Consider all of the men in a small town as members of a set. Now imagine that a barber puts up a sign in his shop that reads "I shave all those men, and only those men, who do not shave themselves."
Obviously, we can further divide the set of men in this town into two further sets, those who shave themselves, and those who are shaved by the barber. To which set does the barber himself belong?
The barber cannot shave himself, because he has said he shaves only those men who do not shave themselves. Further, he cannot not shave himself, because he shaves all men who do not shave themselves!