[ajuc]

> A point has no volume so no matter how many you add together you don't get something with a volume

Not true. If you add uncountably many infinitesimal objects they can add up to noninfinitesimal object, that's how integration works in math, it's pretty confusing cause there's many kinds of infinity and they allow some unintuitive things to happen, but if they didn't worked we couldn't move (see Zeno paradox).

Banach-Tarski is formally correct, you add a finite number of sets with uncountably many points in each so you can get something with volume (depending on how they are positioned).

And yes - a line in math is just a set of points, same with a sphere (but it has 0 volume cause a sphere is just the "skin" without the insides) and a ball (which is what Banach-Tarski talks about). In fact every geometric object is just a set of points.

[stan_rogers]

Points, though, as traditonally defined, are not infinitessimal. They are literally zero in extent, having only a defined location.