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.

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.

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