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The most famous paradox of set theory. Some sets are members of themselves and others are not: for example, the set of all sets is a member of itself, because it is a set, whereas the set of all penguins is not, because it is not a penguin. Now consider the set of all sets that are not members of themselves; is it a member of itself or is it not? If it is a member of itself, then it is not a member of itself, because it is one of those that are not members of themselves; and if it is not a member of itself, then it is a member of itself. It was discovered in 1901 by the Welsh philosopher Bertrand (Arthur William) Russell (1872–1970), and it undermined the attempt by the German logician and mathematician Friedrich Ludwig Gottlob Frege (1848–1925) to derive the whole of arithmetic from logic via set theory. When the second volume of Frege's monumental Grundgesetze der Arithmetik was in press, he received a letter from Russell describing the paradox; he replied that ‘arithmetic totters’ and was forced to add an appendix to the book explaining why the ambitious project had to be abandoned. Russell's paradox is closely related to the barber's paradox.

*Subjects:*
Mathematics — Psychology.

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