But it does fit the description. There is absolutely no statement that can be proven in standard math about the integers whose truth depends on the truth or falseness of Banach-Tarski. (More technically there is a theorem that any statement that can be proven about ZFC can be proven in ZF alone - this falls out of the construction that Goedel used to prove that ZFC is consistent if ZF is. The same construction and result falls out for many other axioms that you can add to ZF, including the closure of ZF is consistent, ZF + the consistency of ZF is consistent, ZF + the consistence of ZF plus the consistency of THAT is consistent, etc.)
Therefore the truth of Banach-Tarski has nothing to do with any statement you can readily make about the integers, which would include any statement that you can make about anything that can be done on a Turing machine.
Incidentally the whole debate about constructivism is much more subtle than you likely appreciate. For instance it is trivial to prove that almost all real numbers that exist, cannot be written down. (Proof, I can easily write out a countably infinite set of statements that could define real numbers. So the set of real numbers that can be written down is countable. But there are an uncountable number of real numbers, so almost all cannot be written down.) In what sense do said real numbers actually exist?
If our axioms say that numbers must exist, which have in a very real sense no actual existence, then are those axioms capturing what we want to mean by "exist"?
If you say it does not, then you're a constructivist who hasn't thought carefully enough yet. If you say it does make sense, then you've been brainwashed by decades of standard mathematical presentations into accepting a definition of "existence" that makes no sense to the average lay person. :-P
Therefore the truth of Banach-Tarski has nothing to do with any statement you can readily make about the integers, which would include any statement that you can make about anything that can be done on a Turing machine.
Incidentally the whole debate about constructivism is much more subtle than you likely appreciate. For instance it is trivial to prove that almost all real numbers that exist, cannot be written down. (Proof, I can easily write out a countably infinite set of statements that could define real numbers. So the set of real numbers that can be written down is countable. But there are an uncountable number of real numbers, so almost all cannot be written down.) In what sense do said real numbers actually exist?
If our axioms say that numbers must exist, which have in a very real sense no actual existence, then are those axioms capturing what we want to mean by "exist"?
If you say it does not, then you're a constructivist who hasn't thought carefully enough yet. If you say it does make sense, then you've been brainwashed by decades of standard mathematical presentations into accepting a definition of "existence" that makes no sense to the average lay person. :-P