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\begin{document}


\begin{center}
{\bf Math 160B - Winter 2022 -
\bf Homework 2} \\  {\bf Due Wednesday, January 19, 11:00pm} \\
(Hand in by uploading to Gradescope)
\end{center}

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\noindent
{\bf 1.}
Let $T$ be the theory of linear orders from
Example~IV.89 using the language~$L=\{<\}$.
A model~$\mathfrakA$ of~$T$ is called \emph{well-founded}
if there is no sequence $a_0,a_1,a_2,\ldots$ in~$|\mathfrakA|$
such that $a_{i+1} <^\mathfrakA a_i$ holds for all~$i\in\mathbb N$.
Show that there is no
set~$\Gamma$ of sentences over~$T$ that expresses the
property of being well-founded. That is, there is no
set~$\Gamma$ of sentences such that for all models~$\mathfrakA$
of~$T$, we have $\mathfrakA \vDash \Gamma$ if and only if $\mathfrakA$ is well-founded.

\noindent
{\bf 2.}
Let $L$~be the language with a unary predicate symbol~$P$ (and, as always, equality).
Let $\Gamma$ be a set of sentences expressing that there infinitely many
objects~$x$ satisfying $P(x)$ and infinitely many
objects~$x$ satisfying $\lnot P(x)$.
\begin{itemize}
\item[(a)] Describe explicitly the formulas in~$\Gamma$.
\item[(b)] Show that $\Gamma$ is $\aleph_0$-categorial. Conclude
that the theory axiomatized by $\Gamma$ is complete.
\item[(c)] Show that $\Gamma$ is
not $\kappa$-categorical for $\kappa > \aleph_0$.
\end{itemize}

\noindent{\bf 3.}
Give an example of a theory which is $\kappa$-categorical for all
infinite~$\kappa$.

\noindent
{\bf 4.}
A theory is called \emph{categorical} if all models of~$T$ are
isomorphic.
\begin{enumerate}
\item Describe what must happen if $T$ is categorical.
\item Suppose $T$ is a categorical theory over a finite language~$L$.
Prove that $T$ is \emph{finitely axiomatizable}, that is, that there is
a finite set of sentences~$\Gamma$ such that $\Gamma \tautEq T$.
\end{enumerate}

\noindent
{\bf 5.}Let $\Sigma = \{a\}$.  Let $f : \Sigma^* \rightarrow \Sigma^*$
be the unary function so that for all $w\in\Sigma^*$,
\[
f(w) ~=~ \left\{ \begin{array}{ll}
\epsilon \quad & \hbox{if the Riemann hypothesis is true} \\[0.5ex]
a & \hbox{if the Riemann hypothesis is false.}
\end{array} \right.
\]
Prove that $f$ is computable.

\noindent
{\bf 6.}Let $\Sigma=\{1\}$. Let $f$ be the unary function defined by
\[
f(1^n) ~=~ \left\{ \begin{array}{ll}
\epsilon &\hbox{if there is a run of $n$ consecutive 7's in the decimal expanson of~$\pi$} \\
1 & \hbox{otherwise.}
\end{array} \right.
\]
Prove that $f$ is computable. (It is an open question whether there are
arbitrarily long (finite) runs of 7's in the decimal expansion of~$\pi$.)

\end{document}
