No less a mind than Bertrand Russell, of Principia Mathematica fame, was responsible for this paradox, which opened a hole in math. It's those barbers. The Barber Paradox shows that barbers are not to be trusted.
In 1901, Bertrand Russell, a philosopher and mathematician, was mulling over set theory. Sets, in mathematics, are any groups of numbers that fit certain conditions. For example, I could declare a set that includes any whole, non-negative, number that ends in two. The set would include, 2, 12, 22, 32 and on upwards forever. Numbers that don't meet those conditions are necessarily excluded from the set. Sets don't have to include numbers. They can be made of anything. Bertrand Russell thought of some that included only men.
Say there was a small Italian town. It was off on its own that was never visited. That town had one male barber. If no one in the town decided to grow a beard, the barber would be busy. He would shave every man in town who didn't shave himself. Thus we have two sets of men in that town; those that shave themselves, and those that the barber shaves. The question that breaks the set is, "Which set is the barber in?" Naturally, the barber has the skill to shave himself so he would be in the set that shaves themselves. Except for the fact that that set necessarily excludes anyone the barber shaves. Since the barber is the only barber in town, he has to be in the set of men shaved by the barber, and therefore not in the set of men that shave themselves. The barber, then, can shave himself, as long as he doesn't shave himself.
Although it seems like someone just trying to be difficult, it exposed a wrong assumption in set theory – that given a certain simple condition, it's possible to construct a set. Instead, it was recognized that sometimes set theory had to start with concrete examples and make sets from them. The town would have to then be knocked into three sets, those that shave themselves, those that are shaved by the barber, and the barber. Or the barber could just let his beard grow.
[Via The Barber Paradox.]