Let
C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function
f defined on
C with the property that, for each set
S in the collection,
f(S) is a member of
S.
In set theory class, we had to prove the equivalence of the Axiom of Choice (AC) to some other theorems, such as the well ordered principle, and Zorn's Lemma.
It is an axiom because it has been proven that the axiom of choice cannot be proven or disproven. One of the interesting consquences of AC is the Banach-Tarski Paradoxical Decomposition, which, using AC, is a proof that it is possible to take a 3 dimensional closed unit ball and partition it into finitely many pieces, and reassemble the pieces into two copies of the original ball. Messes with your head, don't it?