounding''
+Self-Justifying
+% logical
+formalisms of \cite{ww1,ww5,ww21}. This raises the following
+tantalizing issue:
+
+%question:
+
+\begin{description}
+\item{ +++ }  
+\it  Could  the main mysteries about a formalism's inability to
+  simultaneously recognize its semantic tableau consistency and
+  multiplication as a total function be
+ philosophically
+  resolved by
+letting
+``AndMultiply'' replace
+% the
+% traditional
+% role of
+the classic Integer-Multiply operation?
+\end{description}
+
+\noindent
+Furthermore, if ``AndAddition$(x,y,z)$''
+% is
+designates
+the straightforward analog of the
+primitive
+``AndMultiply$(x,y,z)$''
+for Integer-Addition,
+then the AndAddition$(x,y,z)$ operator can be
+easily
+incorporated into the Type-NS Grounding-language
+``ISCE'' system (which did
+appear in both this article and in its year-2006
+predecessor \cite{wwapal} ).
+
+These two  AndAddition$(x,y,z)$
+and AndMultiply$(x,y,z)$ operations are not as powerful as
+the more conventional 
+classic Integer-Addition and Integer-Multiplication
+operations.
+However, their potential in
+several
+pragmatic engineering environments
+should not be under-estimated.
+% overlooked.
+
+Also, it should be mentioned that the results in
+\cite{ww93}-\cite{ww21} can be further
+%extended
+amplified
+by using
+\cite{ww6}'s Floating-Point-With-Rounding Multiplication
+operation (FPWRM). This FPWRM primitive suggests
+\cite{ww6}
+that
+floating point arithmetics, unlike classic  integer arithmetics,
+are  compatible with  self-justifying logics
+using semantic tableau deduction.
+
+\section{Overall Perspective}
+
+%%% 7777777777777777
+
+\label{aaa7}
+
+The  preceding discussion has focused mainly
+on only one of the three legs of a more elaborate
+Tripod Response to Hilbert's Second Problem.
+The other two legs, involving  generalizations
+of G\'{o}del's
+ Second Incompleteness Theorem
+%%% cite?????? 
+and Artemov's
+step-by-step
+SBSIS-like approach \cite{Ar19,Ar21},
+have been surveyed only very much more briefly.
+Together, these three legs
+formulate a triple
+ reply to Hilbert's Second Problem,
+which is
+significantly
+more far-reaching than a more isolated
+type of
+one-or-two leg
+response.
+
+Collectively,
+these three legs
+clarify the nature of the statements  
+$*$ and $**$, which 
+Hilbert and G\"{o}del had  articulated.
+Moreover, the Question +++ (in
+\textsection\ref{aaa6}) is quite tantalizing.
+
+In essence, Hilbert's Second Problem 
+is so complex that
+an isolated one-or-two legged
+reply  is insufficient to
+answer Hilbert's fascinating question.
+Moreover,
+Gerald Sacks
+has
+recalled 
+G\"{o}del 
+expressing
+opinions that were
+{\it ``almost the opposite of what every one
+  else would have expected''} him to make
+(see again \textsection\ref{aaa2} ).
+% for details).
+Thus, he would presumably quite
+directly
+approve the philosophical
+positions that Remark \ref{rm13} and Section \ref{aaa6}
+%% (in the current article)
+have formulated,
+since G\"{o}del
+repeated
+several
+analogs of his
+often-quoted controversial
+1931
+remark $\, **\,$
+during numerous private
+conversations
+ G\"{o}del shared
+with Gerald
+Sacks  \cite{YouSa14}.
+
+
+%\baselineskip = 1.0  \normalbaselineskip
+%\noindent
+%  \medskip
+
+{\bf Acknowledgment:}
+I thank Seth Chaiken for
+helping   improve the
+presentation.  
+
+
+%I thank Robert  Willard for
+%help   improving th
+% presentation.  
+
+
+% \newpage
+
+\footnotesize
+\baselineskip = 0.80 \normalbaselineskip 
+
+
+
+
+ \begin{thebibliography}{99}
+
+  %rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr
+   
+\baselineskip =  1.01 \normalbaselineskip
+
+\parskip 1 pt
+
+
+
+\bibitem{AZ1}
+Adamowicz, Z., 
+Zbierski,  P.:
+  On Herbrand consistency in weak theories. 
+{\it Archive for Mathematical Logic}
+40(6):  399-413 (2001).    
+
+
+\bibitem{Ar1}
+Artemov,  S.:
+Explicit provablity and constructive semantics.
+{\it 
+Bulletin of Symbolic Logic}
+7(1): 1-36 (2001).
+
+\bibitem{Ar19}
+Artemov,  S.:
+    The provability of consistency. {\it
+Cornell Archives arXiv Report}      
+1902.07404v5
+         (2020).
+
+\bibitem{Ar21}
+Artemov,  S.:
+Missing Proofs and the Provability of Consistency. 
+Video presentation at the {\it 2021 T\"{u}bingen conference
+``Celebrating 90 Years of G\"{o}del's Incomppleteness
+Theorem''.} For the video recording see:
+\newline
+https://www.youtube.com/watch?v=Nw4nERAqi2U
+
+
+
+\bibitem{AB5}
+Artemov,  S.,
+Beklemishev,  L. D.:
+Provability logic. In
+{\it
+  Handbook of Philosophical Logic, Second Edition},    
+       pp.     189-360
+  (2005).
+
+\bibitem{Be5}
+Beklemishev,  L. D.:
+Reflection principles and provability algebras in formal arithmetic.
+{\it Russian Mathematical Surveys} 
+60(2):  197-268  (2005).   
+
+
+
+\bibitem{Be14}
+Beklemishev,  L. D.:
+Positive provability for uniform reflection principles.
+{\it   Annals Pure \& Applied Logic} 
+   165(1): 82-105  (2014).
+
+
+   
+\bibitem{BS76}
+Bezboruah, A., 
+ Shepherdson, J. C.:
+G\"{o}del's second incompleteness theorem for Q.
+{\it Journal of Symbolic Logic} 
+ 41(2):  503-512 (1976). 
+
+
+\bibitem{Bu86}
+Buss, S. R.:
+ {\it Bounded Arithmetic.} 
+Studies in Proof Theory, Lecture Notes 3, disseminated
+by  Bibliopolis
+as revised version of Ph. D. Thesis
+    (1986).
+
+
+
+\bibitem{BI95}
+Buss, S. R., 
+Ignjatovi\'{c},  A.:
+Unprovability of consistency statements in fragments of
+bounded arithmetic.
+{\it  Annals Pure \& Applied Logic } 
+ 74(3): 221-244 (1995).
+
+\bibitem{Da97}
+Dawson,   J.  W.:
+         {\it Logical Dilemmas: The Life and Work of
+Kurt G\"{o}del},
+   AKPeters Press
+          (1997).
+
+
+\bibitem{Fe60}
+Feferman, S.:
+ Arithmetization of mathematics in a general setting.
+   {\it Fundmenta Mathematicae}
+49: 35-92 (1960).
+
+
+\bibitem{Fi96}
+Fitting, M.: 
+{\it First Order Logic and Automated Theorem Proving.}
+Springer-Verlag (1996).
+
+
+\bibitem{Fr79a}
+Friedman, H. M.: 
+ On the consistency, completeness and correctness
+problems.
+ Ohio State  Tech Report (1979).
+ See also  Pudl\'{a}k \cite{Pu96}'s summary of this result.
+
+
+
+\bibitem{Go31}
+G\"{o}del, K.:
+ \"{U}ber 
+formal unentscheidbare S\"{a}tze der Principia
+Mathematica und verwandte Systeme I.
+{\it Monatshefte f\"{u}r Mathematik und Physik} 38:  349-360 (1931).
+
+
+
+
+\bibitem{Go33}
+G\"{o}del, K.:
+     The present situation in the foundations of
+mathematics.
+ In  {\it Collected Works Volume III: Unpublished Essays and Lectures}
+(eds. Feferman, S.,
+Dawson, J. W.,
+Goldfarb, W.,
+Parsons, C.,
+Solovay, R. M.),
+        pp.       {45--53},
+     Oxford    University Press (2004).
+
+
+\bibitem{Go5}
+Goldstein,        R.:
+   {\it Incompleteness: The Proof and Paradox of Kurt G\"{o}del}.
+        {Norton} Press
+         (2005).
+
+
+
+\bibitem{Ha11}
+H\'{a}jek, P.:
+              Towards metamathematics of weak arithmetics over
+fuzzy logics.
+{\it Logic Journal of the IPL}
+      19: 467-475
+          (2011).
+
+\bibitem{HP91}
+H\'{a}jek, P.,
+Pudl\'{a}k, P.:
+{\it Metamathematics of First Order Arithmetic.}
+Springer (1991). 
+
+\bibitem{Hil26}
+Hilbert, D.:
+                 \"{U}ber das unendliche.
+{\it
+   Mathematische Annalen}    
+      95: 161-191
+ (1926).
+
+
+ 
+
+\bibitem{Kl38}
+Kleene,  S. C.:
+On  notation for ordinal numbers.
+{\it Journal of Symbolic Logic} 
+3(1): 150-15 (1938).
+
+
+
+\bibitem{KT74}
+Kreisel, G., 
+Takeuti,  G.:
+Formally self-referential propositions  for cut-free classical
+analysis.
+{\it Dissertationes Mathematicae}
+118: 1-50
+(1974).
+
+
+\bibitem{Ne86}
+Nelson, E.:
+ {\it  Predicative Arithmetic.}
+Mathematics Notes,
+ Princeton University Press
+  (1986)
+
+\bibitem{Ne20}
+Nerode, A:
+ Some useful comments by A. Nerode  on Janaury 7, 2020
+when he heard Willard's conference  talk
+at the LFCS 2020  conference. Nerode said
+  he could reinforce
+Gerald Sacks's YouTube
+presentation \cite{YouSa14} (because
+Nerode heard Stanley  Tennenbaum
+make similar comments
+about
+G\"{o}del).
+
+
+\bibitem{Pa71}
+Parikh, R.:
+ Existence and feasibility in arithmetic.
+  {\it Journal of Symbolic Logic} 
+   36(3): 494-508
+(1971).
+
+\bibitem{PD83}
+Paris, J. B.,  
+Dimitracopoulos,  C.:
+A note on the 
+undefinability of cuts. 
+{\it Journal of Symbolic Logic} 
+48(3):
+  564-569
+(1983).
+
+
+\bibitem{Pa72}
+Parsons, C.:
+On $n-$quantifier elimination.
+{\it Journal of Symbolic Logic} 
+37(3):  466-482
+ (1972).
+
+
+\bibitem{Pu85}
+Pudl\'{a}k, P.:
+Cuts, consistency statements and interpretations.
+{\it Journal of Symbolic Logic} 
+50(2):   423-441
+(1985).
+
+\bibitem{Pu96}
+Pudl\'{a}k, P.:
+On the lengths of proofs of consistency.
+ {\it Collegium Logicum: 1996 Annals of the Kurt G\"{o}del
+Society} (Volume 2, pp 65-86).  Springer-Wien-NewYork
+(1996).
+
+
+\bibitem{Ro67}
+Rogers, H. A.:
+ {\it 
+Theory of Recursive Functions and Effective 
+Computibility.} McGrawHill (1967).
+
+
+\bibitem{YouSa14}
+  Sacks, G.:
+  Reflections on G\"{o}del. A YouTube recorded talk
+  delivered 
+  on 11$~$April$~$2007 as Lecture 3 in
+  {\it The Thomas and Yvonne Williams Symposia for
+    the Advancement of Logic}. 
+The easiest way to find a video recording of the
+Upenn-Sacks lecture is to do a google-search on the
+keywords of
+ {\it ``Gerald Sacks,
+   Reflections on Goedel, YouTube''}. This video is also
+ among several recordings stored at the deposit  site of:
+ 
+{\small
+  
+https://itunes.apple.com/us/itunes-u/lectures-events-williams-lecture/id431294044
+
+}
+
+\bibitem{Sm95}
+Smullyan, R.: 
+{\it First Order Logic}. Dover Books (1995).
+
+
+
+
+\bibitem{So88}
+Solovay, R. M.:
+Injecting Inconsistencies into models of PA.
+{\it   Annals Pure \& Applied Logic } 
+  44(1-2): 102-132 (1989).
+
+
+\bibitem{So94}
+Solovay, R. M.:
+Private
+Telephone  
+conversations
+during April of  1994
+between Dan Willard and
+ Robert M. Solovay.
+During those conversations
+Solovay described 
+an unpublished
+ generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85},
+using 
+some 
+methods
+of Nelson and Wilkie-Paris \cite{Ne86,WP87}.
+(The Appendix A of 
+\cite{ww1} offers a 
+4-page summary of
+our interpretation of Solovay's remarks.
+Several other articles 
+\cite{BI95,HP91,Ne86,PD83,Pu85,WP87}
+have also noted
+that Solovay  often
+has  chosen to privately communicate noteworthy
+ insights that he has elected not
+to formally publish.)
+
+
+\bibitem{Sv7}
+\v{S}vejdar, V.:
+ An interpretation of Robinson arithmetic in its
+Grzegorczjk's weaker variant.
+{\it Fundamenta Informaticae} 81:  347-354
+(2007).
+
+\bibitem{Vi5}
+Visser, A.:
+Faith and falsity.
+{\it        Annals Pure \& Applied Logic } 
+131(1-3): 103--131
+     (2005).
+
+
+\bibitem{WP87}  
+Wilkie, A. J.,
+Paris, J. B.:  
+On the scheme of induction for bounded
+ arithmetic. {\it  Annals Pure \& Applied Logic } 
+ 35: 261-302 (1987).
+
+\baselineskip =  1.0 \normalbaselineskip
+
+\parskip 0 pt
+
+
+ 
+\bibitem{ww93}
+Willard, D. E.:
+Self-verifying axiom systems.
+ In Proceedings of
+the   Third Kurt G\"{o}del
+Colloquium
+(with eds.
+Gottlob, G.,
+Leitsch, A.,
+Munduci, D.),
+{\it   LNCS } vol.
+713.
+pp.  325-336,
+Springer, Heidelberg
+(1993).
+https://doi.org/10.1007/BFb0022580.
+
+
+
+  
+
+\bibitem{ww1}
+Willard, D. E.:
+Self-verifying  systems, the incompleteness
+theorem and the tangibility reflection
+principle.
+{\it Journal of Symbolic Logic} 
+66(2):  536-596
+(2001).
+
+
+
+\bibitem{ww2}
+Willard, D. E.:
+How to extend the semantic tableaux and
+cut-free versions of the second
+incompleteness theorem 
+almost 
+to 
+Robinson's arithmetic Q.
+{\it Journal of Symbolic Logic} 
+67(1): 465--496
+(2002). 
+
+
+%% 
+%% \bibitem{wwlogos}
+%% Willard, D. E.:
+%% A version of the
+%% second incompleteness theorem for axiom
+%% systems that recognize addition 
+%% but not multiplication as a total function.
+%% In {\it First Order Logic Revisited}
+%% (eds.
+%%  Hendricks, V.,
+%% Neuhaus, F.,
+%% Pedersen, S. A.,
+%% Sheffler, U.,
+%% Wansing, H.),
+%% pp. 337--368,
+%% Logos Verlag,
+%%    Berlin
+%%  (2004). 
+
+
+\bibitem{ww5}
+Willard, D. E.:
+An exploration of the partial respects in which an axiom
+system recognizing solely addition as a total function can
+verify its own consistency.
+{\it Journal of Symbolic Logic} 
+70(4):  1171-1209  (2005).
+
+\bibitem{wwapal}
+Willard, D. E.:
+A generalization of the second incompleteness 
+theorem and some exceptions to it.
+{\it  Annals Pure \& Applied Logic } 
+141(3):  472-496
+(2006).
+
+
+
+\bibitem{ww6}
+Willard, D. E.:
+On the available partial respects in which
+ an axiomatization
+for real valued  arithmetic can  recognize its 
+consistency.
+{\it Journal of Symbolic Logic} 
+ 71(4):  1189-1199 
+(2006).
+
+
+\bibitem{ww7}
+Willard, D. E.:
+Passive induction and a solution to a Paris-Wilkie 
+open question.
+{\it  Annals Pure \& Applied Logic } 
+146(2):  124-149
+(2007). 
+
+
+\bibitem{ww9}
+Willard, D. E.:
+Some specially formulated axiomizations for
+I$\Sigma_0$ 
+manage to
+evade
+the Herbrandized version of the second incompleteness theorem,
+{\it Information and Computation}
+207(10): 1078-1093
+(2009).
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071
+         (2016).
+
+
+%%% 
+%%% \bibitem{ww18}
+%%% Willard, D. E.:
+%%% About   the 
+%%% chasm separating the goals of
+%%% Hilbert's consistency program from the
+%%% second incompleteness theorem.
+%%% {\it
+%%% Cornell Archives arXiv Report} 
+%%% 1807.04717
+%%%          (2018). (This quite preliminary annnouncement of
+%%% \cite{ww20}'s results
+%%% appears  in an
+%%%  {\it  essentially roughly written} summary-abstract
+%%% form.)
+
+
+\bibitem{ww21}
+Willard, D. E.:
+About the Characterization of a
+     Fine Line That Separates
+      Generalizations and Boundary-Case Exceptions for the
+ Second Incompleteness Theorem under Semantic Tableau
+ Deduction'',
+ {\it Journal of Logic and Computation}
+ 31(1):375-392 (2021).
+
+%%  
+%% OLD DELETED
+%% ``On the tender line
+%% separating generalizations and boundary-case exceptions for the
+%% second incompleteness theorem under semantic tableaux
+%% deduction'',
+%% a talk given 
+%% on January 7 at the LFCS 2020 conference.
+%% It preceded the current article, and
+%% its shorter manuscript can be found on
+%% pp.  268-286 of 
+%% Volume 11972 of
+%%  Springer's LNCS series.
+
+\bibitem{Yo5}
+Yourgrau, P.:
+              {\it A World Without Time: The Forgotten Legacy of
+G\"{o}del and Einstein}.
+(See page 58 for the passages we have quoted.)
+        {Basic Books}
+          ( 2005).
+
+
+
+\end{thebibliography}
+
+\end{document}
+
+
+
+
+\bibliographystyle{abbrv}
+
+\bibliography{bb}
+
+\end{document}
+
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+% modified on 2009 april 26 1am with three 1-word changes
+
+%% Dan Willard's Final Draft (November 10) for
+%% ``Some Specially Formulated Axiomizations for I$\Sigma_0$ Manage to
+%%  Evade the Herbrandized Version of the Second Incompleteness Theorem''
+
+\documentclass{elsart}
+
+% Use the option doublespacing or reviewcopy to obtain double line spacing
+% \documentclass[doublespacing]{elsart}
+
+% if you use PostScript figures in your article
+% use the graphics package for simple commands
+% \usepackage{graphics}
+% or use the graphicx package for more complicated commands
+% \usepackage{graphicx}
+% or use the epsfig package if you prefer to use the old commands
+% \usepackage{epsfig}
+
+% The amssymb package provides various useful mathematical symbols
+\usepackage{amssymb}
+
+\begin{document}
+
+\begin{frontmatter}
+
+% Title, authors and addresses
+
+% use the thanksref command within \title, \author or \address for footnotes;
+% use the corauthref command within \author for corresponding author footnotes;
+% use the ead command for the email address,
+% and the form \ead[url] for the home page:
+% \title{Title\thanksref{label1}}
+% \thanks[label1]{}
+% \author{Name\corauthref{cor1}\thanksref{label2}}
+% \ead{email address}
+% \ead[url]{home page}
+% \thanks[label2]{}
+% \corauth[cor1]{}
+% \address{Address\thanksref{label3}}
+% \thanks[label3]{}
+
+ \title{Some Specially Formulated Axiomizations for
+I$\Sigma_0$ 
+Manage to
+Evade
+the Herbrandized Version of the Second Incompleteness Theorem}
+
+
+\def\f22{\baselineskip = 1.0 \normalbaselineskip}
+\def\xpsi{\Psi}
+\def\iiir{I$\Delta_0^R$}
+\def\iiisr{I$\Delta_0^R~$}
+\def\iiid{I$\Delta_0$}
+\def\iiisd{I$\Delta_0~$}
+\def\iisi{I$\Delta_0~$}
+\def\jjsj{I$\Sigma_0~$}
+\def\jjj{I$\Sigma_0$}
+\def\iiia{I$\Delta_0$'s}
+\def\thsap{Theorem 2's$~$}
+\def\thscp{Theorem 2,$~$}
+\def\thsp{Theorem 2$\,$ }
+\def\rrr{\mbox{\bf R}}
+\def\srrr{\mbox{\bf S}}
+\def\mxm{\mbox{\bf m}}
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+% use optional labels to link authors explicitly to addresses:
+% \author[label1,label2]{}
+% \address[label1]{}
+% \address[label2]{}
+
+\author{Dan E. Willard}
+
+\address{University of Albany }
+
+\thanks[thank]{Email=dew@cs.albany.edu.
+This research was partially 
+supported by NSF Grant CCR  99-02726}
+
+
+
+\begin{abstract}
+In 1981, Paris and Wilkie \cite{PW81}
+indicated  it was an open question whether
+I$\Sigma_0$ 
+would satisfy
+the Second Incompleteness Theorem for  Herbrand deduction.
+We will show that
+some 
+specially formulated axiomizations for
+I$\Sigma_0$  can evade the
+Herbrandized version of the Second Incompleteness Theorem.
+\end{abstract}
+
+
+
+\begin{keyword}
+ G\"{o}del's
+Second Incompleteness Theorem, 
+Herbrand Consistency
+\newline
+MSC:  03B52; 03F25; 03F45; 03H13 
+
+
+
+
+% keywords here, in the form: keyword \sep keyword
+
+
+% PACS codes here, in the form: \PACS code \sep code
+\end{keyword}
+%\begin{pacs}
+%
+%\end{pacs}
+
+\end{frontmatter}
+
+\setlength{\parindent}{1.0 em}
+
+
+\addtolength{\topmargin}{.9in}
+
+\newcommand{\xbar}[1]{\widetilde{\, {#1} \,}}
+\newcommand{\ggd}[1]{ \, \lceil \, {#1} \, \rceil \,}
+\newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+\newcommand{\unders}[1]{\underbrace{\, {#1} \,}}
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+
+\newcommand{\hll}[1]{Herb$- {#1} -$List}
+\newcommand{\hxl}[1]{Herb$- {#1}$}
+\newcommand{\hlxl}[1]{Herb$- {#1}$ }
+
+
+
+\def\xpsi{\Psi}
+\def\iiir{I$\Delta_0^R$}
+\def\iiisr{I$\Delta_0^R~$}
+\def\iiid{I$\Delta_0$}
+\def\iiisd{I$\Delta_0~$}
+\def\iisi{I$\Delta_0~$}
+\def\jjsj{I$\Sigma_0~$}
+\def\jjj{I$\Sigma_0$}
+\def\iiia{I$\Delta_0$'s}
+\def\thsap{Theorem 2's$~$}
+\def\thscp{Theorem 2,$~$}
+\def\thsp{Theorem 2$\,$ }
+\def\rrr{\mbox{\bf R}}
+\def\srrr{\mbox{\bf S}}
+\def\mxm{\mbox{\bf m}}
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+\newcommand{\oveb}[1]{~ \overbrace{\, {#1} \, } } 
+\newcommand{\ovec}[1]{ \overbrace{\, {#1} \, } } 
+
+
+\newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{Theorem}
+
+\newtheorem{definition}{Definition}
+\newtheorem{lemma}{Lemma}
+\newtheorem{corollary}{Corollary}
+\newtheorem{fct}{Fact}
+
+
+\newenvironment{proof}{{\it Proof:}}{$\Box$}
+
+
+
+\setlength{\parindent}{1.0 em}
+\setcounter{page}{1}
+
+
+
+\normalsize
+
+
+\section{Introduction}
+
+
+
+ G\"{o}del's
+Second Incompleteness
+Theorem \cite{Go31} asserts
+that neither Peano Arithmetic, nor any
+consistent extension of it, can prove a theorem
+affirming its own self-consistency under Hilbert
+deduction. There have been numerous generalizations
+and extensions
+of G\"{o}del's seminal result
+\cite{Ad2,AB1,AZ1,BS76,BI95,Da83,DPR61,Fe60,HB39,Ko7,Lo55,Ma93,PW81,Pu84,Pu85,Pu96,Sa1,Sm77,Sm85,So94,Sw3,TMR53,Ta0,Vi93,Vi5,WP87,ww0,ww2,ww5,ww6}.
+For example, Solovay \cite{So94} has 
+combined the work of
+Nelson, Pudl\'{a}k  and Wilkie-Paris
+\cite{Ne86,Pu85,WP87}
+to establish
+that essentially no axiom system that recognizes
+Successor$(x)~=~x+1~$ as a total function
+(and which treats addition and multiplication as 3-way
+relations)
+ can prove a theorem
+affirming its own consistency under 
+the conventional paradigm of
+Hilbert deduction.
+
+
+
+In 1981, Paris and Wilkie \cite{PW81} 
+noticed that it was an open question whether
+the axiom system \jjsj
+did satisfy
+the Second Incompleteness Theorem for 
+semantic tableaux and Herbrand deduction.
+Interestingly
+at the same time,  Paris-Wilkie observed that
+there was an available solution to this problem for Hilbert deduction.
+Thus,  
+\jjj +Exp is unable to prove the Hilbert consistency of even
+an axiom system as simple as Q \cite{WP87}.
+Subsequently.
+ Adamowicz-Zbierski \cite{Ad2,AZ1} showed that \jjj $+ \Omega_1$ was unable
+to verify its Herbrand  consistency,
+and
+Willard \cite{ww0,ww2}
+developed an alternate variant of the
+  Adamowicz-Zbierski formalism that showed at
+least one type of natural encoding of the 
+\jjsj axiom system would satisfy the semantic tableaux version
+of the 
+Second Incompleteness Theorem.
+
+On 16 November 2005, we received a fascinating 
+email communication
+from 
+L.A. Ko{\l}odziejczyk 
+about this subject
+(which was an outgrowth out of some conversations
+he had with
+ Zofia  Adamowicz and
+Konrad  Zdanowksi). 
+It
+observed that there are  two
+natural formalisms 
+for axiomatizing \jjj , henceforth called
+Ax-1 and Ax-2. Both  shall take  the Tarski-Mostowski-Robinson
+axiom system $Q$ as their starting base.
+In a context where $\phi(x,y)$ is
+a $\Delta_0$ formula, these formalisms will use
+respectively 
+Equations \eq{hp} and \eq{wp} to denote
+their induction schemes.
+\begin{equation}
+\label{hp}
+ \forall x  ~~\{~~ \{~~
+ \phi(x,0)\,\,\wedge\,\, \forall y ~[~
+\phi(x,y) \, \Longrightarrow \, \phi(x,y')\, \, ] \,\,  \} \,~  \Longrightarrow \,~
+ \forall y ~ \phi(x,y)~\, \}
+\end{equation}
+\begin{equation}
+\label{wp}
+\forall x  \,  \forall z  \,    \{  \,    
+ \{ \,    \phi(x,0) \,  \, \wedge \,  \,  \forall y \leq z \, [ \, 
+\phi(x,y) \, \Longrightarrow \, \phi(x,y') \,  \, ]   \, \}  \,  \,   \Longrightarrow  \,  \, 
+ \forall y \leq z \,  \phi(x,y)  \, \}~~
+\end{equation}
+ Ko{\l}odziejczyk noticed 
+that
+logically equivalent axiom systems, such as Ax-1 and Ax-2,
+do not necessarily have the same properties with regards to
+the semantic tableaux and Herbrandized versions of the Second Incompleteness
+Theorem. Thus,
+Ko{\l}odziejczyk's email
+asked 
+whether \cite{ww2}'s semantic tableaux version of
+the Second Incompleteness Theorem
+will generalize for Ax-2's 
+unconventional induction scheme?
+It also inquired to what extent are
+generalizations of the Second Incompleteness Theorem
+germane to Herbrand deduction? One reason the second question
+is especially intriguing is that 
+Ko{\l}odziejczyk demonstrated in \cite {Ko6b,Ko7}
+that there exists bounded arithmetics where Herbrand consistency
+and tableaux consistency are provably not logically equivalent.
+
+
+One part of our  answer to this question 
+had appeared in the separate
+paper 
+\cite{ww7}. It  explained how our
+prior results about Ax-1's  
+semantic tableaux
+incompleteness properties
+have direct generalizations  for Ax-2.  
+
+A second type of  answer to 
+Ko{\l}odziejczyk's stimulating 
+open
+question will appear in this
+paper. We will prove that there is a third type of axiomization of
+\jjj, called Ax-3, which is logically equivalent to Ax-1 and Ax-2, but
+which has the property that an extension of Ax-3 is capable of
+recognizing its own Herbrand consistency.
+
+This last  result is likely to raise almost as many questions
+as it does answer. This is because there are many potential logically
+equivalent axiomizations for \jjj.
+$~$Thus,
+one may ask for which 
+potential 
+particular
+axiomizations $~\alpha~$ 
+for \jjsj do
+the Questions (1) and (2) below  have a positive answer?
+\bee
+\item Are all extensions of 
+$~\alpha\,$'s axiomization for \jjsj unable to
+prove a theorem verifying their own semantic tableaux consistency?
+\item Likewise, are   all extensions of
+$~\alpha\,$'s axiomization for \jjsj unable to
+prove a theorem verifying their own Herbrand consistency?
+\ene
+Since the Second Incompleteness Theorem 
+generalizes for most types of axiom systems, it is of course reasonable
+to conjecture that 
+most of
+the answers to the Questions 1 and 2 (above)
+will be
+in a positive direction.
+However, the point of this article is
+that an automatic ``yes'' response to Questions 1 and 2 cannot be
+always secured. Thus,
+Ko{\l}odziejczyk \cite{Ko6b,Ko7} has shown that 
+Herbrand consistency and semantic tableaux consistency are not always
+equivalent to each other, and the current article will 
+actually
+construct a
+particular 
+formalization of \jjj, called
+Ax-3, that manages to evade
+at least
+Question 2's
+Herbrandized 
+version of
+the Second Incompleteness Theorem.
+
+
+\section{  The Definition of a New Version of \jjj }
+\label{s2}
+
+
+
+This section will 
+define  the  Ax-3  axiomatization for
+\jjsj 
+and  provide the
+formal statement of our main theorem
+(which will be subsequently proven in 
+Sections \ref{s3} and \ref{main}). 
+As the reader examines the 
+formal definitions 
+in the current section, 
+it should be kept in mind that Ax-3 was
+deliberately endowed with an unconventional method
+for axiomatizing 
+\jjsj so that its formalism will evade the Herbrandized version
+of the Second Incompleteness Theorem.
+(Moreover
+because
+of
+ a
+technical difference between the definitions of
+Herbrand consistency and
+semantic tableaux consistency,
+it should be kept in mind that
+ Ax-3's evasion of the
+ Second Incompleteness Theorem
+under a Herbrandized definition of consistency
+does not generalize for semantic tableaux deduction.)
+
+
+
+In our discussion,
+a formula will be called 
+ $~\Delta_0^R~$
+iff it has a structure similar to a 
+ $~\Delta_0~$ formula 
+(appearing in for example the H\'{a}jek -Pudl\'{a}k textbook \cite{HP91})
+except that its bounded quantifiers,
+``$~\forall ~v \, \leq \, T~$'' and  ``$~\exists ~v \, \leq \, T,~$'',
+are now
+disallowed from using  the conventional arithmetic functions
+of addition and multiplication  in their terms $~T~$.
+Instead, the terms of a   $~\Delta_0^R~$ formula will
+employ only  the maximum function as the only  permissible operator to define
+a variable's bounded range.
+(Arithmetic functions are allowed to appear elsewhere in the body of
+a   $~\Delta_0^R~$ formula.)
+Thus, \ep{ex1} is an example of a  
+ $~\Delta_0^R~$ formula, and \eq{ex3}
+is  an example of a  
+ $~\Delta_0~$ formula that is not  $~\Delta_0^R~$.
+\begin{equation}
+\label{ex1}
+\forall ~p \leq~ \mbox{Max}(x,y)   ~~~[~~(~ p + y \leq ~ x \, + \, 2*y)~
+\vee ~(~ p * y~ \leq~ y*y*y~)~]~~
+\end{equation}
+\begin{equation}
+\label{ex3}
+\forall ~p \leq~x*y  ~~  ~~~[~~(~ p + y \leq ~ x \, + \, 2*y)~
+\vee ~(~ p * y~ \leq~ y*y*x~)~]
+\end{equation}
+
+
+Let us say a  formula is
+ $\Pi_1^R$ 
+iff it can be written as
+$~\forall \, v_1 \,  \forall \, v_2 \,  ...  \, \forall \, 
+v_n~ \phi(v_1,v_2, \, ... \, ,\, v_n) $ where
+$\phi(v_1,v_2,~ \ldots ,~ v_n)$ 
+is a  $ \Delta_0^R $ formula. 
+Each of  Ax-1, Ax-2 and Ax-3  will contain
+a common set of nine
+ $\Pi_1^R$ 
+ axioms,  called $~Q_0~$ and listed below.
+The main purpose of $~Q_0~$ will be to define
+the constructs 
+of addition, multiplication, integer-successor, maximum
+and also $\, = \,$ and $\, \leq$.
+\begin{equation}
+\label{pw1}
+ 1 \, = \, 0'  ~ \, \wedge~ \, 
+  2 \, = \, 1'  ~ \, \wedge~ \, 
+ 0 \, = \, 0  ~ \, \wedge~ \, 
+0' \, \neq \, 0  ~ \, \wedge~ \, 
+0  \,\leq \, 0 ~ \, \wedge~ \,  
+\neg~[~ 0'  \,\leq \, 0~] 
+\end{equation}
+\begin{equation}
+\label{pw2}
+\forall ~x~~(~~ x+0 \, = \, x ~~ \wedge
+~~ x \cdot 0 \, = \, 0 ~~ \wedge
+~~ x \cdot 1 \, = \, x ~~ )
+\end{equation}
+\begin{equation}
+\label{pw3}
+\forall x ~ \forall y ~~ (~~ x' \, = \, y'~~ \Longleftrightarrow ~~ x \, = \, y ~~)
+\enq
+\begin{equation}
+\label{pw4}
+\forall x ~ \forall y ~~ (~~
+x \, \leq \, y~~ \Longleftrightarrow ~~(~ x' \, \leq \, y ~ \vee ~ 
+x \, = \, y ~~)
+\end{equation}
+\begin{equation}
+\label{pw6a}
+\forall x ~ \forall y ~~~
+  ~~~~  x \cdot y'~=~ (x \cdot y)+x ~~~
+\wedge ~~~ x+y'~=~ (x+y)'
+\end{equation}
+\begin{equation}
+\label{pw6b}
+\forall x ~ \forall y  ~ \forall z ~~~
+[~x=y ~ \wedge ~ y=z ~] ~ \Rightarrow~ [~x=z ~\wedge ~z=x ~]
+\end{equation}
+\begin{equation}
+\label{pw6c}
+\forall x ~ \forall y  ~ \forall z ~~~
+[~x=y ~ \wedge ~  y \leq z  ~] ~ \Rightarrow~ x \leq z
+\end{equation}
+\begin{equation}
+\label{pw6.1}
+\forall x ~ \forall y  ~ \forall z ~~~
+[~x=y ~ \wedge ~  z \leq y  ~] ~ \Rightarrow~ z \leq x
+\end{equation}
+\begin{equation}
+\label{pw6}
+ \forall x  \,   \forall y  \,  
+  \,  \, ( \,  \,  x \, \leq \, y \,  \, 
+\Rightarrow   \,  \,  \mbox{Max}(x,y)=y \,  \, )
+  \,  \,  \wedge \,  \, 
+( \,  \,  y \, \leq \, x \,  \, 
+\Rightarrow   \,  \,  \mbox{Max}(x,y)=x \,  \, )
+\end{equation}
+In the context of the above definition for $Q_0$,
+ the Ax-1 and Ax-2 formalisms
+will be respectively
+defined
+ as the union of $~Q_0~$ with all 
+the instances of the 
+respective
+induction
+schemes in
+ Equations \eq{hp} and \eq{wp},
+where $~\phi(x,y)~$ is a 
+$\Delta_0$ formula.
+Similarly,$~$ \iiisr will be defined as the union of
+ $~Q_0~$ with all instances of 
+\ep{wp}'s induction
+schemas  where $~\phi(x,y)~$ is 
+$\Delta_0^R$. 
+
+
+This paragraph will define a set of
+$\Pi_1^R$ sentences, called {\bf Trivial-R}, that has the property that
+\iiir $\, + \,$Trivial-R proves the same set of theorems as the
+more conventional Ax-1 and Ax-2
+axiomatization for \jjj .
+In our discussion, a
+tuple $ ( a_0 ,  a_1 ,  a_2 , ~ \ldots ,~   a_N ) $
+is called
+a {\bf  Split Representation} of an non-negative 
+integer $~ x~ $
+when the following condition is satisfied:
+\begin{equation}
+\label{r111}
+x \,  \, = \,  \,  \sum_{i=1}^{N} \, a_i  \,  * \,  (a_0+1)^{i-1}
+    \,  \,  \,  \,  \mbox{AND}  \,  \,  \,  \, 
+a_1  \,   \leq \,  a_0 \, \wedge \, a_2  \,   \leq \,  a_0 \, \wedge \, ... \, 
+a_N  \,   \leq \,  a_0 \,  
+\end{equation}
+For a fixed integer $N$,
+ let Split$^N( \, x \, , \, a_0 \, , \, a_1 \, , \, ~ \ldots ,~   \,  \, a_N \, ) \, $
+denote a  $\Delta_0^R$ formula indicating  \eq {r111} is satisfied.
+
+For each of the  arithmetic operators of
+$\, + \,$, $\, * \,$, Max, $\, = \,$ and $\, \leq , \,$ the
+axiom system Trivial-R will have available a family of
+$\Delta_0^R$ predicates 
+and $\Pi_1^R$ axioms
+for simulating the 
+operations of these functions when they
+manipulate Split
+Representations of integers.
+Thus for a fixed triple
+$~(I,J,K)$, let 
+Mult$^{I,J,K}(a_0,a_1,  \, ..., \,  a_I,b_0,b_1, \, ..., \,  b_J,c_0,c_1,  \, ..., \,  c_K)$
+designate a $\Delta_0^R$ predicate
+simulating the action of integer multiplication when its input is the
+two
+split integers of 
+$(a_0,a_1,~ \ldots ,~ a_I) ~$ and $(b_0,b_1,~ \ldots ,~ b_J)~$ and
+its resultant is 
+the multiplicative product of $~(c_0,c_1,~ \ldots ,~ c_K).~$
+The accompanying 
+$\Pi_1^R$ axiom 
+of Trivial-R
+that formalizes this predicate  will
+be:
+
+\bigskip
+
+{\baselineskip = 1.0 \normalbaselineskip
+\begin{center}
+$\forall~x~~\forall~y~~\forall~z~~
+\forall~a_0~~\forall~a_1~... ~\forall~a_I~~
+\forall~b_0~~\forall~b_1~... ~\forall~b_J~~
+\forall~c_0~~\forall~c_1~... ~\forall~c_K~~~$
+\end{center}
+\begin{center}
+$\{~~~[~~ ~\mbox{Split}^I(x,a_0,~ ... \,  , ~a_I)
+~\wedge ~\mbox{Split}^J(y,b_0, ~ ... \,  , ~b_J)
+~\wedge ~\mbox{Split}^K(z,c_0, ~ ... \,  , ~c_K)~~~]
+~~~\Longrightarrow ~~ $
+\end{center}
+\begin{equation}
+\label{trivr-1}
+[~~x*y=z~~~\Longleftrightarrow ~~ 
+\mbox{Mult}^{I,J,K}(a_0, ~ ... \,  , ~a_I,b_0, ~ ... \,  , ~b_J,c_0, ~ ... \,  , ~c_K)   ~]~~\}~~~
+\end{equation}}
+
+\bigskip
+
+\noindent
+Likewise,  Trivial-R 
+will have available  analogs
+of \ep{trivr-1}
+to
+ simulate
+ addition, maximum,  equality, and less-than-or-equals
+among
+split integers.
+Thus, 
+the predicates of
+Add$^{I,J,K}(a_0,a_1, ~ \ldots , ~ a_I,b_0,b_1, ~ \ldots , ~ b_J,c_0,c_1, ~ \ldots , ~ c_K)$,
+$~$Maxim$^{I,J,K}(a_0,a_1, ~ \ldots , ~ a_I,b_0,b_1, ~ \ldots , ~ b_,c_1, ~ \ldots , ~ c_K)$,
+$~$Eq$^{I,J}(a_0,a_1, ~ \ldots , ~ a_I,b_0,b_1, ~ \ldots , ~ b_J)$
+and
+$~$LTE$^{I,J}(a_0,a_1, ~ \ldots , ~ a_I,b_0,b_1, ~ \ldots , ~ b_J)$
+will be the
+$\Delta_0^R$
+ analogs of 
+Mult$^{I,J,K}$
+for these four structural relations.
+ Their counterparts of 
+ \ep{trivr-1}'s formal axiom  will then be:
+
+
+\bigskip
+
+
+{\baselineskip = 1.0 \normalbaselineskip
+
+
+\begin{center}
+$\forall~x~~\forall~y~~\forall~z~~
+\forall~a_0~~\forall~a_1~... ~\forall~a_I~~
+\forall~b_0~~\forall~b_1~... ~\forall~b_J~~
+\forall~c_0~~\forall~c_1~... ~\forall~c_K~~~$
+\end{center}
+\begin{center}
+$\{~~~[~~ ~\mbox{Split}^I(x,a_0...a_I)
+~\wedge ~\mbox{Split}^J(y,b_0...b_J)
+~\wedge ~\mbox{Split}^K(z,c_0...c_K)~~~]
+~~~\Longrightarrow ~~ $
+\end{center}
+\begin{equation}
+\label{trivr-2}
+[~~x+y=z~~~\Longleftrightarrow ~~ 
+\mbox{Add}^{I,J,K}(a_0...a_I,b_0...b_J,c_0...c_K) ~~]~~~\}
+\end{equation}
+
+\bigskip
+
+
+
+
+\begin{center}
+$\forall~x~~\forall~y~~\forall~z~~
+\forall~a_0~~\forall~a_1~... ~\forall~a_I~~
+\forall~b_0~~\forall~b_1~... ~\forall~b_J~~
+\forall~c_0~~\forall~c_1~... ~\forall~c_K~~~$
+\end{center}
+\begin{center}
+$\{~~~[~~ ~\mbox{Split}^I(x,a_0...a_I)
+~\wedge ~\mbox{Split}^J(y,b_0...b_J)
+~\wedge ~\mbox{Split}^K(z,c_0...c_K)~~~]
+~~~\Longrightarrow ~~ $
+\end{center}
+\begin{equation}
+\label{trivr-m}
+[~~z~=~\mbox{Max}(x,y)~~~\Longleftrightarrow ~~ 
+\mbox{Maxim}^{I,J,K}(a_0...a_I,b_0...b_J,c_0...c_K) ~~]~~~\}
+\end{equation}
+
+\bigskip
+
+
+\begin{center}
+$\forall~x~~\forall~y~~
+\forall~a_0~~\forall~a_1~... ~\forall~a_I~~
+\forall~b_0~~\forall~b_1~... ~\forall~b_J~~
+  ~~~[~ \,\mbox{Split}^I(x,a_0...a_I)
+~\wedge ~\mbox{Split}^J(y,b_0...b_J) ~]$
+\end{center}
+\begin{equation}
+\label{trivr-3}
+ ~~~~~~~~\Longrightarrow ~~ 
+[~~x=y~~~\Longleftrightarrow ~~ 
+\mbox{Eq}^{I,J}(a_0...a_I,b_0...b_J) ~~]
+\end{equation}
+
+\bigskip
+
+\begin{center}
+$\forall~x~~\forall~y~~
+\forall~a_0~~\forall~a_1~... ~\forall~a_I~~
+\forall~b_0~~\forall~b_1~... ~\forall~b_J~~
+  ~~~[ \, ~\mbox{Split}^I(x,a_0...a_I)
+~\wedge ~\mbox{Split}^J(y,b_0...b_J) ~]$
+\end{center}
+\begin{equation}
+\label{trivr-5}
+ ~~~~~~~~\Longrightarrow ~~ 
+[~~x \leq y~~~\Longleftrightarrow ~~ 
+\mbox{LTE}^{I,J}(a_0...a_I,b_0...b_J) ~~]
+\end{equation}
+
+}
+
+\noindent
+Henceforth,  {\bf Ax-3} will denote the
+axiom system 
+\iiir $\, + \,$Trivial-R.
+Section \ref{s3} will prove that
+Ax-3 proves the
+same set of theorems as Ax-1 and Ax-2.
+
+
+
+\bigskip
+
+{\bf Definition 1: }
+Let $ \, \alpha \, \supseteq \, \beta \, $ denote that $ \, \alpha$'s
+set of formal axioms
+includes all $ \, \beta \,$'s axioms.
+(This definition of $ \, $``$ \, \supseteq \,$''$ \, $
+is  stronger than the 
+{\it more modest construct} that
+$ \, \alpha \, $ proves all $ \, \beta \,$'s theorems.)
+Also assuming 
+$ \, \alpha \, $ denotes a {\bf consistent} axiom system
+and $ \, D \, $ denotes a deductive method,
+$(\alpha,D)$ will be called a
+{\bf Threshold}
+for the Second Incompleteness Effect iff
+all consistent extensions
+$ \, \alpha^* \, \supseteq \, \alpha \, $ have the property
+that 
+ $ \, \alpha^* \, $ is unable to prove the consistency
+of its proofs using deduction method $D$.
+Otherwise,  $(\alpha,D)$  will be called an 
+{\bf Anti-Threshold}. (It  means that {\it some consistent}
+$ \, \alpha^* \, \supseteq \, \alpha \, $ 
+can prove a theorem affirming its own
+consistency
+under deduction method $D$.)
+
+
+\bigskip
+
+Using Definition 1's notation, the main theorem proven
+in this article  will be
+that Ax-3 is an anti-threshold for the Herbrandized version of the
+Second Incompleteness Theorem.
+This means that there must assuredly exist some {\it consistent}
+system
+$~\alpha^* ~ \supseteq ~$Ax-3, where $\alpha^*$ can prove
+a theorem affirming its own Herbrand consistency.
+
+
+\section{ Basic Framework and  Underlying Intuition}
+\label{s3}
+
+
+This section
+will have two purposes. It
+will formally
+prove  Ax-3 proves the same set of
+theorems as Ax-1 and Ax-2,
+thus establishing that Ax-3 is a
+permissible
+ formal means for
+axiomatizing \jjj . 
+It will also provide an intuitive explanation of why
+Ax-3 is able to evade the Herbrandized version of
+the Second Incompleteness Theorem (by satisfying Definition 1's
+``anti-threshold'' property
+for Herbrand consistency).
+
+
+
+
+
+
+
+\begin{theorem}
+\label{tm4}
+Each of 
+Ax-1, Ax-2 and Ax-3 prove the
+same set of theorems.
+\end{theorem}
+
+
+{\bf Proof Sketch:} It is well known  Ax-1 and Ax-2 prove the same
+set of theorems. 
+Thus to establish \thx{tm4}, we
+need  only
+show  Ax-2 and Ax-3.
+also prove the same
+set of theorems. 
+
+Our proof will use the fact that 
+ Paris and  Dimitracopoulos \cite{PD82} have observed that
+in
+a
+ model-theoretic sense, there is a 1-to-1 correspondence
+between $\Delta_0$ formulae and their equivalent representations
+in a   $\Delta_0^R$ form.
+Let $\psi(x,y)$ 
+denote an arbitrary $\Delta_0^R$ formula. 
+The  Paris and  Dimitracopoulos \cite{PD82} result easily implies
+that
+for any integer $k$, 
+it is possible to construct 
+a $\Delta_0^R$ formula 
+ $\psi^*(x,y_0,y_1,~ \ldots , ~y_k)$ that is its
+logical
+ counterpart of
+ $\psi(x,y)$ under split representations for an integer.
+More precisely, it implies that one
+can map
+$\psi(x,y)$ onto
+a $\Delta_0^R$ formula 
+ $\psi^*(x,y_0,y_1,~ \ldots , ~y_k)$ 
+such that the Ax-2 and Ax-3 representations of
+\jjsj can both trivially prove 
+the following property:
+\begin{center}
+\small
+$\forall~x~
+ ~\forall~y~~
+\forall~y_0~~\forall~y_1~... ~\forall~y_k$
+\end{center}
+\begin{equation}
+\small
+\label{etm4}
+ \{~\mbox{Split}^k(y,y_0,y_1,~ \ldots , ~y_k)~~
+\Longrightarrow ~~
+[~\psi(x,y)~~ \Longleftrightarrow~~
+\psi^*(x,y_0,y_1,~ \ldots , ~y_k)~~]~ \}~~
+\end{equation}
+
+Let Size$_L(y_0,y_1,~ \ldots , ~y_k)$ denote a $\Delta_0^R$ formula indicating
+that $(y_0,y_1,~ \ldots , ~y_k)$ represents an integer $\, \leq \, L.~$
+Then it is not hard to show that
+Ax-3 can use its Trivial-R axioms
+to prove
+ that
+the two $\Delta_0$ formulae of
+$~\exists y \leq x^k~ \, \psi(x,y)~~~$
+and
+$~~~\forall y \leq x^k~ \, \psi(x,y)~~~$
+are equivalent to the respective  $\Delta_0^R$ formulae of:
+\begin{equation}
+\small
+ \exists \,  y_0 \, \leq \, x \, \, 
+\exists \,  y_1 \, \leq \, x \, \, ...~ \, 
+\exists \,  y_k \, \leq \, x \, \, ~~
+\mbox{Size}_{x^k}(y_0,y_1, \, ... \, , ~y_k)\wedge \psi^*(x,y_0,y_1, \, ... \, , ~y_k) ~~
+\end{equation}
+\begin{equation}
+\footnotesize
+ \forall \,  y_0 \, \leq \, x \, \, 
+\forall \,  y_1 \, \leq \, x \, \, ...~ \, 
+\forall \,  y_k \, \leq \, x \, \,~~ 
+\mbox{Size}_{x^k}(y_0,y_1, \, ... \, , ~y_k)\Rightarrow \psi^*(x,y_0,y_1, \, ... \, , ~y_k) ~~
+\end{equation}
+
+
+ Thus by essentially applying $~n~$ iterations of this
+technique
+(and its obvious analogs)
+ to any initial 
+ $\Delta_0$ formula with $ \, n \,$ bounded quantifiers,
+Ax-3 can transform an arbitrary
+ $\Delta_0$ formula into a provably equivalent  $\Delta_0^R$
+formula. It thus follows that although the Ax-3 system
+contains technically only instances of
+\ep{wp}'s axiom schema
+ for   $\Delta_0^R$ 
+formulae, it nevertheless
+has an ability to formally prove as theorems all the
+remaining instances of
+Ax-2's axiom schema for  $\Delta_0$ formulae {\it as well.}
+(In particular if 
+$\phi(x,y)$ 
+is 
+a $\Delta_0$ formula which is not 
+ $\Delta_0^R$ 
+and if
+$\phi^*(x,y)$ is a  
+ $\Delta_0^R$ formula equivalent to 
+$\phi(x,y)$,  
+then Ax-3 can prove a theorem verifying
+the validity of 
+\ep{wp}'s axiom scheme for 
+$\phi(x,y)$  by first observing that 
+this axiom scheme is
+ valid for 
+$\phi^*(x,y)$
+and then observing the
+latter is equivalent to 
+$\phi(x,y)$'s axiom scheme.)
+
+
+Hence although Ax-3 contains technically only a
+proper superset of Ax-2's induction axioms as formal axioms
+within its own induction schema, it has the ability to
+formally prove the validity of all of Ax-2's axioms.
+The proof in the reverse direction (establishing that Ax-2 can prove all
+Ax-3's axioms) is 
+straightforward
+ because Ax-2 can easily prove
+all Ax-3's Trivial-R axioms.
+Hence,
+Ax-2 and Ax-3 will
+prove  identical sets of theorems.
+ $\Box$
+
+
+
+
+
+\bigskip
+
+{\bf Remark 1.} $~$
+Our proof that Ax-3 is an anti-threshold for the 
+Herbrandized version of the Incompleteness Theorem will
+appear in Section \ref{main}. 
+This result is quite surprising because 
+there are of course many more generalizations of 
+G\"{o}del's Second Incompleteness Theorem available in the
+mathematical literature 
+\cite{Ad2,AB1,AZ1,BS76,BI95,Da83,DPR61,Fe60,HB39,Ko7,Lo55,Ma93,PW81,Pu84,Pu85,Pu96,Sa1,Sm77,Sm85,So94,Sw3,TMR53,Ta0,Vi93,Vi5,WP87,ww0,ww2,ww5,ww6}
+than there are
+published 
+examples of boundary-case style
+exceptions to its formalism.
+
+In order to explain  intuitively 
+the core idea about
+why our
+paper is able to achieve this surprising evasion of
+the 
+Second Incompleteness Theorem,
+it is helpful to
+let $~\Upsilon_n~$ denote 
+\ep{crazy}'s
+$\Delta_0$ sentence.
+Note 
+that
+this sentence
+is comprised of
+$~O(~n~)~$ logic symbols.
+It asserts that
+the variables $~v_0 \, , \,v_1 \, , \,v_2 \, , \, ~ \ldots , ~v_n \, , \,~$
+satisfy $~v_i~=~2^{2^i}.~$
+\begin{center}
+$\exists~v_0 \, \leq \, 2~~
+\exists~v_1 \, \leq \, v_0*v_0~~
+\exists~v_2 \, \leq \, v_1*v_1~~...~~
+\exists~v_n \, \leq \, v_{n-1}*v_{n-1}$
+\end{center}
+\beq
+\label{crazy}
+v_0 \, = \, 2~\wedge~
+v_1 \, = \, v_0*v_0~\wedge~
+v_2 \, = \, v_1*v_1~\wedge~ ... \wedge
+v_n \, = \, v_{n-1}*v_{n-1}
+\enq
+It is easy to see  there exists
+some $\Delta_0^R$ sentence, called say $~\Upsilon_n^R~,~$
+that is the counterpart of \ep{crazy} written in a notation
+using split integers. This sentence will
+indicate the existence of a sequence of split integers
+ $~S_0 \, , \,S_1 \, , \,S_2 \, , \, ~ \ldots , ~S_n \, , \,~$
+where $~S_i~$ represents the quantity 
+$~2^{2^i}.~$
+
+However although they in some sense represent equivalent
+concepts, there is a fundamental difference between
+the $\Delta_0$ sentence  $~\Upsilon_n~$
+and
+its $\Delta_0^R$
+counterpart $~\Upsilon_n^R.~$
+This difference is easiest to explain if one uses
+a logical language that has only 3 named constants, 0, 1 and 2,
+and if split integers are encoded as base 2 numbers.
+Then    $~\Upsilon_n^R~$ 
+will be encoded as
+a sequence of at least $~2^n~$ characters, but 
+  $~\Upsilon_n$'s length has a
+sharply smaller O$(n)$ magnitude.
+As a consequence of this distinction
+(and its generalizations), we 
+realized that one could formulate an axiom system that was logically
+equivalent to the more conventional encodings for \jjj ,
+but which was
+an anti-threshold to the Herbrandized version of
+the Second Incompleteness Theorem (using Definition 1's terminology).
+
+In particular,
+it turns out that
+the exponential difference between the lengths of the 
+$\Delta_0$ sentence  $~\Upsilon_n~$
+and
+its $\Delta_0^R$
+counterpart $~\Upsilon_n^R~$
+plays a major role in 
+understanding the 
+central concept that 
+motivated much of the research that stimulated the current article.
+Thus, this paradigm (and its
+formal generalization for arbitrarily more
+complicated 
+sequences of
+sentences) are used by Section \ref{main}'s machinery
+in a much more sophisticated context
+to prove that it is feasible
+to construct awkward encodings for \jjsj , similar to Ax-3,
+that are 
+actual
+anti-thresholds to the Herbrandized version of the
+Second Incompleteness Theorem.
+(A reader
+should not worry if he
+ does not  follow
+fully the intuitive idea,
+sketched in this paragraph, 
+about the significance of logically equivalent statements that
+have exponentially different sizes in length.
+This is
+because a fully formalized
+proof 
+of our main theorem, that uses this effect,
+will be explored in great detail in the next section.)
+
+
+
+\section{ Main Analysis}
+\label{main}
+
+
+A sentence $ \psi $ in the propositional
+calculus will be called
+an {\bf Anti-Tautology} iff  $ \psi $
+ is unsatisfiable (i.e. 
+$ \neg \, \psi $ is a tautology). 
+Our definition of Herbrand deduction
+\cite{He30}
+ will be identical to the definitions
+used by Adamowicz,
+H\'{a}jek-Pudl\'{a}k and 
+Ko{\l}odziejczyk
+\cite{Ad2,HP91,Ko7}, 
+except that we will 
+use a  dual version of this definition that follows from
+De Morgan's Rule, where  disjunctions are  replaced with conjunctions
+and where 
+tautologies
+are
+ replaced with anti-tautologies. In other words,
+our definition will use the well-known identity that
+\begin{equation}
+\label{prot.taut}
+\bigvee_{i=1}^n~\neg \, \phi_i~~~~~=~~~~~
+ \neg~\bigwedge_{i=1}^n~ \phi_i
+\end{equation}
+Our definition of Herbrand deduction
+will differ from its more
+conventional definitions by using the  
+right (instead of left) side of \eq{prot.taut}'s identity. This change in
+notation  
+is 
+unnecessary, but it does help
+to considerably
+ simplify our proofs.
+
+
+
+ Let $~\Psi~$ denote an arbitrary prenex normal sentence such
+as the prototype below, whose open subcomponent is denoted
+as $~\xbar{\psi}~$.
+\begin{equation}
+\label{prot}
+ ~~ \forall~ x_1 ~~ \exists~y_1  ~~ \forall~ x_2 ~~ \exists~y_2 
+ ~~ .... ~~ ~~ \forall~ x_n ~~ \exists~y_n ~~~
+\xbar{\psi} \,
+(x_1,y_1...x_n,y_n)
+\end{equation}
+In a context where
+$~f_1^{\psi}(x_1),~f_2^{\psi}(x_1,x_2)~...~f_n^{\psi}(x_1,x_2,~ \ldots , ~x_n)~$
+are new function symbols,
+\ep{prot2} is called the Skolemization of \ep{prot}.
+\begin{equation}
+\label{prot2} 
+\forall  \, x_1  \,     \forall  \,  x_2
+  \,    ....    \,  \,  \forall  \,  x_n  \,  \,  \,   
+\xbar{\psi} \,[ \, \, x_1  \, , \, f_1^{\psi}(x_1)  \, , \, x_2  \,
+ , \, f_2^{\psi}(x_1,x_2)
+  \, ...  \,  x_n  \, , \, 
+f_n^{\psi}(x_1,x_2,~ \ldots , ~x_n) \, ]~~
+\end{equation}
+In a context where
+$ \, L \, $ is a logical language and
+ $ \, \alpha \, $ is an axiom system, we will let
+$ \, C_L \, $ and $ \, F_L \, $ denote the set of constant and function symbols
+associated with $ \, L \, $. Similarly,
+$ \, F_{\alpha} \, $ will denote the set of
+``Skolemized'' function symbols
+associated with $ \, {\alpha}  $'s axioms. 
+Thus using   \eq{prot}
+and \eq{prot2}'s notation,
+let $\alpha$ denote a system of axioms
+$ \, \Psi_1 \, , \, \Psi_2 \, , \,\Psi_3 \, ...  \, $,
+and
+for an arbitrary 
+index $ \, i \, $ let its Skolemized function symbols 
+carry names such as
+$ \, f_i^{\psi_1}\, , \, f_i^{\psi_2}\, , \, f_i^{\psi_3}\, , \, ... \, $
+The {\bf Herbrandized Terms} for this ordered pair
+$ \, (\alpha,L) \, $ will then be defined to be the set of all terms
+generated by the constants from the set $ \,  C_L \, $
+combined with the functional operations from the set
+ $ \, F_\alpha \cup F_L.$ 
+
+
+A {\bf Herbrandized Instance} of a Skolemized axiom is a sentence identical
+to this axiom except that all its universally quantified variables
+are replaced by Herbrandized terms. For instance
+in a context where $~T_1,T_2,T_3...~$ are
+Herbrandized terms, \ep{prot3} is
+such an instance of \eq{prot2}'s axiom:
+\begin{equation}
+\label{prot3}
+\xbar{\psi} \,[ \,  \, T_1  \, , \, f_1^{\psi}(T_1)  \, , \, T_2  \,
+ , \, f_2^{\psi}(T_1,T_2)
+ ~ ~ \ldots , ~ ~ T_n  \, , \, 
+f_n^{\psi}(T_1,T_2, ~ \ldots , ~T_n)~]
+\end{equation}
+Let $~\bot~$ denote the logical constant of FALSE.
+A {\bf Herbrandized Proof} 
+of  $~\bot~$ 
+from  the axiom system $~\alpha~$ is defined as a finite collection
+of Herbrandized instances 
+of $~\alpha~$
+together with a proof, in the pure propositional
+calculus, that the conjunction of these instances is 
+an anti-tautology.
+
+\medskip
+
+{\bf  Definition 2 : $~$} Using our revised notation convention,
+ the theorem $~\Upsilon~$ will be said to have a {\bf Herbrandized
+Proof} from the axiom system $~\beta~$ if and only if the union of
+the axiom system $~\beta~$ with the
+added sentence $~\neg \, \Upsilon~$
+produces a Herbrandized proof of 
+$~\bot~.~$
+
+\medskip
+
+{\bf More Notation:} $~$
+Let us say that a function
+$G(\, x_1 \, , \, x_2, \,  ~ \ldots , ~ \,x_n \,)$
+is a {\bf Non-Growth Function} iff
+$G(\, x_1 \, , \, x_2, \,  ~ \ldots , ~ \,x_n \,) ~
+ \leq ~$Max$(\, x_1 \, , \, x_2, \,  ~ \ldots , ~ \,x_n \,).~$
+Define a set $~S~$ of functions to be an 
+{\bf Arithmetic Controlled Set} iff $~S~$
+includes the arithmetic functions of addition, multiplication and successor
+and all its other functions are non-growth functions. Also,
+define a term $~t~$ to be an {\bf Arithmetically Controlled Term}
+iff $~t~$ is a term that uses only the symbols of 0, 1 and 2
+as its inputted constants and 
+all its function symbols come from some 
+Arithmetic Controlled Set $~S$.
+Thus if
+ $G_1$ and $G_2$ are non-growth
+functions, \ep{silly} 
+represents
+an arithmetically controlled
+term.
+\begin{equation}
+\label{silly}
+G_1[~(2+1)*(1+1)~,~1+2~]~*~G_2(~2+2+0~,~2+2+1+0~)
+\end{equation}
+Also, in a context where
+$~C_t~$ and $~F_t~$
+denote the number of constant and function
+symbols in $~t,~$  we will use the following notation:
+\bee
+\item
+MinG$(t)$ will denote the quantity $~2^{C_t+F_t}$.
+\item
+Val$(t)$ will denote the quantity represented by the term $~t$.
+\ene
+For example if $~G_1(x,y)~=~|x-y|~$ and $~G_2(x,y)~=~$Min$(x,y)$
+then \ep{silly}'s term $~t~$  will have
+$~Val(t)~=~3*4~=~12~$  
+and MinG$(t)~=~2^{25}~$ (because $~t~$ contains 12 function 
+symbols and 13 constant symbols).
+
+
+\begin{lemma}
+\label{lem1}
+Let  $~t~$ be an arithmetically controlled
+ term which satisfies the inequality $~$Val$(t) \, \geq \, 4$.
+Then
+$~~~$Val$(t)~ < ~$MinG$(t)$.
+\end{lemma}
+
+
+{\bf Proof Sketch:} $~$
+Suppose for some $~k \geq 2,~$ 
+that Val$(t)~=~2^k.~$ Then it easy to see that $~t \,$'s maximally
+compressed representation
+as an {\it arithmetically controlled term} 
+is ``$~2*2*....~2~$''. Thus
+MinG$(t)=2^{2k-1}> \,$Val$(t) \, = \, 2^k$ is valid in this case
+because the preceding product has
+$ \, k \, $ appearances of the constant 2 connected by $ \, k-1 \, $ appearances
+of the multiplication symbol.  
+Moreover, it is easily proven that terms, which are not powers
+of 2, are never represented in a more compressed form than the
+greatest power of 2 that they exceed. Thus Lemma \ref{lem1}
+is valid for all terms where 
+ $ \, $Val$(t) \,  \geq \, 4$. $ \,  \, \Box$
+
+
+\medskip
+
+
+{\bf Definition 3.} $~$
+For a fixed constant $~B>0,~$
+a set $~S~$ of functions is defined to be a 
+{\bf $B-$Bounded Arithmetic Set}  iff $~S~$
+includes the arithmetic functions of addition, multiplication and successor
+and all its other functions $~G~$
+satisfy the constraint that
+\begin{equation}
+\label{bcs}
+G( x_1  ,  x_2,   ~ ... \,  , ~ x_n ) ~
+ \leq ~\mbox{Max}( x_1  ,  x_2,   ~ ... \,  , ~ x_n )~ \, 
+\mbox{when}~ \,
+\mbox{Max}( x_1  ,  x_2,   ~ ... \,  , ~ x_n )~<~B~~~
+\end{equation}
+Also, we will say a term $~t~$ is a {\bf
+B-Bounded  Arithmetic Term}
+iff $~t~$ is a term that uses only the symbols of 0, 1 and 2
+as its inputted constants and 
+all its function symbols come from some B-Bounded 
+Arithmetic  Set $~S~$.
+Lemma \ref{lem2} provides the  generalization of
+ Lemma \ref{lem1} for B-bounded arithmetic terms.
+Its proof is omitted because it
+is an easy generalization 
+(see footnote \footnote{The
+intuitive  reason that
+Lemmas \ref{lem1} and \ref{lem2} have
+similar proofs
+is that arithmetically controlled terms and
+ $B-$bounded arithmetic terms
+have precisely identical growth
+rates until a construction process builds an
+intermediate object $~t~$  with 
+$~$MinG$(t)~\geq~B~$.} ) of  
+Lemma \ref{lem1}'s
+proof.
+
+
+\begin{lemma}
+\label{lem2}
+Suppose that $~t~$ is a $B-$bounded arithmetic
+ term with $~$MinG$(t)~<~B~$
+and Val$(t)~ \geq ~4$. Then 
+$~$Val$(t)~ < ~$MinG$(t)$.
+\end{lemma}
+ 
+
+{\bf Definition 4.} $~$ Let $~\Phi~$ denote the $\Pi_1^R$ sentence below
+whose $\Delta_0^R$ subformula  is defined by $~\xbar{\phi}(~a_1,a_2~...~a_n~)~~$:
+\begin{equation}
+\label{bvp}
+\forall    a_1    \forall    a_2    ...   
+  \forall    a_n \,  \,     
+\xbar{\phi}(  a_1,a_2  ...  a_n  ).
+\end{equation}
+For any $~B \,\geq \, 1,~~$  \ep{bvp} is
+called
+a {\bf $B-$Bounded Valid $\Pi_1^R$ sentence} iff 
+\eq{bvpc} is valid under the standard model
+of the natural numbers
+\begin{equation}
+\label{bvpc}
+\forall ~ a_1 \, < \, B~~\forall ~ a_2  \, < \, B~~...
+  ~~\forall ~ a_n  \, < \, B~~~~
+\xbar{\phi}(~a_1,a_2, ~ \ldots , ~~a_n~).
+\end{equation}
+
+\bigskip
+
+{\bf Definition 5.} $~$
+An axiom system $~\alpha~$ 
+will be said to satisfy
+the {\bf Canonical Arithmetic Condition} 
+when all $\alpha$'s axioms are $\Pi_1^R$ sentences
+and they include $Q_0$'s nine axioms (i.e. Equations
+\eq{pw1}--\eq{pw6} ). 
+
+\bigskip
+
+{\bf Definition 6.} $~$
+Let $~\Theta~$ 
+denote a methodology for assigning G\"{o}del numbers
+to Herbrand proofs
+(which are henceforth denoted as $~P ).~$ 
+Let us recall that MinG$(t)$ was defined by Item (1) in this
+section.
+Define $ ~ \Theta ~$  to be a
+{\bf Conventional Encoding Method} if  
+$~\Theta(P)~>~$MinG$(t)~$ 
+whenever the proof $~P~$ contains the
+Herbrand term $~t~$. (Such encodings  are called
+``conventional'' because all usual methods for encoding Herbrand
+proofs satisfy $~\Theta(P)~>~$MinG$(t)~$.)
+
+\medskip
+
+\begin{theorem}
+\label{tttccc}
+			   Suppose $~\alpha~$ is
+			  a canonical arithmetic
+			axiom system  consisting of
+		 $B-$Bounded Valid $\Pi_1^R$ sentences and
+	  $~\Theta~$ again satisfies Definition 6's Conventional
+			    Encoding property.
+		     Then any Herbrand proof $~P~$ of
+	$~\bot~$ from the axiom system $~\alpha~$ will satisfy the
+		    inequality that $~\Theta(P)~>~B~$.
+\end{theorem}
+
+\medskip
+
+
+
+{\bf General Comments about \thx{tttccc} and its Proof: $~$}
+At an intuitive level, \thx{tttccc} can be viewed as a consequence of
+the machineries of Lemma \ref{lem2} 
+and Definitions 3-6. 
+This is because the B-Bounded validity condition in \thx{tttccc}'s
+hypothesis can be used to show that a Herbrand proof $~P~$ of
+$~\bot~$ must contain some term $~t~$ where
+Val$(t) \geq B \,$. In this context, the combination of  
+Lemma  \ref{lem2}  and Definition 6 will
+imply that such a term will force $~P \,$'s G\"{o}del number to
+exceed the lower bound of $B$.
+
+A  more detailed formal
+proof of \thx{tttccc} 
+appears in Appendix A. It explains the
+precise role that Definition 3 and Lemma \ref{lem2}
+play in establishing this theorem.
+Our recommendation is that 
+a reader postpone
+examining Appendix A
+until after
+he
+finishes
+ the remainder of this section. It
+will explain the significance of
+\thx{tttccc} 
+by showing how it
+ enables us to prove the surprising result that the
+Ax-3 axiomatization for \jjsj is an anti-threshold for the Herbrandized
+version of the Second Incompleteness Theorem.
+
+
+\medskip
+ \bigskip
+
+\begin{theorem}
+\label{tm2}
+For any arbitrary axiom system $\alpha$ and
+deduction method $D$, let
+Diagonal$(\alpha,D)$
+and $~\alpha^D~$ 
+denote the following two constructs:
+\bed
+\item[  A.   ]
+{\rm Diagonal$(\alpha,D)~~$ will denote a logical
+sentence that states: $~$ 
+``There is no proof 
+(using deduction method $~D~$)$~$
+of the {\it falsity sentence} $~\bot~$
+from the  union of
+the axiom system $~\alpha~ $
+with $~this~$
+sentence  Diagonal$(\alpha,D) \,$ (looking at itself).''}
+\item[  B.   ]
+{\rm  $~\alpha^D~$  will denote the formal union of the
+axiom system $~\alpha~$ with the sentence
+Diagonal$(\alpha,D)$.}
+\ennd
+Let
+Diag(Ax-3)
+denote the special variant of 
+Diagonal$(\alpha,D)  $ where $ \, \alpha \, = \, $Ax-3
+and $~D~$ designates Herbrand deduction. 
+Both these
+two
+ constructs 
+and also  $~\alpha^D~$ 
+are well defined. Also,
+Diag(Ax-3) has a
+$\Pi_1^R$ encoding.
+\end{theorem}
+
+\bigskip
+\medskip
+
+
+
+{\bf Abbreviated Sketch of \thx{tm2}'s proof.} $~$
+As early as 1938, Kleene observed 
+\cite{Kl38}
+that
+the Fixed Point Theorem implied that
+ a type of cousin of the sentence 
+ Diagonal$(\alpha,D)$ 
+was well defined. More recently, Willard
+\cite{ww1,ww5} showed how
+ Diagonal$(\alpha,D)$ could
+be formally endowed with a
+ $\Pi_1$ encoding under
+the conventional language of arithmetic. It is 
+reasonably
+straightforward to
+generalize \cite{ww1,ww5}'s result to establish that 
+Diag(Ax-3) also has
+a well defined  $\Pi_1^R$
+encoding (thus completing
+\thx{tm2}'s  proof.)
+The remainder of this proof sketch  
+will summarize the ideas from 
+\cite{ww1,ww5} for the benefit of those readers
+who are unfamiliar with this topic.
+Our discussion will employ the following notation:
+\begin{description}
+\item[  i]   $\mbox{Prf}_{\alpha}^D~( \, t \, , \, p \,)$ 
+will denote a formula  designating
+that $~p~$ is  a proof of the theorem $~t~$ from the axiom
+system
+  $~\alpha~$
+ using the deduction method $~D.~$
+\smallskip
+\item[ ii]  
+$\mbox{ExPrf}_{\alpha}^D~( \, h \, , \, t \, , \, p \,)$  
+will be a formula stating that
+$~p~$ is a proof
+(using the deduction method $~D~)~$
+ of
+a theorem $~t~$ 
+from the union 
+of the axiom system
+  $~\alpha~$
+with the added axiom
+sentence whose G\"{o}del number equals
+$\, h \,$. 
+\smallskip
+\item[iii]  
+ $\mbox{Subst} \, ( \, g \, , \, h \, )$ will denote
+G\"{o}del's
+classic substitution formula --- which yields TRUE when $\, g \,$
+is an encoding of a formula
+and $\, h \,$ is an encoding of a sentence
+that replaces all occurrence of free variables in $\, g \,$ with
+a mathematical term 
+formalizing the
+ G\"{o}del
+number for representing  ``g''.
+\smallskip
+\item[ iv]   
+$\mbox{SubstPrf}_{\alpha}^D~( \, g \, , \, t \, , \, p \,)$  
+will denote the natural 
+hybridizations of the constructs from Items (ii) and (iii)
+which yields a Boolean value of TRUE exactly when there
+exists some integer $~h~$ 
+simultaneously 
+satisfying
+{\it both}
+  $\mbox{Subst} \, ( \, g \, , \, h \, )$ and
+$\mbox{ExPrf}_{~\alpha}^D~( \, h \, , \, t \, , \, p \,)$.
+\end{description}
+Each of (i)--(iv)
+can be encoded as $\Delta_0^R$ formulae
+when $\alpha$ is any recursively enumerable axiom system.
+In particular,
+Appendices C and D of  \cite{ww1}
+essentially established
+(see footnote \footnote{The results of Wrathall
+\cite{Wr78} have been noted by
+H\'{a}jek--Pudl\'{a}k 
+\cite{HP91}
+to imply that every LinH function 
+\cite{HP91,Kr95,Wr78} 
+has a $\Delta_0$ encoding. Using a slightly different
+``$~\Delta_0^-~$''notation, the results from Appendices C and D
+ of  \cite{ww1} 
+explained how this result would imply that the 
+each of 
+$\mbox{Prf}_{\alpha}^D~(  t  ,  p )$,
+$~\mbox{ExPrf}_{\alpha}^D~(  h  ,  t  ,  p )$  
+and $\mbox{SubstPrf}_{\alpha}^D~(  g  ,  t  ,  p)$  
+have ``$~\Delta_0^-~$''encodings. Since the
+$~\Delta_0^R~$ class of formalae is broader than
+$~\Delta_0^-~$, $~$
+ these formulae must also have
+$~\Delta_0^R~$ encodings.} ) 
+that the first three of these predicates can receive
+ $\Delta_0^R$ encodings when one applies 
+the theory of LinH functions from
+\cite{HP91,Kr95,Wr78} 
+in a reasonably
+routine manner.
+In such a context, \ep{encode} illustrates
+one possible  $\Delta_0^R$ encoding for 
+$\mbox{SubstPrf}_{\alpha}^D \,(   g   ,   t   ,   p  )$'s 
+graph. (It is
+equivalent to 
+$~$``$~\exists ~h~[~\mbox{Subst}    (    g    ,    h    )~\wedge~
+\mbox{ExPrf}_{\alpha}^D(    h    ,    t    ,    p   )\, ] \, \,$''$,~$
+  but \ep{encode} is 
+ a $\Delta_0^R$ formula --- {\it unlike} the quoted
+expression.)
+\begin{equation}
+\label{encode}
+\mbox{Prf}_{~\alpha}^D~( \, t \, , \, p \,)~~~\vee~~~\exists ~h\leq p
+~~[~ \mbox{Subst} \, ( \, g \, , \, h \, )~\wedge~
+\mbox{ExPrf}_{~\alpha}^D~( \, h \, , \, t \, , \, p \,)~]  
+\end{equation}
+
+
+
+
+Utilizing \eq{encode}'s
+  $\Delta_0^R$ encoding for 
+$\mbox{SubstPrf}_{\alpha}^D(   g   ,   t   ,   p  )$, it is
+easy to 
+formulate a $\Pi_1^R$ encoding
+for  the
+axiom sentence Diagonal$(\alpha,D)$.
+Thus,  let
+$~\Gamma(g)~$ 
+denote  \ep{encode2}'s formula, and let  $~N~$ denote $~\Gamma(g)$'s
+G\"{o}del number. $~\,$Then  $~\Gamma(~ N~)~$
+is a $\Pi_1^R$  encoding for  Diagonal$(\alpha,D)$.
+\begin{equation}
+\label{encode2}
+ \forall   \, p \,   \,  \neg  \,  \, 
+\mbox{SubstPrf}_{\alpha}^D  (  ~  g ~   , ~ \bot ~       , ~   p ~  )
+\end{equation}
+
+$\Box$
+
+\bigskip
+
+
+
+
+
+{\bf Clarifying Comment: }
+One
+should be somewhat cautious in interpreting the meaning of \thx{tm2}.
+It does not indicate that  Diag(Ax-3)  is a logically valid
+statement under the standard model of the
+natural numbers. Rather, it merely indicates
+ Diag(Ax-3)  is a well defined $\Pi_1^R$ sentence.
+In order to establish prove that 
+ Diag(Ax-3)  is also valid,
+we will need the 
+added
+force of \thx{tm3} below.
+
+
+\begin{theorem}
+\label{tm3}
+Let Ax-3* 
+denote the union of
+the axiom system
+ Ax-3 with the sentence Diag(Ax-3).
+Then Ax-3* is consistent. 
+(Thus,
+Ax-3 is an ``anti-threshold'' for the
+Herbrandized version of the Second Incompleteness Theorem
+under Definition 1's notation convention.) 
+\end{theorem}
+
+{\bf Proof of the Consistency Property of Ax-3* : $~$}
+Suppose for the sake 
+of establishing a
+proof-by-contradiction that Ax-3* was inconsistent.
+Then one could identify a
+proof $~P~$ of  $~\bot~$ 
+whose G\"{o}del number $~\Theta(P)~$ is
+the smallest G\"{o}del number of a Herbrand proof of  
+ $~\bot~$
+from Ax-3*. We will now construct 
+from $~P~$ an alternate Herbrand proof $~R~$ of
+ $~\bot~$ where $~\Theta(R)~<~\Theta(P).~$
+The formal
+construction of such a $~R~$ will suffice for our proof by
+contradiction to reach its desired end because
+such a  $~R~$ cannot possibly exist
+(on account of $P$'s minimality property).
