For about a hundred years, mathematicians like Bertrand Russell were trying to present axioms that would define the entire field of mathematics on an axiomatic basis. Austrian-American mathematician and logician Kurt Gödel (1906–1978) was the first to suggest that any formal system strong enough to include the laws of mathematics is either incomplete or inconsistent; this was called “Gödel’s incompleteness theorem.” Thus, axioms could not define all of mathematics.

Gödel also stated that the various branches of mathematics are based in part on propositions that are not provable within the system itself, although they may be proved by means of logical (metamathematical) systems external to mathematics. In other words, nothing is as simple as it seems; and, interestingly enough, Gödel’s idea also implies that a computer can never be programmed to answer all mathematical questions.