Gödel's Incompleteness Theorem

Russell and Whitehead tried to write all knowledge of arithmetic as logical axioms in a common format in three volumes called Principia Mathematica (1910–1913).

David Hilbert challenged mathematicians to show that all mathematical truths could be proven from a finite set of consistent inference rules (early 1920’s).

In 1931, Czech-born mathematician Kurt Gödel demonstrated that:

Theorem

In any axiomatic mathematical system, there are propositions that cannot be proved or disproved within the axioms of the system

Gödel showed that Hilbert's goal was impossible. Truth and proof were different.

We cannot have a finite set of rules that will let us prove everything in mathematics (that is, sound but incomplete).