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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 Positive set theory ： ウィキペディア英語版 Positive set theory In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension *"$\backslash$ exists" holds for at least the positive formulas $\backslash phi$ (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification). Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact. The set theory $GPK^+\_$ of Olivier Esser consists of the following axioms: * The axiom of extensionality: $x=y\; \backslash Leftrightarrow\backslash forall\; a\backslash ,\; (a\backslash in\; x\; \backslash Leftrightarrow\; a\backslash in\; y)$. * The axiom of empty set: there exists a set $\backslash emptyset$ such that $\backslash ,\backslash neg\; \backslash exists\; x\backslash ;\; x\backslash in\backslash emptyset\backslash ,$ (this axiom can be neatly dispensed with if a false formula $\backslash perp$ is included as a positive formula). * The axiom of generalized positive comprehension: if $\backslash phi$ is a formula in predicate logic using only $\backslash vee$, $\backslash wedge$, $\backslash exists$, $\backslash forall$, $=$, and $\backslash in$, then the set of all $x$ such that $\backslash phi(x)$ is also a set. Quantification ($\backslash forall$, $\backslash exists$) may be bounded. * * Note that negation is specifically not permitted. * The axiom of closure: for every formula $\backslash phi(x)$, a set exists which is the intersection of all sets which contain every ''x'' such that $\backslash phi(x)$; this is called the closure of $\backslash$ and is written in any of the various ways that topological closures can be presented. This can be put more briefly if class language is allowed (any condition on sets defining a class as in NBG): for any class ''C'' there is a set which is the intersection of all sets which contain ''C'' as a subclass. This is obviously a reasonable principle if the sets are understood as closed classes in a topology. * The axiom of infinity: the von Neumann ordinal $\backslash omega$ exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of $\backslash omega$ exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that $\backslash omega$ contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal. == Interesting properties == * The universal set is a proper set in this theory. * The sets of this theory are the collections of sets which are closed under a certain topology on the classes. * The theory can interpret ZFC (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory (Morse-Kelley set theory with the proper class ordinal a weakly compact cardinal). 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「Positive 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.