+
+Our strategy is to
+ use \thx{tttccc}
+to construct $~R~$ from $~P~$.
+\thx{tttccc} is relevant to Ax-3* 
+because all the  formal axiom
+sentences of Ax-3* are
+assuredly  $\Pi_1^R$
+sentences (see 
+footnote \footnote{ \thx{tm2}
+implies that the Diag(Ax-3) axiom of 
+Ax-3* has a $\Pi_1^R$ structure, and 
+Section \ref{s2}'s definition of Ax-3 implies that
+all the other axiom-sentences belonging to Ax-3 are certainly
+also
+$\Pi_1^R$. } for the justification of this claim).  
+In such a context,
+we may apply
+\thx{tttccc} to conclude
+that for some $~B~<~\Theta(P),~~$ at least one of
+the  axiom sentences of Ax-3* 
+must
+fail to be a B-Bounded valid
+$\Pi_1^R$ sentence. Moreover, it is obvious
+that all the axioms of
+Ax-3 possess an unbounded level of validity
+(i.e they are $B-$Bounded valid {\it for all possible B}).
+ Hence, these two
+observations imply 
+Diag(Ax-3) fails to be $B-$bounded valid
+(simply because
+some axiom from Ax-3* must fail to be $B-$bounded valid, and
+ Diag(Ax-3) is the only 
+available
+axiom belonging to Ax-3* 
+that is not  also a member of  Ax-3.)
+
+The latter observation,
+combined with Diag(Ax-3)'s definition implies 
+(see footnote \footnote{ The 
+strictly formalistic definition of
+Diag(Ax-3) as the entity ``$~\Gamma(N)~$''
+(using a self-reference principle)
+can be found in the last sentence of 
+\thx{tm2}'s
+proof. The syntax of its \ep{encode2}
+implies that if 
+Diag(Ax-3) fails to be 
+$B-$Bounded valid then another proof 
+ $R$ must
+assuredly exist for the sentence  $~\bot$
+with  $ \, \Theta(R)< B \,$.} ) that
+ some $~R~$ with
+ $ \, \Theta(R)< B \,$ 
+must 
+be another proof of $~\bot$.
+Hence
+our last two inequalities
+certainly
+ imply that
+  $ \, \Theta(R)< \, B \, < \, \Theta(P) \,$.
+This finishes
+our proof-by-contradiction  because
+ $~P$'s previously 
+presumed minimality has been contradicted by $R$.
+ $~~\Box$
+
+
+\bigskip
+
+{\bf Remark 2.} $~$
+Our discussion in this section had assumed that the terms $~T~$ in
+a $\Delta_0^R$'s formula's bounded quantifiers included
+{\it only the Maximum function symbol.}
+The results of \thx{tm3} would actually also hold if these
+quantifiers were also permitted to include the Addition function
+symbol. (The only reason our discussion had omitted the
+possibility that both 
+the
+addition and maximum
+function symbols
+ appear in
+the $\Delta_0^R$ formula $\phi$'s bounded quantifiers in
+\ep{wp} was for the sake of simplifying the presentation.)
+
+
+\bigskip
+
+{\bf Remark 3.} $~$
+The attached Appendix B discusses 
+a yet further reason  why 
+\thx{tm3} is of interest.
+The 
+anonymous
+referee had suggested we add
+this appendix to the current paper.
+Its methodologies  are related to
+Ko{\l}odziejczyk's observation \cite {Ko6b,Ko7}
+that semantic tableaux and Herbrand deduction can sometimes
+have an exponential difference in their 
+proof
+lengths.
+The  purpose of Appendix B is to
+sketch how one 
+can generalize \cite{ww7}'s
+results for Ax-1 and Ax-2 to establish
+that Ax-3 is also a threshold for the semantic tableaux
+version of the Second Incompleteness Theorem.
+In a context where \thx{tm3}
+had established the polar opposite
+result for Ax-3
+under Herbrand deduction, 
+this contrast is, of course, quite interesting.
+
+
+\section{Discussion of Significance of Results}
+
+A comparison between our research and the prior research
+of  Kreisel-Takeuti and Pudl\'{a}k \cite{KT74,Pu85,Ta53} 
+was  
+postponed until the closing part  of this current article
+because the results from
+ Sections 3 and 4
+were needed to precede this discussion.
+
+
+
+In this section,
+$~S(x)~$ will denote 
+the ``successor'' operation that maps the
+integer $x$ onto $x+1$. 
+A formula
+ $ \varphi(x) $ is called \cite{HP91} a
+{\bf Definable Cut for}
+an axiom system $ \alpha $   iff
+$ \alpha $ can prove:
+\begin{equation}
+\label{initdefx}
+\varphi(0)~\mbox{ AND }~
+\forall~x~~ \{~\varphi(x)\Rightarrow\varphi[S(x)]~ \}
+ ~\mbox{ AND }~
+\forall~x ~\forall~y<x
+~~ \{ ~\varphi(x)\Rightarrow\varphi(y) ~ \}~~
+\end{equation}
+A very extensive literature
+\cite{Ad2,AZ1,BI95,GD82,Ha71,HP91,Ka91,Kr87,Kr95,Ne86,PD83,PW81,Pu83,Pu84,Pu85,Pu96,Sa1,Sm85,Sv83,Vi92,Vi93,Vi5,VH73,WP87,ww2,ww4}
+has studied Definable cuts.
+We have
+published two 6-page summaries of this
+literature
+in the  review chapters of our articles  \cite{ww5,ww6}.
+(Also, Pudlak's full-length survey article
+\cite{Pu96}
+ offers an excellent
+review of the work done in this subject prior to 
+approximately 1990.)
+
+
+All axioms systems,
+strictly weaker than Peano Arithmetic, contain some
+definable cut that is not provably equivalent to the full set
+of integers. 
+In the proof-theory literature,
+the definition of a 
+``Definable Cut'' 
+(see \cite{HP91})
+is
+formally
+ unrelated to Gentzen's notion
+of a sequent calculus
+deductive ``cut rule'',
+despite their very similar sounding names.
+
+Let
+$~\lceil ~ \Psi ~ \rceil ~$
+denote $~\Psi\,$'s
+G\"{o}del number, and
+Prf$_\alpha^D \, \,(t,p)$
+denote
+that $~p~$ is a proof
+of the theorem $~t~$ from the axiom system $~\alpha~$ using the
+deduction method $~D.~$
+Also, let
+ $ \, \varphi(x) \, $
+denote a definable cut.
+Let us say that an axiom system
+$~\alpha~$ can recognize its 
+{\bf Cut-Localized  D-consistency 
+under}
+ $ \, \varphi(x) \, $
+iff $ \, \alpha \, $ can 
+{\it formally prove:}
+\begin{equation}
+\label{dcon}
+\forall~p~~~~~~ \{ ~~~\varphi(p)~~\Rightarrow~~
+\neg~\mbox{Prf}_\alpha^D \, ~(~\lceil 0=1 \rceil~,~p~)~~~ \}
+\end{equation}
+Pudl\'{a}k 
+\cite{Pu85}
+proved that no consistent extension of the Tarski-Mostowski-Robinson
+\cite{TMR53}
+system $~Q~$ 
+can prove the validity of \ep{dcon} for any definable cut
+when $~D~$ denotes either Hilbert deduction or a Gentzen sequent calculus
+system with a deductive cut rule.
+
+ 
+At the same time,
+the literature about Definable Cuts has also defined some
+circumstances where \eq{dcon} is provable 
+from $\alpha$
+when for example
+$~D~$ denotes either  semantic tableaux or Herbrand
+deduction.
+The strongest result of this type was discovered by Pudl\'{a}k \cite{Pu85}.
+For both Herbrand and semantic tableaux deduction,
+\cite{Pu85} showed that
+it is possible to construct a definable cut  
+$~\varphi(p)~$
+where $~\alpha~$ can prove the validity of
+\ep{dcon}'s
+ Cut-Localized  D-consistency property
+ when
+ $~\alpha~$
+is an axiom system with finite cardinality that satisfies a
+relatively minor
+additional
+ constraint called ``sequentiality''.
+
+
+A second  class of
+boundary-like
+ exceptions to
+the Second Incompleteness can be constructed via the
+``Cut Free Analysis'' (CFA) systems of Kreisel and 
+Takeuti \cite{KT74,Ta53}.
+It has consisted of a Second Order Logic generalization
+of the cut-free version of Gentzen's Sequent Calculus that
+is able to prove some types of versions of statements
+about its own consistency
+
+The work of 
+ Kreisel and 
+Takeuti 
+\cite{KT74,Ta53}
+ had chronologically
+preceded most of the literature about Definable Cuts.
+There is a fascinating partial analogy between
+the perspectives of these two approaches.
+This is because the Kreisel-Takeuti papers \cite{KT74,Ta53}
+ can be viewed as using an analog of
+\ep{dcon} in an implicit 
+manner. Thus \cite{KT74} employs
+a set of
+ objects, which we shall call $~I,~$
+that includes all the standard integers 
+plus plausibly {\it non-standard integers}
+that can represent a proof of a contradiction.
+It then uses $~I~$ to
+construct a subset of it, called
+ ``$~N~$'', which can be viewed as a representation
+of  the natural numbers.
+(More precisely, the pages 16-17 of  \cite{KT74} construct $~N~$ from
+$~I~$ in this manner by employing Dedekind's
+and Zermelo's inductive 
+second-order logic
+definitions of the natural numbers.
+They thus formally treat the
+ Dedekind
+and Zermelo inductive 
+definitions 
+of $~N~$ 
+as an analog of $~\varphi(p)~$
+in \ep{dcon}'s Cut-Localized D-Consistency statement.)
+
+
+
+A question which then naturally arises is whether or not one
+can also develop some types of axiom systems which can prove
+their own consistency without  relying upon
+ \ep{dcon}'s
+``Cut-Localized  D-consistency'' machinery
+(or \cite{KT74}'s analog of it for a second-order generalization
+of a Gentzen Cut-Free logic.)
+Such an approach is desirable because one would 
+ideally
+like to 
+characterize the properties of the natural numbers directly
+--- instead of 
+being required to 
+view them as a subset of 
+a potentially
+larger set of objects,
+called $~I~$.
+
+
+It is indeed possible to make some progress in this area by
+using a third approach, relying upon
+the Diagonal$(\alpha,D)$ sentence,
+which was formally defined by the
+\thx{tm2}.
+(Analogs of this Diagonal$(\alpha,D)$ sentence
+have also been used in several of our previous
+papers 
+\cite{ww93,ww1,ww5,ww6}
+in  some contexts
+quite different from
+the paradigm outlined by
+ Theorem \ref{tm3}.)
+
+It is  difficult to make
+more
+  detailed comparisons
+between the partial exceptions to G\"{o}del's Second Theorem
+that use our 
+ Diagonal$(\alpha,D)$ sentence
+ with the methods of Kreisel-Takeuti and Pudl\'{a}k
+\cite{KT74,Pu85,Ta53}. Each technique has its own
+distinct separate advantage. There also 
+appears to be no
+ natural way to hybridize
+these techniques. (The point is that each
+of these different methodologies is looking at
+a different type of problem setting, where a different
+form of solution method is available.)
+
+In particular, our research
+ has treated 
+the sentence 
+ Diagonal$(\alpha,D)$ as an axiom of $~\alpha~$ while
+the 
+ Kreisel-Takeuti and Pudl\'{a}k papers 
+\cite{KT74,Pu85,Ta53} treat their analogs of
+\ep{dcon} as
+derived
+ theorems.
+In this context,
+it is  very natural
+ for a reader to inquire
+whether is is preferable to treat an ``I am consistent'' statement
+as a theorem rather than
+as an axiom of the system $~\alpha~$ that
+generates it?
+
+There is surprisingly no easy answer to this question.
+For instance,  if one's goal is to attempt
+to return to the original objectives of Hilbert's consistency
+program, then the 
+ Kreisel-Takeuti and Pudl\'{a}k 
+approaches
+are quite significant  because they
+indicate some respects in which an axiom system can 
+indeed 
+formally
+prove at least
+a reduced versions of its consistency statement.
+However from an alternate perspective, there is a
+substantial difficulty with
+approaches that treat the statement ``I am consistent''
+and its analogs
+ as a theorem
+rather than as an axiom.
+The difficulty arises when the
+relevant systems do not have an operating modus ponens or
+Gentzen-like cut rule
+(as is the case with
+the axiom systems of \cite{KT74,Pu85}).
+Such
+systems cannot draw inferences when they
+prove a theorem essentially
+declaring  that
+ ``I am consistent''. However,  a similar such
+difficulty does not affect the self-justifying axiom systems
+in our earlier papers 
+\cite{ww93,ww1,ww5,ww5b,ww6}
+or
+in Theorem \ref{tm3} 
+because they treat the statement
+ Diagonal$(\alpha,D)$
+ {\it as a formal axiom rather than
+as a theorem.}
+(The point here is that axiom sentences,
+{\it quite unlike theorem sentences,} can be 
+permissibly
+used as intermediate
+steps to generate other deductions under the inference rules
+of semantic tableaux, the cut-free sequent calculus and/or Herbrand
+deduction
+$~$---$~$ thus causing their presence to have stronger
+implications.)
+
+
+Thus, there are 
+quite
+different insights
+that arise
+ from systems that
+treat 
+variants of
+the ``I am consistent'' statement
+as an axiom
+ instead of as
+a theorem
+(because of the different connotations these two approaches
+carry).
+ Neither approach is uniformly
+preferable
+over the other.
+ Another difference between our research and
+the investigations of  
+ Kreisel-Takeuti
+and Pudl\'{a}k 
+\cite{KT74,Pu85,Ta53} is that the sentence
+ Diagonal$(\alpha,D)$ declares the consistency of
+$~\alpha~$ in a global sense whereas 
+\ep{dcon} refers only to the localized subset of integers
+that lie within the domain of $~\varphi.~$ 
+(In the case of 
+ Kreisel-Takeuti second-order logic system, 
+\ep{dcon} is used implicitly on the pages 16 and 17
+of \cite{KT74}
+to draw the distinction between 
+what we have called
+$I$
+ and $N$ earlier in this section.)
+
+
+When one compares our results in
+  Theorem  \ref{tm3} or in our earlier papers
+\cite{ww93,ww1,ww5,ww5b,ww6}
+with the research of 
+ Kreisel-Takeuti
+and Pudl\'{a}k 
+\cite{KT74,Pu85,Ta53},
+it is best to remember that G\"{o}del's Second Incompleteness
+Theorem precludes an exception to it from becoming too
+powerful. Thus, it is possible to 
+develop
+different types of
+partial evasions
+ to the Second Incompleteness Theorem around its
+periphery that shed different types of useful 
+new perspectives. However, it is awkward to attempt to
+hybridize these different types of partial evasions into
+a stronger uniform evasion.
+This is because if such a hybrid combined the strengths of
+the different approaches X, Y and Z, then it would become
+so strong that its properties could
+potentially violate 
+the statement of G\"{o}del's historic result.
+Thus, there are trade-offs where different types of useful 
+insights, stemming from different approaches
+(that are difficult to hybridize into one unified methodology),
+examine somewhat different problems and produce
+different new perspectives.
+
+
+Part of the reason that  
+Ax-3's evasion of the Herbrandized version of the
+Second Incompleteness Theorem is
+ interesting is that it is
+known that further improvements upon this result are very
+difficult to obtain.
+Thus,
+ Adamowicz-Zbierski \cite{Ad2,AZ1} showed that \jjj $+ \Omega_1$ was unable
+to verify its Herbrand  consistency,
+Willard \cite{ww0,ww2,ww7} 
+modified this result to
+show the
+Second Incompleteness Theorem
+applied also to
+Ax-1 and Ax-2's  versions
+of
+\jjj 's axiomatization
+under semantic tableaux deduction,
+and
+ Salehi \cite{Sa1}
+extended some of
+the earlier results from \cite{Ad2,AZ1,ww0,ww2} to develop 
+some further             incompleteness results for \jjsj under Herbrand
+deduction.
+Thus in a context where several earlier papers have explored
+types of results where the Second Incompleteness Theorem
+holds for 
+ encodings for
+\jjsj and its cousins ,  it is surprising that the
+current article has documented 
+a paradigm where at least the Ax-3 encoding for \jjsj evades 
+the Herbrandized version of the Second Incompleteness
+Theorem. 
+
+
+\section{Broader Perspectives} 
+
+
+Some added notation is useful so that we can examine 
+\thx{tm3}'s
+results from a broader perspective.
+Let us say a  formula $~\Phi~$ is
+ $\Sigma_1^R$ when it can be written in the form
+$~\neg~\Psi~$ where $~\Psi~$ is 
+ $\Pi_1^R$.
+Some other fairly conventional notation  is
+that a sentence will be called
+$~\Pi_{k+1}^R~$
+when it can be written in the form
+$\forall \, v_1~ \forall \, v_2~ ... ~\forall \, v_n~
+\phi(v_1,v_2, ~ \ldots , ~v_n)$ where $ \phi(v_1,v_2, ~ \ldots , ~v_n) $
+is a  $ \Sigma_{k}^R $ formula. 
+Likewise, a sentence will be called
+$\Sigma_{k+1}^R~$
+when  it can be written as
+$\exists \, v_1~ \exists \, v_2~ ... ~\exists \, v_n~
+\phi(v_1,v_2, ~ \ldots , ~v_n)$ where $ \phi(v_1,v_2, ~ \ldots , ~v_n) $
+is a  $ \Pi_k^R $ formula. 
+Also, a sentence will be said to be
+Level$-k$ if it is either 
+ $ \Pi_k^R $ or  $ \Sigma_{k}^R $.
+
+
+\bigskip
+
+{\bf Definition 7. }
+Let
+ $~H~$ denote a sequence of ordered pairs
+$~(t_1,p_1),~(t_2,p_2),~...~(t_n,p_n),~$
+where $~p_i~$ is a Herbrand 
+ proof of the theorem $~t_i.~$
+For an arbitrary integer $~k \geq 1~$,
+this list $~H~$ will be defined
+to be a 
+\hxl{k} 
+proof
+of a theorem $~T~$ 
+from the axiom system $~\alpha~$
+  iff $~T=t_n~$
+and also:
+\begin{enumerate}
+\parskip 3 pt
+\item
+Each axiom in  $ p_i$'s
+proof is
+either one of $ t_1,t_2, \, ... \, , t_{i-1} $ 
+or comes from $\alpha$.
+\item
+Each of the ``intermediately derived theorems'' 
+$~t_1,t_2, \, ... \, , t_{n-1}~$
+must lie within the 
+Level-k class of sentences.
+\end{enumerate}
+Intuitively, 
+\hxl{k} deduction can be viewed as an extension of
+Herbrand deduction that contains a type of Gentzen-like
+deductive cut rule for Level-k sentences.
+
+\bigskip
+\medskip
+
+
+The Definition 7's machinery  is  useful for
+helping to describe both
+the maximal generalizations as well as inherent limitations of
+Section \ref{main}'s formalism.
+Thus using 
+ Definition 7's notation, one can establish the following two
+tightly  complementary negative and
+positive results:
+\bed
+\parskip 3 pt
+\item[   I.  ]
+There exists a 
+logically valid
+$\Pi_1^R$ sentence denoted as $~\Psi~$ such that no
+consistent axiom system can contain $~\Psi~$ as an axiom and prove
+a theorem affirming its own consistency under  
+\hxl{2} deduction.
+\item[  II.  ]
+In contrast for each consistent axiom system $~A~$, there exists
+a consistent axiom system $~I(A)~$ that can prove all $~A\,$'s
+$\Pi_1^R$ theorems and recognize its own consistency under
+\hxl{1} deduction.
+\ennd
+In other words, Items I and II indicate that there is a fundamental difference
+between  \hxl{1} 
+ and \hxl{2} deduction. Thus,
+ \hxl{1} deduction allows for a type of robust evasion of the Second
+Incompleteness Theorem under a formalism that contains a type of
+limited Gentzen-style cut rule.
+However, Result-II  precludes this evasion
+from becoming too strong.
+
+
+We will not prove results I and II here because each has a rather
+long proof. Also,
+partial
+analogs of these two prior results have 
+appeared in our prior work. Thus,
+\cite{ww4} and \cite{ww5} have used the term
+\txl{k} deduction to refer to a construct similar
+to \hxl{k} deduction except for the following two changes:
+\bee
+\item
+Under \txl{k} deduction, the proofs
+$~p_1~,~p_2~,~ ~ \ldots , ~~p_n~$ associated with
+the sequence
+$~(t_1,p_1),~(t_2,p_2),~ ~ \ldots , ~~(t_n,p_n),~$
+are semantic tableaux proofs instead of Herbrand proofs.
+\item
+Also under \txl{k} deduction, the 
+intermediate results
+$~t_1~,~t_2~,~ ~ \ldots , ~~t_{n-1}~$ 
+will  technically represent what \cite{ww4,ww5}
+had called 
+ $ \Pi_k^* $ and  $ \Sigma_{k}^* $ sentences
+(instead of being
+ $ \Pi_k^R $ and  $ \Sigma_{k}^R $ sentences).
+These 
+ $ \Pi_k^* $ and  $ \Sigma_{k}^* $ sentences
+can be intuitively viewed as being
+roughly analogous to 
+ $ \Pi_k^R $ and  $ \Sigma_{k}^R $ sentences 
+---
+except that they contain
+no multiplication function symbol.
+(They instead use a relation primitive
+$M(x,y,z)$ to treat multiplication as a 3-way
+relation.)
+\ene
+In essence Item-I's result can be viewed as being analogous
+to \cite{ww4}'s main theorem.
+Similarly, Item-II's result
+can be viewed as following from a 
+rather natural
+hybridization of  
+\cite{ww5}'s main result  with the machinery 
+that
+we had developed in Section \ref{main}.
+
+
+For further details about this material,
+the reader
+should
+examine \cite{ww4,ww5}'s generic formalisms.
+The reason we had slightly 
+changed the topic from
+semantic tableaux to Herbrand deduction in the current paper
+is because Herbrand deduction and its \hxl{1} generalization
+possess one interesting  quality that semantic tableaux
+and \txl{1} deduction 
+simply
+do not possess. This is that the
+Ax-3 encoding of the axiom system \jjsj is an anti-threshold to
+the Herbrandized (and also \hxl{1} ) versions  
+of the Second Incompleteness Theorem.
+However, the similar anti-threshold
+effect does not  apply
+also to semantic tableaux deduction
+(as is explained in Appendix B).
+Thus, the 
+\hxl{1} deduction method will support certain types of
+evasions of the Second Incompleteness Theorem that have
+no analogs under
+\txl{1} deduction. 
+(The interested reader should also 
+examine
+Ko{\l}odziejczyk
+work \cite {Ko6b,Ko7}, which observed how semantic tableaux and
+Herbrand-styled proofs can sometimes have an exponential
+difference in their lengths.)
+
+In essence
+the over-all goal of our research, 
+in the current paper and in the previous papers
+ \cite{ww93} -- \cite{ww7},
+has
+been to 
+attempt to
+sharpen the
+academic community's understanding of the meaning of 
+ the Second Incompleteness Theorem, by 
+exploring both
+ its maximal generalizations
+and permitted allowed boundary-case exceptions.
+We plan to prepare 
+a new article about this subject in the near future,
+describing the
+underlying philosophical motivation for
+much of this research.
+One must
+clearly  approach this 
+subject matter carefully
+because
+the many
+generalizations of the Second Incompleteness Theorem are
+seemingly more significant
+ than its occasional boundary-case exceptions.
+The reason that the partial exceptions to the
+Second Incompleteness Theorem should not be
+ignored
+is because
+G\"{o}del's Incompleteness Theorem is often
+regarded as 
+the paramount discovery of 20th century mathematics.
+It thus beckons  the academic
+community to  explore both its
+maximum
+generalizations and
+possible  boundary case exceptions,
+so that an
+understanding of the full meaning
+of its historic result
+ can be sharpened and made more
+precise. Within such a
+limited-but-precise framework, the anomalous behavior of Ax-3,
+documented in this article, should be
+of
+scholarly interest.
+
+\bigskip
+
+{\bf Acknowledgments:} I warmly thank
+Leszek Ko{\l}odziejczyk for the interesting questions that he had
+emailed me on November 16, 2005. Those 
+questions played a major role in
+encouraging me to re-examine my 
+earlier work in 
+ \cite{ww2} for one more time.
+I also thank Zofia  Adamowicz and
+Konrad  Zdanowksi, who helped at least partially stimulate
+the emailed questions from
+L. Ko{\l}odziejczyk through their conversations with
+him in Warsaw. 
+Those questions 
+from Leszek Ko{\l}odziejczyk
+had greatly
+stimulated my  research
+in this article. I also thank the anonymous referee for his many
+useful comments, including the  suggestion that I
+add the Appendix B to this article.
+
+
+\section*{Appendix A:  The Proof For  \thx{tttccc}}
+
+This appendix will explain in further detail how
+Definitions 3-6 and Lemma \ref{lem2}
+may be used to prove
+\thx{tttccc}.
+Our discussion will begin
+with one further definition and
+two  further lemmas.
+
+
+\bigskip
+
+{\bf Definition 8. }
+Consider the possibility that $~\Psi~$ is the
+prenex normal sentence, whose open part is formalized
+by $~\xbar{\psi} \,(x_1,y_1...x_n,y_n),~$
+shown in \ep{2prot1} and whose Skolemized normalized
+form is illustrated by \ep{2prot2}.
+\begin{equation}
+\label{2prot1}
+\forall~ x_1 ~~ \exists~y_1  ~~ \forall~ x_2 ~~ \exists~y_2 
+ ~~ .... ~~ ~~ \forall~ x_n ~~ \exists~y_n ~~~
+\xbar{\psi} \,
+(x_1,y_1...x_n,y_n)
+\end{equation}
+\begin{equation}
+\label{2prot2}
+\forall x_1  \,  \,   \forall  x_2
+  \,  \,  ....  \,  \,   \,  \,  \forall  x_n  \,  \,  \,  \, 
+\xbar{\psi} \,[ \,  x_1  \, , \, f_1^{\psi}(x_1)  \, , \, x_2  \,
+ , \, f_2^{\psi}(x_1,x_2),
+  \, ...  \,  x_n  \, , \, 
+f_n^{\psi}(x_1,x_2  ...  ~x_n) \, ]~~
+\end{equation}
+For any $~B \, \geq \, 1,~$ 
+Equations \eq{2prot1} and 
+ \eq{2prot2} 
+will be called
+a {\bf $B-$Bounded Good Skolemization} iff
+one can define 
+ \eq{2prot2}'s Skolem functions
+$ \, f_1^\Psi \, , \, f_2^\Psi, \,  ...  \, f_n^\Psi \, $
+so they satisfy both 
+Definition 3's  $B-$Bounded requirement
+(see footnote \footnote{\label{83} The
+function of Definition 3's  $B-$Bounded requirement
+is that it assures that each of 
+ \ep{2prot2}'s Skolem functions
+of 
+$f_1^\Psi \, , \, f_2^\Psi, \, ... ~$
+satisfy the constraint
+that
+$f_i^\Psi( x_1  ,  x_2,   ~ ... \,  , ~ x_i ) ~
+ \leq ~\mbox{Max}( x_1  ,  x_2,   ~ ... \,  , ~ x_i )~ $
+whenever 
+Max$( x_1  ,  x_2,   ~ ... \,  , ~ x_n )~<~B~.$ } ) 
+ and
+\ep{2prot3}  under the standard model of the natural
+numbers.
+\begin{center}
+$\forall ~x_1 \, < \, B~~  \forall ~ x_2  \, < \, B
+ ~~ .... ~~ ~~ \forall ~ x_n  \, < \, B$
+\end{center}
+\begin{equation}
+\label{2prot3}
+\xbar{\psi} \,[~ x_1  \, , \, f_1^{\psi}(x_1)  \, , \, x_2  \,
+ , \, f_2^{\psi}(x_1,x_2)
+ ~ ~ \ldots , ~ ~ x_n  \, , \, 
+f_n^{\psi}(x_1,x_2 ~ \ldots , ~x_n)~]
+\end{equation}
+Likewise, we will say an axiom system $~\alpha~$ has a 
+ {\bf $B-$Bounded Good Skolemization} iff all its axioms
+are so Skolemized.
+
+
+
+\bigskip
+
+
+
+\begin{lemma}
+\label{lem3}
+Using the notation conventions from Definitions 4 and 8,
+every  $B-$Bounded Valid $\Pi_1^R$ sentence can be 
+rewritten into a logically equivalent form that
+has a  $B-$Bounded Good Skolemization.
+\end{lemma}
+
+
+\medskip
+
+{\bf  Proof. $~$} Follows immediately from the definitions of
+Bounded Validity and Bounded Good Skolemizations 
+(i.e. see Definitions 4 and 8).
+
+\bigskip
+
+{\bf Remark 4:} $~$
+From Lemma \ref{lem3}, one can gain a further
+intuitive
+appreciation of the role that Definition 3 will play
+in our proof
+of
+\thx{tttccc}.
+This lemma indicated that
+every  $B-$Bounded Valid $\Pi_1^R$ sentence 
+had a  $B-$Bounded Good Skolemization, and Definition 8
+indicated that 
+such skolemizations
+satisfied Definition 3's requirement that no
+such 
+invoked
+Skolem function would grow faster than
+the multiplication primitive. 
+Such slow-growth Skolem functions characterize 
+Ax-3 (but not also the
+Ax-1 and Ax-2 systems).
+The  intuitive reason
+for this distinction is that the paradigm
+in the footnote \ref{83} of Definition 8
+applies only to Ax-3.
+
+
+\begin{lemma}
+\label{lllttt}
+Using the notation 
+conventions
+from Definitions 5, 6 and 8, 
+suppose that $~\alpha~$ is
+a canonical arithmetic
+system  consisting
+of prenex sentences with
+ $B-$Bounded Good Skolemizations
+and that $~\Theta~$ satisfies the Conventional
+Encoding property.
+ Then any 
+ Herbrand proof $~P~$ of
+$~\bot~$ from the axiom system $~\alpha~$ will satisfy  $~\Theta(P)~>~B~$.
+\end{lemma}
+
+\bigskip
+
+{\bf Proof-by-contradiction:}
+Consider the contrary possibility that the 
+inequality  $ \Theta(P) > B $ 
+failed and that $ P $ is a 
+Herbrand-proof of $ \bot $ from the system $ \alpha $ where 
+ $ \Theta(P) \leq B $.
+We shall denote this inequality as
+ $~***~~~$.
+
+\medskip
+ 
+Definition 6 had  indicated
+every term $~T~$ in the proof $~P~$ satisfies
+the inequality of
+$~\Theta(P)~>~$MinG$(T.)~$ Also,
+Lemma \ref{lem2} implied Val$(T)<$MinG$(T)$.
+These inequalities
+and *** imply that every term $~T~$
+in the proof $~P~$ satisfies
+\begin{equation}
+\label{nice}
+\mbox{Val}(T)~<~B
+\end{equation}
+\ep{nice}
+ implies all the terms
+ $  T_1,T_2,T_3...  $ in the
+ Herbrandized instances in the proof $  P  $
+satisfy $  $Val$(T_i)    <    B.    $
+The normalized form of an instance
+of a  Skolemized axiom
+is illustrated by  \ep{xprot3}. The combination of
+our
+ $  $Val$(T_i)    <    B  $ inequalities together with
+\eq{2prot3}'s $B-$Bounded constraint on $\alpha$'s axioms 
+implies that {\it each such instance of
+\eq{xprot3} appearing in the proof $  P  $ must
+{\bf be automatically valid} 
+under the standard model
+of the natural numbers.}
+\begin{equation}
+\label{xprot3}
+\xbar{\psi} \,[ \,  \, T_1  \, , \, f_1^{\psi}(T_1)  \, , \, T_2  \,
+ , \, f_2^{\psi}(T_1,T_2)
+ ~, ~ \ldots , ~ ~ T_n  \, , \, 
+f_n^{\psi}(T_1,T_2 ~ \ldots , ~T_n)~]
+\end{equation}
+
+
+
+
+The latter observation completes our proof-by-contradiction
+because it  contradicts the  statement $***$ 
+that had started our  proof.
+More precisely $***$ had asserted that   
+ $~P~$ was a 
+Herbrand-proof of $~\bot~$ 
+from the axiom system $~\alpha.~$
+However, the Footnote
+\footnote{ The point here is simply that a conjunction of
+Skolemized instances
+ can produce a proof of  $~\bot~$ 
+only when there exists no model  $~M~$ where all these instances are
+simultaneously valid. Hence when the preceding paragraph
+shows that all these Skolemized instances are simultaneously valid
+under the Standard Model of the Natural Numbers, it implies that
+certainly
+{\it no such  proof of  $~\bot~$ can feasibly  exist.}}
+shows that  such is impossible when the last sentence of the preceding
+paragraph indicated that 
+each instance of \eq{xprot3}'s Skolemized axiom
+is 
+valid under the standard model of the natural numbers.
+ $~~\Box$
+
+
+\medskip
+
+{\bf Finishing the Proof for \thx{tttccc}}.
+ $~~$
+It is easy to combine the machineries of
+Lemmas  \ref{lem3} and 
+ \ref{lllttt} to complete  the proof of \thx{tttccc}.
+This is because Lemma \ref{lem3} had indicated
+that every  $B-$Bounded Valid $\Pi_1^R$ sentence can be 
+rewritten into a logically equivalent form that
+has a  $B-$Bounded Good Skolemization.
+Thus,  \thx{tttccc}
+follows by simply taking such rewritten forms of
+$~\alpha$'s axioms and then
+applying Lemma \ref{lllttt}'s machinery. $~~\Box$
+
+
+\section*{Appendix B: An Analysis of Ax-3's Semantic Tableaux Properties}
+
+ 
+This appendix will illustrate
+how the methods of \cite{ww7}
+may be extended to prove that
+Ax-3 satisfies the semantic
+tableaux version of the Second Incompleteness
+Theorem. 
+Our discussion  will
+be closely related to
+Ko{\l}odziejczyk's observations \cite {Ko6b,Ko7} that
+(in the context of Buss's Bounded Arithmetic \cite{Bu86})
+semantic tableaux and Herbrand deduction can sometimes
+have an exponential 
+or greater 
+difference in their 
+proof
+lengths.
+In a context where
+\thx{tm3} had showed that Ax-3 was an
+anti-threshold  under Herbrand deduction,
+the results in this appendix are noteworthy because
+they imply  Ax-3 has  
+ polar opposite qualities under semantic tableaux
+and Herbrand deduction.
+
+
+The discussion in this
+abbreviated
+ appendix will 
+assume that the reader is familiar with
+\cite{ww7}'s proof that Ax-1 and Ax-2 satisfy the semantic
+tableaux version of the Second Incompleteness Theorem.
+We will also often rely upon
+the notation convention from
+the second paragraph of 
+ Section 2 of
+\cite{ww7} (which defined semantic tableaux deduction's
+eight elimination rules).
+
+
+It is desirable to
+examine 
+a fourth
+axiomization for \jjj , called Ax-4,
+before considering 
+Ax-3 because such
+an approach will help make
+the underlying intuitions
+behind our methodologies
+easier to appreciate.
+In our discussion, the symbol $~\Psi~$ will denote
+the $\Pi_1^R$ sentence defined by \ep{b.1}. Ax-4
+will be defined to be an encoding of    
+\jjsj that is identical to Ax-3 except that it includes
+\ep{b.1}'s added sentence. (Since Ax-3 can trivially prove
+the validity of \ep{b.1}, the Ax-4 system is clearly
+logically equivalent to Ax-3. Thus while proof lengths
+may differ in these two axiom systems, the final theorems
+that they derive are the same.) 
+\begin{equation}
+\label{b.1} 
+\forall ~z~~~ \forall ~q \, \leq \, z~~~~[~~ 
+q*q~\leq~z~~\Rightarrow ~~ \exists ~r \, \leq~~z~~(~r=~q*q~)~~]
+\end{equation}
+
+\begin{lemma}
+\label{lem.b1}
+The axiom system Ax-4 satisfies the semantic tableaux version
+of the Second Incompleteness Theorem. (In other words using
+Definition 1's notation, Ax-4 is a 
+``threshold
+for the Second Incompleteness Effect'' under 
+semantic tableaux deduction).
+\end{lemma}
+
+{\bf Proof Sketch:}
+Our justification
+of Lemma \ref{lem.b1}
+ will employ the
+definition of
+semantic tableaux deduction
+that had appeared in Section 2 of \cite{ww7}.
+Its second paragraph listed eight elimination rules for
+semantic tableaux deduction. 
+For any term $~s,~$
+its sixth rule applies
+to bounded existential quantifiers appearing in expressions
+similar to:
+\begin{equation}
+\label{b.2}
+\exists~v~\leq~s~~~\Theta(v)
+\end{equation}
+For an arbitrary new constant symbol $~U~$ that does not
+appear in the base axiom system $~\alpha~$ or in any higher
+node in the semantic tableaux proof tree,
+this rule 6 allows 
+\ep{b.3} to be
+ a descendant of \ep{b.2}'s
+node.
+\begin{equation}
+\label{b.3}
+~U~\leq~s~~\wedge ~~\Theta(U)
+\end{equation}
+
+Also 
+consider 
+\ep{b.4}'s
+universally quantified
+ sentence. 
+\begin{equation}
+\label{b.4}
+\forall~v~~~\Phi(v)
+\end{equation}
+In this context for any term $~t~$
+which is free in $\Phi$,
+the seventh elimination rule in
+Section 2 of \cite{ww7}
+indicated that a descendant of \ep{b.4}'s
+sentence in a semantic tableaux proof tree 
+is allowed to be
+any sentence of the form $~\Phi(t)~$.
+In particular if we take $~t~$ to be a term of the
+form ``$~U*U~$'' 
+(where $~U~$ was defined in \ep{b.3} )
+then this universal quantifier elimination
+rule may produce the following reduction from \ep{b.4}.
+\begin{equation}
+\label{b.5}
+\Phi(~U*U~)
+\end{equation}
+
+
+Lastly for any term $ \,\hat{s},~$
+consider 
+\ep{bc.4}'s sentence. 
+\begin{equation}
+\label{bc.4}
+\forall~v~\leq~\hat{s}~~~\Phi(v)
+\end{equation}
+In this context for any term $~t~$
+(again required to be free in $\Phi$),
+the eighth elimination rule 
+from Section 2 of \cite{ww7}
+indicated that 
+\ep{bc.5} is allowed to be
+ descendant of \ep{bc.4}'s
+sentence in a semantic tableaux proof. 
+\begin{equation}
+\label{bc.5}
+t~\leq~\hat{s} ~~~\Rightarrow ~~~\Phi(t)
+\end{equation}
+Since 
+\ep{bc.5}'s
+ rule for eliminating universal quantifiers can apply to
+any term $~t~$
+(free in $\Phi$), it is applicable to the case where $~t~$ is
+a term of the form 
+``$~U~$'' where $~U~$ is a new constant 
+created by \ep{b.3}'s elimination rule.
+In this special case, \ep{bc.5}'s elimination rule can be
+rewritten as:
+\begin{equation}
+\label{bc.6}
+U~\leq~\hat{s} ~~~\Rightarrow ~~~\Phi(~U~)
+\end{equation}
+
+
+ 
+In essence, one may apply
+to \ep{b.1} 
+ $~n~$ iterations
+of the elimination
+rules from
+Equations \eq{b.3}, \eq{b.5} 
+and \eq{bc.6} 
+to construct a sequence of
+constants $ \, U_0 \, , \, U_1 \, , \, U_2 \, , \,  \ldots  \,  U_n \, $
+such that $ \, U_0 \, = \, 2 \, $ and $ \, U_{i+1}  \, = \, U_i*U_i \, $.
