|Fields||Mathematics, mathematical logic, analytic philosophy, physics|
|Institutions||Institute for Advanced Study|
|Thesis||Über die Vollständigkeit des Logikkalküls (1929)|
|Doctoral advisor||Hans Hahn|
Kurt Friedrich Gödel (
Gödel's discoveries in the foundations of mathematics led to the proof of
Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
Early life and education
Gödel was born April 28, 1906, in Brünn,
Gödel automatically became a citizen of
In his family, the young Gödel was nicknamed 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. Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire life.
Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn 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 Gödel 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
Studies in Vienna
At the age of 18, Gödel joined his brother at the
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 Grundzüge der theoretischen Logik (Principles of Mathematical 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 problem became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral
Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement.
In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.
Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (called in English "
- If a (logical or axiomatic formal) omega-consistent, it cannot be syntactically complete.
- The consistency of axioms cannot be proved within their own system.
These theorems ended a half-century of attempts, beginning with the work of
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. Thus there will always be at least one true but unprovable statement. That is, for any
In his two-page paper Zum intuitionistischen Aussagenkalkül (1932) Gödel refuted the finite-valuedness of
Mid-1930s: further work and U.S. visits
Gödel earned his habilitation at Vienna 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 one of his former students, Johann Nelböck. This triggered "a severe nervous crisis" in Gödel. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.
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 was able to present 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
Gödel visited the IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he took a break to recover from a depressive episode. 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 went 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. Gödel's parents had opposed their relationship because she was a divorced dancer, six years older than he was.
Subsequently, he left for another visit to the United States, spending the autumn of 1938 at the IAS and publishing Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory,
Princeton, Einstein, U.S. citizenship
After the Anschluss on 12 March 1938, Austria had become a part of Nazi Germany. Germany abolished the title 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. The University of Vienna turned his application down.
His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the Trans-Siberian Railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the US by train to Princeton. There Gödel accepted a position at the Institute for Advanced Study (IAS), which he had previously visited during 1933–34.
Albert Einstein was also living at Princeton during this time. Gödel and Einstein developed a strong friendship, and were known to take long walks 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".
Gödel and his wife, Adele, spent the summer of 1942 in Blue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but 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.
On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his
Gödel became a permanent member of the Institute for Advanced Study at Princeton 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.
During his time at the institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving
He studied and admired the works of
Awards and honours
Gödel was awarded (with
Later life and death
Later in his life, Gödel suffered periods of
Gödel believed in an afterlife, saying, "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."
In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is
The Kurt Gödel Society, founded in 1987, is an international organization for the promotion of research in logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has held an annual Gödel Lecture each year since 1990. Gödel's Philosophical Notebooks Archived May 14, 2019, at the Wayback Machine are edited at the Kurt Gödel Research Centre Archived May 14, 2019, at the Wayback Machine which is situated at the Berlin-Brandenburg Academy of Sciences and Humanities in Germany.
Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from his Nachlass, and the final two include correspondence.
The Gödel Prize is given annually for an outstanding paper in theoretical computer science.
- 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Mathematik und Physik 37: 349–60.
- 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173–98.
- 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.
- 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.
- 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, Vol. 1, pp. 175–81.
In English translation:
- Kurt Gödel, 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 Gödel, 2000. 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.
- Philosophische Notizbücher / Philosophical Notebooks: De Gruyter: Berlin/München/Boston. Editor: Eva-Maria Engelen [de].
- Gödel machine
- Gödel fuzzy logic
- Gödel–Löb logic
- Gödel Prize
- Gödel's ontological proof
- Infinite-valued logic
- List of Austrian scientists
- List of pioneers in computer science
- Mathematical Platonism
- Original proof of Gödel's completeness theorem
- Primitive recursive functional
- Strange loop
- Tarski's undefinability theorem
- World Logic Day
- "Gödel". Merriam-Webster.com Dictionary.
- For instance, in their "Principia Mathematica " (Stanford Encyclopedia of Philosophy edition).
- Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. V.
- Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. IX.
- Dawson 1997, pp. 3–4.
- Dawson 1997, p. 12
- Procházka 2008, pp. 30–34.
- Dawson 1997, p. 15.
- Balaguer, Mark. "Kurt Godel". Britannica School High. Encyclopædia Britannica, Inc. Retrieved June 3, 2019.
- Kim, Alan (January 1, 2015). Zalta, Edward N. (ed.). Johann Friedrich Herbart (Winter 2015 ed.). Metaphysics Research Lab, Stanford University.
- Dawson 1997, p. 24.
- Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p. 181.
- Dawson 1997, pp. 110–12
- Hutchinson Encyclopedia (1988), p. 518
- Dawson, John W. Jr. "Kurt Gödel at Notre Dame" (PDF). p. 4.
the Mathematics department at the University of Notre Dame was host ... for a single semester in the spring of 1939 [to] Kurt Gödel
- "Kurt Gödel". Institute for Advanced Study. December 9, 2019.
