## Quick Reference

(1906–1978)

Austrian-born US logician and mathematician, who discovered one of the most important mathematical results – the undecidability of mathematics.

Born in Brno (now in the Czech Republic), the son of a businessman, Gödel was educated at the University of Vienna, where he gained his PhD in 1930 and joined the faculty. He left Austria in 1938 and emigrated to the USA, where he joined the staff of the Institute for Advanced Studies at Princeton, becoming professor of mathematics from 1953 to 1976. Gödel became an American citizen in 1948. An extremely cautious and distrustful man, he became intensely hostile to Austria and consistently rejected the many honours they wished to bestow on him in later life.

A previous generation of logicians, especially Gottlob Frege and Bertrand Russell, had sought to derive the whole of mathematics from exclusively logical principles and concepts. While they and their followers had achieved considerable success, it remained a limited success. It is possible to derive parts of mathematics from logic but no one had yet proved that all mathematical truths could be so derived. In 1931 the twenty-five-year-old Gödel published his Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (translated as On Formally Undecidable Propositions of Principia Mathematica and Related Systems, 1962). Gödel demonstrated that in any system, such as Russell's Principia Mathematica, that was consistent and rich enough to contain the laws of simple arithmetic, there would always be true propositions of the system that could never be proved or disproved within the system. They were undecidable. No tinkering with the system, no additions, and no new axioms could overcome this startling defect. There was therefore no way of completing the Frege–Russell programme.

In 1938 Gödel published the solution to a second major mathematical problem. Georg Cantor, towards the end of the nineteenth century, had claimed that there is no set greater than the natural numbers but smaller than the set of real numbers. Known as the continuum hypothesis, it resisted all attempts at proof or disproof until 1938. Gödel then succeeded in showing that the hypothesis was consistent with the axioms, including the axiom of choice, of set theory. Gödel's other important results include the first proof of the completeness of first-order logic in 1930 (his doctoral thesis) and, in 1933, a new axiomatization of modal logic together with a suggestive analysis of the intuitionist logic of Luitzen Brouwer. In the field of cosmology Gödel proposed a number of models of the universe including, in 1949, one that satisfied the equations of general relativity theory without incorporating Mach's postulate.

On his death Gödel left some eighty scientific notebooks running to 5000 pages written in an old-fashioned German shorthand. Although they have yet to be fully analysed it has been reported that they deal at some length with topics as diverse as theology and demonology.

*Subjects:*
Philosophy.