Does the barber shave himself?

I would like to suggest a brain teaser:

The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves. Does the barber shave himself?

  According to the statement above if the barber is gonna shave himself then the barber is not going to shave him because the barber shaves only these who don't shave themselves. If your answer is "it's a paradox!" then you are right. It is called "The barber paradox" and is derived from the Bertrand Russell's paradox.

According to naive set theory any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:

$$\text{Let }R=\{x\mid x\notin x\}\text{, then }R\in R⟺R\notin R$$

(source: Wikipedia)

  By using Russell's paradox you can derive yourself many such paradox. If you are interested in the set theory and its limits then I would recommend you reading this article.