Russell's paradox
The most famous of the paradoxes in the foundations of set theory, discovered by Russell in 1901. Some classes have themselves as members: the class of all abstract objects, for example, is an abstract object. Others do not: the class of donkeys is not itself a donkey. Now consider the class of all classes that are not members of themselves. Is this class a member of itself? If it is, then it is not, and if it is not, then it is.
The paradox is structurally similar to easier examples, such as the paradox of the barber . But it is not so easy to say why there is no such class as the one Russell defines. It seems that there must be some restriction on the kinds of definition that are allowed to define classes, and the difficulty is that of finding a well-motivated principle behind any such restriction. See also types, theory of ; impredicative definitions.

Philosophy dictionary. . 2011.

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