+(The 
+footnote 
+\footnote{The $ \, i-$th iteration of this process will
+have $~z~,~$ $~q~$ and $~r~$ from 
+\ep{b.1} be replaced by respectively $~U_i*U_i~,~$
+$~U_i~$ and $~U_{i+1}~$ via the elimination
+rules from respectively Equations
+\eq{b.5}, \eq{bc.6} and \eq{b.3} to produce
+an essentially thrice-revised form of
+\ep{b.1} which implies that $~U_{i+1}~=~U_i*U_i~$.} summarizes 
+the
+structure of the $~i-$th round of these $~n~$ iterations.)
+In a formal sense, these $~n~$ iterations may thus be simulated
+by a fragment of a semantic tableaux proof tree, denoted as $~F~,~$
+where all
+the branches of $~F~$ are closed except for one
+branch, called the pivotal branch, which contains the
+parameter symbols of  $~U_0~,~U_1~,~U_2~,~ \ldots ~ U_n~$
+together with a collection of sentences,
+appearing in linear order, asserting
+that  $~U_0~=~2~$ and $~U_{i+1} ~=~U_i*U_i~$.
+
+Hence this ``pivotal branch'' 
+of $~F~$
+will imply that $~U_n~=~2^{2^n}~$. Moreover since the 
+fragment $~F~$
+will have only  $O(~n~)$ nodes, it will establish the
+existence of a number  
+ $~U_n~=~2^{2^n}~~,~$ whose binary encoding has a $~2^n~$ length
+that is much larger than $~F \,$'s length.
+
+The above invariant is essentially all that we need to generalize
+\cite{ww7}'s semantic tableaux version of the Second Incompleteness
+Theorem so that it also applies to Ax-4. (We obviously
+have omitted many
+details here.
+However,  they are relatively straightforward
+extensions of \cite{ww7}'s methodologies
+ because the
+super-exponential growth property, established by the prior paragraphs,
+opens an avenue for introducing 
+\cite{ww7}'s proof techniques, whose formal details are too
+lengthy to 
+be
+fully duplicated during this abbreviated proof
+sketch.) $~~\Box$
+
+
+
+The remainder of this appendix will sketch how Lemma \ref{lem.b1}'s
+variant of the semantic tableaux version of the Second Incompleteness
+Theorem can be extended from the axiom system Ax-4 to
+ Ax-3 (itself).
+Before doing so, we wish to introduce one further preliminary lemma.
+
+
+
+\begin{lemma}
+\label{lembb2}
+The axiom system Ax-4 
+satisfies the same
+ anti-threshold
+property 
+ for Herbrandized
+deduction as did Ax-3 (in \thx{tm3} ).
+\end{lemma}
+
+{\bf Proof Sketch} 
+Every aspect of the proof of \thx{tm3} 
+(for Ax-3)
+does generalize also for Ax-4's paradigm. This is because
+\thx{tm3}'s proof generalizes for any extension of Ax-3 that consists
+of
+a finite number of additional 
+logically valid
+$\Pi_1^R$ sentences.
+Thus, if $~\alpha~$ denotes any such an extension of
+Ax-3 and if 
+$~\alpha^*~$ denotes 
+the extension of $~\alpha~$ which contains one additional
+$\Pi_1^R$ sentence, asserting the consistency of 
+$~\alpha^*~$ (analogous to \thx{tm2}'s Diagonal$(\alpha,D)$
+sentence), then every aspect of
+our prior analysis of Ax-3$^*$ applies also to
+$~\alpha^*~$. 
+Thus using the same reasoning as before, it follows that
+$~\alpha~$ (and  hence 
+also Ax-4) must
+be anti-thresholds relative to Herbrand deduction. 
+$~\Box$
+
+
+The combination of Lemmas \ref{lem.b1}
+and \ref{lembb2}
+already shows that semantic tableaux and Herbrand
+deduction have polar opposite threshold properties
+with respect to Ax-4. (This is because the latter satisfies the semantic
+tableaux 
+version of the Second
+Incompleteness Theorem,
+but it does not also
+satisfy
+ its  Herbrandized 
+variant.) The same
+pair of
+ polar opposite qualities
+also applies to \jjj 's Ax-3 axiomization. However, the 
+proof that Ax-3 satisfies the semantic tableaux version of
+the Second Incompleteness Theorem is 
+substantially
+more complicated
+than Lemma \ref{lem.b1}'s analogous 
+result for Ax-4. 
+
+
+The final goal in this appendix 
+will 
+thus be 
+to explain how one may incrementally revise
+Lemma \ref{lem.b1}'s 
+proof-analysis
+ for Ax-4 so that a similar incompleteness property also
+applies to Ax-3.
+In our discussion, $~\xbar{\psi}(v)~$ will denote
+the following $~\Delta_0^R$ formula,
+which is free only in $~v~~$:
+\begin{equation}
+\label{tb.1} 
+\forall ~q \, \leq \, v~~~~[~~ 
+q*q~\leq~v~~\Rightarrow ~~ \exists ~r \, \leq~~v~~(~r=~q*q~)~~]
+\end{equation}
+In this notation, the sentence $~\Psi~$ that constituted 
+Ax-4's one additional $~\Pi_1^R~$ axiom-sentence (defined in
+\ep{b.1} ) can be rewritten as:
+\begin{equation}
+\label{tb.2} 
+\forall ~v~~~ \xbar{\psi}(v)
+\end{equation}
+Note that Ax-3, unlike Ax-4, does not contain 
+\ep{tb.2}'s $\Pi_1^R$ sentence as an axiom of its formalism.
+However using an analog of 
+``passive induction'' from our paper \cite{ww7}, 
+we will show Ax-3 contains a counterpart of \ep{tb.2} 
+within its inductive schema that has comparable properties.
+
+
+
+In particular, let $~\underx{\psi}(z)~$ denote the following
+$\Delta_0^R$ formula:
+\begin{equation}
+\label{apb.wp1}
+ \{ \,  \,  \xbar{\psi}( 0) \,  \, \wedge \,  \,  \forall y \leq z \, [ \, 
+\xbar{\psi}( y) \, \Longrightarrow \, \xbar{\psi}( y') \,  \, ]  \, \, \}  \,  \,   \Longrightarrow  \,  \, 
+ \forall y \leq z \,  \xbar{\psi}( y) 
+\end{equation}
+Then  \ep{tb.3} represents an encoding of one
+of Ax-3's induction axioms.
+\begin{equation}
+\label{tb.3} 
+\forall ~z~~~ \underx{\psi}(z)
+\end{equation}
+In order to explain the significance of this 
+added 
+notation, let $~\Psi~$ again denote
+\ep{b.1}'s axiom sentence (which we saw was identical to
+\ep{tb.2}'s sentence except that the latter uses a
+slightly different
+notation). Also, let $~\Psi^*~$ denote
+\ep{tb.3}'s sentence. In this notation,
+$~\Psi^*~$ 
+can be viewed as a cousin of $~\Psi~$ which has the property
+that although $~\Psi~$ is not an axiom of Ax-3, its cousin
+$~\Psi^*~$ is a formal axiom of Ax-3.  
+
+The latter property is important because we need a vehicle
+for formally translating proofs 
+from the Ax-4 
+system
+to proofs in
+the Ax-3  system.
+Since $~\Psi~$ is the only axiom belonging to Ax-4 which is not also
+in Ax-3, 
+\ep{apb.wp1}'s formal counterpart
+of it,
+called $~\Psi^*~$,
+will provide 
+a means for doing this translation. 
+The implications of this translation mechanism
+is described by our next lemma. 
+
+\begin{lemma}
+\label{lembb3}
+Let $~\Omega~$ 
+denote an arbitrary theorem that is proven
+from the axiom system Ax-4 under a semantic tableaux
+proof $~T~$ that consists of 
+$~n~$ applications of
+the $~\forall ~$ elimination to rule to
+ \ep{tb.2}'s axiom sentence of $~\Psi~$.
+For some term $~t_i~,~$
+let us assume that
+the $~i-$th such application of 
+this rule
+replaces 
+\ep{tb.2} 
+with the reduced sentence of
+\begin{equation}
+\label{tb.2r} 
+\xbar{\psi}(~t_i~)
+\end{equation}
+Then for a constant $~k~$ whose value is
+entirely
+independent of $~n~$, one can construct a proof $~T^*~$
+of the same theorem 
+$~\Omega~$ from the axiom system Ax-3 where the difference
+between the number of node-sentences appearing in the proofs
+of $~T~$ and of $~T^*~$ is bounded by
+the quantity of $~kn~$.
+\end{lemma}
+
+{\bf Proof:}
+The justification of Lemma \ref{lembb3}
+is fairly straightforward. On each occasion
+where $~T \,$'s proof contains a node similar to
+\ep{tb.2r}, the comparable structure in $~T^*~$
+will replace this sentence with a tree-fragment
+that consists of the following  
+four components:
+\bee
+\item
+An initial node-sentence of the form \eq{tb.cute}
+(which is justified because it is an instance of
+\ep{tb.3}'s
+ axiom sentence of $~\Psi^*~$ ).
+\begin{equation}
+\label{tb.cute}
+\underx{\psi}(~ t_i~)
+\end{equation}
+\item
+A branch separation will appear
+directly below \ep{tb.cute}'s
+node that consists of the two
+sibling
+ sentences of
+\eq{tb.cute1} and \eq{tb.cute2}. In light of
+\ep{apb.wp1}'s definition of $~\underx{\psi}~,~$
+this binary separation is justified by the semantic tableaux
+rule for eliminating
+the $~\Rightarrow~$ symbol (which
+was formalized by Item 4 
+from the second paragraph of \cite{ww7}'s 
+ Section 2).
+\begin{equation}
+\label{tb.cute1} 
+\neg ~~ \{ \,  \,  \xbar{\psi}( 0) \,  \, \wedge \,  \,  \forall y \leq t_i \, [ \, 
+\xbar{\psi}( y) \, \Longrightarrow \, \xbar{\psi}( y') \,  \, ]  \, \, \}
+\end{equation}
+\begin{equation}
+\label{tb.cute2}
+ \forall y \leq t_i \,  \xbar{\psi}( y) 
+\end{equation}
+\item
+Since Ax-3 
+can prove 
+\ep{b.1}'s sentence of $~\Psi~$ as a theorem,
+it is trivial to establish that
+for some fixed constant $~k_1~$ which does not depend on $~i~$,
+one may insert a closed semantic tableaux proof
+of length $~k_1~$
+ under \ep{tb.cute1}'s
+sentence, 
+$~$which accomplishes the desired effect of showing that \ep{tb.cute1}
+is inherently contradictory. 
+\item
+Using one more time
+\cite{ww7}'s
+ semantic tableaux rules for eliminating the bounded 
+universal quantifiers and  the $~\Rightarrow~$ symbol,
+one may easily insert a branch below \ep{tb.cute2} 
+that ends with the pair of sibling sentences 
+given in equations 
+\eq{cute3} and \eq{cute4} below. 
+Moreover, it is evident that
+\ep{cute3} is inherently contradictory. Thus for some fixed constant
+$~k_2~$ (which again does not depend on $~i~)~$,
+the net consequence of this step will be to consist of a sequence of
+$~k_2~$ node sentences that includes the sentences given in
+\eq{cute3} and \eq{cute4} and closes the proof-tree that descends 
+from \ep{cute3} with the desired forced contradiction.
+\begin{equation}
+\label{cute3}
+\neg ~~(~~ t_i~ \leq ~ t_i~~)
+\end{equation}
+\begin{equation}
+\label{cute4}
+\xbar{\psi}( ~t_i~) 
+\end{equation}
+\ene
+
+Let $~k~$ denote a constant that equals the quantity
+of $~k_1 \, + \,~k_2 \, + \, 3~$. Then each iteration of
+the above 4-step procedure will
+replace 
+\ep{tb.2r}'s sentence with a subtree fragment
+consisting of $~k~$ 
+new
+sentences.
+Note that the final sentence at the end of each such iteration
+(given in \ep{cute4} )
+contains the
+identical logical statement as 
+was given in 
+\ep{tb.2r}'s initial sentence.
+Since all the other branches of our new sub-structure
+are closed and since Equations
+\eq{tb.2r} and 
+\ep{cute4} are identical
+to each other, it follows that after we finish
+performing $~n~$ iterations of the above process,
+introducing $~kn~$ new sentences, our 
+new revised semantic tableaux tree will 
+prove the theorem $~\Omega~$ from Ax-3 instead of Ax-4. $~~~\Box$
+
+
+
+
+
+
+\begin{theorem}
+\label{tapb}
+The axiom system Ax-3
+(similar to Ax-4)
+ satisfies the semantic tableaux version
+of the Second Incompleteness Theorem. (Thus using
+Definition 1's notation, Ax-3 is also a 
+``threshold
+for the Second Incompleteness Effect'' under 
+semantic tableaux deduction).
+\end{theorem}
+
+{\bf Proof Sketch}: 
+For the sake of brevity, we will only outline
+the intuition behind \thx{tapb}'s proof.
+It will be essentially a consequence of the combination of
+Lemmas \ref{lem.b1} and \ref{lembb3}.
+
+
+In particular, the core idea behind Lemma \ref{lem.b1}'s
+analysis of Ax-4 was to employ a sequence of $\,n\,$ iterated
+applications of \ep{b.1}'s axiom of $\,\Psi\,$
+so as to construct a 
+formal
+sequence of
+constants $\,U_0\,,\,U_1\,,\,U_2\,,\, \ldots \, U_n\,$
+such that $\,U_0\,=\,2\,$ and $\,U_{i+1} \,=\,U_i*U_i\,$.
+By Lemma \ref{lembb3}, the precisely identical 
+sequence of
+$\,U_0\,,\,U_1\,,\,U_2\,,\, \ldots \, U_n\,$
+can be constructed from Ax-3 using only $\,kn\,$ additional
+sentences, $\,$for some fixed constant $\,k\,$ whose value is independent
+of $\,n\,$.
+
+Since $~U_n~$ represents the quantity of $~2^{2^n}~$ whose binary
+encoding has a length of $~2^n~,~$ this binary length is clearly
+much larger than $~kn~$ as $~n~$ approaches of infinity. As a
+result of this exponential difference in lengths, it is relatively routine
+to revise 
+Lemma \ref{lem.b1}'s
+proof of the semantic tableaux version of the Second Incompleteness
+Theorem for Ax-4 so that it may also apply to Ax-3. (The remaining details
+are analogous to the constructions that we used in
+\cite{ww2,ww7} and are omitted 
+for the sake of brevity.) $~~\Box$
+
+
+{\bf Summing Up What Has Been Done in this Appendix :} 
+We have outlined abbreviated proofs
+showing that the
+Ax-3 and Ax-4 encodings for \jjsj satisfy the semantic tableaux
+versions of the Second Incompleteness Theorem. We visited
+this topic twice because the result is easier to establish
+for Ax-4, 
+$~$although
+      it is stronger for Ax-3. (The latter
+is 
+stronger
+because Ax-3 contains one less
+axiom sentence than
+Ax-4.) 
+Both our 
+results for 
+ Ax-3 and Ax-4 are
+interesting 
+ because
+these formalisms are 
+anti-thresholds
+for the Herbrandized version of the Second Incompleteness
+while they are thresholds 
+when the paradigm is changed to focus upon
+semantic tableaux style deduction.
+
+
+We close this appendix by once again reminding the reader that a
+different 
+type of axiomatic setting
+where semantic tableaux and Herbrandized proofs have
+a sharp difference in length was formalized by
+Ko{\l}odziejczyk in \cite {Ko6b,Ko7}.
+(In particular, Ko{\l}odziejczyk  \cite {Ko6b,Ko7}
+focused his discussion on
+Buss's bounded arithmetic
+of \cite{Bu86}.) 
+
+
+\begin{thebibliography}
+\small
+\footnotesize
+
+
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+Arithmetic'', 
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+171 (2002) 279-292.
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+\bibitem{AB1}
+Z. Adamowicz and T. Bigorajska,
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+
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+\bibitem{AZ1}
+Z. Adamowicz and P. Zbierski,
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+A. Bezboruah and J. 
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+P. H\'{a}jek 
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+S.
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+
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+L.A. Ko{\l}odziejczyk,
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+whose slides are available
+at hhtp://www.cs.us.es/glm/jaf25-CMBFD/slides-lk.pdf.
+
+\bibitem{Ko7}
+ L.A. Ko{\l}odziejczyk,
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+of Bound Arithmetic'',
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+71 (2006) pp. 624-638.
+
+
+
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+Y. Matiyasevich, {\it Hilbert's Tenth Problem},
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+J.  Paris and C. Dimitracopoulos,
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+
+
+
+\bibitem{PW81}
+J.  Paris and A.  Wilkie,
+``$\Delta_0$ Sets and Induction'', 
+{\it 1981 Jadswin  Conference Proceedings,}
+(Leeds University Press) 
+pp. 237-248.
+
+
+ \bibitem{Pu83}
+ P. Pudl\'{a}k,
+ Some Prime Elements in the Lattice of
+ Interpretability
+ Types,
+ {\it Transactions of the American Mathematical
+ Society} 280 (1983) pp. 255-275
+ 
+
+
+\bibitem{Pu84}
+P. Pudl\'{a}k,
+``On Lengths of Proofs of Finisitic Consistency Statements in
+First order Theories'',
+{\it Logic Colloquium 84}, North Holland (1986)
+pp. 165-196.
+
+
+
+\bibitem{Pu85}
+P. Pudl\'{a}k,
+``Cuts, Consistency Statements and Interpretations'',
+{\it Journal of Symbolic Logic}
+50 (1985) 423-442
+
+
+\bibitem{Pu96}
+P. Pudl\'{a}k,
+``On the Lengths of Proofs of Consistency'',
+in {\it The Collegium Logicum: The Offical 1996 Annals of  Kurt G\"{o}del
+Society} ( Volume 2),  Springer-Wien-NewYork,
+pp. 65-86.
+
+
+\bibitem{Sa1}
+S. Salehi, Herbrand Consistency in Arithmetics with
+Bounded Induction, Polish Academy of
+Sciences  Ph D thesis, October 2001.
+
+
+
+ \bibitem{Sm77}
+ C.  Smory\'{n}ski, ``The Incompleteness Theorem'',
+{\it The Handbook on Mathematical Logic},
+1977,      {North Holland},
+        pp.      {821--865}.
+
+
+ \bibitem{Sm85}
+ C.  Smory\'{n}ski, ``Non-standard Models and Related Developments
+ in the Work of Harvey Friedman'', in {\it Harvey Friedman's
+ Research in  Foundations of Math.},
+ North Holland 1985, pp. 179-220.
+
+
+
+\bibitem{So94}
+R. Solovay, Private 
+telephone conversations between  
+Robert Solovay and Dan Willard 
+(during April of 1994)
+concerning Solovay's generalization of one of  Pudl\'{a}k's  theorems 
+\cite{Pu85}, using also some of
+Nelson's and Wilkie-Paris's methodologies \cite{Ne86,WP87}.
+Solovay's unpublished theorem
+ shows that essentially no axiom system that recognizes
+Successor$(x)~=~x+1~$ as a total function
+(and which treats multiplication and addition as 3-way relations)
+ can prove a theorem
+affirming its own consistency under Hilbert deduction.
+The Appendix A of 
+\cite{ww1} offers a 4-page interpretation of the  intuition behind
+Solovay's unpublished idea.
+
+
+\bibitem{Sv83}
+V. \v{S}vejdar, Modal Analysis of Generalized
+Rosser Sentences,
+{\it Journal of Symbolic Logic} 48 (1983) pp. 986-999.
+
+
+
+\bibitem{Sw3}
+S. Swierczkowski,
+Finite Sets and G\"{o}del's Incompleteness Theorem
+Dissertationes Mathematicae
+422 (2003)
+pp. 1--58
+
+
+\bibitem{Ta53}
+G. Takeuti,
+``On a Generalized Logical Calculus'',
+{\it Japan Journal of Mathematics}
+23 (1953) pp. 39--96.
+
+
+
+
+\bibitem{Ta0} 
+G. Takeuti,
+``G\"{o}del Sentences of Bounded Arithmetic'',
+{\it Journal of Symbolic Logic} 65 (2000) pp. 1338-1346
+
+ \bibitem{TMR53}
+ A. Tarski, A. Mostowski and R. M. Robinson,
+ {\it Undecidable Theories}, North Holland, 1953.
+
+
+\bibitem{Vi92}
+A. Visser,
+An Inside View of Exp (or The Closed
+Fragment of The Provability Logic I$\Delta_0 +
+\Omega_1$
+With a Propositional Constant for Exp),
+{\it Journal of Symbolic Logic} 57 (1992)
+pp. 131--165
+
+
+\bibitem{Vi93}
+A. Visser,
+``The Unprovability of Small Inconsistency'',
+{\it Archive for Mathematical Logic} 32 (1993) pp. 275-298.
+
+\bibitem{Vi5}
+A. Visser,
+ ``Faith and Falsity'',
+ {\it Annals Pure \& Applied Logic}
+131 (2005), pp. 103-131.
+
+
+\bibitem{VH73}
+P. Vop\v{e}nka and P. H\'{a}jek,
+Existence of  a Generalized Semantic Model of
+G\"{o}del-Bernays Set Theory,
+{\it Bulletin de l'Academie Polonaise des
+Sciences,
+Mathmatiques, Astromiques et Physiques}
+12 (1973) pp.1079-1086.
+
+
+
+
+\bibitem{WP87}  
+A. Wilkie and J. Paris,  
+``On the Scheme of Induction for Bounded
+Arithmetic'', 
+ {\it Annals of Pure and Applied Logic}
+(35) 1987, 261-302
+
+
+\bibitem{ww93}
+D. Willard,
+``Self-Verifying Axiom Systems'', {\it
+   Third Kurt G\"{o}del
+Colloq}
+(1993),
+Springer-Verlag LNCS\#713,  325-336.
+
+
+
+
+\bibitem{ww0}
+D. Willard, 
+``The Semantic Tableaux Version of the Second
+Incompleteness Theorem Extends Almost to 
+Robinson's 
+Arithmetic 
+Q'',
+Springer Verlag LNCS\#1847, July 2000,
+ pp. 415-430. 
+
+\bibitem{ww1}
+D. Willard, ``Self-Verifying  Systems, the Incompleteness
+Theorem and the
+Tangibility 
+Principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+
+
+\bibitem{ww2}
+D. Willard, 
+``How to Extend The Semantic Tableaux And
+Cut-Free Versions of the Second
+Incompleteness Theorem 
+Almost 
+to 
+Robinson's Arithmetic Q'', in $~$
+{\it Journal of Symbolic Logic}
+$~\,67~ (2002)~$ pp. 465--496.
+
+
+\bibitem{ww4}
+D. Willard, 
+``A Version of the
+Second Incompleteness Theorem For Axiom
+Systems that Recognize Addition 
+But Not Multiplication as a Total Function'',
+{\it First Order Logic Revisited,}
+Logos Verlag (Berlin) 2004, pp. 337--368.
+
+
+\bibitem{ww5}
+D. Willard, 
+``An Exploration of the Partial Respects in which an Axiom
+System Recognizing Solely Addition as a Total Function Can
+Verify Its Own Consistency'', 
+{\it Journal of Symbolic Logic} 70  (2005) pp. 1171-1209. 
+
+
+\bibitem{ww5b}
+D. Willard, 
+``On the Available Partial Respects in which
+ an Axiomatization
+for Real Valued  Arithmetic Can  Recognize its 
+Consistency'', 
+{\it Journal of Symbolic Logic} 71 (2006)
+pp. 1189-1199.
+
+
+
+\bibitem{ww6}
+D. Willard,  
+``A Generalization of the Second Incompleteness 
+Theorem and Some Exceptions to It'',
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+
+\bibitem{ww7}
+D. Willard,  
+``Passive Induction and a Solution to a Paris-Wilkie
+Open Question'',
+{\it Annals of Pure and Applied Logic}
+146 (2007) pp. 124-149.
+
+
+
+\bibitem{Wr78}
+C. Wrathall, ``Rudimentary Predicates and Relative Computation'',
+Siam J. on Computing 7 (1978), pp. 194-209.
+
+
+\end{thebibliography}
+
+
+
+\end{document}
diff --git a/nachlass/collected_dew_materials/llncs.cls b/nachlass/collected_dew_materials/llncs.cls
new file mode 100644
index 0000000..06401a9
--- /dev/null
+++ b/nachlass/collected_dew_materials/llncs.cls
@@ -0,0 +1,1218 @@
+% LLNCS DOCUMENT CLASS -- version 2.20 (10-Mar-2018)
+% Springer Verlag LaTeX2e support for Lecture Notes in Computer Science
+%
+%%
+%% \CharacterTable
+%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
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+
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+
+% languages
+\let\switcht@@therlang\relax
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+
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+\def\andname{and}
+\def\lastandname{\unskip, and}
+\def\appendixname{Appendix}
+\def\chaptername{Chapter}
+\def\claimname{Claim}
+\def\conjecturename{Conjecture}
+\def\contentsname{Table of Contents}
+\def\corollaryname{Corollary}
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+\def\exercisename{Exercise}
+\def\figurename{Fig.}
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+\def\listtablename{List of Tables}
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+\def\noteaddname{Note added in proof}
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+\def\partname{Part}
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+\def\proofname{Proof}
+\def\propertyname{Property}
+\def\propositionname{Proposition}
+\def\questionname{Question}
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+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
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+ \def\ackname{Remerciements.}%
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+ \def\proofname{Preuve}%
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+ \def\remarkname{Remarque}%
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+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
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+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
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+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
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+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}
+ \def\indexname{Index}
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+ \def\mailname{{\it Correspondence to\/}:}
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+ \def\notename{Anmerkung}%
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+ \def\proofname{Beweis}%
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+ \def\remarkname{Anmerkung}%
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+  {\centering
+    \ifnum \c@secnumdepth >\m@ne
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+  {\centering
+    \normalfont
+    \interlinepenalty\@M
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+                       {\normalfont\normalsize\bfseries\boldmath
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+{\mbox{\boldmath$\textstyle#1$}}
+{\mbox{\boldmath$\scriptstyle#1$}}
+{\mbox{\boldmath$\scriptscriptstyle#1$}}}
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+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip1pt}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9pt}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-1pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8pt}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3pt}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to0pt{\kern0.3\wd0\vrule height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle     \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle        \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox
+to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle      \rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox
+to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\textstyle Z\kern-0.4em Z$}}
+{\hbox{$\mathsf\scriptstyle Z\kern-0.3em Z$}}
+{\hbox{$\mathsf\scriptscriptstyle Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength\leftmargini  {17\p@}
+\setlength\leftmargin    {\leftmargini}
+\setlength\leftmarginii  {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv  {\leftmargini}
+\setlength  \labelsep  {.5em}
+\setlength  \labelwidth{\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+            \parsep 0\p@ \@plus1\p@ \@minus\p@
+            \topsep 8\p@ \@plus2\p@ \@minus4\p@
+            \itemsep0\p@}
+\let\@listi\@listI
+\@listi
+\def\@listii {\leftmargin\leftmarginii
+              \labelwidth\leftmarginii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus2\p@ \@minus\p@}
+\def\@listiii{\leftmargin\leftmarginiii
+              \labelwidth\leftmarginiii
+              \advance\labelwidth-\labelsep
+              \topsep    0\p@ \@plus\p@\@minus\p@
+              \parsep    \z@
+              \partopsep \p@ \@plus\z@ \@minus\p@}
+
+\renewcommand\labelitemi{\normalfont\bfseries --}
+\renewcommand\labelitemii{$\m@th\bullet$}
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\def\tableofcontents{\chapter*{\contentsname\@mkboth{{\contentsname}}%
+                                                    {{\contentsname}}}
+ \def\authcount##1{\setcounter{auco}{##1}\setcounter{@auth}{1}}
+ \def\lastand{\ifnum\value{auco}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{auco}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+   \addvspace{2em plus\p@}%  % space above part line
+   \begingroup
+     \parindent \z@
+     \rightskip \z@ plus 5em
+     \hrule\vskip5pt
+     \large               % same size as for a contribution heading
+     \bfseries\boldmath   % set line in boldface
+     \leavevmode          % TeX command to enter horizontal mode.
+     #1\par
+     \vskip5pt
+     \hrule
+     \vskip1pt
+     \nobreak             % Never break after part entry
+   \endgroup}
+
+\def\@dotsep{2}
+
+\let\phantomsection=\relax
+
+\def\hyperhrefextend{\ifx\hyper@anchor\@undefined\else
+{}\fi}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+                     {\thechapter}#3}{\thepage}\hyperhrefextend}}%
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}\hyperhrefextend}}%
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}\hyperhrefextend}}%
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmark}%
+  \or    \def\@gtempa{\addcontentsmarkwop}%
+  \fi\@gtempa{toc}{chapter}%
+}
+\def\addtocmark{%
+\phantomsection
+\@ifnextchar[{\@adcmk}{\@adcmk[3]}%
+}
+
+\def\l@chapter#1#2{\addpenalty{-\@highpenalty}
+ \vskip 1.0em plus 1pt \@tempdima 1.5em \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\large\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+      \nobreak
+      \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+      \@dotsep mu$}\hfill
+      \nobreak\hbox to\@pnumwidth{\hss #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@title#1#2{\addpenalty{-\@highpenalty}
+ \addvspace{8pt plus 1pt}
+ \@tempdima \z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ #1\nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu.\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hss #2}\par
+ \penalty\@highpenalty \endgroup}
+
+\def\l@author#1#2{\addpenalty{\@highpenalty}
+ \@tempdima=15\p@ %\z@
+ \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by 0pt plus 2cm
+ \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima %\hskip -\leftskip
+ \textit{#1}\par
+ \penalty\@highpenalty \endgroup}
+
+\setcounter{tocdepth}{0}
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=\z@            % no chapter numbers
+\tocsecnum=15\p@          % section 88. plus 2.222pt
+\tocsubsecnum=23\p@       % subsection 88.8 plus 2.222pt
+\tocsubsubsecnum=27\p@    % subsubsection 88.8.8 plus 1.444pt
+\tocparanum=35\p@         % paragraph 88.8.8.8 plus 1.666pt
+\tocsubparanum=43\p@      % subparagraph 88.8.8.8.8 plus 1.888pt
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listfigurename\@mkboth{{\listfigurename}}{{\listfigurename}}}
+ \@starttoc{lof}\if@restonecol\twocolumn\fi}
+\def\l@figure{\@dottedtocline{1}{0em}{1.5em}}
+
+\def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\section*{\listtablename\@mkboth{{\listtablename}}{{\listtablename}}}
+ \@starttoc{lot}\if@restonecol\twocolumn\fi}
+\let\l@table\l@figure
+
+\renewcommand\listoffigures{%
+    \section*{\listfigurename
+      \@mkboth{\listfigurename}{\listfigurename}}%
+    \@starttoc{lof}%
+    }
+
+\renewcommand\listoftables{%
+    \section*{\listtablename
+      \@mkboth{\listtablename}{\listtablename}}%
+    \@starttoc{lot}%
+    }
+
+\ifx\oribibl\undefined
+\ifx\citeauthoryear\undefined
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \def\@biblabel##1{##1.}
+      \small
+      \list{\@biblabel{\@arabic\c@enumiv}}%
+           {\settowidth\labelwidth{\@biblabel{#1}}%
+            \leftmargin\labelwidth
+            \advance\leftmargin\labelsep
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+\def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+     {\let\protect\noexpand\immediate
+     \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\newcount\@tempcntc
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+  \@tempcnta\z@\@tempcntb\m@ne\def\@citea{}\@cite{\@for\@citeb:=#2\do
+    {\@ifundefined
+       {b@\@citeb}{\@citeo\@tempcntb\m@ne\@citea\def\@citea{,}{\bfseries
+        ?}\@warning
+       {Citation `\@citeb' on page \thepage \space undefined}}%
+    {\setbox\z@\hbox{\global\@tempcntc0\csname b@\@citeb\endcsname\relax}%
+     \ifnum\@tempcntc=\z@ \@citeo\@tempcntb\m@ne
+       \@citea\def\@citea{,}\hbox{\csname b@\@citeb\endcsname}%
+     \else
+      \advance\@tempcntb\@ne
+      \ifnum\@tempcntb=\@tempcntc
+      \else\advance\@tempcntb\m@ne\@citeo
+      \@tempcnta\@tempcntc\@tempcntb\@tempcntc\fi\fi}}\@citeo}{#1}}
+\def\@citeo{\ifnum\@tempcnta>\@tempcntb\else
+               \@citea\def\@citea{,\,\hskip\z@skip}%
+               \ifnum\@tempcnta=\@tempcntb\the\@tempcnta\else
+               {\advance\@tempcnta\@ne\ifnum\@tempcnta=\@tempcntb \else
+                \def\@citea{--}\fi
+      \advance\@tempcnta\m@ne\the\@tempcnta\@citea\the\@tempcntb}\fi\fi}
+\else
+\renewenvironment{thebibliography}[1]
+     {\section*{\refname}
+      \small
+      \list{}%
+           {\settowidth\labelwidth{}%
+            \leftmargin\parindent
+            \itemindent=-\parindent
+            \labelsep=\z@
+            \if@openbib
+              \advance\leftmargin\bibindent
+              \itemindent -\bibindent
+              \listparindent \itemindent
+              \parsep \z@
+            \fi
+            \usecounter{enumiv}%
+            \let\p@enumiv\@empty
+            \renewcommand\theenumiv{}}%
+      \if@openbib
+        \renewcommand\newblock{\par}%
+      \else
+        \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+      \fi
+      \sloppy\clubpenalty4000\widowpenalty4000%
+      \sfcode`\.=\@m}
+     {\def\@noitemerr
+       {\@latex@warning{Empty `thebibliography' environment}}%
+      \endlist}
+      \def\@cite#1{#1}%
+      \def\@lbibitem[#1]#2{\item[]\if@filesw
+        {\def\protect##1{\string ##1\space}\immediate
+      \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+   \fi
+\else
+\@cons\@openbib@code{\noexpand\small}
+\fi
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\hangindent 10\p@}
+
+\def\subitem{\par\setbox0=\hbox{--\enspace}% second order
+                \noindent\hangindent\wd0\box0}% index entry
+
+\def\subsubitem{\par\setbox0=\hbox{--\,--\enspace}% third
+                \noindent\hangindent\wd0\box0}% order index entry
+
+\def\indexspace{\par \vskip 10\p@ plus5\p@ minus3\p@\relax}
+
+\renewenvironment{theindex}
+               {\@mkboth{\indexname}{\indexname}%
+                \thispagestyle{empty}\parindent\z@
+                \parskip\z@ \@plus .3\p@\relax
+                \let\item\par
+                \def\,{\relax\ifmmode\mskip\thinmuskip
+                             \else\hskip0.2em\ignorespaces\fi}%
+                \normalfont\small
+                \begin{multicols}{2}[\@makeschapterhead{\indexname}]%
+                }
+                {\end{multicols}}
+
+\renewcommand\footnoterule{%
+  \kern-3\p@
+  \hrule\@width 2truecm
+  \kern2.6\p@}
+  \newdimen\fnindent
+  \fnindent1em
+\long\def\@makefntext#1{%
+    \parindent \fnindent%
+    \leftskip \fnindent%
+    \noindent
+    \llap{\hb@xt@1em{\hss\@makefnmark\ }}\ignorespaces#1}
+
+\long\def\@makecaption#1#2{%
+  \small
+  \vskip\abovecaptionskip
+  \sbox\@tempboxa{{\bfseries #1.} #2}%
+  \ifdim \wd\@tempboxa >\hsize
+    {\bfseries #1.} #2\par
+  \else
+    \global \@minipagefalse
+    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+  \fi
+  \vskip\belowcaptionskip}
+
+\def\fps@figure{htbp}
+\def\fnum@figure{\figurename\thinspace\thefigure}
+\def \@floatboxreset {%
+        \reset@font
+        \small
+        \@setnobreak
+        \@setminipage
+}
+\def\fps@table{htbp}
+\def\fnum@table{\tablename~\thetable}
+\renewenvironment{table}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@float{table}}
+               {\end@float}
+\renewenvironment{table*}
+               {\setlength\abovecaptionskip{0\p@}%
+                \setlength\belowcaptionskip{10\p@}%
+                \@dblfloat{table}}
+               {\end@dblfloat}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+  ext@#1\endcsname}{#1}{\protect\numberline{\csname
+  the#1\endcsname}{\ignorespaces #2}}\begingroup
+    \@parboxrestore
+    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+  \endgroup}
+
+% LaTeX does not provide a command to enter the authors institute
+% addresses. The \institute command is defined here.
+
+\newcounter{@inst}
+\newcounter{@auth}
+\newcounter{auco}
+\newdimen\instindent
+\newbox\authrun
+\newtoks\authorrunning
+\newtoks\tocauthor
+\newbox\titrun
+\newtoks\titlerunning
+\newtoks\toctitle
+
+\def\clearheadinfo{\gdef\@author{No Author Given}%
+                   \gdef\@title{No Title Given}%
+                   \gdef\@subtitle{}%
+                   \gdef\@institute{No Institute Given}%
+                   \gdef\@thanks{}%
+                   \global\titlerunning={}\global\authorrunning={}%
+                   \global\toctitle={}\global\tocauthor={}}
+
+\def\institute#1{\gdef\@institute{#1}}
+
+\def\institutename{\par
+ \begingroup
+ \parskip=\z@
+ \parindent=\z@
+ \setcounter{@inst}{1}%
+ \def\and{\par\stepcounter{@inst}%
+ \noindent$^{\the@inst}$\enspace\ignorespaces}%
+ \setbox0=\vbox{\def\thanks##1{}\@institute}%
+ \ifnum\c@@inst=1\relax
+   \gdef\fnnstart{0}%
+ \else
+   \xdef\fnnstart{\c@@inst}%
+   \setcounter{@inst}{1}%
+   \noindent$^{\the@inst}$\enspace
+ \fi
+ \ignorespaces
+ \@institute\par
+ \endgroup}
+
+\def\@fnsymbol#1{\ensuremath{\ifcase#1\or\star\or{\star\star}\or
+   {\star\star\star}\or \dagger\or \ddagger\or
+   \mathchar "278\or \mathchar "27B\or \|\or **\or \dagger\dagger
+   \or \ddagger\ddagger \else\@ctrerr\fi}}
+
+\def\inst#1{\unskip$^{#1}$}
+\def\orcidID#1{\unskip$^{[#1]}$} % added MR 2018-03-10
+\def\fnmsep{\unskip$^,$}
+\def\email#1{{\tt#1}}
+
+\AtBeginDocument{\@ifundefined{url}{\def\url#1{#1}}{}%
+\@ifpackageloaded{babel}{%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasfrenchb}{}{\addto\extrasfrenchb{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+\@ifundefined{extrasngerman}{}{\addto\extrasngerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+\providecommand{\keywords}[1]{\def\and{{\textperiodcentered} }%
+\par\addvspace\baselineskip
+\noindent\keywordname\enspace\ignorespaces#1}%
+\@ifpackageloaded{hyperref}{%
+\def\doi#1{\href{https://doi.org/#1}{https://doi.org/#1}}}{
+\def\doi#1{https://doi.org/#1}}
+}
+\def\homedir{\~{ }}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\clearheadinfo
+%
+%%% to avoid hyperref warnings
+\providecommand*{\toclevel@author}{999}
+%%% to make title-entry parent of section-entries
+\providecommand*{\toclevel@title}{0}
+%
+\renewcommand\maketitle{\newpage
+\phantomsection
+  \refstepcounter{chapter}%
+  \stepcounter{section}%
+  \setcounter{section}{0}%
+  \setcounter{subsection}{0}%
+  \setcounter{figure}{0}
+  \setcounter{table}{0}
+  \setcounter{equation}{0}
+  \setcounter{footnote}{0}%
+  \begingroup
+    \parindent=\z@
+    \renewcommand\thefootnote{\@fnsymbol\c@footnote}%
+    \if@twocolumn
+      \ifnum \col@number=\@ne
+        \@maketitle
+      \else
+        \twocolumn[\@maketitle]%
+      \fi
+    \else
+      \newpage
+      \global\@topnum\z@   % Prevents figures from going at top of page.
