The discussion centers on the Banach-Tarski Paradox, which asserts that accepting the Axiom of Choice allows for the division of a sphere into a finite number of non-measurable sets that can be rearranged to form two spheres of the same size. The proof involves advanced concepts from abstract set theory and surgery theory, emphasizing that the intermediate steps utilize sets that lack a defined volume. Consequently, the final configuration does not retain the original volume, challenging conventional notions of measure.
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Hurkyl said:IIRC, the proof uses surgery theory.
One of the main things to emphasize about the construction is that its intermediate steps involve sets that are not measurable. All of the clever work is done with sets for which you cannot define volume, so there isn't any reason to expect that you have the original volume when you're done.
And, incidentally, the big point about the construction is that it only uses three pieces. It's a trivial exercise to prove that you can rearrange all of the points in one sphere to form two spheres of the same size, if you do it one point at a time.