Kurt Gödel (1906-1978) was an Austrian-born U.S. mathematician whose most famous work was the Incompleteness Theorem that stated that any axiomatic system powerful enough to describe integer arithmetic could not be simultaneously complete and consistent. This means that if the system is self-consistent, then there will be propositions that can neither be proven nor disproven from within the axiomatic system.
Kurt Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic) as the son of the manager of a textile factory. In his family little Kurt was know as Der Herr Warum (Mr. Why). He attended German-language primary and secondary school in Brünn and completed them with honors in 1923. Although Kurt had first excelled in learning languages he later become more fond of history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to Medical School at the University of Vienna (UV). Already during his teens Kurt studied Gabelsberger shorthand, Goethe's theory of colors and criticisms of Isaac Newton, and the writings of Kant.
Studying in Vienna
At the age of 18 Kurt joined his brother Rudolf in Vienna and entered the UV. By that time he had already mastered university level mathematics. Although initially intending to study theoretical physics he also attended courses on mathematics and philosophy. During this time he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgrunde Der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Kurt then studied number theory but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to mathematical philosophy he became interested in mathematical logic.
While at UV Kurt met his future wife Adele Nimbursky (née Porkert). He started to publish papers on logic and attended a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems. In 1929 Gödel became an Austrian citizen and later that year he completed his doctoral dissertation under Hans Hahn's supervision. In this dissertation he established the completeness of the first-order predicate calculus (also known as Gödel's completeness theorem).
Working in Vienna
In 1930 a Dr. Philosophy had been granted to Gödel. He added a combinatorial version to his completeness result, which was published by the Vienna Academy of Sciences. In 1931 he published his famous Incompleteness Theorems in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. In this article he proved that for any axiomatic mathematical system that is powerful enough to describe the natural numbers it holds that:
- It cannot be both consistent and complete. (It is this theorem that is generally known as the Incompleteness Theorem.)
- If the system is consistent then the consistency of the axioms cannot be proved within the system.
These theorems ended a hundred years of attempts to establish a definitive set of axioms to put the whole of mathematics on an axiomatic basis such as in the Principia Mathematica and Hilbert's formalism. It also implies that a computer can never be programmed to answer all mathematical questions.
Gödel earned his Habilitation at the UV in 1932 and in 1933 he became a Privatdozent (unpaid lecturer) there. When in 1933 Hitler came to power this had little effect on Gödel's life in Vienna since he had little interest in politics. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a National Socialist student, Gödel was much affected and had his first nervous breakdown.
Visiting the USA
In this year he took his first trip to the USA, during which he met Albert Einstein. He delivered an address to the annual meeting of the American Mathematical Society. During this year he also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using the construction of the Goedel numbers.
In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton entitled On undecidable propositions of formal mathematical systems. Stephen Kleene who had just completed his Ph.D. at Princeton, took notes of these lectures which have been subsequently published.
Gödel would visit the IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he had to recover from a depression. He returned to teaching in 1937 and during this time he worked on the proof of consistency of the Continuum hypothesis. He married Adele on September 20, 1938. In the autumn of 1938 he visited again the IAS. After this he visited the USA once more in the spring of 1939 at the University of Notre Dame.
Working in Princeton
After the Anschluss in 1938 Austria had become a part of Nazi Germany. Since Germany had abolished the title of Privatdozent Gödel would now have to fear conscription into the Nazi army. In January 1940 he and his wife left Europe via the trans-Siberian railway and traveled via Russia and Japan to the USA. When they arrived in San Francisco on March 4, 1940, Kurt and Adele settled in Princeton, where he resumed his membership in the IAS. At the Institute, Gödel's interests turned to philosophy and physics. He studied the works of Gottfried Leibniz in detail and, to a lesser extent, those of Kant and Edmund Husserl. In the late 1940s he demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He also continued to work on logic and in the same year he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory which is a classic of modern mathematics. He became a permanent member of the IAS in 1946 and in 1948 he was naturalized as an U.S. citizen. He became a full professor at the institute in 1953 and an emiritus professor in 1976.
During his life Kurt Gödel received many prestigious awards such as the (shared) first Einstein Award in 1951 and the National Medal of Science in 1974. Towards the end of his life Gödel became convinced that he was being poisoned. To avoid being poisoned he refused to eat and thus starved himself to death. He died January 14, 1978, in Princeton, New Jersey, USA.
- Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931). (Available in English at http://www.ddc.net/ygg/etext/godel/)
- Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1941)