Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the *Theory* of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: *countable* and *uncountable*. It also connects with the Smooth or Bumpy post, which considered differences between the discrete and the continuous.

This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including *Star Trek*). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)

So let’s consider Cantor.