[The puzzle](https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong) that has prisoners wearing black and white hats having to guess their own hat color without communication seems to have such a crazy result that it should indicate the axiom of choice to be false?
So why is the axiom of choice an axiom if it’s so controversial and leads to paradoxical?
In: Mathematics
The simple reason is that it doesn’t actually matter all that much.
– Axiom of choice only affect the infinite world, not the finite world. If you believe that ultimately math is about properties of finite objects, then it doesn’t matter if you use axiom of choice or not, other than a convenience.
– The universe with choice and the universe without choice believe in the consistency of each other, because they can simulate each other. In other word, if you do not believe in axiom of choice, you can see any proof using choice as a proof that is applicable to those simulated universe, even if it does not apply to the “real” universe.
– Choice can frequently, in practice, be removed just by asking for the choice in advance, rather than invoking an axiom.
– Various regularity conditions/constraints could be imposed to make the issue of choice irrelevant. For example, in this problem, you’re interpreting the prisoners as some sorts of powerful magical being who can compute with infinity. By imposing any sorts of constraints on what kind of strategy are the prisoners capable of you can easily ruin it.
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