Russell's Antinomy

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.

Curry, H. B. Foundations of Mathematical Logic, 2nd rev. ed. New York: Dover, p. 4, 1977.

Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 175-177, 1998.

Frege, G. Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd rev. ed. Evanston, IL: Northwestern University Press, 1980.

Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 116, 1998.

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

Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles." Enseign. math. 19, 37-52, 1917.

Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, pp. 79 and 101, 1927.