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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 Von Neumann–Bernays–Gödel set theory ： ウィキペディア英語版 Von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical Zermelo–Fraenkel set theory (ZFC). A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes proper classes, objects having members but that cannot be members of other entities. NBG's principle of class comprehension is predicative; quantified variables in the defining formula can range only over sets. Allowing impredicative comprehension turns NBG into Morse-Kelley set theory (MK). NBG, unlike ZFC and MK, can be finitely axiomatized. ==Ontology== The defining aspect of NBG is the distinction between proper class and set. Let ''a'' and ''s'' be two individuals. Then the atomic sentence $a\; \backslash in\; s$ is defined if ''a'' is a set and ''s'' is a class. In other words, $a\; \backslash in\; s$ is defined unless ''a'' is a proper class. A proper class is very large; NBG even admits of "the class of all sets", the universal class called ''V''. However, NBG does not admit "the class of all classes" (which fails because proper classes are not "objects" that can be put into classes in NBG) or "the set of all sets" (whose existence cannot be justified with NBG axioms). By NBG's axiom schema of Class Comprehension, all objects satisfying any given formula in the first-order language of NBG form a class; if a class is not a set in ZFC, it is an NBG proper class. The development of classes mirrors the development of naive set theory. The principle of abstraction is given, and thus classes can be formed out of all individuals satisfying any statement of first-order logic whose atomic sentences all involve either the membership relation or predicates definable from membership. Equality, pairing, subclass, and such, are all definable and so need not be axiomatized — their definitions denote a particular abstraction of a formula. Sets are developed in a manner very similarly to ZF. Let ''Rp''(''A,a''), meaning "the set ''a'' represents the class ''A''," denote a binary relation defined as follows: :$\backslash mathrm(A,a)\; :=\; \backslash forall\; x(x\; \backslash in\; A\; \backslash leftrightarrow\; x\; \backslash in\; a).$ That is, ''a'' "represents" ''A'' if every element of ''a'' is an element of ''A'', and conversely. Classes lacking representations, such as the class of all sets that do not contain themselves (the class invoked by the Russell paradox), are the proper classes. 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「Von Neumann–Bernays–Gödel 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.