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\begin{fcprop}[]
  \label{prop:15.11}
  Assuming:
  - $T$ a complete $\mathcal{L}$-theory with infinite models
  - $\mathcal{M} \equiv \mathcal{N}$ are both
  countable and saturated
  Then:
  $\mathcal{M} \cong \mathcal{N}$.
\end{fcprop}

\begin{proof}
  Exercise (use back and forth argument).
\end{proof}

\newpage
\section{Omitting Types}

Let $\mathcal{M}$ be an $\mathcal{L}$-structure.
Which types must be realised?

\begin{fcdefn}[]
  \glsadjdefn{isol}{isolated}{$n$-type}%
  \label{defn:16.1}
  We say $p \in \Snm_n^{\mathcal{M}}(A)$ is
  \emph{isolated} if it is an isolated point with
  respect to topology on $\Snm_n^{\mathcal{M}}(A)$
  (i.e. $\{p\}$ is open).
\end{fcdefn}

\begin{example*}
  For $a \in A \subseteq \mathcal{M}$,
  $\tp^{\mathcal{M}}(a / A)$ is isolated by $x =
  a$ ($\{\tp^{\mathcal{M}}(a / M)\} = [x = a]$).
\end{example*}

\begin{fcprop}[]
  \label{prop:16.2}
  Assuming:
  - $p \in \Snm_n^{\mathcal{M}}(A)$.
  Then:
  the following are equivalent:
  \begin{cenum}[(i)]
    \item $p$ is \gls{isol}
    \item $\{p\} = [\varphi(z)]$ for some
      $\mathcal{L}_A$-formula $\varphi(\ol{x})$.
      In this case we say $\varphi(\ol{x})$
      \emph{isolates} $p$.
    \item There is an $\mathcal{L}_A$-formula
      $\varphi(\ol{x}) \in p$ such that for any
      $\psi(\ol{x}) \in p$,
      \[
        \Th_A(\mathcal{M}) \models \forall x
        (\varphi(\ol{x}) \to \psi(\ol{x}))
      .\]
  \end{cenum}
\end{fcprop}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/bbbcfe1cbef84a5e.png}
\end{center}

\begin{proof}
  (i) $\iff$ (ii): Obvious.

  (ii) $\implies$ (iii): Assume $\varphi(\ol{x})$
  isolates $p$. Fix an $\mathcal{L}_A$-formula
  $\psi(\ol{x})$. We want to show $\mathcal{M}
  \models \forall \ol{x} (\varphi(\ol{x}) \to
  \psi(\ol{x}))$. So suppose $\mathcal{M} \models
  \varphi(\ol{a})$. Then $\tp^{\mathcal{M}}(\ol{a}
  / A) \in [\varphi(\ol{x})] = \{p\}$.

  So $\tp^{\mathcal{M}}(\ol{a} / A) = p$, hence $M
  \models \psi(\ol{a})$.

  (iii) $\implies$ (ii): By assumption, for every
  $\mathcal{L}_A$-formula we have $\psi(\ol{x})
  \in p$, $[\varphi(\ol{x})] \subseteq
  [\psi(\ol{x})]$. Thus if $q \sin
  [\varphi(\ol{x})]$, $a \in [\psi(\ol{x})]$. So
  $\psi(\ol{x}) \in q$, so $p \subseteq q$, so $p
  = q$.
\end{proof}

\begin{fcprop}[]
  \label{prop:16.3}
  Assuming:
  - $T$ is complete and consistent
  - $p \in S_n(T)$ is \gls{isol}
  Then:
  $p$ is realised in every $M \models T$.
\end{fcprop}

\begin{proof}
  Fix $p \in S_n(T)$, \gls{isol} by
  $\varphi(\ol{x})$. Fix $M \models T$.

  By \cref{prop:13.4}, there is some $\mathcal{N}
  \esup \mathcal{M}$ realising $p$.

  So $\mathcal{N} \models \exists \ol{x}
  \varphi(\ol{x})$, so $\mathcal{M} \models
  \exists \ol{x} \varphi(\ol{x})$.

  Fix $\ol{a} \in \mathcal{M}^n$ such that
  $\mathcal{M} \models \varphi(\ol{a})$. Then
  $\ol{a} \models p$ as for any $\varphi(\ol{x})
  \in p$ we have
  \[
    \mathcal{M} \models \forall
    \ol{x}(\varphi(\ol{x}) \to \psi(\ol{x}))
  .\]
  So $\mathcal{M} \models \varphi(\ol{a})$.
\end{proof}

\begin{fcthm}[Omitting Types Theorem]
  \label{thm:16.4}
  Assuming:
  - $\mathcal{L}$ is countable
  - $p \in S_n(T)$ is non-\gls{isol}
  Then:
  there is a countable $M \models T$ such that $p$
  is not realised in $\mathcal{M}$ ($\mathcal{M}$
  \emph{omits} $p$).
\end{fcthm}

\begin{proof}
  Let $\mathcal{L}^* = \mathcal{L} \cup C$, with
  $C$ a countably infinite set of new constants.

  An $\mathcal{L}^*$-theory has the \emph{witness
  property} if for any $\mathcal{L}^*$-formula
  $\varphi(\ol{x})$ there is a constant $c \in C$
  such that $T^* \models \exists x \varphi(x) \to
  \psi(c)$.

