Let me mention my favorite intuition pump against the axiom of choice—the prisoners with infinite hats. For any finite number of prisoners, if they can’t communicate they can’t even do better than chance, let alone saving all but a tiny fraction. But as soon as there are infinitely many, there’s some strange ritual they can do that lets them save all but an infinitely small fraction. This is unreasonable.
The issue is that once you have infinite prisoners you can construct these janky non-measurable sets that aren’t subject to the laws of probability theory. There’s an argument to be made that these are a bigger problem than the axiom of choice—the axiom of choice is just what lets you take the existence of these janky, non-constructive sets and declare that they give you a recipe for saving prisoners.
Let me mention my favorite intuition pump against the axiom of choice—the prisoners with infinite hats. For any finite number of prisoners, if they can’t communicate they can’t even do better than chance, let alone saving all but a tiny fraction. But as soon as there are infinitely many, there’s some strange ritual they can do that lets them save all but an infinitely small fraction. This is unreasonable.
The issue is that once you have infinite prisoners you can construct these janky non-measurable sets that aren’t subject to the laws of probability theory. There’s an argument to be made that these are a bigger problem than the axiom of choice—the axiom of choice is just what lets you take the existence of these janky, non-constructive sets and declare that they give you a recipe for saving prisoners.