Just learned about the Banach-Tarski paradox and I'm mad
Conversation
the pieces in the banach-tarski decomposition don't exist in any remotely meaningful sense. they're non-measurable among other things, which means they "don't have a volume / weight." it's esoteric axiom of choice garbage from a "real-world" perspective
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the "most meaningful" part of banach-tarski is a group theory thing that is sort of like how you can take the integers and divide them into the even and odd integers which "have the same size" as the integers. the axiom-of-choice garbage is how to relate this to balls
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can you spell out how the axiom of choice is related to counting infinitesimals? the connection isn’t clear to me
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the axiom of choice is used to construct the weird bizarre pieces the banach-tarski decomposition needs. idk where to find a really accessible exposition, here's terence tao:
math.ucla.edu/~tao/preprints