Thanks for the answer! For other people reading, here's the blue/red hats puzzle that Scott was referring to (edited from https://en.m.wikipedia.org/wiki/Hat_p

Thanks for the answer! For other people reading, here's the blue/red hats puzzle that Scott was referring to (edited from https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hats_puzzle https://en.m.wikipedia.org/wiki/Hat_puzzle#Prisoners_and_hat...): > A countably infinite number of prisoners, each with an unknown and randomly assigned red or blue hat, line up in a single file line. Each prisoner faces away from the beginning of the line, and each prisoner can see all the hats in front of him, and none of the hats behind. Starting from the beginning of the line, each prisoner must correctly identify the color of his hat or he is killed on the spot. The prisoners have a chance to meet beforehand, but once in line no prisoner can hear what the other prisoners say. The question is, is there a way to ensure that only finitely many prisoners are killed? The solution is that it is possible, and you can look on the Wikipedia page to find out how. Even more paradoxical is that the prisoners can still save all but finitely many of themselves even if there are infinitely many possible hat colours. For example you could allow the hat colours to be specified by three real numbers giving an RGB value.