The Info List - Kurt Gödel

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Kurt Friedrich Gödel (UK: /ˈɡɜːrdəl/, US: /ˈɡoʊ-/;[2] German: [ˈkʊɐ̯t ˈɡøːdl̩] ( listen); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered along with Aristotle, Alfred Tarski
Alfred Tarski
and Gottlob Frege
Gottlob Frege
to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[3] Alfred North Whitehead,[3] and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example 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 neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these 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.


1 Early life and education

1.1 Childhood 1.2 Studying in Vienna

2 Career

2.1 Incompleteness Theorem 2.2 Mid-1930s: further work and U.S. visits 2.3 Princeton, Einstein, U.S. citizenship

3 Awards and honours 4 Later life and death 5 Personal life

5.1 Religious views

6 Legacy 7 Bibliography

7.1 Important publications

8 See also 9 Notes 10 References 11 Further reading 12 External links

Early life and education[edit] Childhood[edit] Gödel was born April 28, 1906, in Brünn, Austria-Hungary
(now Brno, Czech Republic) into the ethnic German family of Rudolf Gödel (1874-1929), the manager of a textile factory, and Marianne Gödel (née Handschuh, 1879-1966).[4] Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging.[5] At the time of his birth the city had a German-speaking majority which included his parents.[6] His father was Catholic and his mother was Protestant and the children were raised Protestant. The ancestors of Kurt Gödel
Kurt Gödel
were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".[7] Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[8] He chose to become an Austrian citizen at age 23[citation needed]. When Germany annexed Austria in 1938, 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. Beginning at age four, Gödel suffered from "frequent episodes of poor health," which would continue for his entire life.[9] 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 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. 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[edit] At the age of 18, Gödel joined his brother in Vienna
and entered the University of Vienna. By that time, he had already mastered university-level mathematics.[10] 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 Anfangsgründe der Naturwissenschaft, and participated in the Vienna
Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick
Moritz Schlick
which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."[11] Attending a lecture by David Hilbert
David Hilbert
in Bologna
on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann
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 became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna
Academy of Science. Career[edit] Incompleteness Theorem[edit]

"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." —John von Neumann[12]

In 1931 and while still in Vienna, Gödel published his 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 Zermelo–Fraenkel set theory with the axiom of choice), that:

If a (logical or axiomatic formal) system is consistent, it cannot be 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 Frege
and culminating in Principia Mathematica
Principia Mathematica
and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. 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 computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; 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). Mid-1930s: further work and U.S. visits[edit] 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.[13] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[14] In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.[15] 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 Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published. Gödel visited the IAS again in the autumn of 1935. The traveling 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. Their relationship had been opposed by his parents on the grounds that 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 the spring of 1939 at the University of Notre Dame. Princeton, Einstein, U.S. citizenship[edit] 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
University of Vienna
turned his application down. His predicament intensified when the German army found him fit for conscription. World War II
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
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
Institute for Advanced Study
(IAS), which he had previously visited during 1933-34.[16] 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.[citation needed] 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–Fraenkel axioms for set theory (ZF). This result has had considerable consequences for working mathematicians, as it means that they can assume the axiom of choice when proving the Hahn–Banach theorem. Paul Cohen
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. Albert Einstein
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".[17] 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 U.S. citizenship
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 that could allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. Fortunately, the judge turned out to be Phillip Forman, who 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
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.[18][19] 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.[20] During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed timelike curves, to Einstein's field equations in general relativity.[21] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[22] His "rotating universes" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric
Gödel metric
(an exact solution of the Einstein field equation). 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.[23] To a lesser extent he studied Immanuel Kant
Immanuel 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. Awards and honours[edit] Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.[24] Gödel was elected a Foreign Member of the Royal Society (ForMemRS) in 1968.[1] He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.[25] The Gödel Prize, an annual prize for outstanding papers in the area of theoretical computer science, is named after him.

Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery

Later life and death[edit] Later in his life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.[26] He weighed 29 kilograms (65 lb) 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.[27] He was buried in Princeton Cemetery. Adele's death followed in 1981. Personal life[edit] Religious views[edit] Gödel was a convinced theist, in the Christian tradition.[28] He held the notion that God was personal. He believed firmly in an afterlife, stating: "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]."[29] In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."[30] Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not".[31] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[32] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded".[33] Legacy[edit] The Kurt Gödel
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. The University of Vienna
University of Vienna
hosts the Kurt Gödel
Kurt Gödel
Research Center for Mathematical Logic. The Association for Symbolic Logic has invited an annual Kurt Gödel
Kurt Gödel
lecturer each year since 1990. Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's Nachlass, and the final two include correspondence. A biography of Gödel was published by John Dawson in 2005: Logical Dilemmas: The Life and Work of Kurt Gödel
Kurt Gödel
(A. K. Peters, Wellesley, MA, ISBN 1-56881-256-6). Gödel was also one of four mathematicians examined in the 2008 BBC
documentary entitled Dangerous Knowledge by David Malone.[34] Douglas Hofstadter
Douglas Hofstadter
wrote a popular book in 1979 called Gödel, Escher, Bach to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher
M. C. Escher
and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain. Bibliography[edit] Important publications[edit] In German:

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.

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. 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 1: 175–81

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.[35] On Formally Undecidable Propositions Of Principia Mathematica
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.

"My philosophical viewpoint", c. 1960, unpublished. "The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished. Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.

Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback: ISBN 978-0-19-514720-9, Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback: ISBN 978-0-19-514721-6, Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback: ISBN 978-0-19-514722-3, Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5, Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

See also[edit]

Biography portal Logic portal

Gödel machine Gödel Prize Gödel's speed-up theorem List of Austrian scientists List of pioneers in computer science Original proof of Gödel's completeness theorem Slingshot argument


^ a b Kreisel, G. (1980). "Kurt Godel. 28 April 1906-14 January 1978". Biographical Memoirs of Fellows of the Royal Society. 26: 148–224. doi:10.1098/rsbm.1980.0005.  ^ "Gödel". Collins English Dictionary. ^ a b For instance, in their Principia Mathematica
Principia Mathematica
(Stanford Encyclopedia of Philosophy edition). ^ Dawson 1997, pp. 3–4 ^ Kim, Alan (2015-01-01). Zalta, Edward N., ed. Johann Friedrich Herbart (Winter 2015 ed.).  ^ Dawson 1997, p. 12 ^ Procházka 2008, pp. 30–34. ^ Dawson 1997, p. 15. ^ Kim, Alan (2015-01-01). Zalta, Edward N., ed. Johann Friedrich Herbart (Winter 2015 ed.).  ^ Dawson 1997, p. 24. ^ Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p181. ^ Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394 ^ Casti, John L.; Depauli, Werner; Koppe, Matthias; Weismantel, Robert (2001). Gödel : a life of logic. Mathematics
of Operations Research. 31. Cambridge, Mass.: Basic Books. p. 147. doi:10.1287/moor.1050.0169. ISBN 0-7382-0518-4. . From p. 80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgement that the Schlick assassination was its trigger, are from the Rudolf Gödel quote. Rudolf knew Kurt well in those years. ^ Dawson 1997, pp. 110–112 ^ Hutchinson Encyclopedia
Hutchinson Encyclopedia
(1988), p. 518 ^ IAC biography ^ Goldstein (2005), p. 33. ^ Dawson 1997, pp. 179–180. 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). Archived from the original on December 26, 2014. Retrieved June 14, 2016. CS1 maint: Unfit url (link) ^ " Kurt Gödel
Kurt Gödel
– Institute for Advanced Study". Retrieved December 1, 2015.  ^ Gödel, K., "An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation," Rev. Mod. Phys. 21, 447, published July 1, 1949 [1]. ^ Das Genie & der Wahnsinn, Der Tagesspiegel, January 13, 2008 (in German). ^ John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. A K Peters, Ltd., 2005. P. 166. ^ "The President's National Medal of Science: Recipient Details NSF - National Science Foundation". www.nsf.gov. Retrieved 2016-09-17.  ^ Gödel, Kurt (1950). "Rotating universes in general relativity theory" (PDF). In: Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30–September 6, 1950. vol. 1. pp. 175–181.  ^ Davis, Martin (May 4, 2005). "Gödel's universe". Nature. 435: 19–20. doi:10.1038/435019a.  ^ Toates, Frederick; Olga Coschug Toates (2002). Obsessive Compulsive Disorder: Practical Tried-and-Tested Strategies to Overcome OCD. Class Publishing. p. 221. ISBN 978-1-85959-069-0.  ^ Tucker McElroy (2005). A to Z of Mathematicians. Infobase Publishing. p. 118. ISBN 9780816053384. Gödel had a happy childhood, and was called "Mr. Why" by his family, due to his numerous questions. He was baptized as a Lutheran, and re-mained a theist (a believer in a personal God) throughout his life.  ^ Hao Wang, "A Logical Journey: From Gödel to Philosophy", 1996, pp. 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. ^ "Dangerous Knowledge". BBC. June 11, 2008. Retrieved October 6, 2009.  ^ Kurt Godel (1931). "Über formal unentscheidbare Sätze der Principia Mathematica
Principia Mathematica
und verwandter Systeme, I" [On formally undecidable propositions of Principia Mathematica
Principia Mathematica
and related systems I] (PDF). Monatshefte für Mathematik und Physik. 38: 173–198. doi:10.1007/BF01700692. 


