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 doesn't shave himself, and no one else. Such a town cannot exist:

• If the barber doesn't shave himself, he must bide by the rule and shave himself.
• If he does shave himself, according to the rule he will not shave himself.

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.