|
|
|
An important and fundamental result in Set Theory sometimes called Zermelo's Axiom of Choice. It was formulated by Zermelo in 1904 and states that, given any Set of mutually exclusive nonempty Sets, there exists at least one Set that contains exactly one element in common with each of the nonempty Sets.
It is related to Hilbert's Problem 1b, and was proved to
be consistent with other Axioms in Set Theory in 1940 by Gödel. 
In
1963, Cohen demonstrated that the axiom of choice is independent of the other Axioms in Cantorian Set
Theory, so the Axiom cannot be proved within the system (Boyer and Merzbacher 1991, p. 610).
See also Hilbert's Problems, Set Theory, Well-Ordered Set, Zermelo-Fraenkel Axioms, Zorn's Lemma
References
Boyer, C. B. and Merzbacher, U. C. A History of Mathematics, 2nd ed. New York: Wiley, 1991.
Cohen, P. J. ``The Independence of the Continuum Hypothesis.'' Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963.
Cohen, P. J. ``The Independence of the Continuum Hypothesis. II.'' Proc. Nat. Acad. Sci. U. S. A. 51, 105-110, 1964.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 274-276, 1996.
Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York:
Springer-Verlag, 1982.