Paul Bernays
Paul Bernays  

Autumn 1949 in Oberwolfach  
Born 
London, United Kingdom  17 October 1888
Died 
18 September 1977 88) Zurich, Switzerland  (aged
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 the 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.
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 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 the passing of the Law for the Restoration of the Professional Civil Service 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 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 by Hilbert and Bernays (1934, 1939), discussed in Sieg and Ravaglia (2005). 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 some modifications by Kurt Gödel, is now known as von Neumann–Bernays–Gödel set theory. A proof from the Grundlagen der Mathematik that a sufficiently strong consistent theory cannot contain its own reference functor is now known as the Hilbert–Bernays paradox.
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
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: 331–366. doi:10.2307/421184. JSTOR 421184.
 ↑ Institute for Advanced Study: A Community of Scholars
 ↑ 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
 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, 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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