Paul Bernays's family were Swiss but, after a short time in London, his family moved to Paris. From Paris the family moved to Berlin where Paul attended the Köllnisches Gymnasium from 1895 to 1907. In later life Bernays was to speak of the happy childhood he had during these years. At the Gymnasium he had a strong interest in music. Slightly later came a love for ancient languages and mathematics. As his school days drew to an end he had to make the difficult decision between music and mathematics.
Bernays's decision was to take up engineering and he entered the Technische Hoschule in Charlottenburg and began his studies. However, despite his parents' wish that he put his mathematical talents to practical use, Bernays decided after a year that he must make the change from engineering to pure mathematics.
He undertook his pure mathematics studies first at the University of Berlin where he was taught by Schur, Edmund Landau, Frobenius, Schottky and Planck. From 1910 until 1912 he studied at Göttingen where he attended lectures by Hilbert, Landau, Weyl, Klein, W Voight and Born. It was at Göttingen that he obtained his doctorate in 1912, working with Landau on analytic number theory and binary quadratic forms. His habilitation thesis was on modular elliptic functions.
Bernays was appointed to the University of Zurich as an assistant to Zermelo and worked there until 1917. In 1916 Zermelo left Zurich, partly for health reasons, partly because of a dispute with the university administration. Bernays took over Zermelo's lecture courses after he left. Bernays became friends with Pólya, Einstein and Weyl while in Zurich.
In 1917 Hilbert visited Zurich to lecture and offered Bernays a post as his assistant at Göttingen. There he worked on the lecture notes to Hilbert's course Prinzipien der Mathematik. These lecture notes were later edited by Wilhelm Ackermann and published as Grundzüge der theoretischen Logik. Bernays wrote a second habilitation in which he established the completeness of propositional logic; this was in fact is a study of Russell and Whitehead's Principia Mathematica, and uses ideas from Schröder.
In 1922 Hilbert recommended Bernays for an extraordinary professor at Göttingen. In his letter of recommendation Hilbert wrote:-
Bernays's publications extend over the most diverse fields of mathematics ... and are all marked by thoroughness and reliability ... He is distinguished by a deep-seated love for science as well as a trustworthy character and nobility of thought, and is highly valued by everyone. In all matters concerning fundamental questions in mathematics, he is the most knowledgeable expert and, especially for me, the most valuable and productive colleague.
Bernays was appointed extraordinary professor. When the Nazi regime made its directive against Jews in 1933, Bernays lost his post at Göttingen. Hilbert kept him on as his private assistant for several months but soon he was forced to leave Germany. He was still a Swiss citizen so a move to Zurich was not too difficult.
In Zurich he worked at the Eidgenössische Technische Hochschule (ETH: the Swiss Federal Institute of Technology) in a temporary post from 1934. He visited Princeton in session 1935-36. He obtained a half-time post at the ETH from 1945 and there has been criticism of the ETH for not treating a distinguished academic like Bernays in a more honourable way. However Bernays never saw it that way and he was extremely grateful to the ETH for coming to his rescue at a time of great difficulty.
Bernays is perhaps best known for his joint two volume work Grundlagen der Mathematik (1934-39) with Hilbert. This attempted to build mathematics from symbolic logic. In 1899 Hilbert had written Grundlagen der Geometrie and, in 1956, Bernays revised this work on the foundations of geometry.
Bernays, influenced by Hilbert's thinking, believed that the whole structure of mathematics could be unified as a single coherent entity. In order to start this process it was necessary to devise a set of axioms on which such a complete theory could be based. He therefore attempted to put set theory on an axiomatic basis to avoid the paradoxes.
Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal. He attempted to modify von Neumann's axiom system to include features from Zermelo's. He formulated the principle of dependent choices, a form of the axiom of choice independently studied by Tarski later. He used number theoretic models similar to those used by Ackermann to show the independence of his axioms. In 1958 Bernays published Axiomatic Set Theory in which he brought together all his work on the axiomatisation of set theory.
Bernays's work on an axiomatic basis for mathematics was taken further by Gödel.
Article by: J J O'Connor and E F Robertson