Paul Bernays | Biography

Paul Bernays secured his reputation with a classic treatise on mathematical logic, the Foundations of Mathematics, and through his refinement and consolidation of set theory into the von Neumann-Bernays system. Bernays was a platonic mathematician--one who thought of the world of mathematics as separate from the world of material reality. Although Bernays's concept of mathematics as a mental product meant that no system could be designated as right or wrong, he believed that there were truths within the mathematical realm that allowed for a system to remain consistent and logical within itself.

Paul Isaac Bernays was born in London on October 17, 1888, to Julius Bernays, a Swiss businessman from a prominent Jewish family, and Sara Bernays. Shortly after his birth, the family moved to Paris and then to Berlin, where Bernays studied from 1895 to 1907. While studying engineering at the Technische Hochschule (Technical High School) in Charlottenburg, Germany, he developed an interest in pure mathematics. This led him to transfer to the University of Berlin where he studied for four semesters under a distinguished faculty that included philosopher Ernst Cassirer and physicist Max Planck. Bernays then attended the University of Göttingen where physicist Max Born and mathematician David Hilbert were among his professors. In 1912 Bernays received his doctorate degree from Göttingen under Hilbert.

In 1912 Bernays completed his postdoctoral thesis on modular elliptic functions at the University of Zurich in Switzerland under the German mathematician Ernst Zermelo. Bernays remained at Zurich until 1917 when he was invited by Hilbert to return to Göttingen to assist with a program on the foundations of mathematics. Bernays completed a second postdoctoral thesis at Göttingen in 1918 on propositional logic. In addition to serving as assistant to Hilbert, he gave lectures at Göttingen until the Nazi party's rise to power in 1933 when Bernays's right to lecture was withdrawn because of his Jewish background.

Bernays escaped to Zurich in 1934, eventually teaching at the Eidgenossische Technische Hochschule. In 1935 and 1936 he participated in the Institute for Advanced Study at Princeton University in New Jersey. Bernays published the first volume of his work on mathematical logic, Foundations of Mathematics, in 1934; the second volume was published in 1939. Research for this work was a collaborative effort between Bernays and Hilbert, but Bernays wrote both volumes singlehandedly. In this book Bernays and Hilbert created the mathematical discipline of proof theory, in which the correctness of a mathematical statement or theorem is demonstrated in terms of accepted axioms. E. Specker, a colleague of Bernays at the Eidgenossische Technische Hochschule, remarked in Logic Colloquium '78 that the Foundations of Mathematics is unique because "it does not reduce mathematics to logic, or logic to mathematics--both are developed at the same time."

Over the years Bernays published a series of articles in the Journal of Symbolic Logicthat was published collectively as Axiomatic Set Theory in 1958. Axiomatic set theory applies proof theory to set theory, the study of the properties and relationships of sets. Thus, axiomatic set theory involves the presentation of set theory in terms of fundamental axioms and logical rules of inference, rather than as a formalization of tabulated or intuitive knowledge. Classical set theory was largely established by Zermelo at the turn of the century and improved by the German mathematician Abraham Fraenkel in the 1920s. The Zermelo-Fraenkel (ZF) system was defined exclusively in terms of sets, but it could not address transfinite sets (for example, the set consisting of all possible sets). In the late 1920s the axioms of Hungarian mathematician John von Neumann accomplished many tasks previously left unsolved by the Zermelo-Fraenkel system. However, von Neumann's system was expressed in symbolic logic and was defined in terms of function rather than set, and it was less practical in both pure and applied mathematics.

Bernays's contribution to set theory both improved and simplified von Neumann's system. Bernays introduced a distinction between "sets" and "classes" to set theory. He did not view "classes" as mathematical objects in the normal sense. As G. H. Muller characterizes Bernays's distinction in Mathematical Intelligencer, a set is a collection of elements or members, a "multitude forming a proper thing." A class is a collection of objects that can be manipulated or extended, a "predicate regarded only with respect to its extension." For each set there was a corresponding class, but for each class there need not be a corresponding set. This idea created two axiomatic systems, one for sets and one for classes. The sets in the von Neumann-Bernays system operate similarly to those in the Zermelo-Fraenkel system, and thus a new system was created to allow for the construction of classes.

After World War II Bernays became Extraordinary Professor at the Eidgenossische Technische Hochschule. He also served as visiting professor at the University of Pennsylvania and at Princeton, where he was again a member of the Institute for Advanced Study in 1959-60. Bernays served as the president of the International Academy of the Philosophy of Science, as honorary chair of the German Society for Mathematical Logic and Foundation Research in the Exact Sciences, and as a corresponding member of the Academy of Science of Brussels and of Norway. He also served on the editorial boards of several journals, including Dialectica, Journal of Symbolic Logic, and Archiv fur mathematische Logik und Grundlangenforschung. Bernays received an honorary doctorate from the University of Munich in 1976 for his contributions to proof and set theory. Although he remained based in Zurich until his death from heart disease in 1977, Göttingen was always more of a home for Bernays. Bernays, who never married, lived most of his life with his mother and two sisters.