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In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen. According to naive set theory, any definable collection is a set. Let ''R'' be the set of all sets that are not members of themselves. If ''R'' is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically: : In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Skolem, turned out to be first-order logic) while Russell altered the logical language itself. ==Informal presentation== Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares in the plane. That set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. So it is "normal". On the other hand, if we take the complementary set that contains all non-(squares in the plane), that set is itself not a square in the plane and so should be one of its own members as it is a non-(square in the plane). It is "abnormal". Now we consider the set of all normal sets, ''R''. Determining whether ''R'' is normal or abnormal is impossible: if ''R'' were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if ''R'' were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that ''R'' is neither normal nor abnormal: Russell's paradox. 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「Russell's paradox」の詳細全文を読む スポンサード リンク
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