+      \@maketitle
+    \fi
+    \thispagestyle{empty}\@thanks
+%
+    \def\\{\unskip\ \ignorespaces}\def\inst##1{\unskip{}}%
+    \def\thanks##1{\unskip{}}\def\fnmsep{\unskip}%
+    \instindent=\hsize
+    \advance\instindent by-\headlineindent
+    \if!\the\toctitle!\addcontentsline{toc}{title}{\@title}\else
+       \addcontentsline{toc}{title}{\the\toctitle}\fi
+    \if@runhead
+       \if!\the\titlerunning!\else
+         \edef\@title{\the\titlerunning}%
+       \fi
+       \global\setbox\titrun=\hbox{\small\rm\unboldmath\ignorespaces\@title}%
+       \ifdim\wd\titrun>\instindent
+          \typeout{Title too long for running head. Please supply}%
+          \typeout{a shorter form with \string\titlerunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\titrun=\hbox{\small\rm
+          Title Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@title{\copy\titrun}%
+    \fi
+%
+    \if!\the\tocauthor!\relax
+      {\def\and{\noexpand\protect\noexpand\and}%
+       \def\inst##1{}%    added MR 2017-09-20 to remove inst numbers from the TOC
+       \def\orcidID##1{}% added MR 2017-09-20 to remove ORCID ids from the TOC
+      \protected@xdef\toc@uthor{\@author}}%
+    \else
+      \def\\{\noexpand\protect\noexpand\newline}%
+      \protected@xdef\scratch{\the\tocauthor}%
+      \protected@xdef\toc@uthor{\scratch}%
+    \fi
+    \addtocontents{toc}{\noexpand\protect\noexpand\authcount{\the\c@auco}}%
+    \addcontentsline{toc}{author}{\toc@uthor}%
+    \if@runhead
+       \if!\the\authorrunning!
+         \value{@inst}=\value{@auth}%
+         \setcounter{@auth}{1}%
+       \else
+         \edef\@author{\the\authorrunning}%
+       \fi
+       \global\setbox\authrun=\hbox{\def\inst##1{}%    added MR 2017-09-20 to remove inst numbers from the runninghead
+                                    \def\orcidID##1{}% added MR 2017-09-20 to remove ORCID ids from the runninghead
+                                    \small\unboldmath\@author\unskip}%
+       \ifdim\wd\authrun>\instindent
+          \typeout{Names of authors too long for running head. Please supply}%
+          \typeout{a shorter form with \string\authorrunning\space prior to
+                   \string\maketitle}%
+          \global\setbox\authrun=\hbox{\small\rm
+          Authors Suppressed Due to Excessive Length}%
+       \fi
+       \xdef\@author{\copy\authrun}%
+       \markboth{\@author}{\@title}%
+     \fi
+  \endgroup
+  \setcounter{footnote}{\fnnstart}%
+  \clearheadinfo}
+%
+\def\@maketitle{\newpage
+ \markboth{}{}%
+ \def\lastand{\ifnum\value{@inst}=2\relax
+                 \unskip{} \andname\
+              \else
+                 \unskip \lastandname\
+              \fi}%
+ \def\and{\stepcounter{@auth}\relax
+          \ifnum\value{@auth}=\value{@inst}%
+             \lastand
+          \else
+             \unskip,
+          \fi}%
+ \begin{center}%
+ \let\newline\\
+ {\Large \bfseries\boldmath
+  \pretolerance=10000
+  \@title \par}\vskip .8cm
+\if!\@subtitle!\else {\large \bfseries\boldmath
+  \vskip -.65cm
+  \pretolerance=10000
+  \@subtitle \par}\vskip .8cm\fi
+ \setbox0=\vbox{\setcounter{@auth}{1}\def\and{\stepcounter{@auth}}%
+ \def\thanks##1{}\@author}%
+ \global\value{@inst}=\value{@auth}%
+ \global\value{auco}=\value{@auth}%
+ \setcounter{@auth}{1}%
+{\lineskip .5em
+\noindent\ignorespaces
+\@author\vskip.35cm}
+ {\small\institutename}
+ \end{center}%
+ }
+
+% definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+%     \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or  \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or  \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{}
+\def\@thmcounterend{.}
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+  \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}\@addtoreset{#1}{#3}%
+   \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+     \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+                              \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\@definecounter{#1}%
+   \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+   \expandafter\xdef\csname #1name\endcsname{#2}%
+   \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+                               \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+  \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+  {\expandafter\@ifdefinable\csname #1\endcsname
+  {\newaliascnt{#1}{#2}%
+  \expandafter\xdef\csname #1name\endcsname{#3}%
+  \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+  \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+                    \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+       the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\@spbegintheorem#1#2#3#4{\trivlist
+                 \item[\hskip\labelsep{#3#1\ #2\@thmcounterend}]#4}
+
+\def\@spopargbegintheorem#1#2#3#4#5{\trivlist
+      \item[\hskip\labelsep{#4#1\ #2}]{#4(#3)\@thmcounterend\ }#5}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+   {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+    \expandafter\xdef\csname #1name\endcsname{#2}%
+    \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+       {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+                           \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+      \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+\if@envcntsect
+   \def\@thmcountersep{.}
+   \spnewtheorem{theorem}{Theorem}[section]{\bfseries}{\itshape}
+\else
+   \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+   \if@envcntreset
+      \@addtoreset{theorem}{section}
+   \else
+      \@addtoreset{theorem}{chapter}
+   \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+\if@envcntsame % alle Umgebungen wie Theorem.
+   \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % alle Umgebungen mit eigenem Zaehler
+   \if@envcntsect % mit section numeriert
+      \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[section]{#3}{#4}}
+   \else % nicht mit section numeriert
+      \if@envcntreset
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{section}}
+      \else
+         \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+                                   \@addtoreset{#1}{chapter}}%
+      \fi
+   \fi
+\fi
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\itshape}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\itshape}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\itshape}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+
+\def\@takefromreset#1#2{%
+    \def\@tempa{#1}%
+    \let\@tempd\@elt
+    \def\@elt##1{%
+        \def\@tempb{##1}%
+        \ifx\@tempa\@tempb\else
+            \@addtoreset{##1}{#2}%
+        \fi}%
+    \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+    \expandafter\def\csname cl@#2\endcsname{}%
+    \@tempc
+    \let\@elt\@tempd}
+
+\def\theopargself{\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+      \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+                  \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+      \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}
+      }
+
+\renewenvironment{abstract}{%
+      \list{}{\advance\topsep by0.35cm\relax\small
+      \leftmargin=1cm
+      \labelwidth=\z@
+      \listparindent=\z@
+      \itemindent\listparindent
+      \rightmargin\leftmargin}\item[\hskip\labelsep
+                                    \bfseries\abstractname]}
+    {\endlist}
+
+\newdimen\headlineindent             % dimension for space between
+\headlineindent=1.166cm              % number and text of headings.
+
+\def\ps@headings{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \leftmark\hfil}
+   \def\@oddhead{\normalfont\small\hfil\rightmark\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\def\ps@titlepage{\let\@mkboth\@gobbletwo
+   \let\@oddfoot\@empty\let\@evenfoot\@empty
+   \def\@evenhead{\normalfont\small\rlap{\thepage}\hspace{\headlineindent}%
+                  \hfil}
+   \def\@oddhead{\normalfont\small\hfil\hspace{\headlineindent}%
+                 \llap{\thepage}}
+   \def\chaptermark##1{}%
+   \def\sectionmark##1{}%
+   \def\subsectionmark##1{}}
+
+\if@runhead\ps@headings\else
+\ps@empty\fi
+
+\setlength\arraycolsep{1.4\p@}
+\setlength\tabcolsep{1.4\p@}
+
+\endinput
+%end of file llncs.cls
diff --git a/nachlass/collected_dew_materials/n.pdf b/nachlass/collected_dew_materials/n.pdf
new file mode 100644
index 0000000..093c601
Binary files /dev/null and b/nachlass/collected_dew_materials/n.pdf differ
diff --git a/nachlass/collected_dew_materials/n.tex b/nachlass/collected_dew_materials/n.tex
new file mode 100644
index 0000000..06d2ad6
--- /dev/null
+++ b/nachlass/collected_dew_materials/n.tex
@@ -0,0 +1,1559 @@
+
+%%% Robert Telephone 512 6932 
+
+colin recommended doctors Ron Hoenzsch, John Bashant, 
+
+Marie Pesquera  475 9235
+1240 New Scottland Road, fax 475 0406
+
+
+Bashant Jan 8 2pm 
+
+Dr. Gregory Strizich 
+
+Neal Murray 442 3393
+
+SkyMiles Password 1DC 2RO 3AS 4Dn 1329
+
+Floght Confirm Number GR64UB flying aug 29 12.45 pm 
+returning Sept 7
+
+
+
+Libary Systems 442 3631 HP Laser Jet 4423631 Rebecca Muckridge
+
+computer renaissance 689 0068 
+
+Irina 512 6932
+
+Hwang's letters from Seoul, Lowell, Utah, Pittsburgh and Arizona State
+   where latter said his case was ``not clear cut'' at ASU although he
+   would vote for it. Problem was lack of publications.
+
+capito DIstrict Orthpedic 489 2666
+12.15 wednesday  brian Quinn
+
+discount 718 301 0800 
+New Irina the 0117 921 596 6342
+
+
+gregory dermatology 11.45 robert's appointemnt 
+
+micheal lyons 489 5468 
+
+1800 321 4267 ext 4144 135 room in baltimore
+
+unclaimed properties look below for address
+sigact.org 
+
+437 3874 paid 140.36 in 2012 Lucille 437 4505 
+
+toilet clearner ``The Works'' 
+
+James R Lewis
+    2080 Central Avenue
+    Schenetady NY near Lisha Kill Road
+      after LIsha Kill Road 1 1/2 mile after 155 at red light
+      conitinue 2 or 3 hourses and turn left Large Black Mail Box 
+      2080 is address white house don't pass bill board 
+      next house sells lawn mowers etc. 
+   90 dollars for six 456 3992 130 dollars 
+
+DeBuits Theater Company directed by Jason Ford 542 8422 
+Mickey Luce Lake George 793 3521   
+
+Mimi Clark at Middlebury 802  458 9252
+Rita Patey (assistent director)  at Harvard 617 495 4024 
+Mike Chin 
+
+ Emma Willard 833 1342 Jessa   They will call me... 
+   sharon return aug 23 
+
+429 2300 Albany Academy 
+
+Averill Park 674 7000 8am-noon guidanve vounseling 7025 or better 7021
+      Checkpoint Exam administered by Middle School 
+             674 7100 Middle School 7114
+
+irinash14@yahoo.com
+
+sept8  7.15 9.15 11.35-2.05 41dollars amtrak
+greyhound 2  800 231-2222 27 dollars
+   7.45 8pm 8.45pm 930pm 11pm 1130pm 1201am no more
+
+   after 7pm 
+
+HabernigProductions.com
+   go to menu bar last choice is purchase a DVD
+        click to enter the store 
+
+write a check to you
+     PO Box 3785, Kingston NY 12402  20dollars a dvd and 3 more dollars
+
+     
+ 
+
+caps Shrek 800 520 6303 press 1 habernig production in web
+
+ribert weyman 439 4806
+
+tigerdirect.com
+newegg.com
+
+abraham password compren
+
+%% jhu 410 735 6278 6143 6133 7718 or 1180
+
+%% 1 800 393 6095
+
+% id  1165270 pass word mGV6452
+
+
+la quita san diego 619 291 9100 107+parking 10 and tax 
+     Karina  talked to Alex
+        5 minutes to sea worl 5 minutes to zoo
+96, 81, 81  last three 74 10pernite parking
+
+
+
+irina364 5196 snow 475 0505
+
+
+fabiola 1-800 282 2774 122/nite closed 12.30
+
+ 7 100 bills for irina 
+
+RH 956 597 056 (one)
+
+RH 956 428 483 through 488 (six)
+
+
+Holiday 619 232 3861 130 not refundable
+
+
+SHampton Inn San Diego 1531 Pacific Highway 99/109    619-233-8408
+The Sofia Hotel 800-826 0009
+
+
+Focs 12 888 421 1442 139=tax confrimation first nite 10759642
+    second  nite 326 DW 63N
+hyatt regency new brunswick 732 873 1234 ****
+2 albany street new brunswick nj 08901
+6.30 11am pool
+
+NJ turnpike to exit 9 Hyatt rte 18 north parking complimentary
+
+607 433 9000 Hampton Inn 129 confirm 812-898 39
+    433 2250 Holiday  129 no free brakfast but resteraunt cancel 51417955
+    432 7500 Clarion Down town 135 
+
+holiday inn 607-547-8000
+
+Copy Shop 607 547 5150
+
+88 west to exit 17 
+Route 28 (north) 18 miles
+
+55 mph  breakfast cooperstwon diner, Duoble A cafe, DJ's
+
+
+parkplayhouse 434 2035
+
+Roland Downs 399 9126
+
+Irina 7-812-315-8206
+
+
+LOWELL CENTER HTEL IN MADISON 89 NITE
+   RES NUMB 237 075 
+March 31 - april 3
+
+1-608-256-2621
+
+BUS COMPANY VAN GALDER 
+
+zeltner 437 5700 march 21 wedensday 3.45 
+av1shag maya 364 4885
+
+
+
+George Washington Inn 202-337-6620
+
+Funger Hall Room 221 George Washington University
+corner of G street and 23-street (10-15 minutes)
+distance .3miles (google 7 minutes)
+
+Reservation 759-853 149 per day
+            759-854 249 per day 
+            cancel 766 590
+824 New Hampshire Ave
+
+
+jenny 201 242 7416
+
+irina 364 5196
+      258 3463
+
+
+
+AMS 401-455-4000
+
+Westin Copley 617-262-9600 confirm 252-15009 159+tax parking 46
+10 humtimgton aavenue
+
+
+Notre Dame J
+ournal
+user name =danwillard
+password = willard772
+
+
+stupid survey alfred2ruth2debby password corn1998
+
+comp ren
+ 689 0068
+
+November 10 Drug Pick Up
+
+%% remeber that Ellen Kozgrove Brandeis, Fordhan, ALBANY
+
+Barry Eisenberg Empire College
+
+Benjamin Ginsburg politic science Fall of the Faculty 
+
+Irina arriveing 5p as opposewd to 12.20
+
+Easy Tech .- programs >AutoSave> Instant Restart -> Start Backup
+
+718 301 0800
+irina cell 011-7-921 873 1498
+home 7812 314 5255
+
+Micro Soft Word Key: 7RH7K - 8CV77 - YDJMT - D7FG3 - 77KFJ
+
+
+chair 765 2169 <New Salem426 8946 < 417  south pearl 8.30-4.30
+
+Allen Maikel 487 4679
+Jeong	437 3662
+
+
+irina tel 011-7-812=314-5255
+
+   wdt  1718 301 0800
+
+Irina TC  RH=956-428-483...to 8 AND RH=956-597-056
+
+Lance Nevard Shelly Nevard
+Donald Nevard 718 680 8810
+
+One Source Talent and Acting 212 244 7161 x 1125 sivona hamilton
+                          45th between 6th = 7yh
+
+Monday 7pm inviation number 59983
+
+                      115 west 45th street 7th floor 
+6pm evaluation 45 minutes
+
+Peter Potter, Johan Melanson, Brent Hayward, Dan Connaly, Jamie Humphry
+
+Bradley Cooper, Ryan Gosslin, Eva Mendes, Greta Gertwig, cianfrane
+
+Ward Stone 466 1906
+
+Annul 428 8961
+Lake House
+
+trav check ME =RC451 952-189
+  BOTH = RH 956 428 feb 2012 485 to 488 
+
+wrong 483 to 5
+
+
+Kathleen Lasch McNamee VP PP 669 8142
+Dianne Elysee 845 679 2915 monday 8pm harmony
+sister is 15 years older and has 3 year old child
+
+
+Einstein part 10 disk chpater 12 November 11, 1919 editorial
+
+amit goswami ``the self aware universe'' 
+
+richard capeless 813 8689
+
+women's faculty club 110/night 510-642-4175mar24-mar27
+
+Durant Hotel 139 wed march 23 tel 510-845-8981 confirm 336 3933
+     cancel 336 5966
+
+randolph franklin 396 5003
+rpi   276 6077
+
+williamgoldenberg@gmail.com
+516-770-1020
+
+
+
+Francis Wick Nile's friend  388-6267
+
+Edward Kudlack 966-4598 621 7334
+
+best Western 673 3337 220.15
+
+Deaville 114=tax 305 865 8511 res ?? 1079122 cancel 1087027
+
+Howard Johnson 305 865 6661
+
+YMCA 477-2570 E Greenbusch   8-11
+
+456=3634 Guilderland 8,30-9.15 9,30-10.15 11-124
+
+655 East 14th Street apt 2a NYnY 10009-3139
+
+sophia tell emma to call sophia number = 434 9608
+
+irina cell in hy 364 5196
+
+irina hotel 516 678 1300 room 237
+
+trac phone 202 191 577 829 360 or 380
+
+Irina tel 011 7 921 873 1498 at 718 301 0800
+
+Adita Kulkani 781 730 0220
+
+
+Nine TRav Checks
+
+   Two RH956-597 055 +56
+   Seven RH956-597 428 481 thruogh 88 BUT NOT 83
+
+
+
+Chris friday 9.30am HSBC
+
+MEI HWA 442 4283
+
+Paul Morgan Consultant 518 728 9506
+PVMJR@aol.com 
+
+Ali Akbar Salehi
+
+Doctors
+
+Roy Fruiterman 439 8077
+or Stzrizich
+
+
+NORTON TMHPH-J73RF-H9V6V-4XRHD-DCGFR
+
+Empire Plan 890 100 573
+
+Albany Med 1 866 262 7476
+  Ref 222 059 156
+
+20 Dollar Balance
+Dr. Schwartz for Robert Willard
+
+mail to Albany Med Center Hospital
+        PO BOX 1189 MC-13
+        Albany, Ny 12201
+
+Paid to ALbany Medical Center Hospital
+
+for Reference 1004145166 check
+
+DeaR Dirs THis is for roeert with ref number for
+lab work done by Dr Schwartz
+
+telephoned on 10/6
+
+Columbia 1 800 774 2108
+        
+
+
+
+American 3713 492016 34005 <new checheed sept 2012 
+    old 33007 (old 32009) MArch 2012 NESEXT 43005
+Telephone 1-800 430-1000 800-257-0770
+confrirm s7030 on june 2 for payments
+
+CitiCard Visa 4147 1105 9492 1820 tel 888-766 2484 
+
+francois chelaviv work 833 1370
+    homw  482 1017
+
+carl resek 962 8793
+john resek 718 701 8765
+   cross street north of Venzel Platz
+                         top is Museum just past museum
+                          Vinohradska Blvd
+                           walk up 4 -5 to block Latinska
+                          If back is to museum then house to the right
+                          and uphill side of the street at corner
+                          of 2 streets. It is a 3-4 story building
+                          carl on first floor above ground
+                          and Cohn,s on second and ruth on top floor
+
+                         Entrance to bulding on Vinohradska Blvd
+                         and it was a corner building.                             
+
+
+Monte Carlo Hotel 1-800-311-8999
+Foundations of Computeters
+
+Confirm Number ttpbp
+
+OCtober 5
+
+My rates: 119 119 59-59
+
+Usual is 236 80 and 66 rates
+
+swiss air 877 359 7947 Flight 22
+
+mail to SSS by NAIS
+        PO Box 449 
+        Randolph Mass 02368
+
+Baboulis
+
+5,200 discount
+12,620 tuition
+
+17.820 tuition micheal green   
+
+
+Chis Abbey, Layla Johnson, Brent Abbey
+
+Swiss Air 2.50
+
+vinay Deolalikar wrong: Deolatikar
+
+jeff brown albany academy 429 2300 x 2340
+
+439 7972 wright garbage
+
+larissa telephone
+
+Vincent Electric
+Keith Vincent 518 469 0068
+
+Dr. Rosaline F Zam Barron
+
+Amer Express 1800 430 1000 s8466
+3713 492016 32009 new=34005     fouth each of month
+
+
+confrim s7646 
+
+s8488
+
+Citi AAdvantage 4147 1195 9492 1820
+
+fined 15 -35 dollars 446.66 confirm 2040 5042 0128 559 
+
+
+gold march 23
+
+covirm s 0931
+
+Bob the Plumber macDonal 756 2738
+
+S9combination 35 28 21
+
+
+
+Blue-2-naked 15-8-29
+
+Lock New Purple 38-4-10 
+
+NAIS Password crib1948
+
+Mrs. Olby
+
+Ksanznak
+
+Schwartz dermatolgist
+
+math department 442 4600
+
+
+
+
+Jim Travers cell 206 724 3509
+home 756 7591
+JATRAV@yahoo.com
+Walter Thayer and Lanny Waleter and Anita Thayer
+
+
+Nissim 718 570 6765
+DW795841
+corn22r*
+
+xtel
+
+
+Joane Hanley MIT 253 6054
+
+advisee 22 off hpur 12.40 - 2.40
+ 
+harry lewis 496-2424 thurs noon lewis@harvard.edu
+
+Sali Vadhan 617-496 0439
+
+Jay Walsh assessment 475 1206 dave (steve) 439 9601
+
+Irving House 617 547 4600 135 minus 10% AAA plus 14%
+  Hyatt	617 492 1234 199 plus 14%
+195
+cancel 11009
+ nmnaj7mnymyeee name name name bomn bon-bon-=bon-bgon-bon-bon-bon
+Harding House 617 
+
+dran 383 6720 442 3387
+
+  Harvard Square Htel AAA 161.10 617 864 6200	
+
+
+Ken 716 645 4738
+
+Fedex accout 281 613 936 
+
+1-800-622-1147
+
+  home 716 837 8363 Amherst NY
+
+Lia Toyoto 374 3700 colonie toyo 2pm open until 6pm
+Northway Toyota 783 1951
+
+Daughters of Vienna by Karl Adolph and Jo Sternberg
+
+Igor Zurbenko SUNY Statistics
+Roger FInk rfink@nycap.rr.com
+
+Stony Brook Renion 631-632-6330
+web site www.stonybrook.edu/alumni  October 9 (saturday)
+
+noon 12,30 to 3pm 25 adults 15 adults
+
+register bfore oct 6 save 10 dollars
+
+ffiday night and sunday
+
+stonybrook.edu
+
+Price Chopper Glenmont 433 4711
+
+Hol Inn Exxpress 631 471 8000 Homecoming Price
+3 Village Inn 631 751 0555
+
+stony brook Janet 631 632-6212
+
+
+1-888-930 5930
+
+mike jerrison 442 4287  Econ. 4735 108D 1pm
+
+irina hotel 1-242-373-4000 room c4-205
+
+incarcerated umbellical hernia
+
+
+
+Timothy Wilcox
+
+ATT 1 877 879 1872
+   code 359 650 1822
+
+bike contract 195 165 465 purcn june 9 2008 1-866 257 6545
+   seat raised exactly 2 inches.
+
+New Bike bought Apr 10 2010 contract 3Y2BIKEwar/n purchase number 40000300351
+
+
+women surgeons Linda Weber 482 5620, Barabara Burittos, Kelly Denntin 489 7245
+
+good course but should not be required
+killed eletric Power enigneering 
+       env engineering
+
+bob password 1twrs3tqf
+zeus password 3fpIrina
+
+Control Shift 9 to english
+
+Zeltner  437 5700 monday 4.45
+% Northeast Orthodpedic David Abra... or Fred Fletcher
+%% 
+
+old=459 5273 
+
+Allen Maikels 489 1040
+Middle School 439 7460
+   2.46 dismissal last class 2.01
+high 
+
+School 439 4921 ends 2.07 until high school
+   Ms Atallah
+   Heather Coleman ext 22919    
+out until seek of 19th 
+   superisor Marisa Bel 
+      Jaquie 22048 is her secretary 
+       Eric 5 years once a week PLUS costa rica camp
+      student peer tutoring in french
+
+Robert Willard
+    1 ENGLISH
+    2 FRENCH Ms attalah
+    3 Music Theory
+    4 Spanish 3e
+    5 ADVANCED CHEM
+    6 PE STH
+    7 workd history
+    8 honors alg 8
+
+Ms Atallah
+
+
+Lynn Cohler
+
+11am morley
+
+10am Lenhart
+
+melissa haas 439 7481
+
+
+between 7 and 7.15
+
+arubino@bcsd.neric.org
+
+Mrs. Ms. Rubino
+
+
+combination 35 28 21
+
+
+70dollars Transport Ferry Item 4997
+30 dollars ShangHai Chasse Item 7682 
+60 dollars Winter Toy Shop 10199
+
+richard Lipton 404 894 6438
+   cell phone 404 663 1214
+
+Dr.Goussous 489 7245 Kelly Denntin same office
+3.15 dec 17 thrusday
+ 
+and partner Barbara Brazis
+
+middle school 439 7460
+
+george berg 442 4267
+
+diagnosis abnornal cardiogram
+possible acute MI (you will know that 24 hours)
+Dr, Nappi
+Dr. Torossov 262 5076
+
+debby 938 4935
+
+
+wilcox 458 2488
+
+william 718 836 6600 ext 3124
+
+irina 262 5612
+
+emerg room 262 3131
+
+729 3841 ira katco
+
+434 9842
+
+70dollars Transport Ferry Item 4997
+30 dollars ShangHai Chasse Item 7682 
+60 dollars Winter Toy Shop 10199
+
+dan rosenkrants 561 304 0264 pacemaker no taxes retired 2005
+
+dick stearns  734 - 773 3334
+
+niles 482 0784
+
+NYSHIP pin 271 861
+
+tel 1 888-358-2198
+fax 1-866-371 6213
+wait 3 to 4 days with call
+
+myualbany
+
+DW795841
+corn22r*
+
+my fax letter should have pin and last 4 digits of social security
+
+can white out first five of social security and all
+financial information
+
+date due is Nov 25
+Bonnie at 74720 is suny contact.
+
+Flex Med Sept 21 thru Nov 16
+
+to atlanta 6am united 7449
+           9.40am    return united 7647
+
+Cheap Tickets and Hotwire
+
+radisson 703 920 8600
+Americn express 143 = 89
+reserv 1712-224 rates 179 116 71 71 24hours  
+
+Middle School 439 7460
+
+bobby locks 27-2-24   21-0-23
+
+Chirstion Worm Mortinsen
+
+NSF 703 292 directory 5111
+
+8,30am 4201 wilson blvd
+Dawn  Padterson dpatters@nsf.gov  
+703 292 8910
+
+holiday inn 703 243 9800
+256 for wed and thu 129 fri +11% tax
+
+yahoo questions ``swimming pool'' ``chevy'' 
+
+SEDA travel agency 800 817 5257
+558 for two delta and us air
+US Air 800 428 4322
+
+united 1800 241 6522
+to atlanta 6am united 7449
+           9.40am    return united 7647
+
+
+
+
+4,37 11.21 outbound 800 428 4322 resrvation number 
+confrim Bkty9q arrives 6pm
+
+panel P100147
+24dwaf
+
+703 920 8600
+6am to 7.23 or 11am 755pm exit reservaton locator
+940 am from DC 
+for airplane BGOSPZ
+
+alternate 11.21 to 12.47 to wahington
+from washington 9.40am
+
+free service 7am to 2pm
+
+FOCS 2009 oct 24-27
+
+Inn of Rosslyn
+703-524 3400  99nite seth 442 4282 104.99 confirn 24972 arrive tues
+must contact oct 10
+
+best 703 524 5000
+
+Radisson in Atlanta at 139/nite (frist 3 nights and 129) last 2
+Reservation 902 654 12 1-800-468-3571 
+change to 24 arrival first 2 139 then 129
+
+
+Sheraton 404 659 6500
+
+Hampton telephohne 404 881 0881
+
+Indigo at 117 117 per night at reserve 6257 5905
+cancel 5742 4094	
+
+slightly further away at 115 Hampton with breakfast
+
+
+877 477 8003
+
+booked through Hotel.com
+
+
+albany academy 429 2300 FAX 465 5021 SUMMER CAMP 429 2429
+
+Capitot Eye Care
+274 0657
+2200 Blodgett Ave. Troy
+
+John Brice of Lee Nagel
+303-544-1198
+
+Dina 400 Hudson Ave. 449-7797
+
+Fira Effimona 472-1039
+
+
+res number at Morris 97972 talk to wendy or shanon
+132 + tax
+
+
+Notre Dam Main may 20-23 132 + tax
+574 631 2000
+	232 4000 Yes expensive cancel 243 431 
+	234 5959
+ivy court 277 9682 reserv 29508 cost 79 +tax
+	second res 30371
+
+
+   midway is closer than chicago international 
+   Coach USA	 574 234 6600 website coach.usa.com
+
+irina 258-3463 
+
+Stralkowki 453 9641
+
+Eric Lifshin 437 8765
+		 8686
+		 8687
+
+wrong eric phone 437 8086 8686
+	
+
+
+mom	258 3463
+best westgern 305 673 3337
+
+Pine Ridge Ski 518 283 3652
+
+ravi  608 6741 
+
+Amer Express RC 913 298 907 + 8
+
+Anil gupta 412 624 5771
+
+Mrs. Reilly
+439-4139 13 York Road
+
+melonson johan 439 3328
+father Douglass mother christine
+
+
+Vadim Kozlov 442 4627
+	vk614151@albany.edu
+	mvkozlov@mail.ru
+
+
+Ira J Hecklar probably ``Heckler'' John
+
+
+New Nsfpassword robert98
+NSF robert10 robert12 000185268 000185268 bobby2 robert33 
+password 2468xy
+
+NSF 800 673 6188
+
+Panel  100147
+
+usher 437 4554
+
+new comp 3fiP0g@w
+
+beigel 202 243 1243
+       609 203 2000
+       703 292 7849
+
+TIKI 518 668 5744 irina room 318 
+
+JENNY 201 242 7416 265week300
+
+staples fax 518 426 1334  office 426 00
+
+
+44
+
+fax me 518 433 0375
+  and their teleph is 0374
+
+columbia tel. 800-774-2108
+
+confirm numb 06101013047392 march 12
+
+catherine at NR 212 679 7330
+
+hkay 11am
+
+474 5261 Rodney
+
+286.1 miles and 8.491 gallones 33.64 aver
+
+Science News editor;  Julie REhmeyer jrehmeyer@nasw.org 510-868-0842
+
+irina 258 3463
+
+Rober Lubarsky 561-297-3341
+
+gardener Bob Young 439 6393 Jim Maki 756-6258 cell 461-5003
+	bad gardener mark visat 861 7003
+
+ Logic Student Lelan Belmont
+lb681716@albany.edu
+
+paines and patressa
+
+
+John Silver
+655 East 14 Strees
+Apt 2a
+Ny,NY 10009
+
+Camry hybird key 47569
+
+Radisson hotel march 25 confirm 357-495 104  tel 949 833 0570
+4545 MacArthur Blvd
+newport Beach 92660
+
+confirm 357-495
+
+Kinko's Miami Beach 305-532-4241 Howard Johnson 305-865-6661
+
+24 hour kinko 518 482 9094
+
+nicolson 785 7283 tuesday 10am  wednesday 3pm
+  
+Brad noon on wednesday
+
+tues sep 20 noon mon sep26 12.45 
+
+Office Hours Monday 2.15-3,15 wed 3-4
+
+
+temple israel 438 7858
+
+
+george berg 591 8822  442-4267
+
+advisement jul 28
+
+10.15 albany 1.30pm richard Huhl
+
+veronica 439 4862
+
+Stae farm 1 8-- 884 4415 Ext 3603 claim 5282 12506
+
+irina 258 3463
+
+sophia hotel jan 5 -10 confirm 3 44712 619-234-9200
+		jan 11 confirm 143 confirm 44713
+
+
+password is whatfor and code is 016-206-195
+
+Please describe top 2 strengths
+
+Please describe 2 ways person could perform even better.
+
+
+Medical Savins Plan 1-800-358-7202
+www.flex
+
+November 16-Nov30
+
+Nick Sternberg 5pm
+  santa monica 310 394 1802
+  llano 661 288 1915
+  cell phone 661 645 3000
+
+SPEND/STATE.ny.us other phone 1 800 342 8017 second
+laly's email eulloa@fbmc-benefit.com
+
+ password suny63corn CHANGED TO suny93corn
+
+
+cofirm 2011 172 354
+
+FDMC ID number 0022 0330 7133 7005
+
+confirm number 147 142
+
+i wish to change to 2,000 in 5 payments at start
+second confirm was 106681
+
+dan willard of ojai music 805-646-1468
+          
+strz pediatrician  453-9641
+height 72 1/2 february 19
+
+ 
+userid danwillard19 password=robert98
+
+user id = danwillard
+password = plato22
+
+first confirm 106415  amount = 1,800 in 24 payment of 75
+
+I talked to Laly at extension 2415
+
+Department = 28010  UUP = 08
+
+user-id = danwillard
+
+password = alfredruth22 or alfred22 or ruth22
+
+
+Director of Ethics Commission 110 State Street
+Lawrennce Schantz 473-6318 (diretor)
+
+
+Who' Who 908 673 1001  code WA6322EC
+
+Adreanna Cortez of John Hancock Insurace 617-663-2468
+				fax is 2700
+	1 800 344 1029 Extension 32468
+
+5498 form oct 2007
+
+Gardeners matt High school 864-6269 and nick 209 8861
+	40 dollar professional Scott Sweet 756 9637
+
+rhode Island res 82497370 need to reqest refrig
+
+sfaxChronic Obstructive Pulmonary dissease COPD also called chronic bronchitus
+`\	emphasema 1866 316 COPD
+
+	tax citibank 2,081.90 
+
+	M and T bank 1.04 checking b  c cgfdmfn ,kk;
+
+
+Local Fax 433 0375 (phone 0374)
+
+New York State Comptroller = Unclaimed Property, Albany, New York
+
+Dailey News Monday March 26
+      osc.state.ny.us\out
+      osc.state.ny.us/out
+
+      osc.state.ny.us/ouf
+
+
+Dan's hacket in irina closet near wall (where big hanger is).
+
+internet 1-718-3010800 011 7812 314 5255
+
+Ward B  Stone WAMC FM albany 90.3 8.30 473-3032
+Oppenheimer wed 3.30pm
+
+
+
+tanya and david 449 4782 and 258 9446
+      jeff geene 857 0619
+      enameled trap higher flow toilet
+      pressurized toilet flush clean rating
+      home depot and check
+
+
+2.30pm dr timofeev 243 3387
+
+United telephone for 25 dollars 1-800-241-6522
+UNITED old reservation numb ANJ S4n
+with 953.50 credit for Promotion 13TCVA
+   and Pin Number of 9RJ 99E C89H
+
+friday mlrning 655am sunday 5am 206.50 +25 ARRIVE IN BALTIMORE
+   BALTIMORE 3916 5AM 5007 7.38AM 8.50AM
+    NEW CORNFIRMAT PGZHLL
+
+SUNDAY 9.38AM 1.03pm  2,15PM
+
+
+must be spent by 28 May 2014
+BUT can fly after and transfer.
+
+Baltimore flight via Newark
+  leaves albany on FLight3845 at 6.55am and travels via Newark to
+  Baltimore
+
+
+New Combination 30-36-6
+
+Medical Savings Plan 1 800 358 7202
+	www.fexspend.stat.ny.us
+	www.fex.spend.stat.ny.us
+
+Nicolson 785 7283  wed ap 1 2.45
+may 6 1.30 
+
+dentist 1pm wednesdAY
+
+
+robert kovacs 220-9602 latham 783 0018
+		cell 527 6693
+
+bad credit card 1-800-950 5114
+
+459 5273 zeltner
+DR Roy Fruiterman 439 8077
+
+
+505 1824 belinda 2pm friday 11am  clifton park 
+2.30pm
+
+6,672.62
+
+17,000 insurance paid 15,000
+16,000 and we paid 60dollars one night 
+
+coverdale education savings 6700
+
+45,000 irina
+
+keegan 456 2910 insurance 250.080 208,400,
+
+rob 432 2961 cel 527 6693
+
+December 18,2006 contribution
+
+bleinda clifton park twth373 6504 cl
+
+Lawrence GORDON 2PM RTE 50 GLENVILLE, 113 Saratoga Road, Tuesday 14
+	250  , 350 
+	exit 25 890 west until exit 4c left at end of 4c
+		3rd light right , 10 minutes
+
+All-Side vinyl sliding
+
+Cliff the carpenter 376 5049
+
+tel 463-2736
+   
+windjammer 718 891 6600
+comfort 718 368 3334 200 2 full size
+res 160732243
+cancel 7521348
+
+
+best west 769 5000 179
+743 4000 golden gate inn
+
+
+janey  914 262-9245
+
+Dover Plains NY 12522
+Dover Furnace Inn, 11am
+Rte 22 30 Ore Bed Road 
+tel 845 832 6594
+
+dutchess motor inn 845 832 6400  73 night
+
+Tac Pkwy South to NY 23 East NY22 south turn right onto countryn road 26
+called Dover Furnace Road, turn right Ore Bed Road
+
+
+lucy 914 ow3 1559
+
+irina Debby?? 256 8164
+
+belinda 439 4526
+
+Karen W. Arenson of NY Times
+      areanson@nytimes.com 212-556-3500
+
+Delta Airlines 1800 221-1212
+
+veronica 475 9778
+bigelow
+
++++++++---klima 420-318-581-202
+	241-725-655 or 657
+1 718 301 0800
+ivan's email iklima@volny.cz
+
+helena hklimova@volny.cz
+
+
+selmer bringsjord
+
+janey 914 262 9245
+peter wilhartitz 43-69911-6-07960
+
+cwillis 530-282-8158 
+fawn babydeer37@hotmail.com 
+
+meri sternberg 661 799 7896
+nick sternberg 661 944 2527 njvs@juno.com
+	17112 Fort Tejon Road
+	Llano, california 93544
+
+Sprint
+ 1-800-708-0981  552 999 5234
+
+1-800 971 7409  conde 463 738 9925
+
+
+New ATT card 1-800-487 7646 
+
+code  7881-2011-6329
+     
+
+
+nemo thur aug22 1pm third 
+
+8pm sat 8pm yes sunday 5pm back-up
+
+5pm 
+ 775
+ 746 3630 office home 746 2154
+    1pm this wednesday
+friday noon or monday 1pm
+thurs noon
+
+tues aug31 8 pm
+
+monday 4.15
+
+Wrong Address Probably: 1920 Evergreen Ridgeway Reno, Nevada 89523-3804
+
+Correct Address Probably 9140 Sprcue Creek COurt  Reno, Nevada 89523
+
+
+
+friday 1pm on jul 16
+
+5pm daytime thursday
+
+thur 7pm
+
+9140 Spruce Creek
+Reno Nevada 
+
+439 7700
+
+red eye 212 541 9000 redeye.com
+
+john howell 810-3741
+ashwin 442 4422
+	533 9839
+
+
+holiday inn disney   800-366-5437
+			 407 396 4488
+			buget rental car 396-0928
+www.orlandofamilyfunhotel
+
+Lock 22-37-15
+
+carl friedman 486-6090
+
+legal aid 462 6765
+cap district 438-5511
+new york bar 463 3200
+	alb county	445 7691
+
+533 2843 code   6004 127 6761
+	april 25 2087 630 8384 
+
+
+	april 25,2005 9232 299 0430
+	april 25,2005 ATT 800 325 9370  6306 6926 9478
+
+
+
+800-901-1961 code 993-086-1339
+	901-2031 debby 637-098-4304
+		new 142 933 9914
+
+   901- 8182 kmart 459-797-7730
+		new 922-471-4358
+
+debby lunch 12.45 1.25
+
+1 800 901 2031 sprint code 637-098-4304
+
+449 4782 tanya and david 
+
+lance nervard 878-0930
+
+sternberg's niese is Barbara Meyerson 1-505-892-5542
+	6127 Cotton Tail, Rio Rancho NM 81724 1545
+his son is Nicolos area code 661 Llano
+
+painter bob nuttal 356 4785
+	interior purchased at Latham Paint. It is
+	Pittsburh off white 6-84
+		(Nuttal first said 6-78 but that was wrong.)
