TITLE: Why can't you pick socks using coin flips?
QUESTION [76 upvotes]: I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', which probably means I haven't yet digested it properly.
As far as I can make out, one phrasing of the axiom is: for any family of non-empty, pairwise disjoint sets, there exists a set containing exactly one element from each set in the family.
If that's all the axiom states, why is there so much debate around it? If it were stated as there exists a procedure for constructing such a set, that might help me understand (though is that an incorrect statement of the axiom?), but then again:
To use Russell's classic shoes-and-socks example, why won't a coin flip for each pair of socks suffice? 
I'm sure this must be a stupid question, but please help me understand why.

REPLY [8 votes]: Your instinct is basically correct, you can choose socks arbitrarily ("by flipping a coin") without an axiom that lets you. But only because anything you do is necessarily a finite process.
In fact the Axiom of Choice is not needed for finite sets. Various restricted forms of it are theorems of the other axioms: https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf
AC is controversial when applied to transfinite sets. To over-simplify, you can think of the "controversy" as specifically being related to the fact that it's equivalent to the Well-Ordering Theorem (which my course called the Well-Ordering Principle, but apparently that's ambiguous in other contexts). Nobody ever disputed that finite and countable sets can be well-ordered, it's the rest that are tricky.
There's a joke (that doesn't entirely stand up to analysis, but does reflect the gut instincts of many), that the Axiom of Choice is obviously true, the Well-Ordering Theorem is obviously false, and Zorn's Lemma is obviously incomprehensible. They're all equivalent.