Paul Bernays

Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. Born into a distinguished German-Jewish family, Bernays pursued his studies in mathematics, philosophy, and physics at the University of Berlin and the University of Göttingen. He earned his Ph.D. in mathematics from the University of Berlin in 1912 for a thesis on the analytic number theory of binary quadratic forms. In the same year, he received his habilitation from the University of Zurich for a thesis on complex analysis and Picard's theorem. Starting in 1917, Bernays assisted David Hilbert with his investigations into the foundations of arithmetic. His habilitation thesis on the axiomatization of propositional logic in Whitehead and Russell's Principia Mathematica contained the first known proof of the semantic completeness of propositional logic. Bernays's collaboration with Hilbert led to the two-volume work, Grundlagen der Mathematik (Foundations of Mathematics), published in 1934 and 1939. A proof within this work concerning self-reference in consistent theories is known as the Hilbert–Bernays paradox. In the 1930s and 1940s, Bernays developed an axiomatic set theory based on John von Neumann's earlier work. This theory, with modifications by Kurt Gödel, became known as von Neumann–Bernays–Gödel set theory. Due to Nazi Germany's Law for the Restoration of the Professional Civil Service, Bernays was dismissed from his position at the University of Göttingen in 1933 because of his Jewish ancestry. He then moved to Switzerland, where he continued his work, also spending time as a visiting scholar at the Institute for Advanced Study in the United States.