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\begin{document}

\begin{center}
\begin{Large}
{\bf Math 2001 - Assignment 10}
\end{Large}
\bigskip

Due April 1, 2026
\end{center}

\thispagestyle{empty}

\bigskip

\begin{enumerate}
\item Show that every finite model is $\omega$-saturated.

  
\item
 Let $\LL$ be the language consisting only of constant symbols $c_n$ for $n<\omega$, let $T$ be the theory
 with axioms $c_m\neq c_n$ for $m<n<\omega$.
\begin{enumerate}
\item Characterize the countable models of $T$ up to isomorphism [Hint: There are only countably many].
\item Does $T$ have a countable atomic model? What is it?
\item Does $T$ have a saturated countable model? What is it?  
\end{enumerate}
  

\item \cite[Exercise 4.5.17]{Ma:MT} Show
\begin{enumerate}
\item Every algebraically closed field is homogenous.
\item Every uncountable algebraically closed field is saturated. What about countable algebraically closed fields?
\end{enumerate}

\item
 Let $\A$ be a structure, $B\subseteq A$ and $f\colon B\to A$ a partial elementary map. 
 For $a\in A$ show that
\[ \{ \varphi(x, f(\bar{b})) \ :\ \A \models \varphi(a,\bar{b}), \bar{b} \text{ a tuple over } B \} \]
 is a type, denoted $f(\mathrm{tp}^\A(a/B))$. 
\end{enumerate}


\begin{thebibliography}{1}
%\bibitem{En:MIL}
% Enderton. A Mathematical Introduction to Logic. Harcourt/Academic
%Press, 2nd edition, 2001. 

\bibitem{Ma:MT}
 Marker.
\newblock Model Theory: An Introduction.
\newblock Springer, 2002.

% \bibitem{TZ:CMT}
%  Tent, Ziegler. A Course in Model Theory. Lecture Notes in Logic, Cambridge, 2012. 
\end{thebibliography}

\end{document}





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