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I've heard a lot of differing opinions about the Axiom of Choice (AOC): some don't mind it, some hate it. Yet, so far every time I've seen the axiom of choice being applied, it doesn't seem too crazy to be able to apply a result like that. What are some strange/overreaching results we can reach with AOC? I know AOC is used in proving the Banach-Tarski paradox, but the usage isn't too strange.

For the sake of completeness: The axiom of choice states that for any non-empty collection of sets $\{X_{\alpha}\}_{\alpha}$, we can pick $x_{\alpha} \in X_{\alpha}$ for each $\alpha$. For finite sets, this can be deduced from other axioms of set theory, but for infinite sets, it needs to be an axiom and cannot in fact be deduced from more basic axioms.

gtoques
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  • "Some hate it". I think this is a bit of overreaction. After all, in most branches of mathematics the axiom of choice is always assumed to be true. – Mark Apr 20 '20 at 11:45
  • Did you search the site? There are many answers that I wrote about AC and its absence. – Asaf Karagila Apr 20 '20 at 12:14

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