AoC lets you ignore how bad the set it. Banach-Tarski says: Take a measurable set. Split it up into 2 unmeasurable sets. Recombine them into a set with twice t

AoC lets you ignore how bad the set it. Banach-Tarski says: Take a measurable set. Split it up into 2 unmeasurable sets. Recombine them into a set with twice the measure of the original. OK, you can do that, but you've lost the ability to make any reasonable theorems about measure of measurable sets, unless you add qualifications of the form "this theorem does not apply if you introduce a set of uncountable sets" which. What's the benefit? To get a bunch of "proofs" for things that are only true because of the axiom, and don't model anything else we care about?