>>13230867First of all, start with the question "what is this theorem?", and by that I don't mean reciting the "you can divide a ball into finitely many parts that give you two copies" explanation, but where does it fit in the big picture? If you know Lebesgue measure theory (and I will now mean his measure when I speak of "measure"), you will know the following two things:
>the measure of the real line is infinite>if the sigma algebra of measurable sets is the whole power set of the real numbers, then the measure of the real line is also 0 (assuming AC)The first claim is easy to prove. Try to cover the real line in any way with finitely many intervals of finite length, and you will fail. That's basically it. The second claim is where you pull out the so called Vitali sets. Assume that every set is measurable and use the axiom of choice to pick some nice elements such that, for any pair of such elements, it holds that . What these elements give you are the cosets of the rationals in the reals! The rational numbers give you a countable set, so their measure is 0. These are disjoint sets, so you can just add their measures to get the measure of the whole real line, but now you are adding the measures of countable sets and get 0 for the whole space. Quite non-sensical!
Now, how does this relate to Banach-Tarski? I'd say the relation is the following: just like the Vitali sets show that not all subsets of the real line are measurable (in the sense of Lebesgue), so does the Banach-Tarski decomposition of the ball show the same for ! Instead of calling it a "paradox", it is to be seen as a proof of the theorem (when you are pro-choice, idk how the (in)equality would be without the axiom). This is how I'd approach it, at least! Does this make any sense to you?
