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Eugine_Nier comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong
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The paradoxical decomposition of F2 only gives a decomposition for a dense subset of the sphere, because you have to throw away the (countably many) fixed points of all the rotations involved to make the correspondence between F2 and the orbits of various points. To go the rest of the way and you need to use something other than rotations about the origin, ie something more than just the action of F2. But it's certainly fair to say that Banach-Tarski works because of the structure of F2.
To go the rest of the way you still only use rotations, just not the rotations in F2.
The way I always did it was to use rotations about some fixed line that doesn't go through 0.
We seem to be talking about different things, I'm talking about doubling the surface of the sphere. You're talking about how to get the center once you've doubled the surface.
Ahh yes, you're right.