>>9715502

Firstly, AC leads to ridiculous results, like well-ordering theorem, Banach-Tarski paradox, existence of non-Lebesgue-measurable sets.

Secondly, there's no way to even theoretically choose an element from uncountable family of sets. For sets not greater than [math]\omega[/math] it's at least theoretically possible to finish the process in finite time, with uncountable sets it will always take infinitely long if choosing an element takes non-zero amount of time.

It's impossible to construct anything constructed by means of AC. We "know" there is well-order on reals, but it's impossible to construct it. We "know" we can cut a ball in five parts and reassemble them into two balls identical to the original one, but we can't possibly know how any of these parts looks like. AC brings nothing to mathematics, it just creates pathological counterexamples