Definition:Stability (Model Theory)/Kappa-Stable 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 and only if:

for all models $\mathcal M$ of $T$

for all subsets $A \subseteq \mathcal M$ of cardinality $\kappa$

and:

for all $n \in \N$

the cardinality $\left\vert{ {S_n}^{\mathcal M} \left({A}\right) }\right\vert$ of the set ${S_n}^{\mathcal M} \left({A}\right)$ 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.