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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 Easton's theorem ： ウィキペディア英語版 Easton's theorem In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that : and, for , that :$2^\backslash kappa\backslash le\; 2^\backslash lambda\backslash ,$ are the only constraints on permissible values for 2κ when κ is a regular cardinal. == Statement of the theorem == Easton's theorem states that if ''G'' is a class function whose domain consists of ordinals and whose range consists of ordinals such that # ''G'' is non-decreasing, # the cofinality of $\backslash aleph\_$ is greater than $\backslash aleph\_$ for each α in the domain of G, and # $\backslash aleph\_$ is regular for each α in the domain of G, then there is a model of ZFC such that :$2^\backslash ,$ for each $\backslash alpha$ in the domain of ''G''. The proof of Easton's theorem uses forcing with a proper class of forcing conditions over a model satisfying the generalized continuum hypothesis. The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from König's theorem. In Easton's model the powersets of singular cardinals have the smallest possible cardinality compatible with the conditions that 2κ has cofinality greater than κ and is a non-decreasing function of κ. 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「Easton's theorem」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.