Definition:Stability (Model Theory)

Definition
Let $T$ be a complete $\mathcal{L}$-theory where $\mathcal{L}$ is countable.

Let $\kappa$ be an infinite cardinal.

$T$ is $\kappa$-stable if for every model $\mathcal{M}$ of $T$, for every subset $A$ of the universe of $\mathcal{M}$ of cardinality $\kappa$, and for every $n\in\mathbb N$, the cardinality of the set of complete $n$-types over $A$ is $\kappa$.

That is, $T$ is $\kappa$-stable if for all $\mathcal{M}\models T$, for all $A \subseteq \mathcal{M}$ with $|A|=\kappa$, and for all $n\in\mathbb N$, we have $|$$S_{n}^{\mathcal{M}} (A)$$| = \kappa$.

$T$ is stable if it is $\kappa$-stable for some $\kappa \geq \aleph_0$.

$T$ is unstable if it is not stable.

An $\mathcal{L}$-structure $\mathcal{M}$ is $\kappa$-stable if $\operatorname{Th}(\mathcal{M})$ is $\kappa$-stable, where $\operatorname{Th}(\mathcal{M})$ is the $\mathcal{L}$-theory of $\mathcal{M}$.

Despite $\omega$ usually being used to denote the smallest infinite ordinal, $\aleph_0$-stable is usually written as $\omega$-stable.

Note
When $|A| = \kappa$ and $|\mathcal{L}|=\aleph_0$, there are $2^\kappa$ many subsets of the set of $\mathcal{L}\cup\{a: a\in A\}$-formulas with $n$ free variables. Since types are particular kinds of such subsets, one might expect there to be up to $2^\kappa$ many types over $A$, and there are examples of theories where this happens.

So, a $\kappa$-stable theory can be informally thought of as a theory which has a relatively small number of types over its $\kappa$-sized subsets.