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\begin{document}
\title{6.2}
\author{Andrew Lounsbury}
\date{April 15, 2020}
\maketitle

\section{6.2, p.173}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}
    \item[4.]
    Let $\psi(v_1)$ be $Even(v_1)$, and let $\phi$ be the sentence generated when the Self-Reference Lemma is applied to $\psi(v_1)$. Then, $N \vdash \phi$, and $N \nvdash \neg\phi$. 
    \begin{proof}
        By the Self-Referencing Lemma, $N \vdash \phi \leftrightarrow Even(\ol{\ulcorner \phi \urcorner})$. \\
        $\ulcorner \phi \urcorner$ is a product of powers of the first few primes \\
        $\implies 2 \mid \ulcorner \phi \urcorner$

        *The authors instead use the fact that $\phi$ is a sentence. I'm not sure why.*

        $\implies N \vdash Even(v_1)$ since $Even(v_1)$ is a $\Delta$-formula

        *I know this step follows from Prop. 4.6.1 on p.115, but I don't think I've quite yet wrapped my head around the significance of $\Delta$-formulas in all this. If $Even(v_1)$ were a $\Sigma$-formula, wouldn't this statement still be true by Prop. 4.6.2? And if $Even(v_1)$ were a $\Pi$-formula, then we wouldn't be able to say anything about whether $N \vdash \phi$ or $N \vdash \neg\phi$, right?*

        $\therefore N \vdash \phi$ and $N \nvdash \neg\phi$. 
    \end{proof}
\end{enumerate}
\end{document}