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\begin{center}
MATH 776 HOMEWORK 13 \\
{\it Semester II, 2006-2007} \\
{\it Due May 15}
\end{center}


\bigskip



\bigskip
\textbf{61.}
Let $\AAAA = (A; <)$ be any $\aleph_1$--saturated model,
where $<$ is a strict partial order.  Prove that either
$\RRR$ embeds isomorphically into $A$ or there is a finite $n$
such that all chains in $\AAAA$ have size $< n$.
\textit{Hint.}  Let $C$ be an infinite maximal chain.
For $a,b \in C$, say that $a \sim b$ iff there are only finitely
many elements of $C$ between them, and note that the equivalence
classes are densely ordered.

\bigskip

\textbf{62.} Let $\LL = \{\equiv\}$ and let $\Sigma$ be the
theory of an equivalence relation such that there is exactly
one equivalence class of each size $n$ (for  $0 < n < \omega$).
Prove that $\Sigma$ is $\omega$--stable, and,
whenever $\AAAA\models \Sigma$, $S \in [A]^{\aleph_0}$, and 
$X = \TP_1( \Th(\AAAA_S) )$, then $X''' = \emptyset$, but
$X''$ need not be empty (this depends on $S$).
\textit{Remark.} See HW44 for more on this $\Sigma$.


\bigskip
\textbf{63.} Prove that ``locally omits'' is not of finite character.
Specifically, describe a consistent
$\Sigma = \Sigma_0 \cup \{\delta_r : r < \omega\}$
in a countable $\LL$ 
and a set $\Gamma(x)$ of $\LL$-formulas such that $\Sigma_0$
locally omits $\Gamma$ but $\Sigma$ does not locally omit $\Gamma$.
\textit{Remarks.} You can get such an example where
$\LL$ has only constant symbols.
Note that each $\Sigma_0 \cup \{\delta_r : r < s\}$
locally omits $\Gamma$ because it is a finite extension of $\Sigma_0$.
This problem shows that you really need an induction in $\omega$
steps in the proof of the Omitting Types Theorem; the obvious
Zorn's Lemma ``proof'' fails.

\bigskip
\textbf{64.} Describe a model of the form $\AAAA = (\omega_1; <, F)$,
where $<$ is the usual order and $F$ is a binary function
such that $\AAAA$ has no proper eee.
\textit{Remark.}
You need the $F$, since  by Ehrenfeucht, $(\omega_1; <) \prec (\omega_2; <)$.
\textit{Hint.} Let $\beta \mapsto F(\alpha,\beta)$ be a 1-1
map from $\alpha$ into $\omega$.


\bigskip
\textbf{65.}  Assume $\kappa$ is a measurable cardinal, and let
$\AAAA = (\kappa; <, \cdots)$ be a structure for $\LL$.
Prove that $\AAAA$ has a proper eee.
\textit{Hint.} By definition, $\kappa$ is measurable iff
$\kappa > \omega$ and there is
a $\kappa$--complete non-principal ultrafilter $\UU$ on $\kappa$;
$\kappa$--complete means that the intersection of fewer than $\kappa$
elements of $\UU$ is in $\UU$.  So, take an ultrapower.


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