The Axiom of Choice (AOC) is necessary to construct a Zermelo-Fraenkel set theory that admits the well-ordering property. This version of set theory, ZFC, is the version that we use for everyday math because it has a lot of properties that seem natural. As the post implies, there is actually a bit of controversy here. The axiom of choice implies the existence of some (very pathological and complicated) sets that you can also prove are impossible to construct. Mathematicians tend to be bothered by things that not only exist, but you can prove you are unable to meaningfully examine.
The Axiom of Choice (AOC) is necessary to construct a Zermelo-Fraenkel set theory that admits the well-ordering property. This version of set theory, ZFC, is the version that we use for everyday math because it has a lot of properties that seem natural. As the post implies, there is actually a bit of controversy here. The axiom of choice implies the existence of some (very pathological and complicated) sets that you can also prove are impossible to construct. Mathematicians tend to be bothered by things that not only exist, but you can prove you are unable to meaningfully examine.