Saturated model

-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection.

[1] Let κ be a finite or infinite cardinal number and M a model in some first-order language.

Then M is called κ-saturated if for all subsets A ⊆ M of cardinality less than κ, the model M realizes all complete types over A.

The model M is called saturated if it is |M|-saturated where |M| denotes the cardinality of M. That is, it realizes all complete types over sets of parameters of size less than |M|.

This argument allows us to discuss specific features of the model that we may otherwise miss—for example, a bound on a specific increasing sequence cn can be expressed as realizing the type {x ≥ cn : n ∈ ω}, which uses countably many parameters.

This can be generalized as follows: the unique model of cardinality κ of a countable κ-categorical theory is saturated.

Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories.