I personally thought that the Banach-Tarski theorem was a mess, and these non-measurable sets had to go. So I started studying nonstandard-analysis: analysis wi

I personally thought that the Banach-Tarski theorem was a mess, and these non-measurable sets had to go. So I started studying nonstandard-analysis: analysis with infinities and infinitesimals as genuine, arithmetically viable, quantities. Imagine my distress: the same problem occurs, Banach-Tarski still holds, unless you go into the rather more weird, Smooth Infinitesimal Analysis or Synthetic Differential Geometry; basically, treating geometry as made up of little discrete infinitesimally small points.