Paradox Compilation

The Barber's Paradox

Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:

1. shaving himself; or

2.being shaved by the barber.

Also, "The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves."

From this, asking the question "Who shaves the barber?" results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities are valid: they both result in the barber shaving himself, but he cannot do this because he shaves only those men "who do not shave themselves".

Despite its popular name, however, Barber paradox is not really in the true sense of this world. 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), whose existence cannot be painlessly dismissed as it follows from the very intuitive and widely relied upon axioms of. Axioms means a statement or proposition that is regarded as being established, accepted, or self-evidently true