[The puzzle](https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong) that has prisoners wearing black and white hats having to guess their own hat color without communication seems to have such a crazy result that it should indicate the axiom of choice to be false?
So why is the axiom of choice an axiom if it’s so controversial and leads to paradoxical?
In: Mathematics
The puzzle’s result isn’t really unintuitive or “paradoxical.”
It *seems* that way because it’s appealing to your intuition that real humans in real life occupying real bodies with real brains in the physical world would not be able to accomplish this.
But the puzzle is not really about human prisoners, is it?
You have *infinite* prisoners that can see infinitely far, memorize and process infinite information, but not only that, they have the ability to memorize and compute an *uncountable* number of sequences, the majority of which are *uncomputable* sequences. This theoretical scenario has no overlap with our physical reality.
On top of that, the axiom of choice only guarantees the existence of a function on the set of equivalence classes; it doesn’t tell you how to construct or compute it (if it’s even a computable function, which the majority of functions are not).
Nothing about this puzzles corresponds to the real world, so its results don’t violate any intuition. If you give humans magical powers, magical results will follow.
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