Barber paradox

The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell himself as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him. It shows that an apparently plausible scenario is logically impossible. Specifically, it describes a barber who is defined such that he both shaves himself and does not shave himself.

The barber is the "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself?

Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be a barber. Conversely, if the barber does not shave himself, then he fits into the group of people who would be shaved by the barber, and thus, as the barber, he must shave himself.

Despite its popular name, however, the barber paradox is not really a paradox in the true sense of this word. A man who shaves exactly those men who do not shave themselves simply cannot and does not exist, and there are virtually no reasons to expect the opposite. This is in contrast with the set of all sets that do not contain themselves (from Russell's paradox), whose existence cannot be painlessly dismissed as it follows from the very intuitive and widely relied upon axioms of naive set theory.

This paradox is often attributed to Bertrand Russell (e.g., by Martin Gardner in Aha!). It was suggested to him as an alternative form of Russell's paradox, which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained contradictions. However, Russell denied that the Barber's paradox was an instance of his own:

That contradiction [Russell's paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.

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