[inetsee]

Can someone explain the flaw in my reasoning here?

Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?

[Tomte]

> I divide the sphere into individual iron atoms.

You have highly restricted the act of choosing sets of points here. B-T doesn't say that any "division" results in that unintuitive outcome.

Note that points are infinitesimally small and infinitely many, and atoms in your iron sphere are neither.

[hodgesrm]

Your sets are finite. B-T depends on properties of infinite sets.