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 .
Bertrand Russell discovered this paradox and sent it in a letter to G. Frege just as Frege was completing Grundlagen der Arithmetik.
This invalidated much of the rigor of the work, and Frege was forced to add a note
at the end stating, "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."