+		with walls flat and woodowork semigloss offwhite
+	outside not available but near match in can
+
+    lauretta gastwirth mineola lawyer for debby
+
+ Loretta Gatswirth 516 747 0300 recommends Steve Taub after tuesday call 
+
+515-797-6050
+
+516-728-6150 debby cellphone
+
+Robert Kozlovksy poeidatric pulmonoary
+
+297 dollar computer 800 884 9510
+
+Experediea 1888-397 3742
+	eveirena.corn.22r
+	platoalfred
+erev rosha shonna sept 15 wed nite sept 24 nite yom kippur
+
+grafton 292 4553 direct
+        8910 direct and 8912 office
+
+sandwich, juice granolla
+
+timothy lance 442 4605
+
+irina mobile russia 7-812 8921 351 7560
+			315 8206
+irina nwe american mobile telephone 496 7696 
+
+surgeon = Willix
+
+My machine is named blake
+
+Univ rome tr RT-DIA-80-2003 pp. 68-72
+
+314 5255 5253 irina
+
+david konrad 316 1162
+
+myalbany DW795841 passworn = corn.22r
+
+netzero password roslyn22c danedwillard dancorn
+linux root pw compren
+NSF PASsword science13
+
+
+psych Greg Levine
+Carol Ipsen and Ralph Carotemuto
+
+sica thomas and julia 478 9676
+
+beth
+
+elem 439 4955 4pm
+nov 9 5pm dec 12 4pm probably second week in Jan
+
+alice berke 489 1117
+
+eleanor pearlman 689-0244
+
+carrier Big Zoo
+
+lawn 756 6258
+
+www.net.o.com
+
+439 3265 tri Village summer camp
+
+connection by start > programs > accesoories >  Telnet
+
+500 b-  503 8 518 a- 604 I 508 B- 660 B- and ? (took twice) 661 A
+
+
+Yang Zhaohui  482-3397  cell 229 3709
+      summer 59444  fall 84042 or 92
+
+
+Siobone	459 8565
+
+charless and jenifer and 
+eric  silver 10.30am firday
+  674 1301
+
+cell phonee 598 3736 jenifer and 598 7254
+
+3.30 -6.30 today noon-3 sat 10.30 3.30 sunday
+
+gail taylor 785 8782
+
+irina 258 3463
+
+william goldenberg	516 770 1020
+
+dan silver 301 729 7856 ** professional voice coach or music dept
+           work 877-982-2322   304-434-8000 x 248
+
+** is still Dan's number
+ 
+
+
+  microwave hear
+  hot water
+  therma care large size ask pharamcist
+
+
+
+cell 240-727-7024
+
+250k auto-tech equip
+
+doxycycline 100mg twice day for 10-21 days JUST GOT
+
+cefuroxime axetil 500mg twice day 14-21 days BEFORE
+
+hoyel 888-532-4324  mon inn thurs out 
+
+Physician Referall Services 1-877-262-8008
+
+No GOOD 
+Timmothy J Sellati Albany Medical College (research on it)
+Infectos Diseases PhD
+
+Jon SIlver 260 5980
+655 East 14 Street
+Apt 2a
+NY NY 10009-3139
+
+587-5542 Lee?
+
+Days Inn 619 232 1077
+lawm mowing high school 475-0808
+	service i used	966-5609
+	chris jordan	439-8652
+
+bill augustine 437 3733
+grant 6503
+
+587 5542 lee nagel Exit 14
+2nd real left after bridge exactly 1  mile
+
+
+carrie nemovicher 201-541-7204
+
+burglar alarm 1234 initial password
+my pasword irin
+
+avishagmaya@gmail.com
+
+AVIS RIP-OFF  1-800-678-3029
+	CALLED  ``TRAVELER'S ADVATAGE''
+	cancellation number vvj 3206
+
+brad armour garb 442 4256 (dep 4250) 439 1919
+
+tues noon
+
+Micheal Braun 519-884-8575
+	Wife = julia bora 
+
+	dance = American Dance Center
+	Phylis Latin 584-8733
+
+
+NY Times discount s88 ag1
+	account numb 828 489 013
+
+debby 4836 at ny times
+1.75 5.75 3.80
+
+
+Days Inn Conventin Center 67.15  
+	619-232-1077 res, 516 360 55
+Mariott 8200 9308
+
+Nieba 503-234-7432
+cell phone 503-891=6192
+212 473x-7776
+aarhus on mainland
+
+Lundbee 
+
+April 27 saturday 7pm York Avenue waldorf school at saratoga springs 
+
+peter siegal  510-526 5213
+	work 415- 667 0860
+
+turbinate cauterization
+
+bob sarnoff and Louise 858 535 1638 w 4559100 6pm a week from monday
+2nd cancel 006 010 145 8163
+aug 17-19
+
+
+Jingjan 522 6356   3pm
+
+if you like you to do the surgery
+if you don't mind we would you prefer do we the surgery
+
+john kelly 
+
+rsarnoffmd.aol.com
+
+Jeff Gardlin 	202-267-5091
+		9-awa.tellfaa@faa.gov
+
+
+peter home   510-526 5213
+	cell 415-254-9154
+	w0rk 415 667-0860
+
+Frank Olken 703 292 7350 (direct) and 8900 (receptionist) 8am to 5.30 pm
+Maria Zemakova
+
+sarah calametra fine
+
+Sarah.c.fine@gmail.com
+
+
+dick.tsur
+@gmail.com
+dick tsur 650-968 8003**home  8040 cell 862 6962 after 9pm cal time.
+
+	1076 El Monte Avenue, Mountainview
+	Rte 17 meets calidornia 85 
+	Take 85 north to San Francisco
+		take El Camino Exit
+		proceed north on El camino
+		after 4 or 5 traffic lights left onto El Monte
+		   2 minutes and I'll see gas station on right
+			and then church on left.
+			House on right at 1076, driveway shared by 2 houses
+
+			5 pm 
+
+		1 hour  in dry weather
+
+
+zeynop 1-347-647-0726
+
+2.30 call to peter
+
+
+directions West from University on University Street
+	   1 mile ?? right on San Pablo
+	   1 mile left on Solano Ave.
+	   next street Adam Street right
+	  2 blocks down left sice 720 Adam Street
+%%% home 2009 oct23  10.55 am
+
+%%% DON'T FORGET health insurance
+
+%% FLEX MED  and credit card
+
+%% Must do taxes soon
+
+%% BUDCO 1 888 358 2198 Nov 25
+
+%% NYSHIP 
+
+%% Add Feferman reference to second draft
+
+%% ATT 1-877-879-1872
+%%
+%% ATT code 359 650 1822 
+%%
+%%
+%%cheap 1-877 879 1874
+%%
+%%cide 628 762 1030
+%%
diff --git a/nachlass/collected_dew_materials/nate-cam/june2020/7am-polished-draft b/nachlass/collected_dew_materials/nate-cam/june2020/7am-polished-draft
new file mode 100644
index 0000000..06f3dda
--- /dev/null
+++ b/nachlass/collected_dew_materials/nate-cam/june2020/7am-polished-draft
@@ -0,0 +1,1242 @@
+% 2020 5  june 6 6.1 am ( aftr spell and subsequent corredions)
+ 
+
+
+
+% 2017  august 30 8.5 am MINOR REVISONS PAGE 6
+%%Removing rphraisng  skinny as Robert recomeneded
+ 
+
+
+% vladimir mechanic 265-2212
+
+% ``%ss% = sections seth recommendedI skip
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-1.1in}
+\setlength{\textheight}{9.4 in}
+\setlength{\textwidth}{6.5 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+\title{$Very~~ Very~$
+Informal Notes for Cameron and Nate from Dan}
+
+
+%% \title{On  How
+%%   A Novel Indeterminately Defined
+%% %$~\theta~ \gimel$ 
+%% Function Primitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+%% 
+%% \title{On  How
+%%   An Indeterminately Defined
+%% $~\theta~ \gimel$ Function Prmitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+
+
+
+
+
+
+%%% 
+
+
+%X%% \title{On  How the 
+%X%% Ixntroducing of a 
+%X%%  New $~\theta~$ Function Symbol
+%X%% Into Arithmetic's Formalism Is 
+%X%% %Likely 
+%X%% Germane
+%X%% to Devising Axiom Systems that Can 
+%X%% Appreciate Fragments of Their Own
+%X%% Hilbert Consistency}
+
+%% \title{Why a Small Fragment of Hilbert's Consistency Program
+%% Ought to Be Feasible
+%% for Hilbert-like Deductive Methods
+%% After A New $~\theta~$ Function Primitive
+%% %AFTER A NEW ``$~\theta~$'' Function Primitive
+%% Is Added to Arithmetic's Formalism}
+%% 
+
+%% \title{ A 2-Part Conjecture about
+%% How  a Much-Diluted but Non-Trivial
+%% %Variant
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Is
+%% Likely
+%% %Plausible 
+%% Feasible
+%% for the
+%% % Even the Challenging
+%% Case of
+%% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+\begin{center}
+\large
+June 6, 2020
+\end{center}
+
+\begin{abstract}
+\Large
+\baselineskip = 1.65 \normalbaselineskip  
+These notes are written quite informally
+and they are meant to summarize the types of
+probability distributions that will construct 
+a decently 
+``randomly''
+formalized $\Theta$ function for my Cornell paper. This manuscript was
+written in only a couple of hours time, and I therefore apologize
+for many likely examples of carelessness in my QUITE INFORMAL extension
+of \cite{ww16}'s results.
+\end{abstract}
+
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+\def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \Large \baselineskip = 2.5  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+%fffff
+\def\gvs{ \normalsize \baselineskip = 1.3 \normalbaselineskip  \parskip   5pt}
+
+\def\gvx{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip    3pt}
+
+\def\gvs{ \Large \baselineskip = 1.9 \normalbaselineskip  \parskip    5pt}
+
+\section{Crudely Composed Notes}
+% \section{Scientific Notes of Dan Willard Notarized on Nov 22,2016}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+
+
+\gvs
+
+I will use the phrase ``probability distribution'' quite informally
+in our discussion.  The goal is thus to ``invent'' {\it any type}
+of Lebesgue measure that will assure that there is a probability
+bounded below by some tiny constant $~c~$ where
+\cite{ww16}'s $\Theta$ function will produce a consistent self-justifying
+formalism.  {\it Even if that probability is tiny,} any discovered 
+lower bound  $~c~> ~ 0 ~$, 
+will assure that the  IQFS($\beta$)
+formalism is consistent when $~\beta~$ 
+holds true under the Standard Model. This is 
+ because
+IQFS($\beta$)'s
+ Group 0, 1 and 2 axioms trivially hold true under the standard model
+and its final Group-3 axiom sentence cannot be proven false 
+when our probability framework can generate a model
+where it holds with 
+an explicit
+probability lower bound lower bound of  $~c~> ~ 0 ~$.
+
+(In other words since ZF Set Theory can formalize the validity of
+G\"{o}del's Completeness  Theorem, the existence of some model
+satisfying a  probability lower bound lower bound  $~c~> ~ 0 ~$
+will be sufficient for establishing that
+the
+ Group-3 axiom's 
+self-justification statement will not be contradicted.)
+
+
+Please allow me to be informal here because I am trying to
+quickly
+ compose
+a rough approximation of working notes, without delving into 
+a bevy of 
+tedious details.  
+
+Let us recall that page 11 of \cite{ww16}
+defines the $\zzthe(x)$ function-mapping  
+%% haphazard
+to be an
+operation
+ that maps powers of 2
+onto powers of 2
+subject to the following rules:
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+We want our probability
+ distribution to have the property
+that any recursively defined function has a zero
+ probability of
+occurring.  Thus the countable set of all recursively defined functions
+will also have a probability zero of occurring.   BUT YET 
+the $\zzthe(x)$ primitive will grow at a slow enough rate
+that it is incapable of producing a fatal diagonalizing contradiction,
+while satisfying the crucial constraints in Lines  \eq{walk1}-  \eq{walk3}.
+
+I can immediately think of three likely ways of doing this.
+{\it 
+It is (?)  possible all three 
+methods  will
+work,
+successfully.} Among the three plausible methods,
+Method A is the simplest procedure, and Method C is the most
+complicated. The virtue of Method C is that it is the one that
+I am most confident about, although its procedure is a complex
+hybrid of methods A and B.
+
+All three of these 
+randomized
+methods will first generate the value of  
+$ ~\zzthe(1)~$, and then calculate in chronological
+oder the values of 
+ $ ~\zzthe(2)~$,  $ ~\zzthe(4)~$,  $ ~\zzthe(8)~$ etc., 
+This chronological order is important because
+once  $ ~\zzthe(2^i)~$ 
+is assigned a values of  $~2^K~$
+then all  $~ j \, > \, i~$ are forbidden by rule 3
+from mapping    $ ~\zzthe(2^j)~$ onto $~2^K~$.
+This Rule 3 will be called the {\bf Exclusion Principle}
+during our discussion of Methods A-C below:
+
+
+\subsection{Method A: The ``Pairing Method''}
+
+The Pairing method will begin by finding the two smallest
+powers of 2, 
+at least as large as 2,  where no $ j<i $ has  $ ~\zzthe(2^j)~$ 
+correspond to  one of these two powers of 2,
+which we write as $~2^{K_1}$ and 
+ $~2^{K_2}$. It will then set    $ ~\zzthe(2^i)~$
+ equal to one
+of these two values,
+ with each quantity receiving a 50 \%
+probability of occurring. 
+
+It would not surprise me if this Pairing rule (by itself)
+would 
+be sufficient to produce our
+ needed final result.  It might, however,
+be problematic because it would cause   $ ~\zzthe(2^i)~$
+to always satisfy the following
+possibly excessively
+ tight constraint of:
+\beq
+\label{tight}
+\forall ~~x~~~~~ \zzthe(x)
+  ~ \leq ~ 4 ~ x^2
+\enq
+
+\subsection{Method B: The Almost-Stochastic Independence Method}
+
+This method will assume that we have an infinite set of
+real numbers greater than zero, $~p_1~,~p_2~,~p_3~,... $
+such that $~p_j~$~ is the probability that 
+$~ \zzthe(2^0)~=~ 2^j$.  (
+The easiest example
+\footnote{ \normalsize If
+ $~p_j~=~ 2^{-j~} $  then
+$\sum ~p_j~~=~~ \frac{1}{2}~+~  \frac{1}{4}~+~  \frac{1}{8}~+~...~~~ =~~~1.$}  
+ of this
+is when $~p_j~=~ 2^{-j~} $.) 
+ Then for each 
+subsequent 
+power of 2, denoted as $2^L$, the same probability distribution
+will be used, except that the  ``Exclusion Principle''
+will be followed in precluding
+repetitions from
+occurring.
+  Thus, let  the prior  powers of
+2 be mapped onto the quantities of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$ 
+and let $~S~$ be defined as below:
+\beq
+\label{s-sum}
+S~~=~~ \sum_{i=0}^{L-1}{ p_{k_i}}
+%
+%%% p_{ \small K_0}~+~ p_{ \small K_1}~+~ p_{ \small K_2}~+~.... ~+~
+%%%% p_{ \small K_{L-1}}
+\enq
+Then assuming $~2^j$~ is not one of the previously 
+taken
+       positions of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$, the method B will
+have 
+hold
+$~ \zzthe(2^L)~=~ 2^j$ with an obviously adjusted  probability
+of 
+\beq
+\label{adj-prob}
+\frac{p_j}{1-S}
+\enq
+\subsection{Hybridizations of Methods A and B}
+Again, we would not be surprised if either Methods A or B
+(or both)
+ would 
+establish the conjectured self-justification property.
+However, there are also 
+other
+plausible hybrid methods
+that would establish this  property.
+Here are two examples
+\bed
+\item{$~~  Method-C-1 $}. 
+We follow Method A's approach with probability 0.5 and
+method B's approach with probability 0.5.  
+Thus, either of Method A's two choices occur with a 
+precise
+probability
+        of
+$~\frac{1}{4}~,~$ and Method B's setting
+of 
+$~ \zzthe(2^L)~=~ 2^j$ occurs with an exact  probability of:
+\beq
+\label{adj-prob1}
+\frac{p_j}{2~(1-S)}
+\enq
+\item{$~~  Method-C-2 $}. 
+We follow Method A's approach with probability of $~S~$ and
+method B's approach with probability $~1-S~$.
+Thus, either of Method A's two choices will hold with probability of
+$~\frac{S}{2}~$ and Method B's setting of
+$~ \zzthe(2^L)~=~ 2^J$ occurs with a  probability of
+exactly $~p_j~$.
+\end{description}
+
+\section{Basic Strategy}
+
+The basic strategy is to show that  if
+axiom basis
+$~\beta~$ holds valid in the standard model
+then it is impossible for a minimal proof of $0=1$
+to exist, Providing, just an overview,
+this should be able to be proved,
+{\it roughly,}
+ by contradiction.
+
+In particular, if $~\beta~$ holds valid in the standard model
+then all the Group 0, 1 and 2 axioms will hold valid under the
+Standard Model.   Hence
+if the consistency preservation property fails, then
+the Group-3
+ {\it ``I am consistent"  axiom}
+must be false under the standard model.
+Thus, the proof of the falsification of the
+ Group-3
+ axiom
+ must be able to self-reference itself
+and
+thereby
+ establish its own incorrectness.
+
+At this juncture, one must use a natural encoding for the
+G\"{o}del 
+numbers of a proof
+(such as what was formally given in 
+\cite{ww1,wwapal}'s
+examples of G\"{o}del encodings).
+  The proof of the existence of such a proof
+will exceed the proof's length by a factor $\lambda > 4$
+(or more)
+ under all
+natural encodings of proofs
+(including the examples given in 
+\cite{ww1,wwapal}).
+ But the point  
+is that our formalism
+{\it  has only the} $\Theta$ operation as
+a primitive,
+for
+ representing growth.
+Thus, the natural probability
+distributions from the prior section should establish something
+to the effect that there is a
+probability  lower 
+bound
+of
+ $c > 0$,
+such  that an
+excessively fast
+ growth-rate will be impossible.  
+
+Thus leaving aside many
+messy details, this lower bound will assure that there is stochastic
+model where growth is precluded at a fast enough rate for some
+demonstrated model to form the type of counterexample
+that G\"{o}del's Completeness Theorem needs to show that the
+ {\it ``I am consistent"  axiom} needs for corroborating its claim
+for self-justification.
+
+This result differs from my earlier work in that it needs ZF Set Theory
+(rather than Peano Arithmetic) to corroborate what I call the 
+Consistency Preservation Property.  That is fine and legal
+because the system IQFS($\beta$) 
+affirms its own consistency via
+a 1-sentence axiom.
+Thus, ZF Set Theory's knowledge about
+Lebesgue measures 
+should, likely,
+indicate  IQFS($\beta$) 
+is a competent
+enough
+ formalism to make no false claims.
+
+\newpage
+My apologies that the preceeding
+short summary
+ is not a formal proof.
+It merely outlines,
+{\it very roughly},
+ what I have in mind.
+
+
+
+\begin{thebibliography}{99}
+
+
+
+ \normalsize
+\parskip 5 pt
+\baselineskip =  1.3 \normalbaselineskip
+
+
+
+\bibitem{ww1}
+Willard, D. E.:
+ ``Self-verifying  systems, the incompleteness
+theorem and the tangibiltiy reflection
+principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explaining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+\bibitem{wwapal}
+Willard, D. E.:
+``A generalization of the second incompleteness 
+theorem and some exceptions to it''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071v5
+         (2017).
+
+
+\bibitem{ww20}
+Willard, D. E.:
+``On the Tender Line
+Separating Generalizations and Boundary-Case Exceptions for the
+Second Incompleteness Theorem under Semantic Tableaux
+Deduction'',
+a talk given 
+on January 7 at the LFCS 2020 conference.
+Early version in 
+Volume 11972 of
+ Springer's LNCS series, and longer version sent to
+Cameron and Nate.
+
+
+\end{thebibliography}
+\end{document}
+
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--- /dev/null
+++ b/nachlass/collected_dew_materials/nate-cam/june2020/n-print.tex
@@ -0,0 +1,1242 @@
+% 2020 5  june 6 6.1 am ( aftr spell and subsequent corredions)
+ 
+
+
+
+% 2017  august 30 8.5 am MINOR REVISONS PAGE 6
+%%Removing rphraisng  skinny as Robert recomeneded
+ 
+
+
+% vladimir mechanic 265-2212
+
+% ``%ss% = sections seth recommendedI skip
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-1.1in}
+\setlength{\textheight}{9.4 in}
+\setlength{\textwidth}{6.5 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+\title{$Very~~ Very~$
+Informal Notes for Cameron and Nate from Dan}
+
+
+%% \title{On  How
+%%   A Novel Indeterminately Defined
+%% %$~\theta~ \gimel$ 
+%% Function Primitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+%% 
+%% \title{On  How
+%%   An Indeterminately Defined
+%% $~\theta~ \gimel$ Function Prmitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+
+
+
+
+
+
+%%% 
+
+
+%X%% \title{On  How the 
+%X%% Ixntroducing of a 
+%X%%  New $~\theta~$ Function Symbol
+%X%% Into Arithmetic's Formalism Is 
+%X%% %Likely 
+%X%% Germane
+%X%% to Devising Axiom Systems that Can 
+%X%% Appreciate Fragments of Their Own
+%X%% Hilbert Consistency}
+
+%% \title{Why a Small Fragment of Hilbert's Consistency Program
+%% Ought to Be Feasible
+%% for Hilbert-like Deductive Methods
+%% After A New $~\theta~$ Function Primitive
+%% %AFTER A NEW ``$~\theta~$'' Function Primitive
+%% Is Added to Arithmetic's Formalism}
+%% 
+
+%% \title{ A 2-Part Conjecture about
+%% How  a Much-Diluted but Non-Trivial
+%% %Variant
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Is
+%% Likely
+%% %Plausible 
+%% Feasible
+%% for the
+%% % Even the Challenging
+%% Case of
+%% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+\begin{center}
+\large
+June 6, 2020
+\end{center}
+
+\begin{abstract}
+\Large
+\baselineskip = 1.65 \normalbaselineskip  
+These notes are written quite informally
+and they are meant to summarize the types of
+probability distributions that will construct 
+a decently 
+``randomly''
+formalized $\Theta$ function for my Cornell paper. This manuscript was
+written in only a couple of hours time, and I therefore apologize
+for many likely examples of carelessness in my QUITE INFORMAL extension
+of \cite{ww16}'s results.
+\end{abstract}
+
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+\def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \Large \baselineskip = 2.5  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+%fffff
+\def\gvs{ \normalsize \baselineskip = 1.3 \normalbaselineskip  \parskip   5pt}
+
+\def\gvx{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip    3pt}
+
+\def\gvs{ \Large \baselineskip = 1.9 \normalbaselineskip  \parskip    5pt}
+
+\section{Crudely Composed Notes}
+% \section{Scientific Notes of Dan Willard Notarized on Nov 22,2016}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+
+
+\gvs
+
+I will use the phrase ``probability distribution'' quite informally
+in our discussion.  The goal is thus to ``invent'' {\it any type}
+of Lebesgue measure that will assure that there is a probability
+bounded below by some tiny constant $~c~$ where
+\cite{ww16}'s $\Theta$ function will produce a consistent self-justifying
+formalism.  {\it Even if that probability is tiny,} any discovered 
+lower bound  $~c~> ~ 0 ~$, 
+will assure that the  IQFS($\beta$)
+formalism is consistent when $~\beta~$ 
+holds true under the Standard Model. This is 
+ because
+IQFS($\beta$)'s
+ Group 0, 1 and 2 axioms trivially hold true under the standard model
+and its final Group-3 axiom sentence cannot be proven false 
+when our probability framework can generate a model
+where it holds with 
+an explicit
+probability lower bound lower bound of  $~c~> ~ 0 ~$.
+
+(In other words since ZF Set Theory can formalize the validity of
+G\"{o}del's Completeness  Theorem, the existence of some model
+satisfying a  probability lower bound lower bound  $~c~> ~ 0 ~$
+will be sufficient for establishing that
+the
+ Group-3 axiom's 
+self-justification statement will not be contradicted.)
+
+
+Please allow me to be informal here because I am trying to
+quickly
+ compose
+a rough approximation of working notes, without delving into 
+a bevy of 
+tedious details.  
+
+Let us recall that page 11 of \cite{ww16}
+defines the $\zzthe(x)$ function-mapping  
+%% haphazard
+to be an
+operation
+ that maps powers of 2
+onto powers of 2
+subject to the following rules:
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+We want our probability
+ distribution to have the property
+that any recursively defined function has a zero
+ probability of
+occurring.  Thus the countable set of all recursively defined functions
+will also have a probability zero of occurring.   BUT YET 
+the $\zzthe(x)$ primitive will grow at a slow enough rate
+that it is incapable of producing a fatal diagonalizing contradiction,
+while satisfying the crucial constraints in Lines  \eq{walk1}-  \eq{walk3}.
+
+I can immediately think of three likely ways of doing this.
+{\it 
+It is (?)  possible all three 
+methods  will
+work,
+successfully.} Among the three plausible methods,
+Method A is the simplest procedure, and Method C is the most
+complicated. The virtue of Method C is that it is the one that
+I am most confident about, although its procedure is a complex
+hybrid of methods A and B.
+
+All three of these 
+randomized
+methods will first generate the value of  
+$ ~\zzthe(1)~$, and then calculate in chronological
+oder the values of 
+ $ ~\zzthe(2)~$,  $ ~\zzthe(4)~$,  $ ~\zzthe(8)~$ etc., 
+This chronological order is important because
+once  $ ~\zzthe(2^i)~$ 
+is assigned a values of  $~2^K~$
+then all  $~ j \, > \, i~$ are forbidden by rule 3
+from mapping    $ ~\zzthe(2^j)~$ onto $~2^K~$.
+This Rule 3 will be called the {\bf Exclusion Principle}
+during our discussion of Methods A-C below:
+
+
+\subsection{Method A: The ``Pairing Method''}
+
+The Pairing method will begin by finding the two smallest
+powers of 2, 
+at least as large as 2,  where no $ j<i $ has  $ ~\zzthe(2^j)~$ 
+correspond to  one of these two powers of 2,
+which we write as $~2^{K_1}$ and 
+ $~2^{K_2}$. It will then set    $ ~\zzthe(2^i)~$
+ equal to one
+of these two values,
+ with each quantity receiving a 50 \%
+probability of occurring. 
+
+It would not surprise me if this Pairing rule (by itself)
+would 
+be sufficient to produce our
+ needed final result.  It might, however,
+be problematic because it would cause   $ ~\zzthe(2^i)~$
+to always satisfy the following
+possibly excessively
+ tight constraint of:
+\beq
+\label{tight}
+\forall ~~x~~~~~ \zzthe(x)
+  ~ \leq ~ 4 ~ x^2
+\enq
+
+\subsection{Method B: The Almost-Stochastic Independence Method}
+
+This method will assume that we have an infinite set of
+real numbers greater than zero, $~p_1~,~p_2~,~p_3~,... $
+such that $~p_j~$~ is the probability that 
+$~ \zzthe(2^0)~=~ 2^j$.  (
+The easiest example
+\footnote{ \normalsize If
+ $~p_j~=~ 2^{-j~} $  then
+$\sum ~p_j~~=~~ \frac{1}{2}~+~  \frac{1}{4}~+~  \frac{1}{8}~+~...~~~ =~~~1.$}  
+ of this
+is when $~p_j~=~ 2^{-j~} $.) 
+ Then for each 
+subsequent 
+power of 2, denoted as $2^L$, the same probability distribution
+will be used, except that the  ``Exclusion Principle''
+will be followed in precluding
+repetitions from
+occurring.
+  Thus, let  the prior  powers of
+2 be mapped onto the quantities of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$ 
+and let $~S~$ be defined as below:
+\beq
+\label{s-sum}
+S~~=~~ \sum_{i=0}^{L-1}{ p_{k_i}}
+%
+%%% p_{ \small K_0}~+~ p_{ \small K_1}~+~ p_{ \small K_2}~+~.... ~+~
+%%%% p_{ \small K_{L-1}}
+\enq
+Then assuming $~2^j$~ is not one of the previously 
+taken
+       positions of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$, the method B will
+have 
+hold
+$~ \zzthe(2^L)~=~ 2^j$ with an obviously adjusted  probability
+of 
+\beq
+\label{adj-prob}
+\frac{p_j}{1-S}
+\enq
+\subsection{Hybridizations of Methods A and B}
+Again, we would not be surprised if either Methods A or B
+(or both)
+ would 
+establish the conjectured self-justification property.
+However, there are also 
+other
+plausible hybrid methods
+that would establish this  property.
+Here are two examples
+\bed
+\item{$~~  Method-C-1 $}. 
+We follow Method A's approach with probability 0.5 and
+method B's approach with probability 0.5.  
+Thus, either of Method A's two choices occur with a 
+precise
+probability
+        of
+$~\frac{1}{4}~,~$ and Method B's setting
+of 
+$~ \zzthe(2^L)~=~ 2^j$ occurs with an exact  probability of:
+\beq
+\label{adj-prob1}
+\frac{p_j}{2~(1-S)}
+\enq
+\item{$~~  Method-C-2 $}. 
+We follow Method A's approach with probability of $~S~$ and
+method B's approach with probability $~1-S~$.
+Thus, either of Method A's two choices will hold with probability of
+$~\frac{S}{2}~$ and Method B's setting of
+$~ \zzthe(2^L)~=~ 2^J$ occurs with a  probability of
+exactly $~p_j~$.
+\end{description}
+
+\section{Basic Strategy}
+
+The basic strategy is to show that  if
+axiom basis
+$~\beta~$ holds valid in the standard model
+then it is impossible for a minimal proof of $0=1$
+to exist, Providing, just an overview,
+this should be able to be proved,
+{\it roughly,}
+ by contradiction.
+
+In particular, if $~\beta~$ holds valid in the standard model
+then all the Group 0, 1 and 2 axioms will hold valid under the
+Standard Model.   Hence
+if the consistency preservation property fails, then
+the Group-3
+ {\it ``I am consistent"  axiom}
+must be false under the standard model.
+Thus, the proof of the falsification of the
+ Group-3
+ axiom
+ must be able to self-reference itself
+and
+thereby
+ establish its own incorrectness.
+
+At this juncture, one must use a natural encoding for the
+G\"{o}del 
+numbers of a proof
+(such as what was formally given in 
+\cite{ww1,wwapal}'s
+examples of G\"{o}del encodings).
+  The proof of the existence of such a proof
+will exceed the proof's length by a factor $\lambda > 4$
+(or more)
+ under all
+natural encodings of proofs
+(including the examples given in 
+\cite{ww1,wwapal}).
+ But the point  
+is that our formalism
+{\it  has only the} $\Theta$ operation as
+a primitive,
+for
+ representing growth.
+Thus, the natural probability
+distributions from the prior section should establish something
+to the effect that there is a
+probability  lower 
+bound
+of
+ $c > 0$,
+such  that an
+excessively fast
+ growth-rate will be impossible.  
+
+Thus leaving aside many
+messy details, this lower bound will assure that there is stochastic
+model where growth is precluded at a fast enough rate for some
+demonstrated model to form the type of counterexample
+that G\"{o}del's Completeness Theorem needs to show that the
+ {\it ``I am consistent"  axiom} needs for corroborating its claim
+for self-justification.
+
+This result differs from my earlier work in that it needs ZF Set Theory
+(rather than Peano Arithmetic) to corroborate what I call the 
+Consistency Preservation Property.  That is fine and legal
+because the system IQFS($\beta$) 
+affirms its own consistency via
+a 1-sentence axiom.
+Thus, ZF Set Theory's knowledge about
+Lebesgue measures 
+should, likely,
+indicate  IQFS($\beta$) 
+is a competent
+enough
+ formalism to make no false claims.
+
+\newpage
+My apologies that the preceeding
+short summary
+ is not a formal proof.
+It merely outlines,
+{\it very roughly},
+ what I have in mind.
+
+
+
+\begin{thebibliography}{99}
+
+
+
+ \normalsize
+\parskip 5 pt
+\baselineskip =  1.3 \normalbaselineskip
+
+
+
+\bibitem{ww1}
+Willard, D. E.:
+ ``Self-verifying  systems, the incompleteness
+theorem and the tangibiltiy reflection
+principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explaining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+\bibitem{wwapal}
+Willard, D. E.:
+``A generalization of the second incompleteness 
+theorem and some exceptions to it''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071v5
+         (2017).
+
+
+\bibitem{ww20}
+Willard, D. E.:
+``On the Tender Line
+Separating Generalizations and Boundary-Case Exceptions for the
+Second Incompleteness Theorem under Semantic Tableaux
+Deduction'',
+a talk given 
+on January 7 at the LFCS 2020 conference.
+Early version in 
+Volume 11972 of
+ Springer's LNCS series, and longer version sent to
+Cameron and Nate.
+
+
+\end{thebibliography}
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/nate-cam/june2020/n.aux b/nachlass/collected_dew_materials/nate-cam/june2020/n.aux
new file mode 100644
index 0000000..c4add49
--- /dev/null
+++ b/nachlass/collected_dew_materials/nate-cam/june2020/n.aux
@@ -0,0 +1,30 @@
+\relax 
+\citation{ww16}
+\citation{ww16}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Crudely Composed Notes}{1}\protected@file@percent }
+\newlabel{ss1}{{1}{1}}
+\citation{ww16}
+\newlabel{walk1}{{1}{2}}
+\newlabel{walk2}{{2}{2}}
+\newlabel{walk3}{{3}{2}}
+\newlabel{walk4}{{4}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Method A: The ``Pairing Method''}{4}\protected@file@percent }
+\newlabel{tight}{{5}{4}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Method B: The Almost-Stochastic Independence Method}{4}\protected@file@percent }
+\newlabel{s-sum}{{6}{5}}
+\newlabel{adj-prob}{{7}{5}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Hybridizations of Methods A and B}{5}\protected@file@percent }
+\newlabel{adj-prob1}{{8}{6}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}Basic Strategy}{6}\protected@file@percent }
+\citation{ww1}
+\citation{wwapal}
+\citation{ww1}
+\citation{wwapal}
+\bibcite{ww1}{1}
+\citation{ww1}
+\citation{wwapal}
+\bibcite{wwapal}{2}
+\citation{ww1}
+\citation{wwapal}
+\bibcite{ww16}{3}
+\bibcite{ww20}{4}
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+% 2020 5  june 6 6.1 am ( aftr spell and subsequent corredions)
+ 
+
+
+
+% 2017  august 30 8.5 am MINOR REVISONS PAGE 6
+%%Removing rphraisng  skinny as Robert recomeneded
+ 
+
+
+% vladimir mechanic 265-2212
+
+% ``%ss% = sections seth recommendedI skip
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-1.1in}
+\setlength{\textheight}{9.4 in}
+\setlength{\textwidth}{6.5 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
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+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
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+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+\title{$Very~~ Very~$
+Informal Notes for Cameron and Nate from Dan}
+
+
+%% \title{On  How
+%%   A Novel Indeterminately Defined
+%% %$~\theta~ \gimel$ 
+%% Function Primitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+%% 
+%% \title{On  How
+%%   An Indeterminately Defined
+%% $~\theta~ \gimel$ Function Prmitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+
+
+
+
+
+
+%%% 
+
+
+%X%% \title{On  How the 
+%X%% Ixntroducing of a 
+%X%%  New $~\theta~$ Function Symbol
+%X%% Into Arithmetic's Formalism Is 
+%X%% %Likely 
+%X%% Germane
+%X%% to Devising Axiom Systems that Can 
+%X%% Appreciate Fragments of Their Own
+%X%% Hilbert Consistency}
+
+%% \title{Why a Small Fragment of Hilbert's Consistency Program
+%% Ought to Be Feasible
+%% for Hilbert-like Deductive Methods
+%% After A New $~\theta~$ Function Primitive
+%% %AFTER A NEW ``$~\theta~$'' Function Primitive
+%% Is Added to Arithmetic's Formalism}
+%% 
+
+%% \title{ A 2-Part Conjecture about
+%% How  a Much-Diluted but Non-Trivial
+%% %Variant
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Is
+%% Likely
+%% %Plausible 
+%% Feasible
+%% for the
+%% % Even the Challenging
+%% Case of
+%% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+\begin{center}
+\large
+June 6, 2020
+\end{center}
+
+\begin{abstract}
+\Large
+\baselineskip = 1.65 \normalbaselineskip  
+These notes are written quite informally
+and they are meant to summarize the types of
+probability distributions that will construct 
+a decently 
+``randomly''
+formalized $\Theta$ function for my Cornell paper. This manuscript was
+written in only a couple of hours time, and I therefore apologize
+for many likely examples of carelessness in my QUITE INFORMAL extension
+of \cite{ww16}'s results.
+\end{abstract}
+
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+\def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \Large \baselineskip = 2.5  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+%fffff
+\def\gvs{ \normalsize \baselineskip = 1.3 \normalbaselineskip  \parskip   5pt}
+
+\def\gvx{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip    3pt}
+
+\def\gvs{ \Large \baselineskip = 1.9 \normalbaselineskip  \parskip    5pt}
+
+\section{Crudely Composed Notes}
+% \section{Scientific Notes of Dan Willard Notarized on Nov 22,2016}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+
+
+\gvs
+
+I will use the phrase ``probability distribution'' quite informally
+in our discussion.  The goal is thus to ``invent'' {\it any type}
+of Lebesgue measure that will assure that there is a probability
+bounded below by some tiny constant $~c~$ where
+\cite{ww16}'s $\Theta$ function will produce a consistent self-justifying
+formalism.  {\it Even if that probability is tiny,} any discovered 
+lower bound  $~c~> ~ 0 ~$, 
+will assure that the  IQFS($\beta$)
+formalism is consistent when $~\beta~$ 
+holds true under the Standard Model. This is 
+ because
+IQFS($\beta$)'s
+ Group 0, 1 and 2 axioms trivially hold true under the standard model
+and its final Group-3 axiom sentence cannot be proven false 
+when our probability framework can generate a model
+where it holds with 
+an explicit
+probability lower bound lower bound of  $~c~> ~ 0 ~$.
+
+(In other words since ZF Set Theory can formalize the validity of
+G\"{o}del's Completeness  Theorem, the existence of some model
+satisfying a  probability lower bound lower bound  $~c~> ~ 0 ~$
+will be sufficient for establishing that
+the
+ Group-3 axiom's 
+self-justification statement will not be contradicted.)
+
+
+Please allow me to be informal here because I am trying to
+quickly
+ compose
+a rough approximation of working notes, without delving into 
+a bevy of 
+tedious details.  
+
+Let us recall that page 11 of \cite{ww16}
+defines the $\zzthe(x)$ function-mapping  
+%% haphazard
+to be an
+operation
+ that maps powers of 2
+onto powers of 2
+subject to the following rules:
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+We want our probability
+ distribution to have the property
+that any recursively defined function has a zero
+ probability of
+occurring.  Thus the countable set of all recursively defined functions
+will also have a probability zero of occurring.   BUT YET 
+the $\zzthe(x)$ primitive will grow at a slow enough rate
+that it is incapable of producing a fatal diagonalizing contradiction,
+while satisfying the crucial constraints in Lines  \eq{walk1}-  \eq{walk3}.
+
+I can immediately think of three likely ways of doing this.
+{\it 
+It is (?)  possible all three 
+methods  will
+work,
+successfully.} Among the three plausible methods,
+Method A is the simplest procedure, and Method C is the most
+complicated. The virtue of Method C is that it is the one that
+I am most confident about, although its procedure is a complex
+hybrid of methods A and B.
