The disconnect between reality and Banach-Tarski is deep. Important to the paradox is that it does not present a construction of the division. And more importa

The disconnect between reality and Banach-Tarski is deep. Important to the paradox is that it does not present a construction of the division. And more importantly, it relies on the fact that it is not possible to construct such a division. If you accept some notion of the Universal Church-Turing Thesis as a matter of reality, then it is not possible even in principle for such a division to occur. Note that this is not just a matter of "finding the right way to construct it" -- the theorem critically relies on the fact that given any way of constructing sets using finite representations (like computer programs or layout of molecules), the theorem does not hold.