TITLE: What is a good argument for rejecting the axiom of choice?
QUESTION [5 upvotes]: I have only recently learned set theory as a side hobby (I'm a mathematical physics undergraduate), and I have heard various Profs and friends that are math graduates just ridicule or dismiss the idea of rejecting the axiom of choice. I don't know enough to formulate a good argument for rejecting the axiom of choice (or don't even know if a good argument for rejecting it exists). What is a good one for rejecting it? And if you would like to rebuttal it too that would be helpful.

REPLY [4 votes]: Earnest opposition to the axiom of choice is mostly a thing of the early 20th century. At that time the Zermelo-Fraenkel axioms for set theory were pretty new, and the circumstantial evidence that they succeed in avoiding the set-theoretic paradoxes was not as overwhelming as it is today. In that situation, approaching proposed additional axioms with a sound caution was understandable.
The argument from caution goes mostly like this: On one hand, the axiom of choice has some rather surprising and non-intuitive consequences, such as Banach-Tarski. If something like that -- which certainly doesn't correspond to any practical truth about physical space, which we'd like for $\mathbb R^3$ to model -- can be proves, who's to say that it's not because it proves everything, including contradictions? If it did, that would be bad indeed.
On the other, hand the philosophical motivation for the axiom of choice is somewhat weak. Sure enough it corresponds to a form of reasoning that had long been considered unproblematic until the proper foundation of set theory became urgent. But on the other hand, all the other axioms corresponded (at least morally) to operations where it seems to be pretty clear that there's one and only one correct result. On the the axiom of choice purports to give you something that is -- by design -- extremely under-specified. So in an argument that involves it, it is no longer clear how you would distinguish a correct result from a wrong one.
(The latter part of this argument partly ignores that usual first-order logic does allow us to make single choices along the way in a proof, and those choices can be pretty underspecified too. But making an infinity of them seems to have greater psychological impact, especially if you're already skeptical of "completed infinities" and suspect that the root of the paradoxes was basically related to playing too hard and fast with infinity).
However, this whole argument from caution came tumbling down when Gödel proved (using techniques that don't depend on AC itself, or even on infinitary reasoning at all) that if ZF is consistent then ZFC is consistent too. In other words, assuming AC will not lead to any increased chance of inconsistency.