Banach-Tarski is true if you accept the axiom of choice. Which is not especially interesting: Every theorem depends on the axioms used to prove it; axioms are, in fact, what separates math from the natural sciences, and why 'truth' is available to mathematicians as long as you stipulate that a theorem is only absolutely true if you accept the axioms as absolute.
None of this involves the kind of actual (or falsidical) paradoxes the incompleteness theorems do. (Note that the Banach-Tarski Paradox is a viridical paradox, as it is a true result that is only thought paradoxical because it contradicts naïve intuition. A falsidical paradox is one that shows a true flaw or limit in a logical system, such as Russell's Paradox is to naïve set theory. This terminology is due to Quine.)
not all axioms are created equally. some axioms are simple, like that the empty set exists. to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think? also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions.
that's a good distinction you make in the second paragraph. to me though, the viridical paradoxes are more interesting :-)
>to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think?
It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate.
>also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions.
I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.
None of this involves the kind of actual (or falsidical) paradoxes the incompleteness theorems do. (Note that the Banach-Tarski Paradox is a viridical paradox, as it is a true result that is only thought paradoxical because it contradicts naïve intuition. A falsidical paradox is one that shows a true flaw or limit in a logical system, such as Russell's Paradox is to naïve set theory. This terminology is due to Quine.)