Sure, some mathematics can be developed without AC, but we also lose a lot without it. Without it, to name a few examples:

We don’t have infinite product spaces.

Not every vector space has a basis.

A given set may not have a cardinality.

All of the equivalents to AC, of course, such as Zorn’s Lemma, Well ordering theorem, Hausdorff maximality principle, and the Tychonoff theorem.

Some of the most interesting mathematics being researched today depends on the axiom of choice; I doubt there are any modern mathematicians of significance who reject AC.

Well, either it’s true, or it’s not. Does what I want even matter? ZFC and ZF~C (set theory with and without the axiom of choice) are both equally consistent, so they’re both “good” systems. I happen to like the axiom of choice, so I’m going to use it. And like Cabbage says, most mathematicians these days do accept choice.

I recall reading that there’s still some controversy over the axiom of foundation:

I’m rather bothered by the idea of taking a sphere and then putting it back together again into a larger sphere. Also, any proof that involves AC is by necessity nonconstructive. Just how useful is the statement “A set with these properties exists, but there’s absolutely no way that we can figure out what it is”?

Well, the consistency of ZF set theory implies the consistency of ZF theory plus the axiom of choice. While I (and most other mathematicians) think ZF set theory is probably consistent, it can’t be proven that it’s consistent.
About the axiom of foundation, no mathematics I know of relies on it; it’s more of a “technical” axiom than anything else, and not really useful in any modern math. It basically restricts our universe of sets to the well-founded sets, where we do all of our mathematics, anyway. Plus, it gets rid of bizarre things like sets containing themselves.

The Ryan:

Of course, the catch there is that those pieces you divide the sphere into are so pathological that there’s no meaningful way to define a volume for them. It is surprising, but maybe not so surprising in light of that.

Also, I think the statement, “A set with these properties exists, but there’s absolutely no way that we can figure out what it is” is quite useful. There are lots of things, such as well orderings of sets, infinite products, and ultrafilters, which are extremely interesting mathematically, providing many fruitful areas of research, which simply cannot be constructed algorithmically.

Somewhere around 1968, I took a class in which the Banach-Tarski theorem was proved. I vaguely recall that it used group theory as well as set theory and measure theory. I recall that the subsets are obviously not measurable sets. After 5 years of graduate math courses I finally had enough background to follow the proof. Whoopie!

My answer to the question of how useful the statment is would be, “It’s not useful at all.” As The Ryan implies, since it’s non-costructive, and since the subsets are not measurable, there’s no application to any kind of real-world situation.

Cabbage, I thank you for reminding me about the theorem of every vector space having a basis, which I had long since forgotten. In fact, I’m no longer sure how one defines a basis for an infinite-dimentional vector space. Would it be that each point can be represented by a unique finite linear combination of basis elements?

This vector space result is also non-constructive, I guess, so it would also not be useful for real-world problems. It’s elegant, though.

IMHO the A of C yields many elegant mathematical results, but lacks real-world applicability. Whether to use or not might depend on which of these things you’re seeking

That may or may not be true, I’m afraid I’m not familiar enough with applications of mathematics to be certain. I wouldn’t rule out the possibility of applications having been found (or that they will be found); for example, back with the vector space example, aren’t Hilbert spaces used extensively in quantum mechanics? Existence of bases there could be important. And, in general, it’s very easy for me to believe that certain mathematical objects that depend on AC for their “existence” may have many applications in the real world.

I like the Axiom of Choice and, as has been pointed out, it is quite useful. But I want to comment on ultrafilter’s comment with a slight change to his notation. ZFC and ZF (set theory with and without AoC) are both equally consistent. Yes, but since AoC is independent, the negation is also independent. That is ZF~C (ZF assuming AoC is false) and ZF are equally consistent. I have trouble wrapping my mind around ZF~C – Set theory with at least one set that has no choice function, but it is as consistent as ZF. AoC is rather like the parallel postulate, Godel’s statement, and the Continuum Hypothesis in this regard.

But to return to the hijack:

I give. (Does it have anything to do with Tychonoff theorem?)

There are some areas on the borders of mathematics where the axiom of foundation is annoying. It’d be nice to describe computer programs as sets of potential states a computer could be in. However, since some computer programs return to their intitial states, you’d better not describe them in terms of sets if you want the axiom of foundation.

As far as I can tell, the axiom of foundation is there to avoid getting in trouble with Russel’s paradox. It’s stricter than necessary, but it doesn’t really limit any mathematics. Weaker axioms that still avoid the paradox are of interest to computer scientists and philosophers with mathematical tendencies.