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 memberof 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 endstating, ``A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the workis finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through thepress.''

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.

• Vandermonde Determinant
• Vandermonde Identity
• Vandermonde Matrix
• Vandermonde's Sum
• Vandermonde Theorem
• van der Pol Equation
• van der Waerden Number
• van der Waerden's Theorem
• Vandiver's Criteria
• Vanishing Point
• van Kampen's Theorem
• van Wijngaarden-Deker-Brent Method
• Varga's Constant
• Variance
• Difference of Successes
• Difference Operator
• Difference Quotient
• Difference Set
• Difference Table
• Different
• Differentiable
• Differentiable Manifold
• Differential
• Differential Calculus
• Differential Equation