Kurt Gödel (1906–1978) was an Austrian-American logician widely regarded as the greatest of the twentieth century. In his mid-twenties he proved a result so profound it changed what mathematicians believe is possible: there are true statements that can never be proved. Then he spent his later years as Einstein's favourite walking companion at Princeton.

The famous result

Gödel's incompleteness theorems showed that any logical system rich enough to do ordinary arithmetic must contain true statements it cannot prove — you can never have a set of rules that is both complete and guaranteed consistent. He also worked at the frontier of set theory: he proved that the continuum hypothesis and the axiom of choice cannot be disproved from the standard axioms — half of a puzzle that Paul Cohen would later complete.