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Infinities and Banach Tarski
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My very limited understanding is that Banach-Tarski says we can cut a sphere into pieces, move those pieces using only rigid body transformations, and then re-assemble the pieces into two spheres. How many times can this be repeated? I suspect any finite number will work, so then why not the cardinality of the integers? Does this break down with the cardinality of the reals?
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