Let X be a collection of non-empty sets. Then we can choose a member from each set in that collection.
Stated more formally:
There exists a function f defined on X such that for each set S in X, f(S) is an element of S.
Another formulation of the axiom of choice (AC) states:
Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.
It seems obvious: if you've got a bunch of boxes lying around with at least one item in each of them, the axiom simply states that you can choose one item out of each box. Where's the controversy?
Well, the controversy was over what it meant to choose something from these sets. As an example, let us look at some sample sets.
The axiom of choice has been proven to be independent of the remaining axioms of set theory; that is, it can be neither proven nor disproven from them (unless those remaining axioms contain a contradiction, which we don't know). This is the result of work by Kurt Gödel and Paul Cohen. There are thus no contradictions if you choose not to accept the axiom of choice; however, most mathematicians accept either it, or a weakened variant of it, because it makes their jobs easier. Despite this, there is some study of systems in which the axiom of choice is either not true or at least not assumed (see also axiom of regularity). In these cases it is important to be aware which proofs in mathematics use the axiom of choice and which do not.
One of the reasons that some mathematicians do not particularly like the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach-Tarski Paradox which amounts to saying that it is possible to "carve-up" the 3-dimensional solid unit ball into finitely many pieces, and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell you how to carve up the unit sphere to make this happen, it simply tells you that it can be done.
One of the most interesting aspects of the axiom of choice is the sheer number of places in mathematics that it shows up. There are also a remarkable number of statements that are equivalent to the axiom of choice, most important among them Zorn's lemma and the well-ordering principle: every set can be well-ordered. (In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle.)
Jerry Bona once said: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's Lemma?". In truth, all three of these are mathematically equivalent, but the statement was amusing because it underscored the fact that most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex to form any intuitive feeling about. Several central theorems in various branches of mathematics require the axiom of choice (or one of its weaker versions, such as the ultrafilter lemma, the axiom of countable choice, or the axiom of dependent choice). These branches are:
There are many people still doing work on the axiom of choice and its consequences. If you are interested in more, look up Paul Howard at EMU.