Russell's paradox

Russell's paradox (also known as Russell's antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory.

Related Topics:
Paradox - Bertrand Russell - 1901 - Naive set theory - Cantor - Frege

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Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A.

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: M={Amid A otin A}

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In Cantor's system, M is a well-defined set {{dubious}}. Does M contain itself? If it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, then it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to contradictions (but see Independence from Excluded Middle below).

Related Topics:
Well-defined - Set

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In Frege's system, M corresponds to the concept does not fall under its defining concept. Frege's system also leads to a contradiction: that there is a class defined by this concept, which falls under its defining concept just in case it does not.

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