Have you ever thought that traditional mathematical reasoning could prove everything that is true about numbers? Or that we could prove, using traditional mathematical reasoning, that mathematics was free of contradictions? Well, think again. Gödel proved that these things are not possible. How did he do this?

Kurt Gödel is famous for the following two theorems:

  1. Any formal system (with a finite axiom schema and a computationally enumerable set of theorems) able to do elementary arithmetic is either inconsistent or incomplete.

  2. Any formal system able to express its own consistency can prove its own consistency if and only if it is inconsistent.