Is there a missing "deterministically" in the axiom of choice statement?
I can imagine the common-sense "Of course you can choose one element" assumption breaks down if we add the constraint that you need to in some way be able to have another actor choose the same element. Because for an uncountably-infinite-sized set of uncountably-infinite-sized sets, perhaps there is no way to label individual elements such that if I choose A, someone else can also choose A (how will we know they're the same A?).
But what does "picking at random" mean? If you think it's obvious that you can just make infinitely many random choices at the same time, you're endorsing the axiom of choice.
(If you think you can make one random choice at a time, you're endorsing the axiom of finite choice, which is just true. Determinism doesn't matter here, but "at the same time" matters a lot.)
Ah this is super helpful. I’m here stuck thinking “just pick any element at random”. Needing to have deterministically consistent pickings from infinite sets makes the problem clear and much more interesting.
I think that's why the "infinite socks vs. infinite shoes" are mentioned---because the socks are equivalent so you can't tell by the 'label' which one you grabbed.
I can imagine the common-sense "Of course you can choose one element" assumption breaks down if we add the constraint that you need to in some way be able to have another actor choose the same element. Because for an uncountably-infinite-sized set of uncountably-infinite-sized sets, perhaps there is no way to label individual elements such that if I choose A, someone else can also choose A (how will we know they're the same A?).