Russell's paradox
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Russell's paradox is as follows: Consider the set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is. A contradiction ensues. In an interesting application, this can be exploited to preclude the existence of a bijection between any set and its power set. The result is an elegant proof that the cardinality of the latter is always larger.
Bertrand Russell's paradox is a consequence of the axiom of comprehension and the existence of a universal set. Hence, in ZF set theory, the universal set does not exist. Alternatively, one could avoid Russell's paradox and retain a universal set by rejecting the axiom of comprehension.
Another version of this paradox goes: Does a barber, who shaves all those and only those who do not shave themselves, shave himself? If he shaves himself, then he is not one of those who do not shave themselves, and since he only shaves those who do not shave themselves, then he does not shave himself. If he doesn't shave himself, then he is one of the those who do not shave themselves, and since he shaves all those who do not shave themselves, then he shaves himself.