Definition:Stability (Model Theory)/Kappa-Stable Theory

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

Let $\kappa$ be an infinite cardinal.

$T$ is $\kappa$-stable :
 * for all models $\MM$ of $T$
 * for all subsets $A \subseteq \MM$ of cardinality $\kappa$

and:
 * for all $n \in \N$

the cardinality $\card {\map { {S_n}^\MM} A}$ of the set $\map { {S_n}^\MM} A$ of complete $n$-types over $A$ is $\kappa$.

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