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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 G�del's incompleteness theorem as well as Alan Turing's proof of the undecidability of the halting problem.