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In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.〔; ; .〕 In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in ''V'' are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank. == Definition == The cumulative hierarchy is a collection of sets ''V''α indexed by the class of ordinal numbers; in particular, ''V''α is the set of all sets having ranks less than α. Thus there is one set ''V''α for each ordinal number α. ''V''α may be defined by transfinite recursion as follows: * Let ''V''0 be the empty set: *: * For any ordinal number β, let ''V''β+1 be the power set of ''V''β: *: * For any limit ordinal λ, let ''V''λ be the union of all the ''V''-stages so far: *: A crucial fact about this definition is that there is a single formula φ(α,''x'') in the language of ZFC that defines "the set ''x'' is in ''V''α". The sets ''V''α are called stages or ranks. The class ''V'' is defined to be the union of all the ''V''-stages: :: An equivalent definition sets : for each ordinal α, where is the powerset of . The rank of a set ''S'' is the smallest α such that 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 universe」の詳細全文を読む スポンサード リンク
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