翻訳と辞書 Words near each other ・ Tarsius ・ Tarsius fuscus ・ Tarskavaig ・ Tarski monster group ・ Tarski's axiomatization of the reals ・ Tarski's axioms ・ Tarski's circle-squaring problem ・ Tarski's exponential function problem ・ Tarski's high school algebra problem ・ Tarski's plank problem ・ Tarski's problem ・ Tarski's theorem ・ Tarski's theorem about choice ・ Tarski's undefinability theorem ・ Tarski's World ・ Tarski–Grothendieck set theory ・ Tarski–Kuratowski algorithm ・ Tarski–Seidenberg theorem ・ Tarsky ・ Tarsky District ・ Tarsnap ・ Tarsney ・ Tarso ・ Tarso Genro ・ Tarso Marques ・ Tarso Toh ・ Tarso Voon ・ Tarsocera ・ Tarsocera cassina ・ Tarsocera cassus 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 Tarski–Grothendieck set theory ： ウィキペディア英語版 Tarski–Grothendieck set theory Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than that of conventional set theories such as ZFC. For example, adding this axiom supports category theory. The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs. ==Axioms== Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds "Tarski's axiom". We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that: * Given any set $A$, the singleton $\backslash$ exists. * Given any two sets, their unordered and ordered pairs exist. * Given any family of sets, its union exists. TG includes the following axioms, which are conventional because they are also part of ZFC: * Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC). * Extensionality axiom: Two sets are identical if they have the same members. * Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible. * Axiom schema of replacement: Let the domain of the function $F$ be the set $A$. Then the range of $F$ (the values of $F(x)$ for all members $x$ of $A$) is also a set. It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of infinity, choice,〔Tarski (1938)〕〔http://mmlquery.mizar.org/mml/current/wellord2.html#T26〕 and power set.〔Robert Solovay, (Re: AC and strongly inaccessible cardinals ).〕〔(Metamath grothpw. )〕 It also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC. * Tarski's axiom (adapted from Tarski 1939〔Tarski (1939)〕). For every set $x$, there exists a set $y$ whose members include: - $x$ itself; - every subset of every member of $y$; - the power set of every member of $y$; - every subset of $y$ of cardinality less than that of $y$. More formally: :$\backslash forall\; x\backslash exists\; y\; (y\; \backslash wedge\; \backslash forall\; z\backslash in\; y(\backslash mathcal\; P(z)\backslash subseteq\; y\backslash wedge\backslash mathcal\; P(z)\backslash in\; y)\; \backslash wedge\; \backslash forall\; z\backslash in\backslash mathcal\; P(y)(\backslash neg\; z\backslash approx\; y\backslash to\; z\backslash in\; y))$ where "$\backslash mathcal\; P(x)$" denotes the power class of ''x'' and "$\backslash approx$" denotes equinumerosity. What Tarski's axiom states (in the vernacular) for each set $x$ there is a Grothendieck universe it belongs to. 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「Tarski–Grothendieck set theory」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.