Definition:Stability (Model Theory)

Definition
Let $T$ be a complete $\LL$-theory where $\LL$ is countable.

Let $\kappa$ be an infinite cardinal.

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

Note
When $\card A = \kappa$ and $\card \LL = \aleph_0$, there are $2^\kappa$ many subsets of the set of $\LL \cup \set {a: a \in A}$-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.