Russell's paradox

I came across this interesting paradox on a recent podcast. According to wikipedia:

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:
\[\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R\]

There is a nice commentary on the paradox in SciAm, and a superb entry on the Stanford Encyclopedia of  Philosophy