+
+All three of these 
+randomized
+methods will first generate the value of  
+$ ~\zzthe(1)~$, and then calculate in chronological
+oder the values of 
+ $ ~\zzthe(2)~$,  $ ~\zzthe(4)~$,  $ ~\zzthe(8)~$ etc., 
+This chronological order is important because
+once  $ ~\zzthe(2^i)~$ 
+is assigned a values of  $~2^K~$
+then all  $~ j \, > \, i~$ are forbidden by rule 3
+from mapping    $ ~\zzthe(2^j)~$ onto $~2^K~$.
+This Rule 3 will be called the {\bf Exclusion Principle}
+during our discussion of Methods A-C below:
+
+
+\subsection{Method A: The ``Pairing Method''}
+
+The Pairing method will begin by finding the two smallest
+powers of 2, 
+at least as large as 2,  where no $ j<i $ has  $ ~\zzthe(2^j)~$ 
+correspond to  one of these two powers of 2,
+which we write as $~2^{K_1}$ and 
+ $~2^{K_2}$. It will then set    $ ~\zzthe(2^i)~$
+ equal to one
+of these two values,
+ with each quantity receiving a 50 \%
+probability of occurring. 
+
+It would not surprise me if this Pairing rule (by itself)
+would 
+be sufficient to produce our
+ needed final result.  It might, however,
+be problematic because it would cause   $ ~\zzthe(2^i)~$
+to always satisfy the following
+possibly excessively
+ tight constraint of:
+\beq
+\label{tight}
+\forall ~~x~~~~~ \zzthe(x)
+  ~ \leq ~ 4 ~ x^2
+\enq
+
+\subsection{Method B: The Almost-Stochastic Independence Method}
+
+This method will assume that we have an infinite set of
+real numbers greater than zero, $~p_1~,~p_2~,~p_3~,... $
+such that $~p_j~$~ is the probability that 
+$~ \zzthe(2^0)~=~ 2^j$.  (
+The easiest example
+\footnote{ \normalsize If
+ $~p_j~=~ 2^{-j~} $  then
+$\sum ~p_j~~=~~ \frac{1}{2}~+~  \frac{1}{4}~+~  \frac{1}{8}~+~...~~~ =~~~1.$}  
+ of this
+is when $~p_j~=~ 2^{-j~} $.) 
+ Then for each 
+subsequent 
+power of 2, denoted as $2^L$, the same probability distribution
+will be used, except that the  ``Exclusion Principle''
+will be followed in precluding
+repetitions from
+occurring.
+  Thus, let  the prior  powers of
+2 be mapped onto the quantities of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$ 
+and let $~S~$ be defined as below:
+\beq
+\label{s-sum}
+S~~=~~ \sum_{i=0}^{L-1}{ p_{k_i}}
+%
+%%% p_{ \small K_0}~+~ p_{ \small K_1}~+~ p_{ \small K_2}~+~.... ~+~
+%%%% p_{ \small K_{L-1}}
+\enq
+Then assuming $~2^j$~ is not one of the previously 
+taken
+       positions of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$, the method B will
+have 
+hold
+$~ \zzthe(2^L)~=~ 2^j$ with an obviously adjusted  probability
+of 
+\beq
+\label{adj-prob}
+\frac{p_j}{1-S}
+\enq
+\subsection{Hybridizations of Methods A and B}
+Again, we would not be surprised if either Methods A or B
+(or both)
+ would 
+establish the conjectured self-justification property.
+However, there are also 
+other
+plausible hybrid methods
+that would establish this  property.
+Here are two examples
+\bed
+\item{$~~  Method-C-1 $}. 
+We follow Method A's approach with probability 0.5 and
+method B's approach with probability 0.5.  
+Thus, either of Method A's two choices occur with a 
+precise
+probability
+        of
+$~\frac{1}{4}~,~$ and Method B's setting
+of 
+$~ \zzthe(2^L)~=~ 2^j$ occurs with an exact  probability of:
+\beq
+\label{adj-prob1}
+\frac{p_j}{2~(1-S)}
+\enq
+\item{$~~  Method-C-2 $}. 
+We follow Method A's approach with probability of $~S~$ and
+method B's approach with probability $~1-S~$.
+Thus, either of Method A's two choices will hold with probability of
+$~\frac{S}{2}~$ and Method B's setting of
+$~ \zzthe(2^L)~=~ 2^J$ occurs with a  probability of
+exactly $~p_j~$.
+\end{description}
+
+\section{Basic Strategy}
+
+The basic strategy is to show that  if
+axiom basis
+$~\beta~$ holds valid in the standard model
+then it is impossible for a minimal proof of $0=1$
+to exist, Providing, just an overview,
+this should be able to be proved,
+{\it roughly,}
+ by contradiction.
+
+In particular, if $~\beta~$ holds valid in the standard model
+then all the Group 0, 1 and 2 axioms will hold valid under the
+Standard Model.   Hence
+if the consistency preservation property fails, then
+the Group-3
+ {\it ``I am consistent"  axiom}
+must be false under the standard model.
+Thus, the proof of the falsification of the
+ Group-3
+ axiom
+ must be able to self-reference itself
+and
+thereby
+ establish its own incorrectness.
+
+At this juncture, one must use a natural encoding for the
+G\"{o}del 
+numbers of a proof
+(such as what was formally given in 
+\cite{ww1,wwapal}'s
+examples of G\"{o}del encodings).
+  The proof of the existence of such a proof
+will exceed the proof's length by a factor $\lambda > 4$
+(or more)
+ under all
+natural encodings of proofs
+(including the examples given in 
+\cite{ww1,wwapal}).
+ But the point  
+is that our formalism
+{\it  has only the} $\Theta$ operation as
+a primitive,
+for
+ representing growth.
+Thus, the natural probability
+distributions from the prior section should establish something
+to the effect that there is a
+probability  lower 
+bound
+of
+ $c > 0$,
+such  that an
+excessively fast
+ growth-rate will be impossible.  
+
+Thus leaving aside many
+messy details, this lower bound will assure that there is stochastic
+model where growth is precluded at a fast enough rate for some
+demonstrated model to form the type of counterexample
+that G\"{o}del's Completeness Theorem needs to show that the
+ {\it ``I am consistent"  axiom} needs for corroborating its claim
+for self-justification.
+
+This result differs from my earlier work in that it needs ZF Set Theory
+(rather than Peano Arithmetic) to corroborate what I call the 
+Consistency Preservation Property.  That is fine and legal
+because the system IQFS($\beta$) 
+affirms its own consistency via
+a 1-sentence axiom.
+Thus, ZF Set Theory's knowledge about
+Lebesgue measures 
+should, likely,
+indicate  IQFS($\beta$) 
+is a competent
+enough
+ formalism to make no false claims.
+
+\newpage
+My apologies that the preceeding
+short summary
+ is not a formal proof.
+It merely outlines,
+{\it very roughly},
+ what I have in mind.
+
+
+
+\begin{thebibliography}{99}
+
+
+
+ \normalsize
+\parskip 5 pt
+\baselineskip =  1.3 \normalbaselineskip
+
+
+
+\bibitem{ww1}
+Willard, D. E.:
+ ``Self-verifying  systems, the incompleteness
+theorem and the tangibiltiy reflection
+principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explaining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+\bibitem{wwapal}
+Willard, D. E.:
+``A generalization of the second incompleteness 
+theorem and some exceptions to it''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071v5
+         (2017).
+
+
+\bibitem{ww20}
+Willard, D. E.:
+``On the Tender Line
+Separating Generalizations and Boundary-Case Exceptions for the
+Second Incompleteness Theorem under Semantic Tableaux
+Deduction'',
+a talk given 
+on January 7 at the LFCS 2020 conference.
+Early version in 
+Volume 11972 of
+ Springer's LNCS series, and longer version sent to
+Cameron and Nate.
+
+
+\end{thebibliography}
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/nate-cam/june2020/o.tex b/nachlass/collected_dew_materials/nate-cam/june2020/o.tex
new file mode 100644
index 0000000..06f3dda
--- /dev/null
+++ b/nachlass/collected_dew_materials/nate-cam/june2020/o.tex
@@ -0,0 +1,1242 @@
+% 2020 5  june 6 6.1 am ( aftr spell and subsequent corredions)
+ 
+
+
+
+% 2017  august 30 8.5 am MINOR REVISONS PAGE 6
+%%Removing rphraisng  skinny as Robert recomeneded
+ 
+
+
+% vladimir mechanic 265-2212
+
+% ``%ss% = sections seth recommendedI skip
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-1.1in}
+\setlength{\textheight}{9.4 in}
+\setlength{\textwidth}{6.5 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+\title{$Very~~ Very~$
+Informal Notes for Cameron and Nate from Dan}
+
+
+%% \title{On  How
+%%   A Novel Indeterminately Defined
+%% %$~\theta~ \gimel$ 
+%% Function Primitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+%% 
+%% \title{On  How
+%%   An Indeterminately Defined
+%% $~\theta~ \gimel$ Function Prmitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+
+
+
+
+
+
+%%% 
+
+
+%X%% \title{On  How the 
+%X%% Ixntroducing of a 
+%X%%  New $~\theta~$ Function Symbol
+%X%% Into Arithmetic's Formalism Is 
+%X%% %Likely 
+%X%% Germane
+%X%% to Devising Axiom Systems that Can 
+%X%% Appreciate Fragments of Their Own
+%X%% Hilbert Consistency}
+
+%% \title{Why a Small Fragment of Hilbert's Consistency Program
+%% Ought to Be Feasible
+%% for Hilbert-like Deductive Methods
+%% After A New $~\theta~$ Function Primitive
+%% %AFTER A NEW ``$~\theta~$'' Function Primitive
+%% Is Added to Arithmetic's Formalism}
+%% 
+
+%% \title{ A 2-Part Conjecture about
+%% How  a Much-Diluted but Non-Trivial
+%% %Variant
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Is
+%% Likely
+%% %Plausible 
+%% Feasible
+%% for the
+%% % Even the Challenging
+%% Case of
+%% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+\begin{center}
+\large
+June 6, 2020
+\end{center}
+
+\begin{abstract}
+\Large
+\baselineskip = 1.65 \normalbaselineskip  
+These notes are written quite informally
+and they are meant to summarize the types of
+probability distributions that will construct 
+a decently 
+``randomly''
+formalized $\Theta$ function for my Cornell paper. This manuscript was
+written in only a couple of hours time, and I therefore apologize
+for many likely examples of carelessness in my QUITE INFORMAL extension
+of \cite{ww16}'s results.
+\end{abstract}
+
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+\def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \Large \baselineskip = 2.5  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+%fffff
+\def\gvs{ \normalsize \baselineskip = 1.3 \normalbaselineskip  \parskip   5pt}
+
+\def\gvx{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip    3pt}
+
+\def\gvs{ \Large \baselineskip = 1.9 \normalbaselineskip  \parskip    5pt}
+
+\section{Crudely Composed Notes}
+% \section{Scientific Notes of Dan Willard Notarized on Nov 22,2016}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+
+
+\gvs
+
+I will use the phrase ``probability distribution'' quite informally
+in our discussion.  The goal is thus to ``invent'' {\it any type}
+of Lebesgue measure that will assure that there is a probability
+bounded below by some tiny constant $~c~$ where
+\cite{ww16}'s $\Theta$ function will produce a consistent self-justifying
+formalism.  {\it Even if that probability is tiny,} any discovered 
+lower bound  $~c~> ~ 0 ~$, 
+will assure that the  IQFS($\beta$)
+formalism is consistent when $~\beta~$ 
+holds true under the Standard Model. This is 
+ because
+IQFS($\beta$)'s
+ Group 0, 1 and 2 axioms trivially hold true under the standard model
+and its final Group-3 axiom sentence cannot be proven false 
+when our probability framework can generate a model
+where it holds with 
+an explicit
+probability lower bound lower bound of  $~c~> ~ 0 ~$.
+
+(In other words since ZF Set Theory can formalize the validity of
+G\"{o}del's Completeness  Theorem, the existence of some model
+satisfying a  probability lower bound lower bound  $~c~> ~ 0 ~$
+will be sufficient for establishing that
+the
+ Group-3 axiom's 
+self-justification statement will not be contradicted.)
+
+
+Please allow me to be informal here because I am trying to
+quickly
+ compose
+a rough approximation of working notes, without delving into 
+a bevy of 
+tedious details.  
+
+Let us recall that page 11 of \cite{ww16}
+defines the $\zzthe(x)$ function-mapping  
+%% haphazard
+to be an
+operation
+ that maps powers of 2
+onto powers of 2
+subject to the following rules:
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+We want our probability
+ distribution to have the property
+that any recursively defined function has a zero
+ probability of
+occurring.  Thus the countable set of all recursively defined functions
+will also have a probability zero of occurring.   BUT YET 
+the $\zzthe(x)$ primitive will grow at a slow enough rate
+that it is incapable of producing a fatal diagonalizing contradiction,
+while satisfying the crucial constraints in Lines  \eq{walk1}-  \eq{walk3}.
+
+I can immediately think of three likely ways of doing this.
+{\it 
+It is (?)  possible all three 
+methods  will
+work,
+successfully.} Among the three plausible methods,
+Method A is the simplest procedure, and Method C is the most
+complicated. The virtue of Method C is that it is the one that
+I am most confident about, although its procedure is a complex
+hybrid of methods A and B.
+
+All three of these 
+randomized
+methods will first generate the value of  
+$ ~\zzthe(1)~$, and then calculate in chronological
+oder the values of 
+ $ ~\zzthe(2)~$,  $ ~\zzthe(4)~$,  $ ~\zzthe(8)~$ etc., 
+This chronological order is important because
+once  $ ~\zzthe(2^i)~$ 
+is assigned a values of  $~2^K~$
+then all  $~ j \, > \, i~$ are forbidden by rule 3
+from mapping    $ ~\zzthe(2^j)~$ onto $~2^K~$.
+This Rule 3 will be called the {\bf Exclusion Principle}
+during our discussion of Methods A-C below:
+
+
+\subsection{Method A: The ``Pairing Method''}
+
+The Pairing method will begin by finding the two smallest
+powers of 2, 
+at least as large as 2,  where no $ j<i $ has  $ ~\zzthe(2^j)~$ 
+correspond to  one of these two powers of 2,
+which we write as $~2^{K_1}$ and 
+ $~2^{K_2}$. It will then set    $ ~\zzthe(2^i)~$
+ equal to one
+of these two values,
+ with each quantity receiving a 50 \%
+probability of occurring. 
+
+It would not surprise me if this Pairing rule (by itself)
+would 
+be sufficient to produce our
+ needed final result.  It might, however,
+be problematic because it would cause   $ ~\zzthe(2^i)~$
+to always satisfy the following
+possibly excessively
+ tight constraint of:
+\beq
+\label{tight}
+\forall ~~x~~~~~ \zzthe(x)
+  ~ \leq ~ 4 ~ x^2
+\enq
+
+\subsection{Method B: The Almost-Stochastic Independence Method}
+
+This method will assume that we have an infinite set of
+real numbers greater than zero, $~p_1~,~p_2~,~p_3~,... $
+such that $~p_j~$~ is the probability that 
+$~ \zzthe(2^0)~=~ 2^j$.  (
+The easiest example
+\footnote{ \normalsize If
+ $~p_j~=~ 2^{-j~} $  then
+$\sum ~p_j~~=~~ \frac{1}{2}~+~  \frac{1}{4}~+~  \frac{1}{8}~+~...~~~ =~~~1.$}  
+ of this
+is when $~p_j~=~ 2^{-j~} $.) 
+ Then for each 
+subsequent 
+power of 2, denoted as $2^L$, the same probability distribution
+will be used, except that the  ``Exclusion Principle''
+will be followed in precluding
+repetitions from
+occurring.
+  Thus, let  the prior  powers of
+2 be mapped onto the quantities of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$ 
+and let $~S~$ be defined as below:
+\beq
+\label{s-sum}
+S~~=~~ \sum_{i=0}^{L-1}{ p_{k_i}}
+%
+%%% p_{ \small K_0}~+~ p_{ \small K_1}~+~ p_{ \small K_2}~+~.... ~+~
+%%%% p_{ \small K_{L-1}}
+\enq
+Then assuming $~2^j$~ is not one of the previously 
+taken
+       positions of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$, the method B will
+have 
+hold
+$~ \zzthe(2^L)~=~ 2^j$ with an obviously adjusted  probability
+of 
+\beq
+\label{adj-prob}
+\frac{p_j}{1-S}
+\enq
+\subsection{Hybridizations of Methods A and B}
+Again, we would not be surprised if either Methods A or B
+(or both)
+ would 
+establish the conjectured self-justification property.
+However, there are also 
+other
+plausible hybrid methods
+that would establish this  property.
+Here are two examples
+\bed
+\item{$~~  Method-C-1 $}. 
+We follow Method A's approach with probability 0.5 and
+method B's approach with probability 0.5.  
+Thus, either of Method A's two choices occur with a 
+precise
+probability
+        of
+$~\frac{1}{4}~,~$ and Method B's setting
+of 
+$~ \zzthe(2^L)~=~ 2^j$ occurs with an exact  probability of:
+\beq
+\label{adj-prob1}
+\frac{p_j}{2~(1-S)}
+\enq
+\item{$~~  Method-C-2 $}. 
+We follow Method A's approach with probability of $~S~$ and
+method B's approach with probability $~1-S~$.
+Thus, either of Method A's two choices will hold with probability of
+$~\frac{S}{2}~$ and Method B's setting of
+$~ \zzthe(2^L)~=~ 2^J$ occurs with a  probability of
+exactly $~p_j~$.
+\end{description}
+
+\section{Basic Strategy}
+
+The basic strategy is to show that  if
+axiom basis
+$~\beta~$ holds valid in the standard model
+then it is impossible for a minimal proof of $0=1$
+to exist, Providing, just an overview,
+this should be able to be proved,
+{\it roughly,}
+ by contradiction.
+
+In particular, if $~\beta~$ holds valid in the standard model
+then all the Group 0, 1 and 2 axioms will hold valid under the
+Standard Model.   Hence
+if the consistency preservation property fails, then
+the Group-3
+ {\it ``I am consistent"  axiom}
+must be false under the standard model.
+Thus, the proof of the falsification of the
+ Group-3
+ axiom
+ must be able to self-reference itself
+and
+thereby
+ establish its own incorrectness.
+
+At this juncture, one must use a natural encoding for the
+G\"{o}del 
+numbers of a proof
+(such as what was formally given in 
+\cite{ww1,wwapal}'s
+examples of G\"{o}del encodings).
+  The proof of the existence of such a proof
+will exceed the proof's length by a factor $\lambda > 4$
+(or more)
+ under all
+natural encodings of proofs
+(including the examples given in 
+\cite{ww1,wwapal}).
+ But the point  
+is that our formalism
+{\it  has only the} $\Theta$ operation as
+a primitive,
+for
+ representing growth.
+Thus, the natural probability
+distributions from the prior section should establish something
+to the effect that there is a
+probability  lower 
+bound
+of
+ $c > 0$,
+such  that an
+excessively fast
+ growth-rate will be impossible.  
+
+Thus leaving aside many
+messy details, this lower bound will assure that there is stochastic
+model where growth is precluded at a fast enough rate for some
+demonstrated model to form the type of counterexample
+that G\"{o}del's Completeness Theorem needs to show that the
+ {\it ``I am consistent"  axiom} needs for corroborating its claim
+for self-justification.
+
+This result differs from my earlier work in that it needs ZF Set Theory
+(rather than Peano Arithmetic) to corroborate what I call the 
+Consistency Preservation Property.  That is fine and legal
+because the system IQFS($\beta$) 
+affirms its own consistency via
+a 1-sentence axiom.
+Thus, ZF Set Theory's knowledge about
+Lebesgue measures 
+should, likely,
+indicate  IQFS($\beta$) 
+is a competent
+enough
+ formalism to make no false claims.
+
+\newpage
+My apologies that the preceeding
+short summary
+ is not a formal proof.
+It merely outlines,
+{\it very roughly},
+ what I have in mind.
+
+
+
+\begin{thebibliography}{99}
+
+
+
+ \normalsize
+\parskip 5 pt
+\baselineskip =  1.3 \normalbaselineskip
+
+
+
+\bibitem{ww1}
+Willard, D. E.:
+ ``Self-verifying  systems, the incompleteness
+theorem and the tangibiltiy reflection
+principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explaining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+\bibitem{wwapal}
+Willard, D. E.:
+``A generalization of the second incompleteness 
+theorem and some exceptions to it''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071v5
+         (2017).
+
+
+\bibitem{ww20}
+Willard, D. E.:
+``On the Tender Line
+Separating Generalizations and Boundary-Case Exceptions for the
+Second Incompleteness Theorem under Semantic Tableaux
+Deduction'',
+a talk given 
+on January 7 at the LFCS 2020 conference.
+Early version in 
+Volume 11972 of
+ Springer's LNCS series, and longer version sent to
+Cameron and Nate.
+
+
+\end{thebibliography}
+\end{document}
+
diff --git a/nachlass/collected_dew_materials/nate-cam/june2020/r.tex b/nachlass/collected_dew_materials/nate-cam/june2020/r.tex
new file mode 100644
index 0000000..06f3dda
--- /dev/null
+++ b/nachlass/collected_dew_materials/nate-cam/june2020/r.tex
@@ -0,0 +1,1242 @@
+% 2020 5  june 6 6.1 am ( aftr spell and subsequent corredions)
+ 
+
+
+
+% 2017  august 30 8.5 am MINOR REVISONS PAGE 6
+%%Removing rphraisng  skinny as Robert recomeneded
+ 
+
+
+% vladimir mechanic 265-2212
+
+% ``%ss% = sections seth recommendedI skip
+
+%\documentclass[11pt]{article}
+%\documentclass[10pt]{article}
+% \documentclass[11pt]{article}
+\documentclass[12pt]{article}
+
+
+%%%%%%%%%% \documentstyle[11pt]{article}
+
+
+\usepackage{amssymb}
+
+
+\addtolength{\oddsidemargin}{-1.1in}
+\setlength{\textheight}{9.4 in}
+\setlength{\textwidth}{6.5 in}
+%%%% above paper
+
+%\setlength{\textwidth}{6,0 in}
+%\setlength{\textwidth}{5,5 in}
+
+\addtolength{\topmargin}{-0.75in}
+
+
+%% 
+%% \addtolength{\oddsidemargin}{-0.5 in}
+%% \setlength{\textheight}{10.1 in}
+%% \setlength{\textwidth}{7.5 in}
+%% \addtolength{\topmargin}{-0.5in}
+
+
+
+
+
+
+          \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+              \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+\newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+\newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                        \newenvironment{#1}{\begin{#1q}\rm}}
+
+\newtheorem{theorem}{$~~~~$ Theorem}[section]
+% \newtheorem{corollary}{Corollary}[section]
+%\newtheorem{fact}{Fact}[section]
+\newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:}
+\end{tabbing}}
+
+\newtheorem{example}[theorem]{$~~~~$ Example}
+\newtheorem{themx}[theorem]{$~~~~$ Theorem}
+\newtheorem{corollary}[theorem]{$~~~~$ Corollary}
+\newtheorem{lemma}[theorem]{$~~~~$ Lemma}
+\newtheorem{remark}[theorem]{$~~~~$Remark}
+\newtheorem{definition}[theorem]{$~~~~$Definition}
+\newtheorem{fact}[theorem]{$~~~~$Fact}
+
+
+\newtheorem{dff}[theorem]{$~~~~$ Definition}
+\newtheorem{exx}[theorem]{$~~~~$ Example}
+\newtheorem{lemm}[theorem]{$~~~~$ Lemma}
+\newtheorem{propp}[theorem]{$~~~~$ Proposition}
+\newtheorem{remm}[theorem]{$~~~~$Remark}
+\newtheorem{ccr}[theorem]{$~~~~$Corrolary}
+\newtheorem{coj}[theorem]{$~~~~$Conjecture}
+
+
+\newtheorem{deff}[theorem]{$~~~~$Definition}
+
+
+
+
+
+% \def\Box{ QED}
+\def\nop{ }
+\def\nyp{\newpage }
+% \def\nxp{ }
+\def\nxp{ Here $~$NXP }
+
+
+%  \def\nyp{ }
+% \def\nyp{ }
+
+\def\bigc{$\,$of the unabridged version of this paper \cite{ww12}}
+
+\def\nor1{Normed$\{~2^{ \zzz \theta  \, )} ~$,$~\sqrt{~2^{ \zzz \theta  \, )}}~\}$}
+
+\def\pagxx{Page ?xx?}
+\def\xor2{Normed$\{ ~\sqrt{~2^{ \zzz \theta  \, )}}~,~2~ \} $}
+%\def\fffx{Fact \#}
+\def\fffx{{\bf Fact *}}
+\def\zhz{H }
+%\def\fffx{Fact \#}
+\def\appD{Appendix D }
+\def\appxD{Appendix D}
+\def\fffour{three }
+\def\zazsta{ and EA-stability}
+
+% \def\glamb{\xi}
+\def\glamb{\lambda}
+\def\glamb{P}
+\def\glamb{\theta}
+\def\pag2{Page 2}
+%% \def\zzthe{\zeta}
+
+
+\def\glamb{\zeta}
+\def\zzthe{\theta}
+
+
+
+
+
+\def\gggen{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )~$}
+\def\gggcp{$( L^\xi  ,  \Delta_0^\xi   ,  B^\xi  ,  d  ,  G  )$}
+\def\peta{\sigma}
+\def\zzxz{~ \sharp ( \, }
+\def\zzz{~ \sharp ( ~ }
+\def\zip{\sharp }
+\def\mheta{\theta^\bullet}
+\def\mxi{\xi^\bullet}
+%\def\mbi{\bullet}
+\def\mbi{\bigdot}
+\def\mbi{\bigoplus}
+\def\xxi{$\, \xi^* \,$}
+
+
+
+
+\def\tftt{~ \frac{1}{2}~ }
+\def\sss{ }
+
+\def\goodshit{\triangleright}
+\def\bullshit{\triangleleft}
+\def\foo{footnote \footnote}
+\def\Uxp{\Upsilon}
+
+\def\f55{ \normalsize  \baselineskip = 1.8 \normalbaselineskip } 
+
+
+\def\f55{  \baselineskip = 1.1 \normalbaselineskip } 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+
+\def\f55{  \baselineskip = 0.7 \normalbaselineskip } 
+
+
+
+\def\bbskip{\bigskip}
+
+
+
+\def\axst{\odot}
+\def\rp{ p^\theta } 
+\def\thsp{Theorem $ \, * \,$ }
+\def\thss{Theorem $ \, *\,$'s }
+\def\dexxt{\Delta}
+\newcommand{\co}[1]{Corollary \ref{#1}}
+\newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\cjx}[1]{Conjecture \ref{#1}}
+\newcommand{\phx}[1]{Proposition \ref{#1}}
+ \newcommand{\dfx}[1]{Definition \ref{#1}}
+\newcommand{\lem}[1]{Lemma \ref{#1}}
+
+
+
+%% \newcommand{\co}[1]{Corollary \ref{#1}}
+%%  \newcommand{\thx}[1]{Theorem \ref{#1}}
+\newcommand{\lxem}[1]{Lemma \ref{#1}}
+\newcommand{\overx}[1]{\, \overline{ {#1} } \,} 
+
+
+\newcommand{\el}[1]{Line (\ref{#1})}
+\newcommand{\ex}[1]{Expression (\ref{#1})}
+\newcommand{\ei}[1]{item (\ref{#1})}
+
+\newcommand{\eq}[1]{(\ref{#1})}
+\newcommand{\ep}[1]{Equation (\ref{#1})}
+\newcommand{\thetlam}{ \theta }
+\newcommand{\underx}[1]{\overline{~ {#1} ~}}
+\newcommand{\appaa}{$App \forall$}
+\newcommand{\appee}{$App \exists$}
+\newcommand{\tll}[1]{Tab$- {#1} -$List}
+\newcommand{\txl}[1]{Tab$- {#1}$}
+\newcommand{\tlxl}[1]{Tab$- {#1}$ }
+\newcommand{\sll}[1]{Short$- {#1} -$List}
+\newcommand{\axx}[1]{NS$_D^{\,k,m}( ${#1}$)$}
+
+
+\newenvironment{proof}{{\bf Proof:}}{$\Box$}
+\newenvironment{sketchproof}{{\bf Sketch of Proof:}}{$\Box$}
+
+
+\begin{document}
+
+%\title{Rough Summary of March 2012 Research Before I Attended
+%the AMS March 17-18 Conference}
+
+
+%% \title{ On
+%% How the Revival of a Diluted 
+%% Version of
+%% Hilbert's Consistency Program 
+%% %is 
+%% Should
+%% Likely
+%%  Be
+%% % Plausible and
+%% % Veru
+%%  Germane
+%% to Computer Science}
+
+
+\title{$Very~~ Very~$
+Informal Notes for Cameron and Nate from Dan}
+
+
+%% \title{On  How
+%%   A Novel Indeterminately Defined
+%% %$~\theta~ \gimel$ 
+%% Function Primitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+%% 
+%% \title{On  How
+%%   An Indeterminately Defined
+%% $~\theta~ \gimel$ Function Prmitive 
+%% Enables Some Axiom Systems
+%% To
+%% Appreciate Fragments of Their Own
+%% Hilbert Consistency}
+%% 
+%% 
+
+
+
+
+
+
+
+%%% 
+
+
+%X%% \title{On  How the 
+%X%% Ixntroducing of a 
+%X%%  New $~\theta~$ Function Symbol
+%X%% Into Arithmetic's Formalism Is 
+%X%% %Likely 
+%X%% Germane
+%X%% to Devising Axiom Systems that Can 
+%X%% Appreciate Fragments of Their Own
+%X%% Hilbert Consistency}
+
+%% \title{Why a Small Fragment of Hilbert's Consistency Program
+%% Ought to Be Feasible
+%% for Hilbert-like Deductive Methods
+%% After A New $~\theta~$ Function Primitive
+%% %AFTER A NEW ``$~\theta~$'' Function Primitive
+%% Is Added to Arithmetic's Formalism}
+%% 
+
+%% \title{ A 2-Part Conjecture about
+%% How  a Much-Diluted but Non-Trivial
+%% %Variant
+%% Fragment
+%%  of
+%% Hilbert's Consistency Program 
+%% Is
+%% Likely
+%% %Plausible 
+%% Feasible
+%% for the
+%% % Even the Challenging
+%% Case of
+%% Hilbert Deduction}
+
+
+
+% 
+% \title{ On
+% How  a Much-Diluted but Non-Trivial
+% %Variant
+% Fragment
+%  of
+% Hilbert's Consistency Program 
+% Is
+% Likely
+% %Plausible 
+% Feasible
+% for Even the 
+% Challenging
+% Case of
+% Hilbert Deduction}
+% 
+% %Plausible and
+% % Veru
+% % Germane
+% %to Computer Science}
+% 
+% 
+% %%% \title{\large \bf On the Revival of a Much-Diluted 
+% %%% but Non-Trivial
+% %%% Version of
+% %%% Hilbert's Consistency Program (And Its Applications
+%%% for Computer Science, Mathematics and Philosophy)}
+
+
+
+
+\def\beq{\begin{equation}}
+\def\enq{\end{equation}}
+
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+
+
+\def\bec{\begin{corollary}}
+\def\enc{\end{corollary}}
+
+\def\bed{\begin{description}}
+\def\ennd{\end{description}}
+\def\bee{\begin{enumerate}}
+\def\ene{\end{enumerate}}
+
+
+\def\bxbxd{\begin{definition}}
+\def\bxbxdd{\begin{definition}}
+\def\eedd{\end{definition}}
+\def\bxbxdr{\begin{definition} \rm}
+\def\bel{\begin{lemma}}
+\def\enl{\end{lemma}}
+\def\ent{\end{theorem}}
+
+
+
+\author{  Dan E.Willard }
+%Email = dew@cs.albany.edu.}}
+%\newline
+%Email = dan.willard.albany@gmail.com}}
+
+
+
+
+
+
+
+
+
+
+
+%%%\date{Copyright 2012 by Dan E. Willard}
+
+\date{State University of New York at Albany}
+
+\maketitle
+
+\setcounter{page}{0}
+\thispagestyle{empty}
+
+\normalsize
+
+
+
+
+\baselineskip = 1.3\normalbaselineskip
+
+
+
+\normalsize
+
+
+\baselineskip = 1.0 \normalbaselineskip 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+
+%%\baselineskip = 1.0 \normalbaselineskip 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip } 
+%%\def\bbint{\large \baselineskip = 1.6 \normalbaselineskip }
+%%\def\bbint{\normalsize \baselineskip = 1.3 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.27 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.95 \normalbaselineskip } 
+
+
+
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+
+\def\bbint{\normalsize \baselineskip = 1.7 \normalbaselineskip } 
+
+\def\bbint{\large \baselineskip = 2.3 \normalbaselineskip } 
+\def\bbinm{ \baselineskip = 1.18 \normalbaselineskip }
+
+
+\def\bbint{\large \baselineskip = 2.0 \normalbaselineskip } 
+\def\bbing{ }
+\def\bbins{ }
+\def\bbinm{ }
+\def\bbinr{ }
+
+
+\def\bbint{\normalsize \baselineskip = 1.25 \normalbaselineskip }
+\def\bbina{\normalsize \baselineskip = 1.24 \normalbaselineskip }
+\def\bbinr{ \baselineskip = 1.3 \normalbaselineskip }
+\def\bbing{ \baselineskip = 1.28 \normalbaselineskip }
+\def\bbins{ \baselineskip = 1.21 \normalbaselineskip }
+\def\bbinm{  }
+
+\def\ftl{ \baselineskip = 1.5 \normalbaselineskip }
+
+
+\bbint
+
+\parskip 5 pt
+
+
+
+
+\noindent
+
+
+
+
+
+\small
+
+
+\baselineskip = 1.14 \normalbaselineskip 
+
+
+ 
+\parskip 5pt
+
+\baselineskip = 1.2 \normalbaselineskip 
+
+%\setcounter{page}{0}
+
+
+
+%%%%%%{\small
+
+
+
+
+
+
+
+
+\large
+\normalsize
+
+ \baselineskip = 1.2 \normalbaselineskip 
+
+%mmmmmmmmmmmmm
+
+\begin{center}
+\large
+June 6, 2020
+\end{center}
+
+\begin{abstract}
+\Large
+\baselineskip = 1.65 \normalbaselineskip  
+These notes are written quite informally
+and they are meant to summarize the types of
+probability distributions that will construct 
+a decently 
+``randomly''
+formalized $\Theta$ function for my Cornell paper. This manuscript was
+written in only a couple of hours time, and I therefore apologize
+for many likely examples of carelessness in my QUITE INFORMAL extension
+of \cite{ww16}'s results.
+\end{abstract}
+
+
+
+
+
+
+
+
+
+\def\ww22{\normalsize \baselineskip = 1.21\normalbaselineskip \parskip 4 pt}
+\def\bb22{\normalsize \baselineskip = 1.19\normalbaselineskip \parskip 4 pt}
+\def\zz22z{\normalsize \baselineskip = 1.19 \normalbaselineskip \parskip 3 pt}
+\def\xx22{\normalsize \baselineskip = 1.17\normalbaselineskip \parskip 4 pt}
+\def\vx22s{\normalsize \baselineskip = 1.16 \normalbaselineskip \parskip 3 pt} 
+\def\vv22{\normalsize \baselineskip = 1.17 \normalbaselineskip \parskip 3 pt} 
+\def\aa22{\normalsize \baselineskip = 1.15 \normalbaselineskip \parskip 3 pt} 
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\vspace*{- 1.0 em}
+
+
+\def\waw11{\normalsize \baselineskip = 1.72\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+\def\waw11{\normalsize \baselineskip = 1.85\normalbaselineskip}
+
+
+
+\def\waw11{\normalsize \baselineskip = 1.45\normalbaselineskip}
+
+
+\def\waw11{\normalsize \baselineskip = 1.7\normalbaselineskip}
+
+\def\waw11{\normalsize \baselineskip = 1.12\normalbaselineskip}
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.5 \normalbaselineskip } 
+
+
+\def\g55{  \baselineskip = 1.50 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.50 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\aa22{\normalsize  \waw11 \parskip 6 pt} 
+\def\bb22{\normalsize  \waw11 \parskip 5 pt}
+\def\ww22{\normalsize \waw11 \parskip 4 pt}
+\def\vv22{\normalsize  \waw11 \parskip 3 pt} 
+\def\tt22{\normalsize  \waw11 \parskip 2 pt} 
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 0.9 \normalbaselineskip } 
+
+
+
+
+
+
+\def\mal{ \bf  }
+\def\nal{\mathcal}
+
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+\def\ttt2c{ }
+\def\tttc{ }
+
+\def\tttc{\tiny \baselineskip = 0.8 \normalbaselineskip  \parskip 0pt }
+\def\ttt2c{\tiny \baselineskip = 0.7 \normalbaselineskip  \parskip 0pt }
+\def\tttc{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 2.1 \normalbaselineskip  \parskip 5pt }
+
+\def\tttc{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+\def\ttt2c{ \baselineskip = 1.15 \normalbaselineskip  \parskip 5pt }
+
+
+\def\tttc{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ \baselineskip = 1.12 \normalbaselineskip  \parskip 4pt }
+
+
+\def\tttc{ \baselineskip = 1.14 \normalbaselineskip  \parskip 3pt }
+\def\ttt2c{ \baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\cvt{ \baselineskip = 0.98 \normalbaselineskip }
+\def\cv9{ \baselineskip = 0.99 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.0 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.03 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.00 \normalbaselineskip }
+
+
+\def\cvt{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.6 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.6 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.6 \normalbaselineskip \parskip 3pt }
+
+
+
+
+\def\cvt{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cv9{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvs{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvl{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvh{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvg{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvb{ \baselineskip = 1.22 \normalbaselineskip }
+\def\cvnew{ \baselineskip = 1.4 \normalbaselineskip }
+\def\cvmew{ \baselineskip = 1.35 \normalbaselineskip }
+\def\cvwew{ \baselineskip = 1.4 \normalbaselineskip \parskip 5pt }
+\def\cvrew{ \baselineskip = 1.22 \normalbaselineskip \parskip 3pt }
+
+
+\def\cvt{ }
+\def\cv9{ }
+\def\cvs{ }
+\def\cvl{ }
+\def\cvh{ }
+\def\cvg{ }
+\def\cvb{ }
+\def\cvnew{ } 
+\def\cvmew{ }
+\def\cvwew{ }
+\def\cvrew{ }
+
+
+
+\def\fend{ 
+
+\medskip -------------------------------------------------------}
+
+
+\def\g55{  \baselineskip = 1.0 \normalbaselineskip } 
+\def\s55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\sm55{ \baselineskip = 1.0 \normalbaselineskip } 
+\def\h55{  \baselineskip = 1.08 \normalbaselineskip } 
+\def\b55{  \baselineskip = 1.1 \normalbaselineskip } 
+
+\normalsize
+
+\baselineskip = 1.85 \normalbaselineskip 
+
+
+
+%% Sleepy  
+
+%\cvlpm %% Sleepy  
+%\cvnew
+
+
+%\small
+
+%\parskip 0p
+
+\parskip 2pt
+
+\vspace*{- 1.0 em}
+
+% \newpage
+
+%\large
+
+%\setcounter{page}{0}
+\baselineskip = 1.04 \normalbaselineskip 
+\parskip 2pt
+
+\baselineskip = 0.96 \normalbaselineskip 
+%\baselineskip = 0.90 \normalbaselineskip 
+
+%\parskip 1pt
+% 
+\baselineskip = 2.16 \normalbaselineskip 
+\baselineskip = 2.3 \normalbaselineskip 
+
+\baselineskip = 0.95 \normalbaselineskip 
+%\baselineskip = 0.95 \normalbaselineskip 
+\baselineskip = 0.88 \normalbaselineskip 
+\parskip 0pt
+ 
+
+
+\noindent
+
+% 
+% 
+% NNEW COMMENT
+% 
+% 
+% The pdf version of this draft is verbatim identical to August's Version 3.
