“The problem with Banach-Tarski is that we tend to think of it less in mathematical terms, and more in concrete terms. It’s often described as something like “You can slice up an orange, and then re-assemble it into two identical oranges“. Or “you can cut a baseball into pieces, and re-assemble it into a basketball.” Those are both obviously ridiculous. But they’re ridiculous because they violate one of our instinct that derives from the conservation of mass. You can’t turn one apple into two apples: there’s only a specific, finite amount of stuff in an apple, and you can’t turn it into two apples that are identical to the original. But math doesn’t have to follow conservation of mass in that way. A sphere doesn’t have a mass. It’s just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.” MarkCC on the Banach-Tarski non-paradox


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