Russell's paradox

Russell's paradox (also known as Russell's antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory.

Applied versions

There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: for example, the Barber paradox supposes a barber who shaves everyone who does not shave himself, and no one else. When you start to think about whether he should shave himself or not, the paradox begins to emerge.

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As illustrated below, consider five lists of encyclopedia entries within that same encyclopedia:

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If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.

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While appealing, these "layman's" versions of the paradox share a drawback: an easy refutation of, for example, Barber's paradox seems to be: "Such a barber does not exist". The whole point of Russell's paradox is that the answer "such a set does not exist" means that the definition of the notion of "set" within a given theory is unsatisfactory. Notice the subtle difference between the statements: "such a set does not exist" and "such a set is empty".

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