Russell's paradox

Russell's paradox is a paradox 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 The result is an elegant proof that the cardinality of the latter is always larger.

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.

This can also be considered more practically via ledgers that track books in a library. One ledger might list all books about geology; another ledger might list all books that mention battles in World War II. Ledgers might keep track of other ledgers; eg. one ledger might keep track of all ledgers that cover books about animals. Now, imagine an uber-ledger that lists all ledgers that do not list themselves. Based on that criteria, should the uber-ledger list itself?

Resolution
Bertrand Russell's paradox is a consequence of the axiom of comprehension and the existence of a universal set.

The issue let to a reworking of set theory in the early 20th century, from naive set theory to axiomatic set theory. In particular, the Zermelo–Fraenkel axioms (usually, plus the axiom of choice) are now the standard foundation of set theory. The axioms are designed to avoid the Russell paradox: 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.