\documentclass[a4paper,10pt,draft]{article}
\usepackage{amsmath}
\input{../../math/abbreviations}
\input{../../math/format}

\title{Model-theory exercises}
\author{Math 406}
\date{2004.12.30}

\newtheorem{problem}{Problem}
\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}

\begin{document}
\maketitle
\thispagestyle{empty}

\begin{problem}
  Suppose $\lang=\{P\}$, where $P$ is a singulary predicate.  Let
  $\str A$ be a \emph{finite} $\lang$-structure, and suppose $\str
  B\equiv\str A$.  Prove $\str B\cong\str A$.
\end{problem}

\begin{problem}
  Let $\lang=\{P_n:n\in\varN\}$, where each $P_n$ is a singulary
  predicate.  Let $T$ be axiomatized by the sentences
  \begin{gather*}
    \Forall x P_0x,\\
\Forall x(P_{n+1}x\to P_n x),\\
\Exists x\lnot (P_nx\to P_{n+1}x),\\
\Forall x\Forall y(\lnot (P_nx\to P_{n+1}x)\to (\lnot (P_ny\to
P_{n+1}y)\to x=y)),
  \end{gather*}
where $n\in\varN$.
Prove that $T$ is complete.
\end{problem}

\begin{problem}
  Let $\Sigma$ and $\Sigma'$ be sets of sentences of some $\lang$ such
  that
  \begin{equation*}
    \Sigma\models\lnot\sigma\implies\Sigma'\nmodels\sigma
  \end{equation*}
for all sentences $\sigma$ of $\lang$.  Prove that $\Sigma\cup\Sigma'$
has a model.
\end{problem}

\begin{problem}
    Suppose $\Phi$ and $\Psi$ are $1$-types of a consistent theory $T$.
  Prove that $T$ has a model in which both $\Phi$ and $\Psi$ are
  realized.  (Use only Compactness and the appropriate definitions.)
\end{problem}

\begin{problem}
  Assuming $T$ is $\varN$-categorical, prove that $T$ has only
  countably many complete types.
\end{problem}



\end{document}