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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 -saturated, meaning that every descending nested sequence of internal sets has a nonempty intersection, see Goldblatt (1998).
==Definition==
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 -saturated; that is, it realizes all complete types over countable sets of parameters. According to others, it is countably saturated if it is -saturated; i.e. realizes all complete types over finite parameter sets.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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