If one doesn’t accept the Axiom of Choice but uses instead Dependent Choice the paradox no longer holds. Is it the case in this situation that the 4 desirable properties hold?
You don't get new theorems if you remove assumptions. Rather, you get the ability to add different assumptions.
The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability.
There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets.
I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
Assuming the usual consistency caveats, the paradox is no longer a theorem of ZF+DC, but its complement isn't either. So in that case the analogue to the fifth postulate is even stronger, as there are both models in which you get the counterintuitive results of unmeasurable sets and those in which you don't, and the axioms are not strong enough to distinguish the two.
In ZF+DC is it true that measures satisfy the desirable properties mentioned by Colin? I think the sticking point is isometry invariance. Are there measures in ZF+DC of R^3 that are finitely (countably?) additive and isometry invariant?
