The Barber's paradox was discovered by mathematician and philosopher Bertrand Russell.
Suppose one day you walk past a barber's shop, and see a sign that says "Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and nobody else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself? If he does, then he mustn't, because he doesn't shave men who shave themselves, but then he doesn't, so he must, because he shaves every man who doesn't shave himself... and so on. Both possibilities lead to a contradiction.