+% The prior draft's abstract was incorrectly broadcast by Arxiv on the 
+% Internet, after I pressed a wrong computer button. Thus, 
+% Version 4 was issued.
+
+
+\newpage
+
+\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \large \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip    5pt}\def\gvs{ \normalsize \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{   \baselineskip = 1.74 \normalbaselineskip  \parskip    5pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.44 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.04 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize \baselineskip = 2.64 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \Large \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ }
+
+
+\gvs
+
+\footnotesize
+
+
+\def\gvs{ }
+  
+
+\normalsize \baselineskip = 0.98 \normalbaselineskip
+\normalsize \baselineskip = 1.0 \normalbaselineskip
+\normalsize \baselineskip = 1.01 \normalbaselineskip
+
+
+\def\gvs{ \normalsize \baselineskip = 1.25 \normalbaselineskip  \parskip 4pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip 5pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4  \normalbaselineskip  \parskip 6pt}
+
+%\def\gvs{ \large \baselineskip = 1.5  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \Large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \large \baselineskip = 1.6  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+\def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 1.5 \normalbaselineskip  \parskip 3pt}
+
+% \def\gvs{ \large \baselineskip = 1.8  \normalbaselineskip  \parskip 6pt}
+
+% \def\gvs{ \normalsize \baselineskip = 1.65 \normalbaselineskip  \parskip 3pt}
+
+\def\gvs{ \large \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.1  \normalbaselineskip  \parskip 6pt}
+
+%%%old 
+
+ \def\gvs{ \normalsize \baselineskip = 1.227 \normalbaselineskip  \parskip 3pt}
+
+ \def\gvs{ \large  \baselineskip = 1.6 \normalbaselineskip  \parskip 5pt}
+%% march 31
+
+\def\gvs{ \Large  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \LARGE  \baselineskip = 1.8 \normalbaselineskip  \parskip 5pt}
+\def\gvs{ \normalsize  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \Large  \baselineskip = 2.0 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \large  \baselineskip = 2.2 \normalbaselineskip  \parskip 5pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.4  \normalbaselineskip  \parskip 6pt}
+
+\def\gvs{ \normalsize \baselineskip = 2.6  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 2.2  \normalbaselineskip  \parskip 6pt}
+\def\gvs{ \normalsize \baselineskip = 1.8  \normalbaselineskip  \parskip 5pt}
+% \def\gvs{ \normalsize \baselineskip = 1.5  \normalbaselineskip  \parskip 5pt}
+
+\def\sgvs{ \small \baselineskip = 1.33  \normalbaselineskip  \parskip 1pt}
+\def\tttc{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+\def\ttt2c{ }
+%\baselineskip = 1.14 \normalbaselineskip  \parskip 4pt }
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+
+\def\gvs{ }
+
+
+\def\gvs{ \normalsize \baselineskip = 2.1 \normalbaselineskip  \parskip 7pt}
+\def\gvs{ \normalsize \baselineskip = 1.8 \normalbaselineskip  \parskip    7pt}
+
+%\def\gvs{ \normalsize \baselineskip = 1.4 \normalbaselineskip  \parskip    5pt}
+
+ \def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 9pt}
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 9pt}
+
+\def\gv2{ \normalsize \baselineskip = 1.30  \normalbaselineskip  \parskip 3pt}
+
+
+\def\gvs{ \large \baselineskip = 1.7  \normalbaselineskip  \parskip 5pt}
+
+
+\def\gvs{ \normalsize \baselineskip = 2.0  \normalbaselineskip  \parskip 8pt}
+\def\gvs{ \Large \baselineskip = 2.5  \normalbaselineskip  \parskip 8pt}
+
+
+%fffff
+
+
+%fffff
+\def\gvs{ \normalsize \baselineskip = 1.3 \normalbaselineskip  \parskip   5pt}
+
+\def\gvx{ \normalsize \baselineskip = 1.23 \normalbaselineskip  \parskip    3pt}
+
+\def\gvs{ \Large \baselineskip = 1.9 \normalbaselineskip  \parskip    5pt}
+
+\section{Crudely Composed Notes}
+% \section{Scientific Notes of Dan Willard Notarized on Nov 22,2016}
+%%%%%%%%%% 1111111111111111}
+\label{ss1}
+
+
+
+
+
+\gvs
+
+I will use the phrase ``probability distribution'' quite informally
+in our discussion.  The goal is thus to ``invent'' {\it any type}
+of Lebesgue measure that will assure that there is a probability
+bounded below by some tiny constant $~c~$ where
+\cite{ww16}'s $\Theta$ function will produce a consistent self-justifying
+formalism.  {\it Even if that probability is tiny,} any discovered 
+lower bound  $~c~> ~ 0 ~$, 
+will assure that the  IQFS($\beta$)
+formalism is consistent when $~\beta~$ 
+holds true under the Standard Model. This is 
+ because
+IQFS($\beta$)'s
+ Group 0, 1 and 2 axioms trivially hold true under the standard model
+and its final Group-3 axiom sentence cannot be proven false 
+when our probability framework can generate a model
+where it holds with 
+an explicit
+probability lower bound lower bound of  $~c~> ~ 0 ~$.
+
+(In other words since ZF Set Theory can formalize the validity of
+G\"{o}del's Completeness  Theorem, the existence of some model
+satisfying a  probability lower bound lower bound  $~c~> ~ 0 ~$
+will be sufficient for establishing that
+the
+ Group-3 axiom's 
+self-justification statement will not be contradicted.)
+
+
+Please allow me to be informal here because I am trying to
+quickly
+ compose
+a rough approximation of working notes, without delving into 
+a bevy of 
+tedious details.  
+
+Let us recall that page 11 of \cite{ww16}
+defines the $\zzthe(x)$ function-mapping  
+%% haphazard
+to be an
+operation
+ that maps powers of 2
+onto powers of 2
+subject to the following rules:
+
+\vspace*{- 0.6 em}
+{\parskip -6 pt
+\beq
+\label{walk1}
+\forall ~~x~~~~~ \mbox{Power}(x) ~~~ \Rightarrow  ~~
+\mbox{Power}(~ \zzthe(x)~)
+\enq
+\beq
+\label{walk2}
+\forall ~~x~~~~~ \zzthe(x)~ \neq ~ 1
+\enq
+\beq
+\label{walk3}
+\forall ~~x~~~ \forall ~~y ~~~~[~ x ~ \neq~ y ~
+\wedge ~\mbox{Power}(x)~]
+ \Rightarrow ~~ 
+\zzthe(x)~ \neq ~\zzthe(y)
+\enq
+\beq
+\label{walk4}
+\forall ~~x~~~~~ \neg ~ ~\mbox{Power}(x)~~~~ \Rightarrow ~~~~ 
+\zzthe(x)~=~0
+\enq}
+We want our probability
+ distribution to have the property
+that any recursively defined function has a zero
+ probability of
+occurring.  Thus the countable set of all recursively defined functions
+will also have a probability zero of occurring.   BUT YET 
+the $\zzthe(x)$ primitive will grow at a slow enough rate
+that it is incapable of producing a fatal diagonalizing contradiction,
+while satisfying the crucial constraints in Lines  \eq{walk1}-  \eq{walk3}.
+
+I can immediately think of three likely ways of doing this.
+{\it 
+It is (?)  possible all three 
+methods  will
+work,
+successfully.} Among the three plausible methods,
+Method A is the simplest procedure, and Method C is the most
+complicated. The virtue of Method C is that it is the one that
+I am most confident about, although its procedure is a complex
+hybrid of methods A and B.
+
+All three of these 
+randomized
+methods will first generate the value of  
+$ ~\zzthe(1)~$, and then calculate in chronological
+oder the values of 
+ $ ~\zzthe(2)~$,  $ ~\zzthe(4)~$,  $ ~\zzthe(8)~$ etc., 
+This chronological order is important because
+once  $ ~\zzthe(2^i)~$ 
+is assigned a values of  $~2^K~$
+then all  $~ j \, > \, i~$ are forbidden by rule 3
+from mapping    $ ~\zzthe(2^j)~$ onto $~2^K~$.
+This Rule 3 will be called the {\bf Exclusion Principle}
+during our discussion of Methods A-C below:
+
+
+\subsection{Method A: The ``Pairing Method''}
+
+The Pairing method will begin by finding the two smallest
+powers of 2, 
+at least as large as 2,  where no $ j<i $ has  $ ~\zzthe(2^j)~$ 
+correspond to  one of these two powers of 2,
+which we write as $~2^{K_1}$ and 
+ $~2^{K_2}$. It will then set    $ ~\zzthe(2^i)~$
+ equal to one
+of these two values,
+ with each quantity receiving a 50 \%
+probability of occurring. 
+
+It would not surprise me if this Pairing rule (by itself)
+would 
+be sufficient to produce our
+ needed final result.  It might, however,
+be problematic because it would cause   $ ~\zzthe(2^i)~$
+to always satisfy the following
+possibly excessively
+ tight constraint of:
+\beq
+\label{tight}
+\forall ~~x~~~~~ \zzthe(x)
+  ~ \leq ~ 4 ~ x^2
+\enq
+
+\subsection{Method B: The Almost-Stochastic Independence Method}
+
+This method will assume that we have an infinite set of
+real numbers greater than zero, $~p_1~,~p_2~,~p_3~,... $
+such that $~p_j~$~ is the probability that 
+$~ \zzthe(2^0)~=~ 2^j$.  (
+The easiest example
+\footnote{ \normalsize If
+ $~p_j~=~ 2^{-j~} $  then
+$\sum ~p_j~~=~~ \frac{1}{2}~+~  \frac{1}{4}~+~  \frac{1}{8}~+~...~~~ =~~~1.$}  
+ of this
+is when $~p_j~=~ 2^{-j~} $.) 
+ Then for each 
+subsequent 
+power of 2, denoted as $2^L$, the same probability distribution
+will be used, except that the  ``Exclusion Principle''
+will be followed in precluding
+repetitions from
+occurring.
+  Thus, let  the prior  powers of
+2 be mapped onto the quantities of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$ 
+and let $~S~$ be defined as below:
+\beq
+\label{s-sum}
+S~~=~~ \sum_{i=0}^{L-1}{ p_{k_i}}
+%
+%%% p_{ \small K_0}~+~ p_{ \small K_1}~+~ p_{ \small K_2}~+~.... ~+~
+%%%% p_{ \small K_{L-1}}
+\enq
+Then assuming $~2^j$~ is not one of the previously 
+taken
+       positions of
+ $2^{K_0}$   $2^{K_1}$, ...   $2^{K_{L-1}}$, the method B will
+have 
+hold
+$~ \zzthe(2^L)~=~ 2^j$ with an obviously adjusted  probability
+of 
+\beq
+\label{adj-prob}
+\frac{p_j}{1-S}
+\enq
+\subsection{Hybridizations of Methods A and B}
+Again, we would not be surprised if either Methods A or B
+(or both)
+ would 
+establish the conjectured self-justification property.
+However, there are also 
+other
+plausible hybrid methods
+that would establish this  property.
+Here are two examples
+\bed
+\item{$~~  Method-C-1 $}. 
+We follow Method A's approach with probability 0.5 and
+method B's approach with probability 0.5.  
+Thus, either of Method A's two choices occur with a 
+precise
+probability
+        of
+$~\frac{1}{4}~,~$ and Method B's setting
+of 
+$~ \zzthe(2^L)~=~ 2^j$ occurs with an exact  probability of:
+\beq
+\label{adj-prob1}
+\frac{p_j}{2~(1-S)}
+\enq
+\item{$~~  Method-C-2 $}. 
+We follow Method A's approach with probability of $~S~$ and
+method B's approach with probability $~1-S~$.
+Thus, either of Method A's two choices will hold with probability of
+$~\frac{S}{2}~$ and Method B's setting of
+$~ \zzthe(2^L)~=~ 2^J$ occurs with a  probability of
+exactly $~p_j~$.
+\end{description}
+
+\section{Basic Strategy}
+
+The basic strategy is to show that  if
+axiom basis
+$~\beta~$ holds valid in the standard model
+then it is impossible for a minimal proof of $0=1$
+to exist, Providing, just an overview,
+this should be able to be proved,
+{\it roughly,}
+ by contradiction.
+
+In particular, if $~\beta~$ holds valid in the standard model
+then all the Group 0, 1 and 2 axioms will hold valid under the
+Standard Model.   Hence
+if the consistency preservation property fails, then
+the Group-3
+ {\it ``I am consistent"  axiom}
+must be false under the standard model.
+Thus, the proof of the falsification of the
+ Group-3
+ axiom
+ must be able to self-reference itself
+and
+thereby
+ establish its own incorrectness.
+
+At this juncture, one must use a natural encoding for the
+G\"{o}del 
+numbers of a proof
+(such as what was formally given in 
+\cite{ww1,wwapal}'s
+examples of G\"{o}del encodings).
+  The proof of the existence of such a proof
+will exceed the proof's length by a factor $\lambda > 4$
+(or more)
+ under all
+natural encodings of proofs
+(including the examples given in 
+\cite{ww1,wwapal}).
+ But the point  
+is that our formalism
+{\it  has only the} $\Theta$ operation as
+a primitive,
+for
+ representing growth.
+Thus, the natural probability
+distributions from the prior section should establish something
+to the effect that there is a
+probability  lower 
+bound
+of
+ $c > 0$,
+such  that an
+excessively fast
+ growth-rate will be impossible.  
+
+Thus leaving aside many
+messy details, this lower bound will assure that there is stochastic
+model where growth is precluded at a fast enough rate for some
+demonstrated model to form the type of counterexample
+that G\"{o}del's Completeness Theorem needs to show that the
+ {\it ``I am consistent"  axiom} needs for corroborating its claim
+for self-justification.
+
+This result differs from my earlier work in that it needs ZF Set Theory
+(rather than Peano Arithmetic) to corroborate what I call the 
+Consistency Preservation Property.  That is fine and legal
+because the system IQFS($\beta$) 
+affirms its own consistency via
+a 1-sentence axiom.
+Thus, ZF Set Theory's knowledge about
+Lebesgue measures 
+should, likely,
+indicate  IQFS($\beta$) 
+is a competent
+enough
+ formalism to make no false claims.
+
+\newpage
+My apologies that the preceeding
+short summary
+ is not a formal proof.
+It merely outlines,
+{\it very roughly},
+ what I have in mind.
+
+
+
+\begin{thebibliography}{99}
+
+
+
+ \normalsize
+\parskip 5 pt
+\baselineskip =  1.3 \normalbaselineskip
+
+
+
+\bibitem{ww1}
+Willard, D. E.:
+ ``Self-verifying  systems, the incompleteness
+theorem and the tangibiltiy reflection
+principle'', in
+{\it Journal of Symbolic Logic}
+$~66~ (2001)\,$ pp. 536-596.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explaining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+\bibitem{wwapal}
+Willard, D. E.:
+``A generalization of the second incompleteness 
+theorem and some exceptions to it''.
+{\it Annals of Pure and Applied Logic}
+141 (2006)
+pp. 472-496.
+(Unlike our later articles, \cite{ww1,wwapal}
+spends some time explining how we generate
+our analogs of
+ G\"{o}del numbers
+ for itemized sentences and proofs.)
+
+
+
+\bibitem{ww16}
+Willard, D. E.:
+On  how the introducing of a 
+ new $~\theta~$ function symbol
+into arithmetic's formalism is 
+germane
+to devising axiom systems that can 
+appreciate fragments of their own
+Hilbert consistency.
+{\it
+Cornell Archives arXiv Report}      
+1612.08071v5
+         (2017).
+
+
+\bibitem{ww20}
+Willard, D. E.:
+``On the Tender Line
+Separating Generalizations and Boundary-Case Exceptions for the
+Second Incompleteness Theorem under Semantic Tableaux
+Deduction'',
+a talk given 
+on January 7 at the LFCS 2020 conference.
+Early version in 
+Volume 11972 of
+ Springer's LNCS series, and longer version sent to
+Cameron and Nate.
+
+
+\end{thebibliography}
+\end{document}
+
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new file mode 100644
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--- /dev/null
+++ b/nachlass/collected_dew_materials/robert.tex
@@ -0,0 +1,318 @@
+ % 2020 July 26 6.22 pm  spell done 
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf       Talk from Daddy to Robert}, $~~~$  
+  July 26, 2020. This statement expresses my sentiments, and I wanted to write them  down.
+ % explicitly,
+ to avoid any ambiguity.
+
+
+  % 266
+
+ \large
+  \baselineskip =1.35 \normalbaselineskip
+ \parskip 4 pt
+
+
+
+
+
+
+Many people travel the World during their young 20's
+(Even your  father did some of that!  )
+
+Because it is for most people the FREE-st Time in their lives.
+I understand why  you are tempted to go to Turkey.
+
+But I will add a few notes to yesterday's talk before starting lesson
+
+I already promised to give you 2K for every month you decide to stay home.
+
+It is NOT RATIONAL to travel abroad in midst of a world-wide
+epidemic --- an opinion of both your mother and me.
+
+You are offering to be cheap labor for a country (Turkey) that is a
+cheap labor supply for Germany (and which has surplus of refugees
+from the Syrian Civil-Religious War).
+
+Airplane Travel ``could'' be again BLOCKED between Turkey and USA.
+
+And while the above sentence with its word ``could''  is speculative,
+IT IS NOT A SPECULATIVE POINT that most Ivy League schools will be
+3/4 closed in their  Fall Semesters
+(If they open ???? EVEN PARTIALLY ? )
+
+In contrast, I could teach you a real logic course in 3 months time,
+and you could in future teach logic in philosophy departments.
+
+Let me close this one page note with an ADDED PERSPECTIVE.
+It turns out that almost all of the
+most recognized
+mathematicians and logicians were also textbook writers.  The
+most famous textbook writer of all times was probably Euclid.
+
+No one knows what  Euclid did besides writing a geometry textbook
+that was quoted over the ages. You and I could do the same in Logic
+with a Willard-and-Willard textbook..... And everyone would want
+to hire you then as the field expert.
+
+The point is your language abilities and my math knowledge would
+make this textbook FLY HIGH AND CLEAR.
+
+
+
+ \end{document}
+
+ 
+\newpage
+
+OLD TEXT
+
+No wife, No responsibilities, No children, No Job, No Mortgage.
+Who cares if there is  little money?
+
+Able to live cheaply in a context where few older people can
+similarly  do.
+
+Able to eat cheaply. Can of baked beans can be quite satisfying.
+
+JFK, for example, went to Europe before WW2 broke out and returned
+from France to England  LITERALLY  on the EXACT day the war started.
+
+Many rich aristocrats did similar (e.g. FDR when he was young)
+
+Willing to stay in crude hostels and camp sites AND ROUGH IT.
+
+When I was in my mid-20's, I used to sleep in my car  near beaches.
+When I was 30 and had my first job, I still traveled to two California
+conferences and slept in a RENTED CAR.
+
+I didn't travel to EUROPE via bicycle, such as the Super-RICH in their 20's
+but I did take two 11-day tours with a supervised teenage group in New England
+when I was 15-16 (camping in cramped hostels).
+
+A famous logician called Brower, who lived in Holland, used to take his
+notebook and take a 90-day walk through France every summer.  One year,
+he lost his notebook in the middle of his walk, and then constructed
+an entirely new theory,  WHICH WAS HIS GREATEST LIFE-TIME ACHIEVEMENT.
+
+Your mother also likes wondering around (more than most women).
+
+The UNUSUAL  ASPECT of the current time is that experts about Covid-19,
+such as Dr. Faucci, are predicting the next six months will be a challenging
+time, with the disease exponentially growing in size, before a vaccine is
+available and MASS PRODUCED for the public to use it.
+
+Instead of hundreds of thousands of people dying, it is going to be possibly
+Hundreds of millions and possibly LITERALLY BILLIONS of people dying.
+
+Also, there could be an epidemic of violence, robbery and possibly civil
+war in impoverished countries.... EVEN IN USA TO LESSER EXTENT.
+
+The closest event to the the coming six months is the start of WW2.
+Your grand-mother was on the last train from Prague to Amsterdam before
+Jews were forbidden.  Her cousin was on next train, watched Nazis beat her
+father to death when he argued with them, and was sterilized and forced
+into prostitution to survive WW2 barely. (JFK left France on EXACTLY the
+day War broke out.)
+
+It may sound like a selfish,  foolish  remark from a 72-year old man, 
+but to go to Turkey before this 6-month wait for the Covid vaccine is
+over could be a fatal mistake, similar to the Last Train from Amsterdam.
+
+And you, perhaps with or without me, could write a book, similar to the
+Diary of Anne Frank, about your experiences during the Epidemic and
+possibly coming Civil War, during the next six months.  At THE VERY LEAST,
+you can have your friend Kira publish it, and probably you can go to a
+commercial publishing house and maybe even make it into a movie.
+
+You are in an unusual  situation because at age age 22, you have a 72-year
+old father and 61-year old mother.... And you need to think about how to
+survive the next 6 months before a vaccine arrives and while riots and
+horrible theft and stealing could become a second epidemic around
+the World.
+
+Let me make a confession  (say in a gentle voice):
+
+When I was 27, my  mother was dying of cancer and I was sadly reluctant
+to come home from Harvard when my grandmother lectured me about my
+mother dying.
+
+My psychologist then subsequently reinforced her lecture by telling
+me that I DID NOT WANT TO admit and RECOGNIZE how horrible events 
+were occurring.  He told me my mother ALSO DID NOT WANT TO ADMIT the
+reality, BUT SHE HAD NO CHOICE but to face it because events were happening
+TOO SUDDENLY .... with her  health deteriorating.
+
+I suspect you, as a young guy feeling your oats and seeing your world
+as invulnerable, are also not admitting the reality. The good news is
+that it is only a 6-month crisis lying ahead, and they are talking about
+a vaccine being ready before December (although its mass manufacturing
+might take 2 months longer).
+
+Partially because I was so young, and did not want my mother's illness
+to interfere with my life in Boston and search for a job, and my search
+for a wife and future mother of my children, I did not want to go home
+to visit my mother.  In the end after my grandmother's lecture,
+ I started coming home every
+4-6 weeks to visit my mother during the last ten months of her life.
+
+The Covid Crisis is only a 6-month interlude before the vaccine becomes
+prevalent. You should know that two of my best friends were diagnosed
+with cancer during the last four months. (They are Lee Nagel and Jon Silver,
+both of whom you know).  Also two other of my firends
+had died in the last 20 years,
+who you don't know.)
+
+ \baselineskip =1.5 \normalbaselineskip
+
+Debby (my sister) is also dying and unlikely to live 3-5 more years.
+
+I think I am healthy, BUT YOU SHOULD KNOW I have avoided taking a cancer
+test THAT EVERYONE OVER THE AGE OF 65  is recommended to take.
+
+I confess I was delinquent in not taking the cancer test, and I am
+unable to take it during the next week while you are getting ready to
+leave.
+
+My grandmother took the same cancer test at age 72. They found cancer and
+removed it successfully. She then lived for 17 more years and never had
+cancer again.  BUT I CAN'T have a cancer removed, if I have it (?), without
+my family
+% being 
+present.
+
+And I am not suggesting you avoid travel --- ONLY POSTPONE it until the
+Covid vaccine is available.  It is less certain, but a second epidemic
+of riots and civil wars 
+and violent crimes
+is also possible during the next 6 months.
+
+I would be happy to pay you 2K per month, if you don't trust me (?) about
+allowing you to postpone your departure until the 6-month Covid waiting period
+is over. (You may not want this money, BUT I AM OFFERING it to show you how
+seriously I am about considering a minor postponement  ( I will
+put my money behind my excessively loud and cranky voice
+and mouth.)
+
+Also, if you don't come with me to a minor 100 dollar visit to lawyer to sign
+the needed documents, you might forfeit the entire Willard inheritance.
+(You need to hear legal details to understand this issue.)
+
+Josh Silver  (who is  45) is trying to persuade his father Jon to move
+to Brooklyn. Jon has cancer, and Josh wants Jon to live nearby Josh
+in Brooklyn. (I am not even asking you to necessarily live at 14 Grafton
+Road during the most troubling crisis since WW2. BUT to be prudently
+cautious...,   for you to at least  consider postpone going  quickly abroad  
+during
+ this
+dangerous
+6-month period ! )
+
+\newpage
+
+{\bf Added Comments:}
+
+Doctors have recently speculated that even
+young children
+  who do not
+come with Covid symptons can possibly experience brain damage
+and other harm to their body.
+
+Don't forget to talk about NP.  (Good guessing is the subtle cure point) 
+
+Talk about Brower.
+
+Jon Silver predicted that instead   of you spending 90 minutes in church,
+you would go for the entire wedding PLUS MORE because you can't say no
+to your mother (who will persuade you to stay longer).
+
+
+\end{document}
+
+
+
diff --git a/nachlass/collected_dew_materials/sep10,2020/#n.tex# b/nachlass/collected_dew_materials/sep10,2020/#n.tex#
new file mode 100644
index 0000000..e7042a1
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/#n.tex#
@@ -0,0 +1,186 @@
+sparsk% 2020 september 10 1.20 pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 4 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/1pm b/nachlass/collected_dew_materials/sep10,2020/1pm
new file mode 100644
index 0000000..013f950
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/1pm
@@ -0,0 +1,186 @@
+% 2020 september 10 1.1pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 4 pt
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must left itself
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/done b/nachlass/collected_dew_materials/sep10,2020/done
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/done
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
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+ %\parskip 1 pt
+
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+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/done-sep10 b/nachlass/collected_dew_materials/sep10,2020/done-sep10
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/done-sep10
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/n.pdf b/nachlass/collected_dew_materials/sep10,2020/n.pdf
new file mode 100644
index 0000000..01d5933
Binary files /dev/null and b/nachlass/collected_dew_materials/sep10,2020/n.pdf differ
diff --git a/nachlass/collected_dew_materials/sep10,2020/n.tex b/nachlass/collected_dew_materials/sep10,2020/n.tex
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/n.tex
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/n.tex~ b/nachlass/collected_dew_materials/sep10,2020/n.tex~
new file mode 100644
index 0000000..ae1532d
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/n.tex~
@@ -0,0 +1,187 @@
+% 2020 september 10 1.20 pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 4 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/o.tex b/nachlass/collected_dew_materials/sep10,2020/o.tex
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/o.tex
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/r.tex b/nachlass/collected_dew_materials/sep10,2020/r.tex
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/r.tex
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/restart b/nachlass/collected_dew_materials/sep10,2020/restart
new file mode 100644
index 0000000..bb7439d
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/restart
@@ -0,0 +1,139 @@
+% 2020 september 10 11.54am   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+eeeeeeeeeeeeeee
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/start b/nachlass/collected_dew_materials/sep10,2020/start
new file mode 100644
index 0000000..ebeacaa
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/start
@@ -0,0 +1,137 @@
+% 2020 Aug 25 3.10   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on August  25,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/v1 b/nachlass/collected_dew_materials/sep10,2020/v1
new file mode 100644
index 0000000..ae07995
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/v1
@@ -0,0 +1,162 @@
+% 2020 september 10 11.54am   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+% \Large
+% \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+xxxxx
+
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept if it is viewed as one leg of the 3-part
+formalism for reviving logic, in view of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My thory cannot stand by itself because it is
+too weak in isolation. Artemov's results, although nice,
+suffer a different type of weakness because they replace
+on unified theorem with pluarlistic schemeS method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unaccepbible
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and ** (as well
+as in the Sacks-Goedel quotes).
+
+
+eeeeeeeeeeeeeee
+
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/v2 b/nachlass/collected_dew_materials/sep10,2020/v2
new file mode 100644
index 0000000..b62a603
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/v2
@@ -0,0 +1,182 @@
+% 2020 september 10 noon+30   Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.5in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \Large
+ \baselineskip =1.6 \normalbaselineskip
+\parskip 4 pt
+
+xxxxx
+
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept if it is viewed as one leg of the 3-part
+formalism for reviving logic, in view of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unaccepbible
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and ** (as well
+as in the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground it must left itself
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce.  Human Beings
+have used an anology of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes waht was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/v4 b/nachlass/collected_dew_materials/sep10,2020/v4
new file mode 100644
index 0000000..013f950
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/v4
@@ -0,0 +1,186 @@
+% 2020 september 10 1.1pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 4 pt
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must left itself
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/v5 b/nachlass/collected_dew_materials/sep10,2020/v5
new file mode 100644
index 0000000..5f05b07
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/v5
@@ -0,0 +1,186 @@
+% 2020 september 10 1.15 pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 5 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 4 pt
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/sep10,2020/v7 b/nachlass/collected_dew_materials/sep10,2020/v7
new file mode 100644
index 0000000..e036047
--- /dev/null
+++ b/nachlass/collected_dew_materials/sep10,2020/v7
@@ -0,0 +1,187 @@
+% 2020 september 10 1.25  pm    Final Version      pm  after spell
+
+% From 2020 Aug 20 415  pm
+
+
+ % 2018 pesquera
+
+ \documentstyle[12 pt]{article}
+ \addtolength{\oddsidemargin}{-1.1in}
+ %\addtolength{\textwidth}{1.0in}
+ \addtolength{\textwidth}{1.9in}
+ \addtolength{\topmargin}{-1.2 in}
+ \addtolength{\textheight}{2.0in}
+ %\parskip 1 pt
+
+ \newcommand{\newthmwithin}[3]{\newtheorem{#1q}{#2}[#3]
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+
+ \newcommand{\newthm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\sf}{\end{#1q}}}
+ \newcommand{\newthmm}[3]{\newtheorem{#1q}[#2q]{#3}
+                         \newenvironment{#1}{\begin{#1q}\rm}}
+
+ \newtheorem{theorem}{Theorem}
+ %\newtheorem{theorem}{Theorem}[section]
+ \newtheorem{lemma}{Lemma}
+ \newtheorem{corollary}{Corollary}
+ \newtheorem{fact}{Fact}[section]
+ \newcommand{\makenewheading}[1]{\begin{tabbing} {\bf #1:} \end{tabbing}}
+ \newcommand{\set}[1]{\{ #1 \}}
+
+ \newcommand{\cp}[1]{\mbox{\sc cap}(#1)}
+ \newcommand{\dem}[1]{\mbox{\sc dem}(#1)}
+
+ %\newenvironment{proof}{{\it Proof:}}
+ \newenvironment{proof}{{\it Proof:}}{$\Box$}
+ \newenvironment{sketchproof}{{\it Sketch of Proof:}}{$\Box$}
+
+ \newcommand{\order}[1]{$\mbox{O}\left(#1\right)$}                   
+
+ \newcommand{\rootm}{\sqrt{m}}
+
+ \newcommand{\floorz}{$ (\lfloor|Z|/2  \rfloor -1) $}
+ \newcommand{\mfloorz}{ (\lfloor|Z|/2  \rfloor -1)}
+ \newcommand{\mceilz}{ (\lceil|Z|/2 \rceil + 1)}
+ \newcommand{\ceilz}{$ (\lceil|Z|/2 \rceil ) $}
+
+ \newcommand{\floor}[1]{\lfloor #1 \rfloor}
+ \newcommand{\lbound}[1]{$\Omega\left(#1\right)$}
+
+ \newcommand{\mod}[1]{\left| #1 \right|}
+ \newcommand{\belongs}{$\in$}
+
+ \def\bed{\begin{description}}
+ \def\ennd{\end{description}}
+ \def\bee{\begin{enumerate}}
+ \def\ene{\end{enumerate}}
+
+
+ \newcommand{\co}[1]{Corollary \ref{#1}}
+ \newcommand{\th}[1]{Theorem \ref{#1}}
+ \newcommand{\lem}[1]{Lemma \ref{#1}}
+ \newcommand{\eq}[1]{(\ref{#1})}
+ \newcommand{\ep}[1]{Equation (\ref{#1})}
+ \newcommand{\underx}[1]{\underbrace{~ {#1} ~}}
+
+
+
+
+
+
+
+
+
+
+
+ \begin{document}
+ \thispagestyle{empty}
+ \large
+ \normalsize
+ % \small
+ \baselineskip = 1.6 \normalbaselineskip
+
+ \parskip 0 pt
+ %\begin{center}
+ $.$
+ \small
+ \bigskip
+ \noindent
+ {\bf  NNN Notarized Scientific Notes of Dan E. Willard on September 10,2020}, $~~~$  
+
+
+  % 266
+
+ \Large
+\normalsize
+\footnotesize
+\baselineskip =1.0 \normalbaselineskip
+
+ \LARGE
+ \baselineskip =1.8 \normalbaselineskip
+\parskip 0 pt
+\footnotesize
+
+xxxxx
+$~$
+On September 9, I listened to Artemov's Indiana talk from
+4-6pm.  It helped me better understand his arXiv paper,
+and its relationship to my work.  His Complete-Induction (CI)
+is a useful concept {\it if it is viewed as one leg of the 3-part
+  formalism} for reviving logic
+within the aftermath of Goedel's 1931
+paradox.  The other two legs are my self-justifying systems
+and the Pudlak-Solovay Theorem (that many of my papers
+have called Result ++).
+
+All three of these components are necessay and needed to
+interact with each other to formulate a fully comprehensive theory
+of Logic. My theory cannot stand by itself because it is
+TOO WEAK in isolation. Artemov's results, although nice,
+suffers a different type of weakness because they replace
+one unified theorem with a pluarlistic ``SCHEMES'' method, whose
+multiplicity is unfortunate. Likewise,the Second Incompleteness
+Theorem (whose strongest form appears in ++) is unacceptable
+because it is an excessivly negative result (as indicated
+by the Hilbert Godel complaints given in * and **
+together with  the Sacks-Goedel quotes).
+
+The word `` TRIPOD '' is an excellant way of summarizing my
+3-part
+theory. It is well known that a camera cannot stand up
+in a steady manner on planet Earth because gravity will
+cause it to collapse in one direction or another.  But if
+it has 3 legs, gravity will cause it to stand steady
+because before collapsing to the ground
+the camera
+must rise
+up before the force of gravity causes it to become unstable.
+
+Likewise, the 3-legs of the ``Tripod'' theory will
+form a stable configuration. It is not a perfect theory:
+It will force philosophical compromises, BUT IT IS A CONFIGURATION
+that can be made stable under the influnce
+of gravity. Human Beings
+have used an analogy of this improvised 3-leg approach
+since the dawn of human thought (we conjecture). My 3-part
+``Tripod'' formalizes what was done informally, since at
+least the time when literacy began.
+
+\tiny
+eeeeeeeeeeeeeee
+$~~~~$
+OLD NOTES
+$~~~~$
+On August 20, I reported that after I gave
+Robert a class about logic (over Internet),
+ I developed a much refined understanding about the Propositional
+ Calculus version of the compactness theorem (appearing in Enderton's
+ textbook). Now after an August 24 class, I developed a second improved
+ proof of preceding, where the
+  ``unknown'' symbol (for when a Boolean value is unknown) is
+ called
+``Zip'' or
+ ``Zippy'', and a fourth symbol is a never-employed
+ symbol,   called ``Oops''
+ Here False, True, Zippy and Oops carry
+ the numeral values of 0, 1, 2, and 3 in a
+ ``quadruple'' analog of a decimal encoding.
+
+ 
+ I also
+ want to record that I plan to telephone Peter Bloniarz tomorrow
+ about a glitch in New York State security that he may not know about.
+ It is that workers in the NYS bureaucracy are using their insecure home
+ computers when servicing customers via transferred telephone calls.
+ Since these workers have access to social security numbers, an
+ obvious security breach is possible.
+
+ 
+ I also developed after talking to Robert an improved
+ proof of  Enderton's
+ A-version of the Completeness Theorem.
+ I don't have time to go into more  details here
+ about any of these topics   because I need to do my taxes today.
+ This records a continuation of my on-going research,
+
+ \end{document}
diff --git a/nachlass/collected_dew_materials/source.tex b/nachlass/collected_dew_materials/source.tex
new file mode 100644
index 0000000..29b0c77
--- /dev/null
+++ b/nachlass/collected_dew_materials/source.tex
@@ -0,0 +1,138 @@
+1empire Health Card 890 100 573
+
+pamphlet called ``Welcpme to EBD''
+
+     NYSHIP New York State Health Insurance Program
+     talk to Civil Service Office
+
+Empire phone 877 7 NYSHIP
+
+civil service phone = 518 457 5754
+
+social security tel = 800 772 1213
+
+NYS december   2020 to Rhannon DiMatinp
+
+NYS december   2020 to ???? 
+
+civil service phone = 518 457 5754
+
+CALL 9 am EXACTLY to get fastest service
+
+press 1 English 
+      2 retiree
+      1  billing
+      1 additional assistance about cost of health insurance 
+Don,t press 2   2 how make payment
+      1 additional help by human
+          estimated wait time less than 5 minutes when call at 9am 
+          They still ask if I requested call-back in context of 5minute wait
+          at time of 9.09am 9.21am. 
+    (On dec 8, I called at 10.25am to get info about who check is payable to)
+
+Info:
+
+Who am I talking to: Misty
+
+      Monthly Premium  568.64
+sick Leave Credit = 347.64  
+ 
+2022 medicare 170.10
+
+Net  = 50.90 per month
+
+Annual 12X above = 610.80
+
+
+note indicate multiple months of premiums
+
+balance is currently 48.36 which I want to continue.
+Indicate that in my correspondence.
+
+always contact office the  Monday after first Friday in December
+
+mail letter to 
+
+New York State Department of Civil Service
+Employee Benefits Division
+PO Box 645 475
+Cincinatti, Ohio 45264 - 5475 (dash should be here)
+
+Can mail regularly or however you feel comfortable
+
+Don't Mail to Albany, NY
+
+
+(Any way you wish to send it. including Express Mail)
+
+To Whom It May Concern
+
+This letter is from Dan E Willard whose empire insurace  ID  = 
+and whose billing id =         
+Enclosed is a check for xyz to cover all 12 months of my
+2022 health insurance  premiums of xyz per month. For safety sake,
+please also keep in you possession my balance of 48.36
+from prior years. Sincerely Yours 
+
+QUESTIONS 1) WHO IS CHECK PAYABLE TO????  Need to call on Tuesday?
+          2) how should    mail (express mail, registered mail?) 
+          3) Three month rule?
+          4) Check amount
+          5) Robert extension?
+Questions
+
+0) Contact Retiree Unit for more details and Misty will do
+   that right now. Call them 9am 
+
+1) At 65, MY wife must be enrolled Parts A and B of Medicare
+
+2) Always stay away from Part D Medicare
+
+3) When my wife is 65, ask Retiree unit how to handle it if she is not
+enrolled in Medicare
+
+4) If i have at least one dependent, NO CHANGE in my premium
+
+5) Beyond 26, i can purchase extra converage for my son
+              until age ???? (unknown). Contact 
+         Cobra Application for 36 month of coverages
+             (2022 rate = 938 per month. If cancelled cannot
+              be restored).
+
+
+6) If I pass and have family coverage, NYSHIP offers 3 free months
+   of extended coverage to beneficiaries, but after that they receive
+   information about how to do their own seperate coverage.
+
+7) xyz is standard medicare deduction. And Misty never saw anything
+   different this year.
+
+8) Wife can enroll in Medicare without getting Social Security by
+   NOT taking money out of Social Security by GIVING BANK ACCOUNT
+   for DIRECT deductions.
+
+9)  To find providers outside NYS, click on
+        empireplanproviders.com
+           enter zip code information
+
+
+
+i have to find out who check is payable to
+abd ALSO contact social security office
+or call 800 772 1213 to discover maximal
+medicare payment.
+
+
+They told me 35 days for reissuing cards in alpabetical order
+
+8.59 am called
+
+talked to Morgan in albany 
+
+check payable to NYS Employee Insurance Pending Account
+
+memo should billing ID = 100 111 8610 Just number on memoline
+
+That number SAME EACH YEAR   
+xxx
+