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\begin{document}

\begin{center}
\begin{Large}
{\bf Math 2001 - Assignment 8}
\end{Large}
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Due March 11, 2026
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\bigskip

\begin{enumerate}

\item
 Show that a structure $\A$ realizes all $1$-types over $A$ if and only if $\A$ realizes all $n$-types over $A$.
   
\item
 \cite[Ex 4.5.7]{Ma:MT}
 Let $T$ be the theory of $(\N,+,\cdot,0,1)$. Show that $|S_1(T)| = 2^{\aleph_0}$.

 Hint: Let $p_n$ be the $n$-the prime number. For $A\subseteq\N$, consider
 $\Gamma_A(x) := \{p_n \text{ divides } x | n\in A \} \cup \{p_n \text{ does not divide } x | n\not\in A \}$.

\item
 Show for any structure $\A$ that the expansion $\A_A$ is an atomic model of $\Th(\A_A)$. 

\item
 \cite[Ex 4.5.3]{TZ:CMT}
 Let $T$ be the theory of $(\R, <, Q)$ where $Q$ is a unary predicate for
the rational numbers. Does $T$ have a prime model?

\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}

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