[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 puzzle’s result isn’t really unintuitive or “paradoxical.”
It *seems* that way because it’s appealing to your intuition that real humans in real life occupying real bodies with real brains in the physical world would not be able to accomplish this.
But the puzzle is not really about human prisoners, is it?
You have *infinite* prisoners that can see infinitely far, memorize and process infinite information, but not only that, they have the ability to memorize and compute an *uncountable* number of sequences, the majority of which are *uncomputable* sequences. This theoretical scenario has no overlap with our physical reality.
On top of that, the axiom of choice only guarantees the existence of a function on the set of equivalence classes; it doesn’t tell you how to construct or compute it (if it’s even a computable function, which the majority of functions are not).
Nothing about this puzzles corresponds to the real world, so its results don’t violate any intuition. If you give humans magical powers, magical results will follow.
An axiom doesn’t have to be universally accepted to be an axiom. An axiom is just a statement that is assumed to be true in order to make further arguments. You can reject an axiom, but that also means rejecting all the proofs that rely on it and you may have to come up with new proofs or systems.
That said, the consensus view among mathematicians is definitely that the axiom of choice is true, and a lot of other math relies on it.
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.
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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