Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert. Bernays spent his childhood in Berlin. Bernays attended the Köllner Gymnasium, 1895-1907.
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- Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert. Bernays spent his childhood in Berlin. Bernays attended the Köllner Gymnasium, 1895-1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Frobenius, and Schottky; philosophy under Riehl, Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Gottingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Landau, on the analytic number theory of binary quadratic forms. That same year, the University of Zurich awarded him the Habilitation for a thesis on function theory and Picard's theorem. The examiner was Ernst Zermelo. Bernays was Privatdozent at the University of Zurich, 1912-17, where he came to know George Pólya. Starting in 1917, David Hilbert employed Bernays to assist him with his investigations of the foundations of arithmetic. Bernays also lectured on other areas of mathematics at the University of Gottingen. In 1919, that university awarded him a second Habilitation, for a thesis on the axiomatics of the propositional calculus of Principia Mathematica. In 1922, Gottingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. In 1933, he was dismissed from this post because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to Switzerland, whose nationality he had inherited from his father, and where the ETH employed him on occasion. He also visited the Institute for Advanced Study in Princeton, USA, and the University of Pennsylvania. Bernays's collaboration with Hilbert culminated in the two volumes Hilbert and Bernays (1934; 1939), discussed in Sieg and Ravaglia (2005) and not translated into English. In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic, Bernays set out an axiomatic set theory whose starting point was a related theory John von Neumann had set out in the 1920s. Von Neumann's theory took the notion of function as primitive; Bernays recast Von Neumann's theory so that sets and proper classes were primitive. Bernays's theory, with some modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory.
- Paul Bernays war ein Mathematiker und Logiker.
- Paul Bernays byl matematik švýcarského původu.
- Paul Bernays né le 17 octobre 1888 à Londres et mort le 18 septembre 1977 à Zurich, est un mathématicien suisse qui a joué un rôle crucial dans le développement de la logique mathématique au XX siècle. Il est longtemps l'assistant et le collaborateur de David Hilbert. Son nom est lié à la théorie des ensembles de von Neumann-Bernays-Gödel (dite NBG). En 1926, il montre le théorème de complétude du calcul des propositions. De 1945 à 1958, il est professeur à l'ETH de Zurich.
- パウル・ベルナイス（Paul Bernays, 1888年10月17日 ロンドン - 1977年9月18日 チューリヒ）はドイツの数学者・論理学者。 ヘルシンキ大学、チューリヒ大学の教授をつとめる。
- Paul Bernays was een Zwitsers wiskundige en logicus, Hij is bekend geworden door het boek Grundlagen der Mathematik uit 1934 dat hij in samenwerking met David Hilbert heeft geschreven. Daarna werkte hij jarenlang aan de axiomatische verzamelingenleer, die mede naar hem, de Von Neumann–Bernays–Gödel-verzamelingenleer, is genoemd.
- Paul Isaac Bernays, szwajcarski matematyk i logik. Pracował na stanowisku profesora w Zurychu i Getyndze. Zasłynął jako współtwórca teorii dowodu i jednego z aksjomatycznych ujęć teorii mnogości, znanego jako teoria mnogości von Neumanna-Bernaysa-Gödla.
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- Paul Bernays in 1976
(photo by George Bergman)
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- Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert. Bernays spent his childhood in Berlin. Bernays attended the Köllner Gymnasium, 1895-1907.
- Paul Bernays war ein Mathematiker und Logiker.
- Paul Bernays byl matematik švýcarského původu.
- Paul Bernays né le 17 octobre 1888 à Londres et mort le 18 septembre 1977 à Zurich, est un mathématicien suisse qui a joué un rôle crucial dans le développement de la logique mathématique au XX siècle. Il est longtemps l'assistant et le collaborateur de David Hilbert. Son nom est lié à la théorie des ensembles de von Neumann-Bernays-Gödel (dite NBG). En 1926, il montre le théorème de complétude du calcul des propositions. De 1945 à 1958, il est professeur à l'ETH de Zurich.
- パウル・ベルナイス（Paul Bernays, 1888年10月17日 ロンドン - 1977年9月18日 チューリヒ）はドイツの数学者・論理学者。 ヘルシンキ大学、チューリヒ大学の教授をつとめる。
- Paul Bernays was een Zwitsers wiskundige en logicus, Hij is bekend geworden door het boek Grundlagen der Mathematik uit 1934 dat hij in samenwerking met David Hilbert heeft geschreven. Daarna werkte hij jarenlang aan de axiomatische verzamelingenleer, die mede naar hem, de Von Neumann–Bernays–Gödel-verzamelingenleer, is genoemd.
- Paul Isaac Bernays, szwajcarski matematyk i logik. Pracował na stanowisku profesora w Zurychu i Getyndze. Zasłynął jako współtwórca teorii dowodu i jednego z aksjomatycznych ujęć teorii mnogości, znanego jako teoria mnogości von Neumanna-Bernaysa-Gödla.
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- Paul Bernays
- Paul Bernays
- Paul Bernays
- Paul Bernays
- パウル・ベルナイス
- Paul Bernays
- Paul Bernays
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