Russell's Antinomy

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. Symbolically, let $R=\{x: x\not\in x\}$. Then $R\in R$ Iff $R\not\in R$.

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.''

See also Grelling's Paradox

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.