Definition:Stability (Model Theory)

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Definition

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

Let $\kappa$ be an infinite cardinal.


$\kappa$-Stable

$T$ 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 $\kappa$.


Stable

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


Unstable

$T$ is unstable if it is not stable.


$\kappa$-Stable Structure

Let $\mathcal M$ be an $\mathcal L$-structure.

Let $\operatorname {Th} \left({\mathcal M}\right)$ be the $\mathcal L$-theory of $\mathcal M$.


$\mathcal M$ is $\kappa$-stable if $\operatorname{Th} \left({\mathcal M}\right)$ 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


When $\left\vert{A}\right\vert = \kappa$ and $\left\vert{\mathcal L}\right\vert = \aleph_0$, there are $2^\kappa$ many subsets of the set of $\mathcal L \cup \left\{{a: a \in A}\right\}$-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.