You are spot on! The paradox uses a, from the point of view of everyday life, highly perverse definition of "piece". The definition of "piece" used is: "Arbitr

You are spot on! The paradox uses a, from the point of view of everyday life, highly perverse definition of "piece". The definition of "piece" used is: "Arbitrary collection of points which need to bear any relation." This is in contrast to pieces of things in everyday life, which are firstly not made of infinitesimal points, but even if we keep rolling with that, satisfy the restriction that with every included point also some neighboring points are included. This is not so with the "pieces" appearing in the Banach–Tarski paradox. In fact, they are so extremely rigged/fractal/dislocated/non-contiguous, that it is not possible to assign a meaningful measure of volume to these. For instance, the unit cube has volume 1. The empty set of points has volume 0, and so do sets which contain just a finite number of infinitesimal points. But the "pieces" appearing in the paradox are so weird that we cannot meaningfully attach any volume at all to them. Also, there can never be any formula or any other explicit description of which points the five pieces consist of. This is because the proof employs the axiom of choice.