THE AXIOM OF CHOICE
There are many equivalent statements of the Axiom of Choice. The following version gave rise to its name:
For any set X there is a function f, with domain , so that f(x) is a member of x for every nonempty x in X.Such an f is called a ``choice function" on X. [Note that means X with the empty set removed. Also note that in Zermelo-Fraenkel set theory all mathematical objects are sets so each member of X is itself a set.]
The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate. The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate:
The question of whether AC is consistent with the other axioms (question  above) was answered by Goedel in 1938. Goedel showed that if the other axioms are consistent then AC is consistent with them. This is a ``relative consistency" proof which is the best we can hope for because of Goedel's Second Incompleteness Theorem.
The third question, ``Should we accept it as an axiom?", moves us into the realm of philosophy. Today there are three major schools of thought concerning the use of AC:
Underlying the schools of thought about the use of AC are views about truth and the nature of mathematical objects. Three major views are platonism, constructivism, and formalism.
A platonist believes that mathematical objects exist independent of the human mind, and a mathematical statement, such as AC, is objectively either true or false. A platonist accepts AC only if it is objectively true, and probably falls into school A or C depending on her belief. If she isn't sure about AC's truth then she may be in school B so that once she finds out the truth about AC she will know which theorems are true.
A constructivist believes that the only acceptable mathematical objects are ones that can be constructed by the human mind, and the only acceptable proofs are constructive proofs. Since AC gives no method for constructing a choice set constructivists belong to school C.
A formalist believes that mathematics is strictly symbol manipulation and any consistent theory is reasonable to study. For a formalist the notion of truth is confined to the context of mathematical models, e.g., a formalist would say "The parallel postulate is false in Riemannian geometry." but she wouldn't say "The parallel postulate is false." A formalist will probably not allign herself with any school. She will comfortably switch between A, B, and C depending on her current interests.
So: Should you accept the Axiom of Choice? Here are some arguments for and against it.
The acceptance of AC is based on the belief that our intuition about finite sets can be extended to infinite sets. The main argument for accepting it is that it is useful. Many important, intuitively plausible theorems are equivalent to it or depend on it. For example these statements are equivalent to AC:
Alternatives to AC
Test Yourself: When is AC necessary?
If you are working in Zermelo-Fraenkel set theory without the Axiom of Choice, can you choose an element from...
Benacerraf, Paul and Putnam, Hilary. "Philosophy of Mathematics: Selected Readings, 2nd edition." Cambridge University Press, 1983.
Dauben, Joseph Warren. "Georg Cantor: His Mathematics and Philosophy of the Infinite." Princeton University Press, 1979.
A. Fraenkel, Y. Bar-Hillel, and A. Levy with van Dalen, Dirk. "Foundations of Set Theory, Second Revised Edition." North-Holland, 1973.
Johnstone, Peter T. "Tychonoff's Theorem without the Axiom of Choice." Fundamenta Mathematica 113: 21-35, 1981.
Leisenring, Albert C. "Mathematical Logic and Hilbert's Epsilon-Symbol." Gordon and Breach, 1969.
Maddy, "Believing the Axioms, I", J. Symb. Logic, v. 53, no. 2, June 1988, pp. 490-500, and "Believing the Axioms II" in v.53, no. 3.
Moore, Gregory H. "Zermelo's Axiom of Choice: Its Origins, Development, and Influence." Springer-Verlag, 1982.
Rubin, Herman and Rubin, Jean E. "Equivalents of the Axiom of Choice II." North-Holland, 1985.
This section of the FAQ is Copyright (c) 1994 Nancy McGough. Send comments and or corrections relating to this part to firstname.lastname@example.org. The most up to date version of this section of the sci.math FAQ is accesible through http://www.jazzie.com/ii/math/index.html