[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 Axiom of Choice is not controversial or paradoxical. Almost every mathematician uses it, or variations of it, without thinking. Anyone trying to sell you on the idea that it’s somehow some big and crazy controversy is lying or doesn’t know what they’re talking about. It’s normal to use it.
In fact, I’d say it’s controversial to *not* use it. Because not using it opens the door to a whole bunch of actually wild things. For instance, I can look at pairs of real numbers like (2,3). I can look at triples (2,3,4). I can look at quadruples like (3,-1,0,309). And I can keep going. Can I look at infinituples, such as (1,2,3,4,5,…)? Without the Axiom of Choice you actually can’t (or, the Axiom of Countable Choice before someone *aktually*s me, and there is [some nuance](https://math.stackexchange.com/a/856350/52449) to how I’ve framed it). You need the axiom of choice to say that there exists infinituples, which is weird because there should be a *lot* of them. Another unruly consequence of rejecting the Axiom of Choice is that there are sets whose sizes are not comparable. That is, I could find sets A and B so that the question “Which set is bigger, A or B?” simply does not have an answer. The Axiom of Choice gives math coherent structure that other mathematical fields can build upon – you don’t want to be working with infinite dimensional spaces for number theory and then suddenly be left without any objects to manipulate.
There was question about the Axiom of Choice historically, because it can provide unintuitive results while also being necessary for math to have any meaningful structure. But almost nobody cares about these questions anymore and if they do then it is purely academic and out of curiosity rather than trying to impact mathematical norms. And some proofs can be done *without* using the Axiom of Choice, and these can sometimes be nicer proofs as the Axiom of Choice asserts the existence of things without giving a recipe to make those things. But outside of logicians who like to play with rules, no mathematician gives a second thought about using the Axiom of Choice and anyone trying to drum things up is probably a crank. The person in the link isn’t a crank, I should add, they’re just talking about a new example that highlights some of its interesting idiosyncrasies to an audience of people who “get” it. The author will continue to regularly use the Axiom of Choice without a passing thought. In fact, he works with objects that would become near impossible (at least extremely tedious) to define without the Axiom of Choice.
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