Russell's Paradox is a headache of logic and naive set theory.

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.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics.

To understand that example (which made my head spin for a few seconds), you have to take everything completely literally. "The set of all teacups" is literally referring to all of the teacups in the universe. A set can contain anything, however you define it, including other sets. That set is not a member of itself because a set of teacups is not a teacup, so doesn't fit the set's definition.

The "set of all non-teacups" is just as literal. This is referring to everything, everywhere, that isn't a teacup. That set includes all sets, also, since a set is not a teacup.

Since it includes all sets, "the set of all non-teacups" is a member of itself.

That's all to explain the types of sets we're talking about, and how sets can contain other sets. Now imagine a set that contains only sets that are not members of themselves. (Like the first set of teacups.) This new set is the paradox. If that new set is a member of itself, then it is excluded from being a member of itself by the very rule that makes the set. Yet, if it's not a member of itself, then it meets the rule that defines the set and becomes a member of itself...

Ow. Now you see where the headache comes from. I love this stuff!

Incidentally, there is no official solution to this paradox. What is commonly done now is to limit the creation of sets to those which don't send logicians into infinite loops. (They now use "ZF", the axiomization of set theory.)