An extended hat puzzle

Shortly after hearing about the hat puzzle I wrote about last month I came across an interesting extension of the problem, which replaces the 100 wizards with an infinite number of wizards:

A countably infinite number of wizards are each given a red or blue hat with 50% probability. Each wizard can see everyone’s hat except their own. The wizards have to guess the colour of their hat without communicating in any way, but will be allowed to devise a strategy to coordinate their guesses beforehand. How can they ensure that only a finite number of them guess incorrectly? You may assume the axiom of choice.

This seems paradoxical since somehow knowing about other wizard’s hats—which are chosen independently from a wizard’s own hat—allows each wizard to conclude that they will almost surely guess their hat colour correctly.

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