The Barber's Paradox

Somewhere far far away there is an island, and on the island many men and women live happily. Nobody ever visits the island and nobody ever leaves it. All the men on the island are clean shaven, some always shave themselves and the others always visit the island's Barber, of which there was only one and he was also a man. Therefore the Barber only shaves those men who do not shave themselves.
Does the Barber shave himself?

So this is a Paradox from Betrand Russell (the Mathematician). He used it to show that Frenge's set theory has a contradiction and so push forward the branch of Mathematics called Set Theorem.