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 that I haven't quite yet digested it properly.
As far as I can make out, one phrasing of it is that 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, then 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.