翻訳と辞書 Words near each other ・ Grotesquerie ・ Groth ・ Grothe ・ Grothendieck category ・ Grothendieck connection ・ Grothendieck construction ・ Grothendieck duality ・ Grothendieck existence theorem ・ Grothendieck group ・ Grothendieck inequality ・ Grothendieck local duality ・ Grothendieck space ・ Grothendieck spectral sequence ・ Grothendieck topology ・ Grothendieck trace formula ・ Grothendieck universe ・ Grothendieck's connectedness theorem ・ Grothendieck's Galois theory ・ Grothendieck's relative point of view ・ Grothendieck's Tôhoku paper ・ Grothendieck–Katz p-curvature conjecture ・ Grothendieck–Ogg–Shafarevich formula ・ Grothendieck–Riemann–Roch theorem ・ Grothendieck–Teichmüller group ・ Grothusenkoog ・ Grotius Lectures ・ Grotius Society ・ Grotiusomyia ・ Grotki ・ Grotkowo Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grothendieck universe ： ウィキペディア英語版 Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' are both elements of ''U'', then is an element of ''U''. # If ''x'' is an element of ''U'', then ''P''(''x''), the power set of ''x'', is also an element of ''U''. # If $\backslash \_$ is a family of elements of ''U'', and if ''I'' is an element of ''U'', then the union $\backslash bigcup\_\; x\_\backslash alpha$ is an element of ''U''. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.) Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry. The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals. Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieck universe. The concept of a Grothendieck universe can also be defined in a topos. 〔 〕 ==Properties== As an example, we will prove an easy proposition. :Proposition. If $x\; \backslash in\; U$ and $y\; \backslash subseteq\; x$, then $y\; \backslash in\; U$. :Proof. $y\; \backslash in\; P(x)$ because $y\; \backslash subseteq\; x$. $P(x)\; \backslash in\; U$ because $x\; \backslash in\; U$, so $y\; \backslash in\; U$. The axioms of Grothendieck universes imply that every set is an element of some Grothendieck universe. It is similarly easy to prove that any Grothendieck universe ''U'' contains: * All singletons of each of its elements, * All products of all families of elements of ''U'' indexed by an element of ''U'', * All disjoint unions of all families of elements of ''U'' indexed by an element of ''U'', * All intersections of all families of elements of ''U'' indexed by an element of ''U'', * All functions between any two elements of ''U'', and * All subsets of ''U'' whose cardinal is an element of ''U''. In particular, it follows from the last axiom that if ''U'' is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe. 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「Grothendieck universe」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.