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Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. == The axioms of Zermelo set theory == : AXIOM I. Axiom of extensionality (''Axiom der Bestimmtheit'') "If every element of a set ''M'' is also an element of ''N'' and vice versa ... then ''M'' ''N''. Briefly, every set is determined by its elements." : AXIOM II. Axiom of elementary sets (''Axiom der Elementarmengen'') "There exists a set, the null set, ∅, that contains no element at all. If ''a'' is any object of the domain, there exists a set containing ''a'' and only ''a'' as an element. If ''a'' and ''b'' are any two objects of the domain, there always exists a set containing as elements ''a'' and ''b'' but no object ''x'' distinct from them both." See Axiom of pairs. : AXIOM III. Axiom of separation (''Axiom der Aussonderung'') "Whenever the propositional function –(''x'') is definite for all elements of a set ''M'', ''M'' possesses a subset ''M' '' containing as elements precisely those elements ''x'' of ''M'' for which –(''x'') is true." : AXIOM IV. Axiom of the power set (''Axiom der Potenzmenge'') "To every set ''T'' there corresponds a set ''T' '', the power set of ''T'', that contains as elements precisely all subsets of ''T'' ." : AXIOM V. Axiom of the union (''Axiom der Vereinigung'') "To every set ''T'' there corresponds a set ''∪T'', the union of ''T'', that contains as elements precisely all elements of the elements of ''T'' ." : AXIOM VI. Axiom of choice (''Axiom der Auswahl'') "If ''T'' is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ''∪T'' includes at least one subset ''S''1 having one and only one element in common with each element of ''T'' ." : AXIOM VII. Axiom of infinity (''Axiom des Unendlichen'') "There exists in the domain at least one set ''Z'' that contains the null set as an element and is so constituted that to each of its elements ''a'' there corresponds a further element of the form , in other words, that with each of its elements ''a'' it also contains the corresponding set as element." 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「Zermelo set theory」の詳細全文を読む スポンサード リンク
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