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Barber paradox

The Barber paradox is a paradox with importance to mathematical logic and set theory. The paradox considers a town with a male barber who daily shaves every man who does not shave himself, and no one else. Such a town cannot exist: Thus the rule results in an impossible situation.

This paradox is attributed to the British logician Bertrand Russell, who in 1901 constructed Russell's paradox to demonstrate the self-contradictory nature of Cantor's elementary set theory by formalizing the Barber paradox. The paradox also underlies the proof of Gödel's incompleteness theorem as well as Alan Turing's proof of the undecidability of the halting problem.