Russell's 'Paradox' is pretty famous (in some circles). It predates the Principia Mathematica by a few years. Wrestling with it provoked refinements in set theory, which were to the good.
I have The Ascent of Man work, which has some really good stuff in it, including a proof of the Pythagorean theorem which is excessively clever (and does not require pencil and paper). I don't know if Russell's paradox is in there, or not.
Anyway--
"Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.
Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those which don't.
The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.
However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves."
Source: http://en.wikipedia.org/wiki/Russell%27s_paradox#Applied_versions
