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\begin{document}


\begin{center}
{\bf Math 160B - Winter 2022 -
\bf Homework 8} \\  {\bf Due Sunday, March 13 11:00pm} \\
(Note revised due date --- Hand in by uploading to Gradescope)
\end{center}

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Problems 1 and 5 depend on the proof of the G\"odel Second Incompleteness Theorem which
will be covered in class on Tuesday (March 8).

\noindent
{\bf 1.}
Let $A$ be an arbitrary $L_\PA$-sentence. Prove that
$\PA$ does not prove $\lnot \scThm_\PA(\underline{\lgn A \rgn})$.


\noindent
{\bf 2.}
Let $E$ be a self-referential formula such that $\TR$ proves
$E \liff\penalty10000 \scThm_\PA(\underline{ \lgn \lnot E \rgn } )$.
Is $E$ true or false? Justify your answer.

\noindent
{\bf 3.}
Let $F$ be a self-referential formula such that $\TR$ proves
$F \liff\penalty10000 \lnot \scThm_\PA(\underline{ \lgn \lnot F \rgn } )$.
Is $F$ true or false? Justify your answer.

\noindent
{\bf 4.}
Let $T\supseteq \TR$ be a consistent, axiomatizable theory.
This exercise asks you to carry Rosser's construction of an
independent sentence. A \emph{Rosser proof} of a formula~$A$ is
defined to be a proof~$P$ of~$A$ such that there is no proof~$P^\prime$
of $\lnot A$ with $\lgn P^\prime \rgn < \lgn P \rgn$.  More formally,
let $\scNeg(x_1,x_2)$ represent the mapping
$\lgn A \rgn \mapsto \lgn \lnot A \rgn$ in~$\TR$ and define
$\scRprf_T(w,x)$ to be the formula
\[
\scRprf_T(w,x) ~:=~
\scPrf_T(w,x) \land \forall v\, \forall y\, ( v<w \land \scNeg(x,y) \limplies \lnot \scPrf_T(v,y) ).
\]
Let $\scRthm_T(x)$ be the formula $\lnot \exists w \, \scRprf_T(w,x)$. Finally, let
$D_R$ be a self-referential formula such that $\TR$
proves $D_R \liff \lnot \scRthm_T(\underline{\lgn D_R\rgn})$.
Also, let $A$ be an arbitrary formula.
\begin{enumerate}[nosep]
\item There is a Rosser $T$-proof of~$A$ if and only if there is
a $T$-proof of~$A$.
\item If $T\vdash A$, then $T\vdash \scRthm_T(\underline{\lgn A \rgn })$. \newline
[Hint: You will need to use $T\supseteq R$ and Axiom~$\TR_{\le}^\prime$.]
\item If $T \vdash \lnot A$, then $T\vdash \lnot \scRthm_T(\underline{ \lgn A \rgn })$.
\item Prove that $T \nvdash D_R$.
\item Prove that $T \nvdash \lnot D_R$.
\item Conclude that $D_R$ is independent of~$T$.
\end{enumerate}

\noindent
{\bf 5.}
Suppose $T\supseteq R$ is consistent, axiomatizable and satisfies the Hilbert-Bernays-L\"ob
conditions.
Let $D$ be a self-referential formula such that
$\TR\vdash D \liff \lnot \scThm_T(\underline{\lgn D \rgn})$.
\begin{enumerate}[nosep]
\item Prove $T \vdash D \liff \scCon_T$.
\item Suppose that $E$ is another sentence such that
$\TR\vdash E \liff \lnot \scThm_T(\underline{\lgn E \rgn})$.
Prove that $T \vdash D \liff E$.
\item Show that there is a theory~$T$ such that $T\vdash \lnot D$.  (This shows
that the assumption of $\omega$-consistency in Theorem~IX.31
cannot be weakened to the assumption of consistency.)
\end{enumerate}


\end{document}
