The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves. So, who shaves the barber?
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His wife.
His wife, of course, or his children
I take it back
NO, he shaves himself in a mirror, so he’s not shaving himself, he’s shaving his reflection.
He doesn’t shave…
There is no answer. That proposition is logically equivalent to “this statement is false.” If the barber does not shave himself, then he is one of the “men in town who do not shave themselves,” so therefore he must be shaved by the barber, so therefore the barber shaves himself. If the barber shaves himself, he is not one of the men who do not shave themselves, so therefore the barber does not shave him, therefore the barber does not shave himself.
Paradox.
Yes and no and yes. Yes, this is a paradox. Specifically, “the Barber paradox” first articulated by Bertrand Russell as an illustration of the more general contradiction that arises in set theory (as used by Cantor and Frege) as a product of self-referentialism. But No, this is also not a paradox in that there is a resolution. Russell’s answer to this would-be paradox is his theory of types. But Yes, this is still a paradox in that Godel used this type of move in one of his proofs of incompleteness. This statement is unproveable.