Saturated model

In mathematical logic, and particularly in its subfield model theory, a saturated model M is one which realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is ${\displaystyle \aleph _{1}}$-saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection, see Goldblatt (1998).

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, 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|. According to some authors, a model M is called countably saturated if it is ${\displaystyle \aleph _{1}}$-saturated; that is, it realizes all complete types over countable sets of parameters. According to others, it is countably saturated if it is ${\displaystyle \aleph _{0}}$-saturated; i.e. realizes all complete types over finite parameter sets.

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