Axiom of Choice

An important and fundamental axiom 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 disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.

In Zermelo-Fraenkel set theory (in the form omitting the axiom of choice), Zorn's lemma, the trichotomy law, and the well ordering principle are equivalent to the axiom of choice (Mendelson 1997, p. 275). In contexts sensitive to the axiom of choice, the notation "ZF" is often used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC" is used if the axiom of choice is included.

In 1940, Gödel proved that the axiom of choice is consistent with the axioms of von Neumann-Bernays-Gödel set theory (a conservative extension of Zermelo-Fraenkel set theory). However, in 1963, Cohen (1963) unexpectedly demonstrated that the axiom of choice is also independent of Zermelo-Fraenkel set theory (Mendelson 1997; Boyer and Merzbacher 1991, pp. 610-611).

REFERENCES:

Boyer, C. B. and Merzbacher, U. C. A History of Mathematics, 2nd ed. New York: Wiley, 1991.

Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, pp. 178-179, 1958.

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.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.