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

This can be expressed symbolically as:
R = { x : x is not a member of* x }
This leads to R is a member of R iff not R is a member of R.

This paradox was discovered just as Gottlob Frege was completing his work Grundlagen der Arithmetik and invaildated much if it. Frege added a note at the end:

A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
*: I can't get that neat 'e' like character from set notation. ∈ shows up as ∈ and ∈ shows up as ? - oh well.

The best English version of this paradox can be thought of as:

The US Library of Congress requested all the libraries in the United States to index all the books they had and send this index back to the Library of Congress.

Some libraries included the index in the list of books, others did not.

A librarian at the Library of Congress was then instructed to make an index of all the indexes that included themselves in the list of books at that library, and an index of all the index that do not include themselves in the list of books at that library.

The first index (all indexes that include themselves) posed no problem, and the index was included in itself.

Upon composing the second index, however, the librarian had a problem. If he fails to include the index itself in the index, then the index is not complete - for it fails to be an index of all indexes that do not include themselves. On the other hand, if it is included, it is no longer an index that does not include itself, and thus should not be included in itself.