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Kurt Gödel
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Kurt Gödel (April 28, 1906 Brno – January 14, 1978 Princeton, New Jersey) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna.
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Kurt Gödel (April 28, 1906 Brno – January 14, 1978 Princeton, New Jersey) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
Life
Childhood
Kurt Friedrich Gödel was born April 28, 1906, in Brno , Moravia, Austria-Hungary (now the Czech Republic) into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight German-speaking majority, and this was the language of his parents. The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".
Although he spoke very little Czech himself, Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I. He later told his biographer John W. Dawson that he felt like an "exiled Austrian in Czechoslovakia" ("ein Österreicher im Exil in der Tschechoslowakei") during this time. He chose to become an Austrian citizen at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.
In his family, young Kurt was known as Herr Warum ("Mr. Why") because of his insatiable curiosity.
According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.
Gödel attended the Evangelische Volksschule, a Lutheran school in Brno from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt 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 Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna (UV). During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel 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, Kurt also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe 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, Kurt became interested in mathematical logic.
Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course.
In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (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 Hans Hahn's supervision. In it, Gödel established the completeness of the first-order predicate calculus (this result is 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.
Working in Vienna
In 1931, Gödel published his famous incompleteness theorems in "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" (called in English "On formally undecidable propositions of Principia Mathematica and related systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or ZFC), that:
- If the system is consistent, it cannot be complete. (This is generally known as the incompleteness theorem.)
- The consistency of the axioms cannot be proved within the system.
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.
In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the fact that in a consistent system, provable statements are always true.
Thus there will always be at least one true but unprovable statement.
That is, for any humanly constructible set of axioms for arithmetic, there is a formula which obtains in arithmetic, but which is not provable in that system.
To make this precise, however, Gödel needed to solve several technical issues, such as encoding statements, proofs, and the very concept of provability into the natural numbers. He did this using a process known as Gödel numbering.
In his two-page paper "Zum intuitionistischen Aussagenkalkül" (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).
Gödel earned his habilitation at the UV in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians.
In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel.
He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanatarium for nervous diseases.
Visits to the USA
In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel 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 Gödel numbering.
In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, 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 traveling and the hard work had exhausted him and the next year he had to recover from a depression. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
He married Adele Nimbursky (née Porkert, 1899-1981), 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 had no children.
Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.
Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.
Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.
Princeton
After the Anschluss in 1938, Austria had become a part of Nazi Germany.
Germany abolished the title of Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him.
His predicament precipitated when he was found fit for military service and was now at risk of being conscripted into the German army. World War II started in September 1939.
In January 1940, Gödel and his wife left Europe. Due to the difficulty of an Atlantic crossing, they took the trans-Siberian railway and passed through Japan en route to the U.S.. Arriving in San Francisco, California on March 4, 1940, they crossed the U.S. by train so that Gödel could take up a position at the Institute for Advanced Study (IAS) in Princeton, New Jersey.
Gödel very quickly resumed his mathematical work. In 1940, 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. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo-Frankel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.
In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.
During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof.
Gödel became a permanent member of the IAS in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.
Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award, in 1951, and was also awarded the National Medal of Science, in 1974.
Death
In later life, Gödel suffered periods of mental instability and illness.
He had an obsessive fear of being poisoned; he wouldn't eat unless his wife, Adele, tasted his food for him.
Late in 1977, Adele was hospitalized for six months and could not taste Gödel's food anymore.
In her absence, he refused to eat, eventually starving himself to death.
He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.
Legacy
The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics.
Gödel's friendship with Einstein
Albert Einstein and Gödel had a legendary friendship, shared in the walks they took together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. 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".
On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.
Important publications
In German:
- 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," Monatshefte für Mathematik und Physik 38: 173-98.
- 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.
In English:
- 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.
- Kurt Godel, 2000. http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
- 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.
- Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
- Volume I: Publications 1929-1936 ISBN 0195039645,
- Volume II: Publications 1938-1974 ISBN 0195039726,
- Volume III: Unpublished Essays and Lectures ISBN 0195072553,
- Volume IV: Correspondence, A-G ISBN 0198500734.
- Volume V: Correspondence, H-Z ISBN 0198500750
See also
Further reading
- John L. Casti and Werner DePauli, 2000. Gödel: A Life of Logic, Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.
- John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. AK Peters, Ltd., 1996.
- Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
- Rebecca Goldstein, 2005. Incompleteness: The Proof and Paradox of Kurt Gödel. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.
- Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
- Jaakko Hintikka, 2000. On Gödel. Wadsworth.
- Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
- Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint ca. 2001.
- J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
- Ernst Nagel and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
- Procházka, Jirí, 2006, 2006, 2008, 2008. Kurt Gödel: 1906–1978: Genealogie. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl.
- Ed Regis, 1987. Who Got Einstein's Office? Addison-Wesley Publishing Company, Inc.
- Raymond Smullyan, 1992. Godel's Incompleteness Theorems. Oxford University Press.
- Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press.
- Wang, Hao. 1996. A Logical Journey: From Godel to Philosophy. 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.
External links
- Kennedy, Juliette. In Stanford Encyclopedia of Philosophy.
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- - an article about the relationship between Gödel and Einstein by Jim Holt
- - Authored by Gregory Chaitin
- by John W Dawson Jr. (June 2006)
- Kurt Gödel Centenary Issue
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- Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
- Google Video of a BBC documentary featuring Kurt Gödel and other revolutionary mathematical thinkers.
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