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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 Woodin cardinal ： ウィキペディア英語版 Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number λ such that for all functions :''f'' : λ → λ there exists a cardinal κ < λ with : ⊆ κ and an elementary embedding :''j'' : ''V'' → ''M'' from the Von Neumann universe ''V'' into a transitive inner model ''M'' with critical point κ and :Vj(f)(κ) ⊆ ''M''. An equivalent definition is this: λ is Woodin if and only if λ is strongly inaccessible and for all $A\; \backslash subseteq\; V\_\backslash lambda$ there exists a $\backslash lambda\_A$ < λ which is -$A$-strong. $\backslash lambda\; \_A$ being -$A$-strong means that for all ordinals α < λ, there exist a $j:\; V\; \backslash to\; M$ which is an elementary embedding with critical point $\backslash lambda\; \_A$, $j(\backslash lambda\; \_A)\; >\; \backslash alpha$, $V\_\backslash alpha\; \backslash subseteq\; M$ and $j(A)\; \backslash cap\; V\_\backslash alpha\; =\; A\; \backslash cap\; V\_\backslash alpha$. (See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact. == Consequences == Woodin cardinals are important in descriptive set theory. By a result〔(A Proof of Projective Determinacy )〕 of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that $\backslash Theta\; \_0$ is Woodin in the class of hereditarily ordinal-definable sets. $\backslash Theta\; \_0$ is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)). Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on ω1 is $\backslash aleph\_2$-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an $\backslash aleph\_1$-dense ideal over $\backslash aleph\_1$. 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「Woodin cardinal」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.