I recently re-reviewed some of my undergrad analysis text and read the sketch of the proof of Banach-Tarski presented on Wikipedia, starting with a proof that the free group with two generating elements can be cut and merged into two copies of itself by first accepting a partition of the group into members starting with each element/inverse, divying those by element, prepending one side by its inverse element $aS(a^-1) \bigcup S(a) $, and noticing that $aS(a^-1)$ is going to be a sequence that starts with anything but $a$, so you end up by symmetry with the same thing on the $b$ side, and you get $2$ copies of the free group of two generating elements.
I have never felt the unease that others attribute to the paradox, despite having been aware of it at a rather vapid but conceptual level from the age of 14 or 15. I am curious to know whether my instincts are just wrong (but have made me feel okay) or whether there is any merit to them.
My first instinct when younger was to attribute the change in volume to a change in how "tightly packed" the points of the interior were to each other, assuming multiple configurations were possible and meaningful under the assumptions of the theorem.
As I got older, I started to think about it as an analogous to how you can fold and wrinkle paper into a still-planar manifold, and considered whether it could have less surface area. I discovered that at least one case is obvious - if you end up gluing edge to edge, you lose an edge worth of area. So there was nothing intrinsic about surface area for this piece of paper, why would volume work differently?
Once I reached analysis, I had convinced myself of a different physical theory - that somehow you could "pull a string out of the ball" of a single point radius, and that your choice of reconstruction would be a matter of how wasteful you were in spooling the core of the new ball. Since points had no dimensionality, I thought, it should somehow be possible to thread the core for arbitrarily long string, and yet not fill a continuum of points through any ray in 3D from the origin.
Today, I have to say I am no smarter than I was when I was younger, and in fact I don't feel comfortable that any of these pseudo explanations are relevant to the interplay between the paradox and the notion of embedding some space with measure.
So I would appreciate to know:
- Which if any of my past conceptualizations was a reasonable thought model for the paradox?
- Whether any physical explanation is likely to be a reasonable model
- Whether any material, temporal, or otherwise constructive in a layman's sense, notion might be useful
- Any recommended articles, areas of study, problems, related to these questions