Kurt Gödel

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Kurt Gödel

Kurt Gödel (April 28, 1906 – January 14, 1978) was one of the most significant logicians of all time, whose work had an immense impact on 20th century philosophy, logic, and mathematics.

He is best known for his two incompleteness theorems, which he published in 1931 at 25 years of age, a year after finishing his doctorate at the University of Vienna. The theorems and the groundbreaking techniques they employed forced a rethinking of the foundations of mathematics, and initiated the field of metamathematics. The theorems showed that no consistent axiomatic system was capable of capturing all the truths of arithmetic, and this result effectively put an end to nearly a half-century of efforts by mathematicians and logicians, beginning with Gottlob Frege, through Bertrand Russell, Alfred North Whitehead, and David Hilbert, to explain the foundations of mathematics solely in the terms of logic and set theory. Gödel went on in his mathematical work to establish important theorems in set theory and to clarify the connections between classical logic, intuitionistic logic, and modal logic.

Gödel also made direct contributions to philosophy, and was influenced in his thinking by the writings of Immanuel Kant, Edmund Husserl, and, perhaps most strongly, Gottfried Leibniz. He defended mathematical realism and, though he did not make his religious views public, was critical of materialism and described himself as a philosophical theist. Gödel recorded an unpublished revision of the Ontological Argument for the existence of God that makes use of modern advances in modal logic, and which has spawned much discussion among logicians and philosophers since it's publication after his death.

Life of Gödel

Kurt Friedrich Gödel was born April 28, 1906, in what was then Brünn, Austria-Hungary (now Brno, Czech Republic), to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh), a well-educated homemaker. At the time of his birth the town had a slight German-speaking majority, and Gödel grew up speaking German. His father was only formally Catholic, while his mother was raised in a Protestant household, and in her childhood attended church regularly. She did not practice her faith as an adult, however, so that Kurt and his older brother Rudolf were not raised religious, though they studied religion in primary and secondary school. Rudolf remained an agnostic his entire life, but Kurt became interested in theology early on and developed unorthodox religious beliefs. In his later private writings he described himself as "theistic rather than pantheistic, following Leibniz rather than Spinoza."

Growing up Kurt was very attached to his mother, to the point that he was known to cry when she would leave the house. His relationship with his father, who was often preoccupied with his work, was less close, though not forbidding. In his family, young Kurt was known as Der Herr Warum ("Mr. Why") because of his insatiable curiosity. At the age of six or seven, Kurt suffered from rheumatic fever; he completely recovered, but after reading about possible complications in a medical book at age eight, he became increasingly preoccupied with his health. He developed hypochondria, and for the rest of his life remained convinced that his heart had suffered permanent damage.

Kurt attended German language primary and secondary school in Brünn and completed them with honors in 1923. Although he had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother left to go to medical school at the University of Vienna. By the time he joined his brother in Vienna and entered the University at age 18, he had already mastered university-level mathematics. Although he initially intended to study theoretical physics at university, Kurt chose to pursue mathematics after being impressed by the lectures of the mathematician P. Furtwängler. He studied primarily under Hans Hahn, who was at the time an active participant in the Vienna Circle, a group of logical positivists organized around another of Gödel's teachers, Moritz Schlick. Gödel attended their meetings frequently, but remained critical of positivist doctrines.

Attending a lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published "Principles of Theoretical Logic," an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system? This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hahn's supervision. In it he established the completeness of the first-order predicate calculus (a result known as Gödel's Completeness Theorem). He was awarded the doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

In 1931, Gödel published his famous incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme." In that article, he proved that any consistent axiomatic system that is powerful enough to describe arithmetic on the natural numbers must be incomplete (i.e. there are arithmetic truths that cannot be proved from those axioms). Among the things that cannot be proved within an axiomatic system is the consistency of the system itself. These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Hilbert's formalism, to find a set of axioms sufficient for all mathematics. They also imply that it is impossible to program a computer to discover all mathematical truths.

Gödel earned his habilitation at the Universtiy of Vienna in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. Hitler's 1933 ascension in Germany had little effect on Gödel in Vienna, as he took little interest in politics. From 1933-4, Gödel visited the United States for the first time to give a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. During this year, Gödel also developed the ideas of computability and recursive functions and delivered a lecture on general recursive functions and the concept of truth. He also met Albert Einstein for the first time, and the two became lifelong friends.

Following his return to Europe, Gödel turned his attention to set theory, and in 1935 proved the consistency of the Axiom of Choice with the other axioms of Zermelo-Fraenkel set theory. During this period he also published work on intuitionist and modal logic. In 1936 Gödel suffered a case of severe depression, likely triggered by the murder of his former teacher, Moritz Schlick, by a student. He was able to return to teaching at the University of Vienna in 1937 and at this time proved the consistency of Cantor's Continuum Hypothesis with the axioms of Zermelo-Fraenkel set theory.

He married Adele Nimbursky (née Porkert), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he. They remained happily married for the rest of their lives and had no children.

