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⎯ effectively destroying the conclusions Hilbert had intuitively begun from when he originated his program. Gödel’s work is generally taken to show that Hilbert’s Program cannot be carried out. The latter has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. Gödel first announced his Incompleteness Theorem in 1930 to Carnap in Café Reichsrat in Vienna, a habitat of the Vienna Circle. The work on incompleteness was published early in 1931, and defended as a Habilitationschrift at the University of Vienna in 1932. The title of
happened he delivered lectures in Vienna only intermittently during the following years. In 1933–1934 his unsalaried position in Vienna was supplemented by income from visiting positions in the United States of America. Gödel’s first visit was to the Institute for Advanced Study in Princeton where he gave lectures on incompleteness results. The Institute had been formally established in 1930, with Albert Einstein and Oswald Veblen appointed its first professors. At that time Gödel apparently began to work on problems in axiomatic set theory. In the following years he felt rather depressed and lonely, particularly at Princeton. He had several nervous attacks of mental depression and exhaustion. In 1936 he spent almost the whole year in a sanatorium on account of mental illness. On September 20 th , 1938 Kurt Gödel and Adele Nimbursky finally got married and their marriage proved to be a warm and enduring one. Adele was a source of constant support for Kurt in the difficult times ahead. In March, 1939, after the occupation of Austria by Hitler, Gödel’s unpaid position of
examination, and much to his surprise found fit for the duty. On 27 th of November he wrote a letter to Osvald Veblen in Princeton asking for help. Somehow German exit permits were arranged, and Kurt and Adele managed to leave Vienna in January 1940. They travelled by train through Eastern Europe, then via the Trans-Siberian Railway across Russia and Manchuria to Yokohama where they took a ship to San Francisco. In March 1940 they finally came by train to Princeton. Gödel was never to return to Europe. So it was in 1940 Gödel was made an Ordinary Member of the Institute for Advanced Study, and he and his wife settled in Princeton. Among his closest friends there were Albert Einstein and Oskar Morgenstern; the latter was another ex-Viennese, an economist who emigrated from Austria in 1938. At the Institute Gödel had no formal duties and was free to pursue his research and studies. In the springtime 1941 he gave a series of lectures, and on April 15 th he gave a lecture at the Yale University on “In which sense is intuitionistic logic constructive”? He continued his work in mathematical logic; in particular he made efforts to prove the independence of the axiom of choice and the continuum hypothesis. He partially succeeded on this problem. His masterpiece Consistency of the axiom of choice and of the
mathematics. In this he proved that if an axiomatic system of set theory of the type proposed by Russell and Whitehead in Principia Mathematica is consistent, then it will remain so when the axiom of choice and the generalized continuum-hypothesis are added to the system. This did not prove that these axioms were independent of the other axioms of set theory, but when this was finally established by Cohen in 1963 he built on these ideas of Gödel. Another achievement early in this period (published only in 1958) was a new constructive interpretation of arithmetic that proved its consistency, but via methods going beyond finitary means in Hilbert’s sense.
4 From 1943 on, Gödel devoted himself almost entirely to philosophy, first to the philosophy of mathematics and then to general philosophy and metaphysics. Gödel is noted for his support of mathematical realism and Platonism 7 . In this general direction he joins such noted mathematicians and logicians as Cantor, Frege, Zermelo and Church, and the implicit working conceptions of most practicing mathematicians. An expository paper on Cantor’s continuum problem in 1947 brought out Gödel’s Platonist views quite markedly in the context of set theory. As for general philosophy, Gödel continued his long-pursued study of Kant and Leibniz. Beginning in 1951, Gödel received many honours. Particularly noteworthy was his sharing of the first Einstein Award (with Julian Schwinger) in 1951. John von Neumann, one of the first to understand Gödel’s incompleteness results, compared Gödel’s contribution in the field of logic with the work of Aristotle; von Neumann died on January 8 th , 1957, Einstein died on April 18 th 1955. This was Gödel’s best friend and regular companion on their walk home from the Institute. Einstein and Gödel seemed very different in almost every personal way
⎯ Einstein full of laughter and common sense and Gödel solemn, serious and solitary ⎯ but they shared a fundamental feature: both went directly and rigorously to the fundamental questions at the very heart of things. From 1959 on, in addition to Gödel’s primary interest in logic, philosophy and, to a lesser extent, mathematics and physics, he was interested in phenomenology. Gödel’s notes are preserved in his Nachlass (inheritance), and many of them are concerned with the phenomenology of Edmund Husserl. These notes are unexpectedly wide-ranging, revealing interests in history and theology. A logical attempt at the proof of God’s existence is found here. The ‘proof’ was written in 1970 and it reminds a sacral text: it has no introduction, no motivation, and no explication of the modal system used; just axioms, definitions, and the proof. It is an ontological proof, based on Anselm principle, but Gödel does not refer to St. Anselm, or to other philosophers and theologians. In the last fifteen years of his life, Gödel was busy with Institute business 8 and his own philosophical studies; during this time he returned to logic only occasionally, devoting some efforts to revision and translation of his old papers. He translated and revised his 1958 paper 9 ,
published. On April 21–23, 1966, a 60 th birthday symposium was organised at Ohio State University; but the invitation to attend was declined by Gödel. On July 23 rd Marianne (Handschuh) Gödel (mother) died in Vienna, and in August Gödel refused an honorary membership in Austrian Academy of Sciences. In fact, Gödel’s health was poor from the late 1960s on. His wife Adele was not able to help him as before, being herself partially incapacitated, and for a time moved to a nursing home. Gödel’s depressions returned accompanied by paranoia; he developed fears about being poisoned and would not eat. Kurt Gödel died in Princeton Hospital on January 14 th , 1978 of “malnutrition and inanition caused by personality disturbance”. Adele survived him by three years. Kurt and Adele had no children, leaving Kurt’s brother Rudolf as the sole surviving member of the Gödel family. In 1987 an international Kurt Gödel Society was established in Vienna, the first president of which was Gödel’s student and friend Hao Wang. In 1992 the Society of Kurt Gödel was
7 See, e.g., Köhler (2002a) 8 He was made a Permanent Member of the Institute in 1946, and promoted to Professor in 1953 9 The last published paper apart from revisions of earlier works: ‘Über eine bisehr noch nicht benützte Erweiterung des finiten Standardpunktes”, Dialectica 1958. Yüklə 246,82 Kb. Dostları ilə paylaş:
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