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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 Martin's maximum ： ウィキペディア英語版 Martin's maximum In set theory, a branch of mathematical logic, Martin's maximum, introduced by , is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent. Martin's maximum (MM) states that if ''D'' is a collection of $\backslash aleph\_1$ dense subsets of a notion of forcing that preserves stationary subsets of ω1, then there is a ''D''-generic filter. It is a well known fact that forcing with a ccc notion of forcing preserves stationary subsets of ω1, thus MM extends MA($\backslash aleph\_1$). If (''P'',≤) is not a stationary set preserving notion of forcing, i.e., there is a stationary subset of ω1, which becomes nonstationary when forcing with (''P'',≤), then there is a collection ''D'' of $\backslash aleph\_1$ dense subsets of (''P'',≤), such that there is no ''D''-generic filter. This is why MM is called the maximal extension of Martin's axiom. The existence of a supercompact cardinal implies the consistency of Martin's maximum.〔Jech (2003) p.684〕 The proof uses Shelah's theories of semiproper forcing and iteration with revised countable supports. MM implies that the value of the continuum is $\backslash aleph\_2$〔Jech (2003) p.685〕 and that the ideal of nonstationary sets on ω1 is $\backslash aleph\_2$-saturated.〔Jech (2003) p.687〕 It further implies stationary reflection, i.e., if ''S'' is a stationary subset of some regular cardinal κ≥ω2 and every element of ''S'' has countable cofinality, then there is an ordinal α<κ such that ''S''∩α is stationary in α. In fact, ''S'' contains a closed subset of order type ω1. ==References== * (correction ) * * 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「Martin's maximum」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.