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\begin{center}
   A. Miller  \\
   M776 Model Theory\\
   April 18,1988
\end{center}

\begin{theorem}(Morley)
Suppose $T$ is a theory in a countable language $L$ with built in Skolem
functions and $\langle\Sigma_n\;:\;n\in\omega\rangle$ is a countable family of types
such that for every $\alpha<\omega_1$ $T$ has a model $\a$ omitting each
$\Sigma_n$ and $|A|=\Beth_{\alpha}$. Then $T$ has a model with an
infinite set of indiscernibles omitting each $\Sigma_n$.
\end{theorem}
Hence by the expanding technique of indiscernibles $T$ has arbitrarily large
models omitting each $\Sigma_n$ and so the Hanf number of omitting types
is $\Beth_{\omega_1}$.


\begin{lemma} Suppose $N$ is a model of ZFC such that $\omega^N$ is standard
but $\omega_1^N$ contains an infinite descending sequence.  Suppose
that $T$ and the $\Sigma_n$ are in $N$ and
$N$ models:
\begin{quote}
   ``$T$ is a theory in a countable language $L$ with built in Skolem
functions and $<\Sigma_n\;:\;n\in\omega>$ is a countable family of types
such that for every $\alpha<\omega_1$ $T$ has a model $\a$ omitting each
$\Sigma_n$ and $|A|=\Beth_{\alpha}$.''
\end{quote}
Then $T$ has a model with an
infinite set of indiscernibles omitting each $\Sigma_n$.
\end{lemma}
proof: For simplicity assume we just have a single 1-type, $\Sigma(x)$.
Let $\alpha_n$ for $n\in\omega$ be such that
$N\models ``\omega_1>\alpha_n>\alpha_{n+1}+\omega$''.
Hence for every $n\in\omega$ and $k\in\omega$
$$N\models \Beth_{\alpha_n}\rightarrow (\Beth_{\alpha_{n+1}})^k_{\omega}$$
Let $N\models ``\a \;\mbox{models}\; T, \;\mbox{omits}\; \Sigma(x),\;
|A|=\Beth_{\alpha_0}$''.  Build a sequence $X_n$
 of elements of $N$ such that
 $N\models``X_{n+1}\subset X_{n}\subset A;\;|X_n|=\Beth_{\alpha_n}$''
and also for every $L$-formula $\theta(\vec{x})$ there exists $n\in\omega$
such that
$$N\models\forall \vec{a}\in [X_n]^{<\omega} \a\models \theta(\vec{a})$$
or
$$N\models\forall \vec{a}\in [X_n]^{<\omega} \a\models \neg\theta(\vec{a})$$
and for every $L$-term $\tau(\vec{x})$ there exists $n\in\omega$ and
$\theta_{\tau}(x)\in\Sigma$ such that
$$N\models\forall \vec{a}\in [X_n]^{\omega}
\a\models \neg\theta_{\tau}(\tau(\vec{a}))$$
Let $I=\{c_n\;:\;n\in\omega\}$ and consider the theory:

$$T\union\{\theta(\vec{a})\iff\theta(\vec{b})\;:\;\vec{a},\vec{b}\in
 I^{<\omega}\}\union \{\neg\theta_{\tau}(\tau(\vec{a}))\;
 :\;\vec{a}\in I^{<\omega}\} $$
Each finite subset of this is consistent and it gives an E-M type which
does the job.


\begin{lemma} Suppose $M$ is a countable model of ZFC. 
Then there exists a model $N$ such that $M\elemsub N$ and
$$\{n\in M\;:\;M\models n\in\omega\}=\{n\in N\;:\;N\models n\in\omega\}$$
but $\omega_1^N$ contains an infinite descending sequence.
\end{lemma}
proof: Let
$$T=D_{elem}(M)\union\{c_0<\omega_1^M,c_{n+1}<c_n\;:\;n\in\omega\}
\union\{c_n>\alpha\;:\;\alpha<\omega_1^M,n\in\omega\}$$
and let
$$\Sigma(x)=\{x<\omega^M\}\union\{x\not=n\;:\;n<\omega^M\}$$
By the omitting types theorem it is enough to show that
$\Sigma$ is not isolated over T.

{\bf Claim} For any sentence $\theta(\vec{c})$
$$T\union\{\theta(c_0,c_1,\ldots,c_n)\}\;\mbox{is consistent}$$
\centerline{iff}
\centerline{$M$ models:}
$$\forall \alpha<\omega_1\exists x_0,x_1,\ldots,x_n
[(\alpha<x_n<\ldots<x_1<x_0<\omega_1)\conj (\theta(x_0,x_1,\ldots,x_n))]$$

The proof is left to reader. Abbreviate the last statement:
$$\exists^{cof\omega_1}\vec{x}\;\theta(\vec{x})$$

Suppose that $\psi(x,\vec{c})$ isolates the type $\Sigma(x)$.
Hence $T\proves \psi(x,\vec{c})\implies x<\omega$.  By the claim
$$M\models \exists^{cof\omega_1}\vec{y}\;\exists x<\omega\;\psi(x,\vec{y})$$
Since $\omega_1$ is uncountable $\exists n\in M$ such that
$$M\models n<\omega\conj\exists^{cof\omega_1}\vec{y}\; \psi(n,\vec{y})$$
But then $T\union\{ \psi(n,\vec{c}) \}$ is consistent, contradiction.


To prove Morley's theorem, first note that both lemmas are true
for any sufficiently nice fragment of ZFC.  For example let
ZFC$^*$ be the theory of $(V_{\kappa},\in)$ where $\kappa$ is any
regular cardinal greater than $\Beth_{\omega_1}$.
So given any
$T$  a theory in a countable language $L$ with built in Skolem
functions and $<\Sigma_n\;:\;n\in\omega>$  a countable family of types
such that for every $\alpha<\omega_1\;\;T$ has a model $\a$ omitting each
$\Sigma_n$ and $|A|=\Beth_{\alpha}$, use the Lowenheim-Skolem theorem
to find a countable model $M\elemsub V_{\kappa}$ which contains $T$
and $<\Sigma_n\;:\;n\in\omega>$.  Apply  Lemma 3
to get a model $N$ and then apply Lemma 2 to get the conclusion
of the theorem for $T$ and $<\Sigma_n\;:\;n\in\omega>$.

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