Paul Isaac Bernays ( ; ; 17 October 1888 – 18 September 1977) was a

Swiss Swiss most commonly refers to: * the adjectival form of Switzerland * Swiss people Swiss may also refer to: Places * Swiss, Missouri * Swiss, North Carolina * Swiss, West Virginia * Swiss, Wisconsin Other uses * Swiss Café, an old café located ...

mathematician who made significant contributions to

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

atic

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, and the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

. He was an assistant and close collaborator of

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

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. Bernays spent his childhood in Berlin, and attended the Köllnische Gymnasium, 1895–1907. At the

University of Berlin The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...

, he studied mathematics under

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...

, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under

Max Planck Max Karl Ernst Ludwig Planck (; ; 23 April 1858 – 4 October 1947) was a German Theoretical physics, theoretical physicist whose discovery of energy quantum, quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial con ...

. At the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...

, he studied mathematics under

, Edmund Landau,

Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

, and

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

; physics under Voigt and

Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...

; and philosophy under Leonard Nelson. In 1912, the

awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on the

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...

of binary quadratic forms. That same year, the

University of Zurich The University of Zurich (UZH, ) is a public university, public research university in Zurich, Switzerland. It is the largest university in Switzerland, with its 28,000 enrolled students. It was founded in 1833 from the existing colleges of the ...

awarded him

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...

for a thesis on

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

and Picard's theorem. The examiner was

Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel set theory, Z ...

. Bernays was Privatdozent at the University of Zurich, 1912–1917, where he came to know

George Pólya George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...

. His collected communications with

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

span many decades. Starting in 1917,

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 The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

of ''

Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...

''. 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 The Law for the Restoration of the Professional Civil Service (, shortened to ''Berufsbeamtengesetz''), also known as Civil Service Law, Civil Service Restoration Act, and Law to Re-establish the Civil Service, was enacted by the Nazi Party, Na ...

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 Switzerland, officially the Swiss Confederation, is a landlocked country located in west-central Europe. It is bordered by Italy to the south, France to the west, Germany to the north, and Austria and Liechtenstein to the east. Switzerland ...

, whose nationality he had inherited from his father, and where the

ETH Zurich ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ran ...

employed him on occasion. He also visited the

University of Pennsylvania The University of Pennsylvania (Penn or UPenn) is a Private university, private Ivy League research university in Philadelphia, Pennsylvania, United States. One of nine colonial colleges, it was chartered in 1755 through the efforts of f ...

and was a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

in 1935–36 and again in 1959–60.

Mathematical work

His habilitation thesis was written under the supervision of Hilbert himself, on the topic of the axiomatisation of propositional logic in Whitehead and Russell's

. It contains the first known proof of semantic completeness of propositional logic, which was reproved independently also by

Emil Post Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Govern ...

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 reference

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

is known as the Hilbert–Bernays paradox. In seven papers, published between 1937 and 1954 in the ''

Journal of Symbolic Logic The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zent ...

'' (republished in Müller 1976), Bernays set out an axiomatic set theory whose starting point was a related theory

John von Neumann John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, in ...

had set out in the 1920s. Von Neumann's theory took the notions of function and

argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...

as primitive. Bernays recast von Neumann's theory so that

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

es and sets were primitive. Bernays's theory, with modifications by

, is known as von Neumann–Bernays–Gödel set theory.

Publications

* * * * *

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''. * * *

External links

Hilbert Bernays Project
*
''Paul Bernays: A Short Biography'' (1976)
{{DEFAULTSORT:Bernays, Paul 1888 births 1977 deaths 20th-century Swiss philosophers Institute for Advanced Study visiting scholars Jewish philosophers Jewish scientists Mathematical logicians Philosophers of mathematics Set theorists Swiss Ashkenazi Jews Swiss mathematicians Swiss philosophers Academic staff of ETH Zurich