Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make. It's not a "wider context", it's a different thing altogether.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.
> Euclidean geometry and its fifth axiom are interesting, but unrelated to the Banach-Tarski paradox. I don't get the point you're trying to make.
Let me try to make the analogy more explicit.
The interesting thing about the fifth postulate is that we show we can't prove it from the other axioms because there are models where the first four hold and the fifth doesn't.
The interesting thing about the Banach-Tarski Theorem is that it shows we can't have all four desirable properties of a metric because there are constructions that show they they can't all hold at once.
> ... you're completely glossing over the theory of measurable sets.
I'm not glossing over it, I'm showing why it is necessary and important.
I think there's an important difference between the situations.
With (non-)Euclidean geometry, you have a bunch of axioms and it turns out you don't have to accept the parallel postulate even if you accept all the others.
With measure theory, you have a bunch of things you'd like to be true and it turns out you can't accept all of them at once.
Those are quite different.
On the other hand, the analogy between geometry and, say, set theory is closer. There are a bunch of axioms for Euclidean geometry, the parallel postulate seems a bit dicey, and it turns out that you can accept the others and lose that one. There are a bunch of axioms for set theory, the axiom of choice seems a bit dicey, and it turns out that you can accept the others and lose that one.
From this perspective, the role of the Banach-Tarski paradox is to help show why the axiom of choice seems a bit dicey, maybe more so than the other commonly-adopted set-theoretic axioms.
(In set theory, too, there are situations where we can write down a bunch of axioms we would like to be true but that actually can't all be true at once, just like with measure theory. Russell's paradox is the best-known example, and it led set theorists to abandon the otherwise very attractive axiom of "unrestricted comprehension".)
The difference is that it's very easy to construct a non-Euclidean geometry in the physical world, at home, on your table, but it's impossible to construct an object anywhere in the real Universe where the Uncountable Axiom of Choice applies. It's purely a mathematical game of pretend.
They are as far apart from each other as possible, as similar as polar opposites.
Banach-Tarski is resolved by deciding that "the real numbers" aren't real, and nothing is lost except for dubious overly-simple proofs.
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?
I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bnach-Tarski follows from the Axiom of Choice which is independent of the rest of set theory.
Also, as someone pointed out in the linked thread, you're completely glossing over the theory of measurable sets.