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\begin{document}
\begin{center} {\bf Problem Set VI}\\
Math 609  \end{center}

\begin{enumerate}
\item A structure $\F = (F, P ; +, \cdot \,; 0, 1)$ is an {\em ordered field} if it
satisfies the axioms for a field (in the reduct $(+, \cdot \,; 0,1)$) and the following
axioms involving the unary relation, that is, predicate, symbol $P.$ This symbol is
intended to interpret the positive elements of the ordered field $\F.$
\begin{description}
\item[OF1.] $P(x) \and P(y) \implies P(x+y).$
\item[OF2.] $P(x) \and P(y) \implies P( x \cdot y ).$
\item[OF3.] $x \neq 0 \implies (P(x) \or P(-x))$
\item[OF4.] $\neg (P(x) \and P(-x))$
\end{description}
Show that every ordered field has characteristic $0,$ that is, for every 
nonzero natural number $n,$ $OF \, \deduces \, nx \doteq 0 \implies x \doteq 0.$
Prove that the binary relation $<$ on the structure
$\F = (F, P ; +, \cdot \,; 0, 1)$ defined by $x < y = P(y-x)$ is a dense linear order without endpoints. 

\item An ordered field $\F = (F, P ; +, \cdot \,; 0, 1)$ is called {\em Archimidean}
if for every $a \in F,$ there is a natural number $n$ such that $|a| < n.$ 
Prove the following.
\begin{itemize}
\item Every ordered field is elementarily equivalent to a non-Archimedean field.
\item The set of elements $R = \{ a \in F : |a| < n \mbox{ for some natural number } n \}$ 
is a subring of $(F, +, \cdot \,; 0, 1)$
\item The set of non-units of $R$ is the ideal of infinitesimal elements
$$I := \{ a : |a| < 1/n \mbox{ for some positive natural number } n  \}.$$
\item $R/I$ is an Archimedean field.
\end{itemize}
\bigskip

\noindent In the remaining exercises, assume that $\La$ is a countable language and that
$T$ is a complete theory in $\La.$

\item Prove that if a model $\A \models T$ is atomic and countable, then it is the prime
model of $T.$
\bigskip

\noindent {\bf Definition.} A countable model $\A \models T$ is {\em homogeneous} if given
$a, a_1, \ldots, a_n, b_1, \ldots, b_n \in \A$ such that 
$\tp_{\A} (a_1, \ldots , a_n) = \tp_{\A} (b_1, \ldots , b_n),$ there is a $b \in \A$ such that
$\tp_{\A} (a, a_1, \ldots , a_n) = \tp_{\A} (b, b_1, \ldots , b_n).$ \bigskip


\item Verify that if $\A \models T$ is countable and atomic, then it is homogeneous.
\item  Verify that if $\A \models T$ is countable and saturated, then it is homogeneous.
\item Prove that if $\A \models T$ is homogeneous and $a_1, \ldots, a_n, b_n, \ldots, b_n \in \A$ 
are such that  $\tp_{\A} (a_1, \ldots , a_n) = \tp_{\A} (b_1, \ldots , b_n),$ then there is an automorphism
$\gre : \A \to \A$ such that $\gre : a_i \mapsto b_i.$
\item Prove that if $\A,$ $\B \models T$ are countable homogeneous models of $T,$ then $\A \cong \B$ if and
only if they realize the same $n$-types for all $n < \gro.$
\end{enumerate}
\end{document}