>>10869245It doesn't. Finite set AoC is a consequence of the other ZFC axioms. Countable set AoC is strictly stronger and logically independent of the other axioms, but does not imply general AoC. People who disagree with axiom of choice likely don't understand when to use it or why they should. There is a tendency in math to only use axioms when they are needed rather than convenient. In the case of axiom of choice, it can make long proofs using all other axioms trivially short to the point of being unenlightening.
I don't see the harm with invoking AoC for use in theoretical work, but it does have the disadvantage of being strictly non-constructive. If you avoid it, you may end of proving something by writing down how to compute it. This is seen as preferable in many mathematical communities.
