Godel demonstrated that no complex mathematical system was complete. In other words, no matter what axioms are chosen, meaningful mathematical statements can be made whose truth or falseness can never be demonstrated within the system.

Godel's second discovery was even more devastating. He demonstrated that it was impossible to prove that any given complex mathematical system was consistent. In other words, you can never be sure that the set of axioms won't lead to a contradiction. On the Richter scale of mathematical discoveries, Godel's was a 10.

As Andre Weil, number theorist extraordinaire, put it: "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

— The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth di Paul Hoffman (Pagina 117)