- Goldstein 2005, p. 33
- Dawson 1997, pp. 179–80. The story of Gödel's citizenship hearing is repeated in many versions. Dawson's account is the most carefully researched, but was written before the rediscovery of Morgenstern's written account. Most other accounts appear to be based on Dawson, hearsay or speculation.
- Oskar Morgenstern (September 13, 1971). "History of the Naturalization of Kurt Gödel" (PDF). Retrieved April 16, 2019.
- "Kurt Gödel – Institute for Advanced Study". Retrieved December 1, 2015.
- "Das Genie & der Wahnsinn". Der Tagesspiegel (in German). January 13, 2008.
- "The President's National Medal of Science: Recipient Details | NSF – National Science Foundation". www.nsf.gov. Retrieved September 17, 2016.
- "APS Member History". search.amphilsoc.org. Retrieved January 28, 2021.
- Gödel, Kurt (1950). "Rotating universes in general relativity theory" (PDF). In: Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, August 30–September 6, 1950. Vol. 1. pp. 175–81. Archived from the original (PDF) on December 28, 2013. Retrieved December 4, 2017.
- "Tragic deaths in science: Kurt Gödel - looking over the edge of reason - Paperpile".
- Dawson, John W. (June 1, 2006). "Gödel and the limits of logic". Plus. University of Cambridge. Retrieved November 1, 2020.
- Wang 1996, p. 8.
- Wang 1996, p. 104-105.
- Gödel's answer to a special questionnaire sent him by the sociologist Burke Grandjean. This answer is quoted directly in Wang 1987, p. 18, and indirectly in Wang 1996, p. 112. It's also quoted directly in Dawson 1997, p. 6, who cites Wang 1987. The Grandjean questionnaire is perhaps the most extended autobiographical item in Gödel's papers. Gödel filled it out in pencil and wrote a cover letter, but he never returned it. "Theistic" is italicized in both Wang 1987 and Wang 1996. It is possible that this italicization is Wang's and not Gödel's. The quote follows Wang 1987, with two corrections taken from Wang 1996. Wang 1987 reads "Baptist Lutheran" where Wang 1996 has "baptized Lutheran". Wang 1987 has "rel. cong.", which in Wang 1996 is expanded to "religious congregation".
- Wang 1996, p. 316.
- Wang 1996, p. 51.
- Wang 1996, p. 148, 4.4.3. It is one of Gödel's observations, made between 16 November and 7 December 1975, which Wang found hard to classify under the main topics considered elsewhere in the book.
- "Times Critics' Top Books of 2021". The New York Times. December 15, 2021. Retrieved July 5, 2022.
- "Dangerous Knowledge". BBC. June 11, 2008. Retrieved October 6, 2009.
- Dawson, John W (1997), Logical dilemmas: The life and work of Kurt Gödel, Wellesley, MA: AK Peters.
- ISBN 978-0-393-32760-1.
- ISBN 0-262-73087-1
- ISBN 0-262-23189-1
- Stephen Budiansky, 2021. Journey to the Edge of Reason: The Life of Kurt Gödel. W.W. Norton & Company.
- Casti, John L; DePauli, Werner (2000), Gödel: A Life of Logic, Cambridge, MA: Basic Books (Perseus Books Group), ISBN 978-0-7382-0518-2.
- Dawson, John W Jr(1996), Logical Dilemmas: The Life and Work of Kurt Gödel, AK Peters.
- Dawson, John W Jr (1999), "Gödel and the Limits of Logic", Scientific American, 280 (6): 76–81, PMID 10048234.
- Franzén, Torkel (2005), Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, Wellesley, MA: AK Peters.
- Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
- Hämeen-Anttila, Maria (2020). Gödel on Intuitionism and Constructive Foundations of Mathematics (Ph.D. thesis). Helsinki: University of Helsinki. ISBN 978-951-51-5922-9.
- Jaakko Hintikka, 2000. On Gödel. Wadsworth.
- Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
- Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint c. 2001.
- Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 978-0-923891-57-2)
- J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
- Ernest Nagel and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
- Ed Regis, 1987. Who Got Einstein's Office? Addison-Wesley Publishing Company, Inc.
- Raymond Smullyan, 1992. Godel's Incompleteness Theorems. Oxford University Press.
- Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.
- Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
- Yourgrau, Palle, 2004.
- Kennedy, Juliette. "Kurt Gödel". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
- Time Bandits: an article about the relationship between Gödel and Einstein by Jim Holt
- Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue
- Paul Davies and Freeman Dyson discuss Kurt Godel (transcript)
- "Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
- It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.
- Gödel photo gallery. (archived)
- Kurt Gödel MacTutor History of Mathematics archivepage
- National Academy of Sciences Biographical Memoir