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In set theory, a minimal model is a minimal standard model of ZFC. Minimal models were introduced by . The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a ''set'' W in the von Neumann universe V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible sets of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element ''s'' of the minimal model can be named; in other words there is a first order sentence φ(''x'') such that ''s'' is the unique element of the minimal model for which φ(''s'') is true. gave another construction of the minimal model as the strongly constructible sets, using a modified form of Godel's constructible universe. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded. If there is no standard model then the minimal model cannot exist as a set. However in this case the class of all constructible sets plays the same role as the minimal model and has similar properties (though it is now a proper class rather than a countable set). ==References== * * * * 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「Minimal model (set theory)」の詳細全文を読む スポンサード リンク
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