Why does the infinite prisoners and hats puzzle require the axiom of choice? Infinite prisoners puzzle. The link to Wikipedia describes the puzzle, and the solution. The axiom of choice is used to pick a sequence from each equivalence class, which the prisoners memorize beforehand. However, the answer says " _When they are put into their line, each prisoner can see what equivalence class the actual sequence of hats belongs to._ " Why not wait until the prisoners see which equivalence class they are in. Then, pick a sequence from this equivalence class? This would avoid having to use the axiom of choice, or am I mistaken?

If each prisoner individually picks an arbitrary representative of the equivalence class of the actual sequence of hat colors, and guesses their own hat color according to this arbitrarily selected one, then it is possible that _every single prisoner_ guesses their hat color _wrong_. There is no limit.

They need to coordinate and agree on one representative for every single equivalence class and stick to the chosen representative, because then only finitely many of them can be wrong instead of arbitrarily many of them.