[IngoBlechschmid]

There are three ways to resolve this paradox:

1. Accept that our intuition about volumes is off when dealing with point clouds so weird that they cannot actually be described, but require the axiom of choice to concoct them.

2. Reject the axiom of choice and adopt the axiom of determinacy. This axiom restores our intuition about volumes to all subsets of Euclidean space, at the expense of which sets can be formed. (That said, the axiom of determinacy allows other sets to be formed which are not possible with the axiom of choice, so it wouldn't be correct to state that the axiom of determinacy causes the set-theoretic universe to shrink.)

3. Keep logic and set theory as it is, but employ locales instead of topological or metric spaces. Locales are an alternative formalization of the intuitive notion of spaces. For many purposes, there are little differences between locales and more traditional sorts of spaces. But, crucially, a locale can be nontrivial even if it does not contain any points. Locale-theoretically, the five pieces appearing in the Banach–Tarski paradox have a nontrivial overlap (even though no points are contained in the overlapping regions), hence you wouldn't expect the volumes to add up.

I tried to give a varied account on the axiom of choice at the Chaos Communication Congress once, the slides are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...

[constantcrying]

Why didn't you list the actual "standard" approach of accepting that certain sets have no definable volume? This is the basis of measure theory, which is perfectly accepting of the fact that measures don't need to be defined everywhere.

[IngoBlechschmid]

Sorry, I was in a hurry before, the standard approach is exactly what I wanted to refer to with option 1!