\documentclass[twoside,a4paper,12pt,draft]{article}
\usepackage{amssymb,amsmath,amsthm}
\title{Model-theory homework}
\date{2001, fall}
\author{Math 736---David Pierce}
\pagestyle{myheadings}
 \markboth{Model-theory homework IV}{Math 736, fall 2001}

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\begin{document}
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\thispagestyle{empty}

\noindent\emph{Homework IV, Math 736, Model-Theory.}

Elements $a_0$, \dots, $a_{k-1}$ of an abelian group are called
\tech{additively independent} if
\begin{equation*}
\sum_{i<k}n_ia_i\neq0
  %n_0a_0+\dots +n_{k-1}a_{k-1}=0
  %\implies n_0=\dots=n_{k-1}=0
\end{equation*}
for all integers $n_0$ \dots, $n_{k-1}$, not all of which are 0.

\begin{problem}
Let $\Q$ be the \emph{abelian group} of rational numbers.  Show that
there is an abelian group $\str G$ such that:
\begin{enumerate}
\item
 $\str G\equiv\Q$, and
\item
 $\str
G$ contains $n$ additively independent elements for every $n$ in
$\omega$.
\end{enumerate}
Then show that any two such \emph{countable} groups are isomorphic.
\end{problem}

Now let $\lang$ be an arbitrary signature.  If $\str M,\str
N\in\Mod(\lang)$ and $\str M\included\str N$, let us write
\begin{equation*}
  \str M\preccurlyeq_1\str N
\end{equation*}
if the inclusion of $M$ in $N$ preserves \emph{universal} formulas of
$\lang$.

\begin{problem}
Prove that the following are equivalent:
\begin{enumerate}
  \item
  $\str M\preccurlyeq_1\str N$
  \item
  there is $\str R$ in $\Mod(\lang)$ such that $\str M\preccurlyeq\str R$
  and $\str N\included\str R$.
\end{enumerate}
\end{problem}

Suppose $\{\str M_n:n\in\omega\}$ is a subset of $\Mod(\lang)$ forming a
\tech{chain}, that is, $\str M_n\included\str M_{n+1}$ for all $n$ in
$\omega$.  Then the \tech{union} of this chain is defined to be the
structure $\str N$, where:
\begin{enumerate}
\item
 $N=\bigcup_{n\in\omega}M_n$, and
\item
 for all basic formulas $\phi$, if
$\tuple a$ is a tuple from $M_n$, and $\str M_n\models\phi(\tuple a)$,
then $\str N\models\phi(\tuple a)$.
\end{enumerate}
  (You should verify that $\str N$ is
well-defined, but you need not submit the verification.)  The chain is
called \tech{elementary} if $\str M_n\preccurlyeq\str M_{n+1}$ for all
$n$.

\begin{problem}
Show that the union of an elementary chain is an elementary extension of
each structure in the chain.
\end{problem}

Recall that a theory $T$ of $\lang$ is called \tech{model-complete} if
$\str M\preccurlyeq\str N$ whenever $\str M\included\str N$ and both
structures are models of $T$.  A formally weaker notion is
\tech{1-model-completeness}: $T$ is 1-model-complete if $\str
M\preccurlyeq_1\str N$ whenever $\str M\included\str N$ and both
structures are models of $T$.

\begin{problem}
Prove that 1-model-completeness and model-completeness coincide.
\end{problem}

\end{document}
