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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Critical point (set theory) ： ウィキペディア英語版 Critical point (set theory) In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself.〔 p. 323〕 Suppose that j : ''N'' → ''M'' is an elementary embedding where ''N'' and ''M'' are transitive classes and j is definable in ''N'' by a formula of set theory with parameters from ''N''. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(ω)=ω. If j(α)=α for all α<κ and j(κ)>κ, then κ is said to be the critical point of j. If ''N'' is ''V'', then κ (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a <κ-complete, non-principal ultrafilter over κ. Specifically, one may take the filter to be $\backslash \; \backslash ,.$ Generally, there will be many other <κ-complete, non-principal ultrafilters over κ. However, j might be different from the ultrapower(s) arising from such filter(s). If ''N'' and ''M'' are the same and j is the identity function on ''N'', then j is called "trivial". If transitive class ''N'' is an inner model of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial. ==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「Critical point (set theory)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.