@Twirlip,
The axiom say there is a choice function F for a set X of non-empty sets s .
If X is finite, and the sets in X in finite, then There is an enumerable number of such F. Surely, such F exist, but the problem shift to uniqueness. Which one is it? On the other hand, for the sets in X that might be infinite. You can ` t enumerate it, so that can be a problem.
simple case:
X= { {0, 1}, { 2, 3}}
has 4 candidate for F. They are { (0, 2), ( 0, 3), ( 1, 2), (1, 3)}.
while X= { N, C}
N is the natural numbers,
and
C is the complex numbers.
if we say:
1. { (x, y) | x in N, y in C}
This would solve the problem of existence( but not uniqueness), but 1 commit us to infinite sets.
The statement of the problem largely suppose existence of infinite sets, so i think the axiom of choice is largely to do with uniqueness, and not existence. They are separate questions.