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\begin{fcdefn}[Atomic, prime]
  \glsadjdefn{atomic}{atomic}{model}%
  \glsadjdefn{prime}{prime}{model}%
  \label{defn:16.5}
  Fix $\mathcal{M} \models T$.
  \begin{citems}
    \item We say $\mathcal{M}$ is \emph{atomic} if
      every $n$-type over $\emptyset$ realised in
      $\mathcal{M}$ is \gls{isol}.
    \item We say $\mathcal{M}$ is \emph{prime} if
      for any $\mathcal{N} \models T$ there is an
      elementary embedding $\mathcal{M} \to
      \mathcal{N}$.
  \end{citems}
\end{fcdefn}

\begin{example*}
  Let $K \models \acf_0$. Then $\ol{\Qbb} =
  \Qbb^{\text{alg}} \subseteq K$, and $\ol{\Qbb} <
  \kappa$ by \gls{qe}. So $\ol{\Qbb}$ is the prime
  model of $\acf_0$.
\end{example*}

Assume $\mathcal{L}$ is countable.

\textbf{Fact:} $\mathcal{M}$ is \gls{prime} if and only
if $\mathcal{M}$ is countable and \gls{atomic}.

\begin{fcthm}[]
  \label{thm:16.6}
  Assuming:
  - $\mathcal{L}$ countable
  Then:
  the following are equivalent:
  \begin{cenum}[(i)]
    \item $T$ has a \gls{prime} model.
    \item $T$ has an \gls{atomic} model.
    \item For all $n \ge 1$, the \gls{isol} types
      are dense.
  \end{cenum}
\end{fcthm}

\begin{fcthm}[]
  \label{thm:16.7}
  \begin{cenum}[(a)]
    \item Suppose $|S_n(T)| < 2^{\aleph_0}$ for
      all $n$. Then $T$ has a \gls{prime} model
      and a countable saturated model.
    \item If $T$ has a countable saturated model,
      then it has a \gls{prime} model.
  \end{cenum}
\end{fcthm}

\begin{example*}
  What if $|S_n(T)| = 2^{\aleph_0}$?

  $\Th(\Zbb, +, 0)$ has no countable saturated
  model, no \gls{prime} model.

  $\Th(\Zbb, +, 0, 1)$ has a \gls{prime} model,
  but no countable saturated model.
\end{example*}

\begin{fcdefn}[]
  \glssymboldefn{Icount}%
  \label{defn:16.8}
  For $\kappa \ge \aleph_0$, let $I(T, \kappa)$ be
  the number of models of $T$ of size $\kappa$
  (modulo isomorphism).
\end{fcdefn}

What size can $\Icount(T, \kappa)$ be?
\begin{itemize}
  \item $1 \le \Icount(T, \kappa) \le 2^{\kappa}$.
  \item Vaught's conjecture (still open):

    If $\Icount(T, \aleph_0) < 2^{\aleph_0}$, then
    $\Icount(T, \aleph_0) \le \aleph_0$.

    Morley got $\le \aleph_1$.
\end{itemize}

\begin{fcthm}[Ryll-Nardsewski / Engeler / Svenonius 59]
  \label{thm:16.9}
  Assuming:
  - $\mathcal{L}$ countable
  - $T$ is a complete $\mathcal{L}$-theory with
  infinite models
  Then:
  the following are equivalent:
  \begin{enumerate}[(i)]
    \item $T$ is $\aleph_0$-\gls{kcat}.
    \item For all $n \ge 1$, every type in
      $S_n(T)$ is \gls{isol}.
    \item For all $n \ge 1$, $S_n(T)$ is finite.
    \item For all $n \ge 1$, the number of
      $\mathcal{L}$-formulas with $x_1, \ldots,
      x_n$ free variables is finite, modulo $T$.
  \end{enumerate}
\end{fcthm}

\begin{fccoro}[]
  \label{coro:16.10}
  Assuming:
  - $G$ an infinite group
  - $\Th(G)$ is $\aleph_0$-\gls{kcat} (in
  $\mathcal{L}_{\text{groups}}$)
  Then:
  $G$ has finite exponent (there exists $n \in
  \Nbb$ such that $\forall g \in G$, $g^n = 1$).
\end{fccoro}

\textbf{Fact:} Any abelian group with finite
exponent has an $\aleph_0$-\gls{kcat} complete
theory.

