Paul Bernays Photos:

Paul Bernays
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Paul Bernays
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Paul Bernays
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Paul Bernays
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Paul Bernays Basic Informations:

Biography
2> Bernays spent his childhood in Berlin. Bernays 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 function theory 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 1919, that university awarded him a second Habilitation, for a thesis on the axiomatics of the propositional calculus of Principia Mathematica. In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen. 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 Institute for Advanced Study in Princeton, USA, and the University of Pennsylvania. [edit]

Tags:Zurich,Mathematics,University Of Berlin,Edmund Landau,Issai Schur,David Hilbert,Ferdinand Georg Frobenius,Friedrich Schottky,Alois Riehl,Carl Stumpf,Ernst Cassirer,Max Planck,University Of Göttingen,Hermann Weyl,Felix Klein,Max Born,Leonard Nelson,Binary Quadratic Forms,University Of Zurich,Habilitation,Function Theory,Picard's Theorem,Ernst Zermelo,George Pólya,Propositional Calculus,Principia Mathematica,Gerhard Gentzen,Switzerland,Eth,Institute For Advanced Study,University Of Pennsylvania,Function,Philosophy,Axiom,Theorem,Theory,Proposition,Gentzen,Hilbert,
Mathematical work
2> 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 notion of function as primitive; Bernays recast Von Neumann's theory so that sets and proper classes were primitive. Bernays's theory, with some modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory. [edit]

Tags:Axiomatic Set Theory,Journal Of Symbolic Logic,John Von Neumann,Sets,Proper Classes,Kurt Gödel,Von Neumann–bernays–gödel Set Theory,Set Theory,Set,Class,Symbol,Gödel,
Publications
2> Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, 40, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04134-4, JFM 60.0017.02, MR0237246, http://www.ags.uni-sb.de/~cp/p/hilbertbernays/goal.htm  Hilbert, David; Bernays, Paul (1939), Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, 50, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05110-7, JFM 65.0021.02, MR0272596, http://www.ags.uni-sb.de/~cp/p/hilbertbernays/goal.htm  Bernays, Paul (1958), Axiomatic set theory, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-486-66637-2, MR0106178, http://books.google.com/books?isbn=0486666379  Bernays, Paul (1976) (in German), Abhandlungen zur Philosophie der Mathematik, Darmstadt: Wissenschaftliche Buchgesellschaft, ISBN 978-3-534-06706-0, MR0444417  [edit]

Tags:Springer-verlag,978-3-540-04134-4,978-3-540-05110-7,978-0-486-66637-2,978-3-534-06706-0,Foundations Of Mathematics,
References
2> 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 Mathematic. Müller, Gert H., ed. (1976), Sets and classes. On the work by Paul Bernays, Studies in Logic and the Foundations of Mathematics, 84, Amsterdam: North-Holland, ISBN 978-0-444-10907-1, MR0414355, http://books.google.com/books?id=7vDATn64DOUC  Lauener, Henri (1978), "Paul Bernays (1888--1977)", Zeitschrift für Allgemeine Wissenschaftstheorie 9 (1): 13–20, doi:10.1007/BF01801939, ISSN 0044-2216, MR546580  Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Mathematik", in Grattan-Guinness, Ivor, Landmark writings in western mathematics 1640--1940, Elsevier B. V., Amsterdam, pp. 981–99, doi:10.1016/B978-044450871-3/50158-3, ISBN 978-0-444-50871-3, MR2169816  Bernays and Set Theory, Akihiro Kanamori, The Bulletin of Symbolic Logic, Vol. 15, No. 1 (Mar., 2009), pp. 43-69. [edit]

