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\begin{document}
\title{6.3-6.6}
\author{Andrew Lounsbury}
\date{April 20, 2020}
\maketitle

\section{6.3, p.174}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
    \item[proof of Lemma 6.3.5]
    The authors use Lemma 6.3.3 to assert the existence of the $\Sigma$-formula $AxiomOfA(x)$. In Lemma 6.3.3, the $\Sigma$-formula in question is 
    \[
        \phi(v_1) :\equiv \ex z \ex y \ex c (Num(v_1,z) \land Sub(\ol{k},\ol{8},z,y) \land Deduction(c,y)).
    \]
    Are they saying that this is what $AxiomOfA(x)$ is? \\
    Also, is the replacement of the $\Delta$-formula $AxiomOfN(e)$ with $AxiomOfA(e)$ the only change that turns $Deduction_A(c,v_1)$ into a $\Sigma$-formula? 
\end{enumerate}


\section{6.4, p.182}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
    \item[Thm 6.4.5] Given Proposition 6.4.3, doesn't Theorem 6.4.5 follow from Propositions 6.4.3 and 6.4.4? 

    6.4.4: $A$ is an $\omega$-consistent and recursive set of axioms extending $N \\
    \stackrel{\text{6.4.3}}\implies A$ is a consistent and recursive set of axioms extending $N \\
    \stackrel{\text{same as 6.4.5}}\implies A$ is incomplete

\end{enumerate}


\section{6.5, p.185}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
    \item[p.185]
    Is this finite restriction of $\Ell_{NT}$ only used for ``naming'' the outline of the proof of Theorem 6.5.1? 

    \item[p.186]
    In $Q \vdash ((\fa x)(\phi(x) \leftrightarrow x = \ol{n}))$, why is there a universal quantifier when it just ends up saying that $x$ has to be some specific natural number $n$? \\
    Do all formulas have to name a number? I'm tempted not to think so since the definition of ``naming'' seems to say (as in the last paragraph of the section) that a number named by a formula is the ``number that makes the formula true.'' But surely there are formulas that are true or provable for not just one natural number. 

    Is ``naming'' used for anything other this one outline of proof? They really make it stand out but seem to use it for just this one thing.  
\end{enumerate}

\section{6.6, p.187}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
    \item[p.191]
    At the end of the paragraph preceding Corollary 6.6.4, the authors seem to suggest that we're about to assume $PA$ is inconsistent for the corollary, but then the corollary starts by saying ``If $PA$ is consistent,...'' Is it the ``...$\cup \{\neg Con_{PA}\}$'' that says $PA$ is inconsistent? So, the task at hand is to show $PA \cup \{\neg Con_{PA}\}$ is consistent? 

\end{enumerate}

\section{Somewhat random questions}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

I'm just now remembering how in matrix algebra we say a system is inconsistent if the reduced echelon form has a row of zeros ending in a nonzero number. For instance,
\[
    \left[
        \begin{array}{cc|c}
            3 & 27 & 9 \\
            2 & 18 & 7
        \end{array}
    \right]
    \sim
    \left[
        \begin{array}{cc|c}
            1 & 9 & 3 \\
            2 & 18 & 7
        \end{array}
    \right]
    \sim
    \left[
        \begin{array}{cc|c}
            1 & 9 & 3 \\
            0 & 0 & 1
        \end{array}
    \right]
\]
is an inconsistent system of equations because the last row is saying $0x+0y = 0 = 1$. \\
Does this directly parallel the discussion of consistency and inconsistency in this book? In other words, are we in a way taking $3x+27y = 9$ and $2x+18y = 7$ as axioms and seeing whether that set of axioms is consistent or not? Also, the authors have repeatedly said that if a set of axioms can prove $\perp$ (or any contradiction?) that we would then be able to prove anything, which certainly sounds like something that would result from $0 = 1$. But then again, I suppose it may just be borrowed verbiage, since this could be thought of as a regular old contradiction, but it just happened to cross my mind. \\

Apart from Chapter 7 and the works referenced throughout this book, do you have any other books you can recommend?

\end{document}