Banach-Tarski says that given a glass ball, we can break it into two glass balls of equal volume to the original (plus other generalizations). The explanations I have found for this paradoxical notion is that volume is not an invariant when we do these operations.
But, surely volume should be an invariant, right? "In reality," is it not realistic to expect that the volume doesn't change (unless we are dealing with some chemical property here, which I assume not since it is a mathematical paradox)?
So, my question is: is it possible, given whatever machinery we want to invent (something to break the glass exactly into what shape pieces we want, etc.), to actually perform this "paradox?" If not, then what is the "trick" behind the paradox which makes the math work out? What abstraction from reality do the hypotheses assume?
N.B. I've done extensive googling (especially on this site) of the subject and none of the answers have satisfied my question. I do realize there are a lot of questions about Banach-Tarski on this website already, but I do believe that my question is not a "duplicate."
Best Answer
The main crux of the Banach-Tarski construction is that you can break up a measurable set into non-measurable sets. Measure is, in a certain sense, analogous to volume. Non-measurable sets are somewhat strange and it can be shown that under mild axiomatic assumptions that it is consistent that there are no non-measurable sets. So in that sense, there is a "reality" where the Banach-Tarski paradox doesn't exist.