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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 Cichoń's diagram ： ウィキペディア英語版 Cichoń's diagram In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to $\backslash aleph\_1$, the smallest uncountable cardinal, and they are bounded above by $2^$, the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets (first category sets). ==Definitions== Let ''I'' be an ideal of a fixed infinite set ''X'', containing all finite subsets of ''X''. We define the following "cardinal coefficients" of ''I'': *$(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\; \backslash bigcup\backslash notin\; I\backslash big\backslash \}$. ::The "additivity" of ''I'' is the smallest number of sets from ''I'' whose union is not in ''I'' any more. As any ideal is closed under finite unions, this number is always at least $\backslash aleph\_0$; if ''I'' is a σ-ideal, then add(''I'')≥$\backslash aleph\_1$. *$(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\backslash bigcup=X\backslash big\backslash \}$. :: The "covering number" of ''I'' is the smallest number of sets from ''I'' whose union is all of ''X''. As ''X'' itself is not in ''I'', we must have add(''I'') ≤ cov(''I''). *$(I)=\backslash min\backslash$, :: The "uniformity number" of ''I'' (sometimes also written $(I)$) is the size of the smallest set not in ''I''. By our assumption on ''I'', add(''I'') ≤ non(''I''). *$(I)=\backslash min\backslash \backslash subseteq\; I\; \backslash wedge\; (\backslash forall\; A\backslash in\; I)(\backslash exists\; B\backslash in\; )(A\backslash subseteq\; B)\backslash big\backslash \}.$ :: The "cofinality" of ''I'' is the cofinality of the partial order (''I'', ⊆). It is easy to see that we must have non(''I'') ≤ cof(''I'') and cov(''I'') ≤ cof(''I''). Furthermore, the "bounding number" or "unboundedness number" $$ and the "dominating number" $$ are defined as follows: * * where "$\backslash exists^\backslash infty\; n\backslash in$" means: "there are infinitely many natural numbers ''n'' such that...", and "$\backslash forall^\backslash infty\; n\backslash in$" means "for all except finitely many natural numbers ''n'' we have...". 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「Cichoń's diagram」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.