"If you take the axiom of choice in your system, Banach-Tarski is simply true"
- The exact same is true if you take the continuum hypothesis as an axiom. Both the axiom of choice and the continuum hypothesis are independent of the other standard ZF axioms
"If you do not take the axiom of choice, it is simply false"
- That is incorrect. It is only independent. To say it is false, another axiom(s) would need to contradict it.
"while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe"
- neither are the real numbers. would you like to claim that pi, or the square root of 2, for that matter, also don't exist? How about the number 1?
- The exact same is true if you take the continuum hypothesis as an axiom. Both the axiom of choice and the continuum hypothesis are independent of the other standard ZF axioms
"If you do not take the axiom of choice, it is simply false"
- That is incorrect. It is only independent. To say it is false, another axiom(s) would need to contradict it.
"while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe"
- neither are the real numbers. would you like to claim that pi, or the square root of 2, for that matter, also don't exist? How about the number 1?