Tags:Mathematical Logic,978-0-444-10907-1,978-0-444-50871-3,Akihiro Kanamori,
External links
2> Hilbert Bernays Project MacTutor biography Paul Bernays, Paul Bernays: A Short Biography (1976) Paul Bernays at the Mathematics Genealogy Project. v d e Logic   Overview Academic areas Argumentation theory Axiology Critical thinking Computability theory Formal semantics History of logic Informal logic Logic in computer science Mathematical logic Mathematics Metalogic Metamathematics Model theory Philosophical logic Philosophy Philosophy of logic Philosophy of mathematics Proof theory Set theory Foundational concepts Abduction Analytic truth Antinomy A priori Deduction Definition Description Entailment Induction Inference Logical consequence Logical form Logical implication Logical truth Name Necessity Meaning Paradox Possible world Presupposition Probability Reason Reasoning Reference Semantics Statement Strict implication Substitution Syntax Truth Truth value Validity   Philosophical logic Critical thinking and Informal logic Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness Theories of deduction Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism   Metalogic and metamathematics Cantor's theorem Church's theorem Church's thesis Consistency Effective method Foundations of mathematics Gödel's completeness theorem Gödel's incompleteness theorems Soundness Completeness Decidability Interpretation Löwenheim–Skolem theorem Metatheorem Satisfiability Independence Type–token distinction Use–mention distinction   Mathematical logic General Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory Traditional logic Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram Propositional calculus and Boolean logic Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Predicate First-order Quantifiers Predicate Second-order Monadic predicate calculus Set theory Set Empty set Enumeration Extensionality Finite set Function Subset Power set Countable set Recursive set Domain Range Ordered pair Uncountable set Model theory Model Interpretation Non-standard model Finite model theory Truth value Validity Proof theory Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax Computability theory Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function   Non-classical logic Modal logic Alethic Axiologic Deontic Doxastic Epistemic Temporal Intuitionism Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory Fuzzy logic Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations Substructural logic Structural rule Relevance logic Linear logic Paraconsistent logic Dialetheism Description logic Ontology Ontology language   Logicians Anderson Aristotle Averroes Avicenna Bain Barwise Bernays Boole Boolos Cantor Carnap Church Chrysippus Curry De Morgan Frege Geach Gentzen Gödel Hilbert Kleene Kripke Leibniz Löwenheim Peano Peirce Putnam Quine Russell Schröder Scotus Skolem Smullyan Tarski Turing Whitehead William of Ockham Wittgenstein Zermelo   Lists Topics Outline of logic Index of logic articles Mathematical logic Boolean algebra Set theory Other Logicians Rules of inference Paradoxes Fallacies Logic symbols Portal Category Outline WikiProject Talk changes Persondata Name Bernays, Paul Alternative names Short description Date of birth 17 October 1888 Place of birth London Date of death 18 September 1977 Place of death Zurich Retrieved from "http://en.wikipedia.org/w/index.php?title=Paul_Bernays&oldid=473954772" Categories: Jewish scientistsSwiss mathematicians1888 births1977 deathsMathematical logiciansSet theoristsPhilosophers of mathematicsHidden categories: Persondata templates without short description parameter Personal tools Log in / create account Namespaces Article Talk Variants Views Read Edit View history Actions Search Navigation Main page Contents Featured content Current events Random article Donate to Wikipedia Interaction Help About Wikipedia Community portal Recent changes Contact Wikipedia Toolbox What links here Related changes Upload file Special pages Permanent link Cite this page Print/export Create a bookDownload as PDFPrintable version Languages Česky Deutsch Français Italiano Nederlands 日本語 Piemontèis Polski Русский Slovenčina Slovenščina This page was last modified on 30 January 2012 at 00:51. 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Tags:Swiss,Philosophy Of Mathematics,London,Mathematics Genealogy Project,Argumentation Theory,Axiology,Critical Thinking,Computability Theory,Formal Semantics,History Of Logic,Informal Logic,Logic In Computer Science,Metalogic,Metamathematics,Model Theory,Philosophical Logic,Philosophy Of Logic,Proof Theory,Abduction,Analytic Truth,Antinomy,A Priori,Deduction,Definition,Description,Entailment,Induction,Inference,Logical Consequence,Logical Form,Logical Implication,Logical Truth,Name,Necessity,Meaning,Paradox,Possible World,Presupposition,Probability,Reason,Reasoning,Reference,Semantics,Statement,Strict Implication,Substitution,Syntax,Truth,Truth Value,Analysis,Ambiguity,Argument,Belief,Bias,Credibility,Evidence,Explanation,Explanatory Power,Fact,Fallacy,Inquiry,Opinion,Parsimony,Premise,Propaganda,Prudence,Relevance,Rhetoric,Rigor,Vagueness,Theories Of Deduction,Constructivism,Dialetheism,Fictionalism,Finitism,Formalism,Intuitionism,Logical Atomism,Logicism,Nominalism,Platonic Realism,Pragmatism,Cantor's Theorem,Church's Theorem,Church's Thesis,Consistency,Effective Method,Gödel's Completeness Theorem,Gödel's Incompleteness Theorems,Soundness,Completeness,Decidability,Interpretation,Löwenheim–skolem Theorem,Metatheorem,Satisfiability,Independence,Type–token Distinction,Formal Language,

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