Paul Bernays | |
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| Born | (1888-10-17)17 October 1888 London, English |
| Died | 18 September 1977(1977-09-18) (aged 88) |
| Alma mater | University of Berlin |
| Known for | Mathematical logic Axiomatic set theory Philosophy of mathematics Axiom of adjunction Axiom of dependent choice Grundlagen der Mathematik Second-order arithmetic Bernays class theory Bernays–Schönfinkel class Bernays–Tarski axiom system Hilbert–Bernays provability conditions Hilbert–Bernays paradox Von Neumann–Bernays–Gödel set theory |
| Scientific career | |
| Fields | Mathematics |
| Thesis | Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante (1912) |
| Doctoral advisor | Edmund Landau |
| Doctoral students | Corrado Böhm Julius Richard Büchi Haskell Curry Erwin Engeler Gerhard Gentzen Saunders Mac Lane |
| Other notable students | Hao Wang |
Paul Isaac Bernays (/bɜːrˈneɪz/bur-NAYZ;Swiss Standard German:[bɛrˈnaɪs]; 17 October 1888 – 18 September 1977) was aSwiss mathematician who made significant contributions tomathematical logic,axiomaticset theory, and thephilosophy of mathematics. He was an assistant and close collaborator ofDavid Hilbert.
Bernays was born into a distinguishedGerman-Jewish family of scholars and businessmen. His great-grandfather,Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849.[1]
Bernays spent his childhood in Berlin, and attended theKöllnische Gymnasium, 1895–1907. At theUniversity of Berlin, he studied mathematics underIssai Schur,Edmund Landau,Ferdinand Georg Frobenius, andFriedrich Schottky; philosophy underAlois Riehl,Carl Stumpf andErnst Cassirer; and physics underMax Planck. At theUniversity of Göttingen, he studied mathematics underDavid Hilbert,Edmund Landau,Hermann Weyl, andFelix Klein; physics under Voigt andMax Born; and philosophy underLeonard Nelson.
In 1912, theUniversity of Berlin awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on theanalytic number theory ofbinary quadratic forms. That same year, theUniversity of Zurich awarded himhabilitation for a thesis oncomplex analysis andPicard's theorem. The examiner wasErnst Zermelo. Bernays was Privatdozent at the University of Zurich, 1912–1917, where he came to knowGeorge Pólya. His collected communications withKurt 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 thepropositional calculus ofPrincipia Mathematica.[2]
In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there wasGerhard Gentzen. After Nazi Germany enacted theLaw 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 toSwitzerland, whose nationality he had inherited from his father, and where theETH Zurich employed him on occasion. He also visited theUniversity of Pennsylvania and was a visiting scholar at theInstitute for Advanced Study in 1935–36 and again in 1959–60.[3]
His habilitation thesis was written under the supervision of Hilbert himself, on the topic of the axiomatisation of propositional logic inWhitehead andRussell'sPrincipia Mathematica. It contains the first known proof ofsemantic completeness of propositional logic, which was reproved independently also byEmil Post later on.
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 referencefunctor is known as theHilbert–Bernays paradox.
In seven papers, published between 1937 and 1954 in theJournal of Symbolic Logic (republished in Müller 1976), Bernays set out an axiomatic set theory whose starting point was a related theoryJohn von Neumann had set out in the 1920s. Von Neumann's theory took the notions offunction andargument as primitive. Bernays recast von Neumann's theory so thatclasses andsets were primitive. Bernays's theory, with modifications byKurt Gödel, is known asvon Neumann–Bernays–Gödel set theory.
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