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Bertrand Russell Paradox
The Bertrand Russell Paradox (also called the Zermelo Paradox), in logic, is a problem formulated independently, about 1902-1903, by Ernst Friedrich Ferdinand Zermelo (1871-1953) and Bertrand Russell.
It deals with collections (sets) of all sets of objects which are not members of themselves. Suppose that we have a set S: Is S a member of itself?
In 1919 Bertrand Russell popularized this problem by posing the following paradox: There is a village barber who shaves all those residents of the village who do not shave themselves. Does this barber shave himself? If he does not shave himself, then he should shave himself. But if he shaves himself, then he must not shave himself.
Another variant of this paradox is as follows: A man asserts that all persons of his nationality consistently lie. If all persons of his nationality are liars, is the man himself telling the truth or lying?
Zermelo, a German mathematician, is known mainly for his contributions to the logical theory of the foundations of arithmetic and to set theory, not the least of these contributions being the paradox which bears his name.