\newpage
\section{Whistle Stop Tour of Stability Theory}

\begin{fcdefn}[]
  \label{defn:17.1}
  Given $\kappa \ge |\mathcal{L}| + \aleph_0$, we
  say $T$ is \emph{$\kappa$-stable} if for any
  $\mathcal{M} \models T$, $|\mathcal{M}| =
  \kappa$ we have $|S_1(M)| = \kappa$.

  We say $T$ is \emph{stable} if it is
  $\kappa$-stable for some $\kappa$.
\end{fcdefn}

\begin{example*}
  \phantom{}
  \begin{enumerate}[(1)]
    \item $\acf_p$, $\tfdag$ ($\Qbb$-vector
      spaces) are $\kappa$-stable for all $\kappa
      \ge \aleph_0$.
    \item (Exercise) $T = \Th(\Zbb, +, 0, 1,
      (\equiv_n)_{n \ge 2})$ (where $\equiv_n$ is
      congruence modulo $n$). This is
      $\kappa$-stable for $\kappa > 2^{\aleph_0}$.
    \item If $\mathcal{M} \models \RG$ then
      $|S_1(M)| = 2^{|M|}$.
  \end{enumerate}
\end{example*}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/e2695fe1148041e0.png}
\end{center}

\textbf{Fact:} $\aleph_0$-stable theories have
saturated models of all infinite cardinalities.

\begin{fcdefn}[]
  \label{defn:17.2}
  Let $\varphi(x, y)$ be an $\mathcal{L}$-formula,
  $x, y$ types of finite length.

  We say $\varphi(x, y)$ has the \emph{order
  property} with respect to $T$ if there is some
  $\mathcal{M} \models T$, $(a_i)_{i \ge 0}$,
  $(b_j)_{j \ge 0}$ such that $\mathcal{M} \models
  \varphi(a_i, b_j)$ if and only if $i < j$.
\end{fcdefn}

\begin{example*}
  $\dlo$ has the order property, choose $(\Qbb, <)
  = \mathcal{M}$ and $a_i = b_i = i$ as your
  sequence.
\end{example*}

\begin{fcthm}[Fundamental Theorem of Stability (light)]
  \label{thm:17.3}
  The following are equivalent:
  \begin{cenum}[(i)]
    \item $T$ is stable.
    \item No $\mathcal{L}$-formula has the order
      property with respect to $T$.
    \item For any $\mathcal{M} \models T$, every
      $p \in S_n(\mathcal{M})$ is definable.
    \item \emph{Non-forking} is an independence relation.
  \end{cenum}
\end{fcthm}

\begin{fcdefn}[]
  \label{defn:17.4}
  A theory $T$ is \emph{strongly minimal} if
  $\forall \mathcal{M} \models T$ every definable
  subset of $\mathcal{M}$ is finite or cofinite.
\end{fcdefn}

\begin{remark*}
  $T$ strongly minimal implires $T$ is stable
  (count types).
\end{remark*}

\begin{fcdefn}[]
  \label{defn:17.5}
  Let $\mathcal{M} \models T$, $A \subseteq
  \mathcal{M}$. Then $b \in \operatorname{acl}(A)$
  if there is an $\mathcal{L}_A$-formula $\phi(x)$
  such that
  \[
    \mathcal{M} \models \exists^{= n}_x \phi(x)
  \]
  and $\mathcal{M} \models \phi(b)$.
\end{fcdefn}

\begin{example}
  \label{eg:17.6}
  Let $T$ be strongly minimal. Then
  $\operatorname{acl}$ has the exchange property:
  \[
    a \in \operatorname{acl}(Bc) \setminus
    \operatorname{acl}(B)
    \implies c \in \operatorname{acl}(Ba)
  .\]
\end{example}