After Austria had become a part of Nazi Germany in 1938, Gödel's former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him and he had difficulty finding an academic position. His predicament precipitated when he was found fit for military service and was now at risk of being conscripted into the German army. In 1940, soon after World War II started, Kurt and Adele emigrated to the USA. Kurt took up a position at the IAS, where he eventually was granted a full professorship and remained until he retired in 1976.

The 1940s proved to be a fruitful decade for Gödel. In 1940, he published 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. In that work he introduced the notion of a constructible universe, a model of set theory in which the only sets which exist are those that can be constructed from simpler sets. Gödel showed that both the Axiom of Choice and the Continuum Hypothesis are true in the constructible universe, and therefore must be consistent. After 1942, he turned his attention away from mathematics toward philosophy and physics. In the late 1940s, he demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. In philosophy, Gödel studied in detail the works of Leibniz, whom he came to admire, and to a lesser extent, those of Kant and Husserl. Late in his career he circulated among his friends an elaboration of Leibniz's Ontological Proof of God's existence. This is now known as Gödel's Ontological Proof.

Einstein and Gödel had a legendary friendship, shared in the walks they took together to and from the IAS. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that "his own work no longer meant much, that he came to the Institute merely…to have the privilege of walking home with Gödel."

He was awarded (with Julian Schwinger) the first Albert Einstein Award, in 1951, and was also awarded the National Medal of Science in 1974.

Gödel battled psychological disorders for much of his life. Adele took close care of him and placated his many eccentricities, which included dressing in winter clothing in the summertime and leaving open all the windows in their home out of his fear of being assassinated. He also feared being poisoned, and for this reason would only eat his wife's cooking. When Adele became incapacitated due to illness in 1977, Gödel refused to eat any food at all and thus died of malnutrition in Princeton Hospital on January 14, 1978.

In 1987, the Kurt Gödel Society, an international organization for the promotion of research in logic, philosophy, and the history of mathematics, was founded in his honor.

Bibliography

Primary Sources

Collected Works:

• Gödel, Kurt, 1986, Collected Works. I: Publications 1929–1936. S. Feferman, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford: Oxford University Press.
• Gödel, Kurt, 1990, Collected Works. II: Publications 1938–1974. S. Feferman, J. Dawson, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford: Oxford University Press.
• Gödel, Kurt, 1995, Collected Works. III: Unpublished essays and lectures. S. Feferman, J. Dawson, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford: Oxford University Press.
• Gödel, Kurt, 2003a, Collected Works. IV: Correspondence A-G. S. Feferman, J. Dawson, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford: Oxford University Press.
• Gödel, Kurt, 2003b, Collected Works. V: Correspondence H-Z. S. Feferman, J. Dawson, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.), Oxford: Oxford University Press.

Important Publications:

• 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," Monatshefte für Mathematik und Physik 38: 173-98.
• 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
• 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515-25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470-85.

In English translation:

• Kurt Godel, 1992. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
• Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
  • 1930. "The completeness of the axioms of the functional calculus of logic," 582-91.
  • 1930. "Some metamathematical results on completeness and consistency," 595-96. Abstract to (1931).
  • 1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596-616.
  • 1931a. "On completeness and consistency," 616-17.

Links and references

Secondary Sources

• Dawson, John W., 1997. Logical dilemmas: The life and work of Kurt Gödel. Wellesley MA: A K Peters.
• Depauli-Schimanovich, Werner, and Casti, John L., 19nn. Gödel: A life of logic. Perseus.
• Franzén, Torkel, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
• Goldstein, Rebecca, 2005. Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries). W. W. Norton.
• Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
• Hintikka, Jaako, 2000. On Gödel. Wadsworth.
• Hofstadter, Douglas, 1980. Gödel, Escher, Bach. Vintage.
• Kleene, Stephen, 1967. Mathematical Logic. Dover paperback reprint ca. 2001.
• Kreisl, Georg, 1980. "Kurt Gödel". Biographical Memoirs of Fellows of the Royal Society, Vol. 26. (Nov., 1980), pp. 148-224.
• Nagel, Ernest and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
• Smullyan, Raymond, 1992. Godel's Incompleteness Theorems. Oxford University Press.
• Wang, Hao, 1987. Reflections on Kurt Gödel. MIT Press.
• Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
• Yourgrau, Palle, 2004. A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books.

See also

• Gödel dust, an exact solution of the Einstein field equation
• Gödel Prize - named after Kurt Gödel
• Gödel programming language - named after Kurt Gödel
• Gödel, Escher, Bach
• Gödel's Slingshot
• List of Austrian scientists

External links

• John J. O'Connor and Edmund F. Robertson. Kurt Gödel at the MacTutor archive
• Kurt Gödel at the Mathematics Genealogy Project
• Time Bandits - an article about the relationship between Gödel and Einstein by Jim Holt
• International Journal of Computers, Communications & Control, ISSN 1841-9844 (online), ISSN 1841-9836 (print), Vol. I (2006), No.1, pp. 69-71 / "Logic will never be the same again"- Kurt Gödel Centenary, by Gabriel Ciobanu
• Eric W. Weisstein, Gödel, Kurt (1906-1978) at ScienceWorld.
• "Gödel and the limits of logic" by John W Dawson Jr. (June 2006)
• Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue

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