#MathsInMinutes Day 23: The barber paradox

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A paradox is a seemingly true statement that contradicts itself, or leads to a situation that seems to defy logic. In 1901, British mathematician Bertrand Russel used the barber paradox to expose the flaws in elementary set theory:

All the men in a village either shave themselves or are shaved by a barber (himself a man from the village). The barber claims to shave only the male villagers who do not shave themselves. So who shaves the barber?

If the barber shaves himself, then his claim that he shaves only those who do not shave themselves is false. If the barber does not shave himself then his claim is that he does shave himself! Whichever way you phrase it there is a contradiction.

Restated in terms of sets, the paradox asks us to consider a set containing all those subsets which do not have themselves as an element. Is this set an element of itself? The immediate solution to such paradoxes was to restrict set theory with a series of rules or axioms, creating a hierarchy of sets that are allowed to be elements only of sets above them in the hierarchy. Although no the most elegant of solutions, axiomatic set theories have become widely accepted

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