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RolfAndreassen comments on Open Thread March 31 - April 7 2014 - Less Wrong Discussion
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With guessing in order, I observe that for every finite subset there are either an even or an odd number of red hats, and prisoner 1 can indicate which it is by his guess; then everyone in that subset can count the red hats and figure out which colour his own must be to make the total number even or odd. Let the size of the finite subset go to infinity.
This is a good idea, and solves the similar problem with finitely many prisoners getting at most one guess incorrect. But...
I don't see how this immediately gives a solution; any finite set of hats has either an even or an odd number of red hats, but an infinite set of hats may have an infinite number of red hats and an infinite number of black hats, and infinities are neither odd nor even.