**Paul Bernays** was a Swiss mathematician, who made important contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He is the **B** in the **NBG** axiom system.

- Let us make clear at this point that Bernays, in a CV that he wrote later in life, says that he was of the Jewish faith and a citizen of Switzerland.
- Bernays's decision was to take up engineering and he entered the Technische Hoschule in Charlottenburg where he began his studies in 1907.
- However, despite his parents' wish that he put his mathematical talents to practical use, Bernays decided after one semester (the 1907 summer semester) that he must make the change from engineering to pure mathematics.
- From 1910 until 1912 he studied at Göttingen where he attended mathematics lectures by David Hilbert, Edmund Landau (who moved to Göttingen at the same time as Bernays), Hermann Weyl and Felix Klein, physics lectures by Woldemar Voight and Max Born, and he studied philosophy with Leonard Nelson (1882-1927).
- At the University of Zürich, Bernays was appointed as a privatdocent and an assistant to Ernst Zermelo.
- Bernays took over Zermelo's lecture courses after he left.
- In many ways Zürich was a good time for Bernays for he became friends with Georg Pólya, made several visits to Albert Einstein and enjoyed frequent social occasions at Hermann Weyl's.
- In 1917 Hilbert visited Zürich to lecture and offered Bernays a post as his assistant at Göttingen.
- The other was a reserved, shy and somewhat nervous man named Paul Bernays.
- To the surprise of Polya and Bernays the subject of conversation on the walk to the top of the Zürichberg was not mathematics but philosophy.
- Bernays, however, had studied some philosophy and, during his student days at Göttingen, had been close to Leonard Nelson.
- Now, in spite of his quietness, Bernays had much more to say than the usually voluble Polya.
- At the end of the walk Hilbert asked Bernays to come to Göttingen as his assistant.
- The walk took place, as Reid writes, when Hilbert visited Zürich in the spring of 1917 but Bernays only received the offer to become Hilbert's assistant when Hilbert returned to Zürich in September 1917.
- At Göttingen, Bernays worked on the lecture notes to Hilbert's course 'Prinzipien der Mathematik'.
- Bernays wrote a second habilitation in which he established the completeness of propositional logic; this was in fact a study of Russell and Whitehead's Principia Mathematica, and uses ideas from Schröder.
- Bernays was no young student but a man in his middle thirties, a mature mathematician.
- In addition to preparing his own lectures, Bernays helped Hilbert prepare his lectures, accompanied him to class and often took over the teaching for part of the hour, supervised Hilbert's students who were working for the doctoral degree, studied and digested the literature necessary for their work, and did a great deal of writing on their joint book, which was entitled 'Grundlagen der Mathematik'.
- In Bernays, Hilbert had found someone as interested in the foundations of mathematics as he was.
- Bernays attributes the emotional quality of these arguments to a fundamental "opposition" in Hilbert's feelings about mathematics.
- Bernays did not always agree with Hilbert about their programme, but he appreciated the fact that, passionate though Hilbert was in their disputation, he never held it against his assistant personally when he took the opposite side.
- Bernays loved music and had played "four hands" with Hurwitz when he was in Zürich.
- In 1922 Hilbert recommended Bernays for an extraordinary professorship at Göttingen.
- Bernays' knowledge is extraordinarily broad and deep, and it extends into the domains of philosophy, physics and biology.
- Bernays was appointed as an extraordinary professor without tenure.
- When the Nazi regime made its directive against Jews in 1933, Bernays lost his post at Göttingen.
- He obtained a half-time permanent post at the ETH from 1945 but 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.
- Today Bernays is best known for his joint two volume work Grundlagen der Mathematik Ⓣ(Foundations of mathematics) (1934-39) with Hilbert.
- Although the book was a joint publication, the two authors made very different contributions with all the text being written by Bernays and much of the content being Bernays' working out answers to, often rather vague, questions from Hilbert.
- Recent re-evaluation of Bernays' role actually places him at the centre of the development of mathematical logic and Hilbert's program.
- In 1899 Hilbert had written Grundlagen der Geometrie Ⓣ(Foundations of geometry) 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.
- Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal.
- In 1958 Bernays published Axiomatic Set Theory in which he brought together all his work on the axiomatisation of set theory.
- Now Bernays' work in Zürich was influenced by his colleague Ferdinand Gonseth (1890-1975).
- Bernays' work on an axiomatic basis for mathematics was taken further by Kurt Gödel.
- As we have already indicated, Bernays held relatively minor posts considering the magnitude of his achievements.

Born 17 October 1888, London, England. Died 18 September 1977, Zürich, Switzerland.

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Origin England, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive