The Barber paradox is attributed to the British philosopher Bertrand Russell. It highlights a fundamental problem in mathematics, exposing an inconsistency in the basic principles on which mathematics is founded.

The barber paradox asks us to consider the following situation:

In a village, the barber shaves everyone who does not shave himself, but no one else.

The question that prompts the paradox is this:

Who shaves the barber?

If the barber shaves himself, then he does not shave himself, for the barber shaves only those who do not shave themselves.

If the barber does not shave himself, then he does shave himself, for the barber shaves everyone who does not shave himself.

Both cases, then, are impossible; there can be no such barber.

Paradoxes of this form present a problem for mathematical set theory. In set theory, we would expect all sets to either be members of themselves or not. The set of all paradoxes is not a member of itself; it is not itself a paradox. The set of all sets, on the other, hand, is a member of itself; the set of all sets is itself a set.

The barber paradox raises the possibility of a set that both is and is not a member of itself: the set of all sets that are not members of themselves.

If this set is a member of itself, then it is not a member of itself, for it is only contains sets that are not members of themselves.

If this set is not a member if itself, then it is a member of itself, for it is the set of all sets that are not members of themselves.

The set of all sets that are not members of themselves, then, both is and is not a member of itself.

This, though, is a breach of the principle of non-contradiction, a principle at the very foundation of mathematics. Mathematicians must therefore choose between set theory at the principle of non-contradiction; both cannot be true.

OK, I\'ll start off with my ridiculous answer: who says the barber must shave? The barber may still exist because the barber must not neccissarily shave himself.

Anyways, I\'m not exactly sure to go from this one. Alright, squares and rectangles are both polygons, right? Both 4 sided. So they each belong to that group. All squares are also rectangles. But all rectangles are not squares, because they do not all have exactly proportianate sides and angles. Bah, I can\'t form a theory, but I think that has some relevence to it.

This kind of paradox is kind of interesting, but has a small problem: it\'s completely pointless in reality.

The Barber is a female?
;D

Ok let\'s bring time into this. When the barber isn\'t shaving himself he must shave himself. The instant his razor touches his face, or a facial hair, he is shaving himself and therefore must refrain from shaving himself. He must then again try to shave himself because he is not shaving himself anymore. Once again he will have to stop because he is shaving himself. The barber would find himself in an endless loop of shaving and not shaving himself.

The barber can\'t be female because it says the barber is referred to as himself. (Anyway a female barber is a hairdresser)

This a good paradox. Even having multiple barbers wouldn\'t solve this. Damn...