Definition:Stability (Model Theory)

Definition

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

Let $\kappa$ be an infinite cardinal.

$\kappa$-Stable

$T$ is $\kappa$-stable if and only if:

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 $\kappa$.

Stable

$T$ is stable if and only if it is $\kappa$-stable for some $\kappa \ge \aleph_0$.

Unstable

$T$ is unstable if and only if it is not stable.

$\kappa$-Stable Structure

Let $\MM$ be an $\LL$-structure.

Let $\map {\operatorname{Th} } \MM$ be the $\LL$-theory of $\MM$.

$\MM$ is $\kappa$-stable if $\map {\operatorname{Th} } \MM$ is $\kappa$-stable.

Also denoted as

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

Note

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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.