An astonishing application of the AC

A countable infinite number of prisoners are placed in a line, facing forward so they can see everyone in front of them in line. The warden will place either a black or white hat on each prisoner’s head, and then starting from the back of the line, he will ask each prisoner what the color of his own hat is (ie, he first asks the person who can see all other prisoners). Any prisoner who is correct may go free; however, prisoners cannot hear previous guesses or whether they were correct. If all the prisoners can agree on a strategy beforehand, prove that there's a way that all but a finite number of them can go free.

The solution makes use of the axiom of choice and is used as an argument against its convenience (not very different than Banach-Tarski or others have been used for that purpose) in the following link:

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

It's also interesting to take a look at Terence Tao's comments on the issue and its relation to probability and measure theory.
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Relevant Pages

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