The discussion centers around the Axiom of Choice and its implications, particularly in relation to the Banach-Tarski Paradox. Participants explore the mathematical concepts involved, the nature of the proof, and the logical implications of dividing a sphere into parts that can be rearranged to form a larger sphere.
Participants express differing views on the specifics of the proof, particularly regarding the number of pieces involved in the construction. There is also a general uncertainty about the implications of the Axiom of Choice and the nature of the sets used in the proof.
Limitations include the dependence on the definitions of measurability and volume, as well as unresolved mathematical steps regarding the proof's construction.
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.