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.
Coin flips don’t work because you need to decide which sock goes for “heads” and which one for “tails”. Once you’ve made that assignment you don’t need the coin anymore; just assume you always get heads.