TITLE: Why does the infinite prisoners and hats puzzle require the axiom of choice?
QUESTION [10 upvotes]: 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?

REPLY [2 votes]: In addition to the fact that the prisoners are not allowed to communicate between them after they are given the hats, it should be added that not every infinitely many choices require the axiom of choice, however this particular choice does.
It is consistent that there is no choice function from the set of equivalence classes of binary sequences modulo finite differences. In such universe the prisoners are doomed to fail.