Explain SMBC — the wiki for Saturday Morning Breakfast Cereal

The Joke

A mathematician tries to explain different sizes of infinity (Cantor's theorem that some infinities are larger than others) to a non-mathematician. The non-mathematician's brain rebels at every step. The mathematician is clearly enjoying the other person's distress.

The Humor

Cantor's proof that the real numbers are "more infinite" than the natural numbers is one of the most counterintuitive results in mathematics. Most people's intuition says "infinity is infinity" — the idea that infinity comes in different sizes sounds like nonsense. The comic gets humor from the non-mathematician's completely reasonable confusion colliding with the mathematician's delight in counterintuitive truths.

Mathematical Context

Georg Cantor showed in the 1870s that the set of real numbers (all decimals) cannot be put into one-to-one correspondence with the natural numbers (1, 2, 3, ...), meaning the reals are a "larger" infinity. This is proven via the diagonal argument, one of the most elegant proofs in mathematics. The result was so controversial that some of Cantor's contemporaries considered it literally insane.