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*{{Citation | last1=Bernays | first1=Paul | title=Axiomatic set theory | url=https://books.google.com/books?isbn=0486666379 | publisher=North-Holland | location=Amsterdam | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-486-66637-2 | year=1958 | mr=0106178}} |
*{{Citation | last1=Bernays | first1=Paul | title=Axiomatic set theory | url=https://books.google.com/books?isbn=0486666379 | publisher=North-Holland | location=Amsterdam | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-486-66637-2 | year=1958 | mr=0106178}} |
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*{{Citation | last1=Bernays | first1=Paul | title=Abhandlungen zur Philosophie der Mathematik | publisher=Wissenschaftliche Buchgesellschaft | location=Darmstadt | language=de | isbn=978-3-534-06706-0 | year=1976 | mr=0444417}} |
*{{Citation | last1=Bernays | first1=Paul | title=Abhandlungen zur Philosophie der Mathematik | publisher=Wissenschaftliche Buchgesellschaft | location=Darmstadt | language=de | isbn=978-3-534-06706-0 | year=1976 | mr=0444417}} |
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*{{Citation | last1=Bernays | first1=Paul | last2=Schonfinkel | first2=Moses | title=''Zum Entscheidungsproblem der mathematischen Logik'' | date=1928 | url=https://doi.org/10.1007/BF01459101 | journal=Mathematische Annalen | number=99 | pages=342-372}} |
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==Notes== |
==Notes== |
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Revision as of 16:12, 3 March 2021
Paul Bernays | |
|---|---|
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 |
Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a
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 German-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 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
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 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
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
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
- MR 0272596, archived from the originalon 2011-05-17
- Bernays, Paul (1958), Axiomatic set theory, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, MR 0106178
- Bernays, Paul (1976), Abhandlungen zur Philosophie der Mathematik (in German), Darmstadt: Wissenschaftliche Buchgesellschaft, MR 0444417
- Bernays, Paul; Schonfinkel, Moses (1928), "Zum Entscheidungsproblem der mathematischen Logik", Mathematische Annalen (99): 342–372
Notes
References
- .
- 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.
- S2CID 147959212
- Müller, Gert H., ed. (1976), Sets and classes. On the work by Paul Bernays, Studies in Logic and the Foundations of Mathematics, vol. 84, Amsterdam: North-Holland, MR 0414355
- Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Mathematik", in MR 2169816
External links
Wikiquote has quotations related to Paul Bernays.
Wikimedia Commons has media related to Paul Bernays (mathematician).
- Hilbert Bernays Project
- O'Connor, John J.; Robertson, Edmund F., "Paul Bernays", MacTutor History of Mathematics Archive, University of St Andrews
- Paul Bernays: A Short Biography (1976)
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