> 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.
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.