What is Russell's paradox?

Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Let  be the set of all sets which are not members of themselves. Then  is neither a member of itself nor not a member of itself. Symbolically, let . Then  iff .