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In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe ''V'', or sometimes of a generic extension of ''V''. Inner model theory studies the relationships of these models to determinacy, large cardinals, and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory. == Examples == *The class of all sets is an inner model containing all other inner models. *The first non-trivial example of an inner model was the constructible universe ''L'' developed by Kurt Gödel. Every model ''M'' of ZFC has an inner model ''L''M satisfying the axiom of constructibility, and this will be the smallest inner model of ''M'' containing all the ordinals of ''M''. Regardless of the properties of the original model, ''L''''M'' will satisfy the generalized continuum hypothesis and combinatorial axioms such as the diamond principle ◊. *The sets that are hereditarily ordinal definable form an inner model *The sets that are hereditarily definable over a countable sequence of ordinals form an inner model, used in Solovay's theorem. *L(R) *L() (see zero dagger) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「inner model theory」の詳細全文を読む スポンサード リンク
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