Paul Bernays  

Born  
Died  18 September 1977  (aged 88)
Nationality  Swiss 
Alma mater  University of Berlin 
Known for  Mathematical logic Axiomatic set theory Philosophy of mathematics 
Scientific career  
Fields  Mathematics 
Thesis 

Doctoral advisor  Edmund Landau 
Doctoral students  Corrado Böhm Julius Richard Büchi Haskell Curry Erwin Engeler Gerhard Gentzen Saunders Mac Lane 
Influences  Issai Schur, Edmund Landau 
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.
Biography
Bernays was born into a distinguished GermanJewish family of scholars and businessmen. His greatgrandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849.^{[1]}
Bernays spent his childhood in Berlin, and attended the Köllner Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, 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 habilitation for a thesis on complex analysis 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. His collected communications with Kurt Gödel span many decades.
Starting in 1917, David Hilbert employed Bernays to assist him with his investigations of the foundation of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second habilitation for a thesis on the axiomatics of the propositional calculus of Principia Mathematica.^{[2]}
In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. After Nazi Germany enacted the Law for the Restoration of the Professional Civil Service in 1933, the university fired Bernays 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 Zurich employed him on occasion. He also visited the University of Pennsylvania and was a visiting scholar at the Institute for Advanced Study in 1935–36 and again in 1959–60.^{[3]}
Mathematical work
Bernays's collaboration with Hilbert culminated in the two volume work, Grundlagen der Mathematik (English: Foundations of Mathematics) published in 1934 and 1939, which is discussed in Sieg and Ravaglia (2005). A proof in this work that a sufficiently strong consistent theory cannot contain its own reference functor is known as the Hilbert–Bernays paradox.
In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic (republished in Müller 1976), 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 notions of function and argument as primitive. Bernays recast von Neumann's theory so that classes and sets were primitive. Bernays's theory, with modifications by Kurt Gödel, is known as von Neumann–Bernays–Gödel set theory.
Publications
 Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, 40, Berlin, New York: SpringerVerlag, ISBN 9783540041344, JFM 60.0017.02, MR 0237246, archived from the original on 20110517^{[4]}
 Hilbert, David; Bernays, Paul (1939), Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, 50, Berlin, New York: SpringerVerlag, ISBN 9783540051107, JFM 65.0021.02, MR 0272596, archived from the original on 20110517
 Bernays, Paul (1958), Axiomatic Set Theory, Studies in Logic and the Foundations of Mathematics, Amsterdam: NorthHolland, ISBN 9780486666372, MR 0106178
 Bernays, Paul (1976), Abhandlungen zur Philosophie der Mathematik (in German), Darmstadt: Wissenschaftliche Buchgesellschaft, ISBN 9783534067060, MR 0444417
 Bernays, Paul; Schonfinkel, Moses (1928), "Zum Entscheidungsproblem der mathematischen Logik", Mathematische Annalen (99): 342–372
Notes
 ^ O'Connor, John J.; Robertson, Edmund F., "Paul Isaac Bernays", MacTutor History of Mathematics archive, University of St Andrews
 ^ Zach, Richard (1999). "Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic". Bulletin of Symbolic Logic. 5 (3): 331–66. doi:10.2307/421184. JSTOR 421184.
 ^ "Paul Bernays". Institute for Advanced Study. n.d. Retrieved July 1, 2021.
 ^ MacLane, Saunders (1935). "Review: Grundlagen der Mathematik, Volume I. By D. Hilbert and P. Bernays" (PDF). Bull. Amer. Math. Soc. 41 (3): 162–165. doi:10.1090/s000299041935060483.
References
 Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15: 43–69, doi:10.2178/bsl/1231081769.
 Kneebone, Geoffrey, 1963. Mathematical Logic and the Foundation of Mathematics. Van Nostrand. Dover reprint, 2001. A gentle introduction to some of the ideas in the Grundlagen der Mathematik.
 Lauener, Henri (1978), "Paul Bernays (18881977)", Zeitschrift für allgemeine Wissenschaftstheorie, 9 (1): 13–20, doi:10.1007/BF01801939, ISSN 00442216, MR 0546580, S2CID 147959212
 Müller, Gert H., ed. (1976), Sets and classes. On the work by Paul Bernays, Studies in Logic and the Foundations of Mathematics, 84, Amsterdam: NorthHolland, ISBN 9780444109071, MR 0414355
 Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Mathematik", in GrattanGuinness, Ivor (ed.), Landmark writings in western mathematics 16401940, Elsevier B. V., Amsterdam, pp. 981–99, doi:10.1016/B9780444508713/501583, ISBN 9780444508713, MR 2169816
External links
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