  \textbf{Fact:} Suppose $T^*$ is a complete,
  satisfiable $\mathcal{L}^*$-theory with the
  witness property.

  Define $\sim$ on $C$ such that $c \sim d$ if and
  only if $T^* \models c = d$. Let $M =
  C / \sim$, and define an
  $\mathcal{L}^*$-structure on $M$ such that:
  \begin{itemize}
    \item $c^M = [c]$
    \item $f^M([c_1], \ldots, [c_n]) = [d]$ if and
      only if
      \[
        T^* \models f(c_1, \ldots, c_n) = d
      .\]
    \item $R^M = \{([c_1], \ldots, [c_n]) \in
      \mathcal{M}^n : T^* \models R(c_1, \ldots,
      c_n)\}$.
  \end{itemize}
  Then $\mathcal{M}$ is a well-defined
  $\mathcal{L}^*$-structure and $\mathcal{M}
  \models T^*$.

  Note we have $\mathcal{M} \models \varphi([c_1],
  \ldots, [c_n])$ if and only if $T^* \models
  \varphi(c_1, \ldots, c_n)$. We call
  $\mathcal{M}$ the Henkin model of $T^*$.

  Fix $p \in S_n(T)$ non-\gls{isol}.

  Aim: build a complete, satisfiable
  $\mathcal{L}^*$-theory $T^* \supseteq T$, with
  the witness property, .

  Such that for all $c_1, \ldots, c_n \in C$ there
  is \emph{some} $\varphi(\ol{x}) \in p$ such that
  $T^* \models \neg \varphi(c_1, \ldots, c_n)$.
  Then the Henkin model of $T^*$ omits $p$.

  Enumerate all the $\mathcal{L}^*$-sentences
  $\varphi_0, \varphi_1, \ldots$ and all $c^(n =
  \{\ol{c}_1, \ol{c}_2, \ldots\}$. We build a
  satisfiable $\mathcal{L}^*$-theory $T \cup
  \{\theta_1, \theta_2, \ldots\}$ such that
  \begin{enumerate}[(0)]
    \item $\models \theta_i \to \theta_j$ for all
      $i > j$.
    \item (Completeness): Either $\models
      \theta_{3i + 1} \to \varphi_i$ or $\models
      \theta_{3i + 1} \to \neg \varphi_i$.
    \item (Witnessing property): If $\varphi_i$ is
      $\exists v \psi(v)$ for some $\psi$ and
      $\models \theta_{3i + 1} \to \varphi_i$ then
      $\models \theta_{3i + 2} \to \psi(c)$ for
      some $c$. (check this does ensure the
      witness property).
    \item (Omit $p$): $\models \theta_{3i + 3} \to
      \neg \psi(c_i)$ for some $\psi(\ol{x}) \in
      p$.
  \end{enumerate}
  Let $\theta_0$ be $\forall v(v =v)$, and suppose
  we have $\theta_0, \ldots, \theta_m$.

  \textbf{Case 1:} $m + 1 = 3i + 1$ for some $i$.

  If $T \cup \{\theta_m, \varphi_i\}$ is
  satisfiable then $\theta_{m + 1} = \theta_m
  \wedge \varphi_i$. Otherwise $\theta_{m + 1} =
  \theta_m \wedge \neg \varphi$.

  So $T \cup \{\theta_{m + 1}\}$ is satisfiable by
  construction.

  \textbf{Case 2:} $m + 1 = 3i + 2$ for some $i$.

  Suppose $\varphi_i$ is $\exists v, \psi(v)$ for
  some $\psi$ an $\mathcal{L}^*$-formula, and
  $\models \theta_i \to \varphi_i$ (otherwise, let
  $\theta_{m + 1} = \theta_m$).

  Choose a $c \in C$ not used in $\theta_m$. Let
  $\theta_{m + 1}$ be $\theta_m \wedge \psi(i)$.

  Exercise: check $T \cup \{\theta_{m + 1}\}$ is
  satisfiable.

  \textbf{Case 3:} $m + 1 = 3i + 3$ for some $i$.

  Let $\ol{c}_i = (c_1, \ldots, c_n)$. Without
  loss of generality assume $x_1, \ldots, x_n$ not
  used in $\theta_m$. We build an
  $\mathcal{L}$-formula as follows:
  \begin{itemize}
    \item Replace $c_t$ by $x_t$ ($t \in \{1,
      \ldots, n\}$).
    \item Replace any $c \in C \setminus \{c_1,
      \ldots, c_n\}$ by new variables $v_c$ and
      add $\exists v_c$ to the front.
  \end{itemize}
  Then $\varphi(\ol{x})$ doesn't \gls{isol} $p$.

  By \cref{prop:16.2}, there is some $\psi(\ol{x})
  \in p$ with
  \[
    T \not\models \forall x(\varphi(\ol{x}) \to
    \psi(\ol{x}))
  .\]
  Let $\theta_{m + 1}$ be $\theta_m \wedge \neg
  \psi(c_1, \ldots, c_n)$. Check $\theta_{m + 1}$
  is satisfiable.

  TODO
\end{proof}