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.''

**References**

Courant, R. and Robbins, H. ``The Paradoxes of the Infinite.'' §2.4.5 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, p. 78, 1996.

Frege, G. *Foundations of Arithmetic.* Evanston, IL: Northwestern University Press, 1968.

Hofstadter, D. R. *Gödel, Escher, Bach: An Eternal Golden Braid.* New York: Vintage Books, pp. 20-21, 1989.

© 1996-9

1999-05-25