This is a good intuition pump, but to me it is no more troubling than the existence of well-ordering of real numbers.
I think the confusion is between existence and computability. I am not troubled by existence of strategy for uncountably many hat cases, because it seems obviously uncomputable. It is actually a theorem of ZFC that well-ordering of real numbers, which exists, is uncomputable. I think when people say “existence of strategy”, they implicitly assume computability of strategy.
I find it hard to accept both the well-ordering of the reals and the example here: out of uncountably many sets, each of countable size, pick one value (and have all of the inmates memorise this choice, lol).
This just seems weird, way too big.
I don’t exactly reject the axiom of choice but when a result depends on such a ginormous ass-pull, I kind of lose interest and look for more manageable things.
Yeah, the Axiom of Choice is never necessary for practical calculations. If weird results arise from its application, it’s a sign that the attempt to approximate a real-world situation with infinite sets has broken down somehow.
A troubling aspect of the Axiom of Choice is either affirming or denying it results in some pretty unintuitive implications (but different ones). Here’s a list of some bizarre results if you deny AoC. The basic problem is that AoC is provably equivalent to dozens of other statements, so in taking a position on it, you implicitly take a position on all those others too as a package deal.
Meh, denying AC is also pretty weird, but you can just refuse to include that too.
Discussions of infinity hints at why it’s nonsensical to say that the axiom of choice is “right” or “wrong”. It’s either depending on context. It’s related to the closed vs open world assumption. The reality is that, in practice, you have alternating layers of closed systems and open systems. Axioms such as that of choice, or the law of the excluded middle, are useful in closed contexts and invalid in open, extensible contexts. Use the tools that are applicable to your situation.
It’s not literally right or wrong, it’s just a matter of taste if you want literally undescribable Lovecraftian monsters in your results or not. Personally, I don’t consider them “useful”.
“Wrong” implies there’s such a thing as a right axiom…