Dawson, John W., 1997. Logical dilemmas: The life and work of Kurt Gödel. 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.

Further reading[edit]

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. John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", Scientific American, vol. 280 num. 6, pp. 76–81 Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters. 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. Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. 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. Procházka, Jiří, 2006, 2006, 2008, 2008, 2010. 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. Volume V, Brno, Princeton 2010, ISBN 80-903476-9-X. In Germ., Engl. Procházka, Jiří, 2012. "Kurt Gödel: 1906–1978: Historie". ITEM, Brno, Wien, Princeton. Volume I. ISBN 978-80-903476-2-5. In Ger., Engl. 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. Gödel, Alois, 2OO6. Brünn 1679-1684.ITEM, Brno
2OO6, edited by Jiří Procházka, ISBN 8O-9O2297-8-6

Procházka, Jiří 2017. "Kurt Gödel: 1906-1978: Curriculum vitae".ITEM, Brno, Wien, Princeton 2017. Volume I. ISBN 978-80-903476-9-4. In Ger., Engl. Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press. Hao Wang, 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. Book review by John Stachel in the Notices of the American Mathematical Society
American Mathematical Society
(54 (7), pp. 861–868):

External links[edit]

Wikimedia Commons has media related to Kurt Gödel.

Wikiquote has quotations related to: Kurt Gödel

Weisstein, Eric Wolfgang (ed.). "Gödel, Kurt (1906–1978)". ScienceWorld.  Kennedy, Juliette. "Kurt Gödel". In Zalta, Edward N. Stanford Encyclopedia of Philosophy.  Time Bandits: an article about the relationship between Gödel and Einstein by Jim Holt "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 Paul Davies and Freeman Dyson discuss Kurt Godel "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. Kurt Gödel
Kurt Gödel
at Find a Grave National Academy of Sciences Biographical Memoir O'Connor, John J.; Robertson, Edmund F., "Kurt Gödel", MacTutor History of Mathematics
archive, University of St Andrews . Guerra-Pujol, Enrique (2013). "Gödel's Loophole". Capital University Law Review. University of Central Florida; Pontifical Catholic University of Puerto Rico. 41: 637–673. SSRN 2010183 . 

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Set theory



countable dependent

Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom


replacement specification


Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union

Concepts Methods

Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element

ordered pair tuple

Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram

Set types

Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal


Alternative Axiomatic Naive Cantor's theorem



Principia Mathematica

New Foundations


von Neumann–Bernays–Gödel


Kripke–Platek Tarski–Grothendieck

Paradoxes Problems

Russell's paradox Suslin's problem Burali-Forti paradox

Set theorists

Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

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United States National Medal of Science
National Medal of Science

Behavioral and social science


1964: Roger Adams Othmar H. Ammann Theodosius Dobzhansky Neal Elgar Miller


1986: Herbert A. Simon 1987: Anne Anastasi George J. Stigler 1988: Milton Friedman


1990: Leonid Hurwicz Patrick Suppes 1991: Robert W. Kates George A. Miller 1992: Eleanor J. Gibson 1994: Robert K. Merton 1995: Roger N. Shepard 1996: Paul Samuelson 1997: William K. Estes 1998: William Julius Wilson 1999: Robert M. Solow


2000: Gary Becker 2001: George Bass 2003: R. Duncan Luce 2004: Kenneth Arrow 2005: Gordon H. Bower 2008: Michael I. Posner 2009: Mortimer Mishkin


2011: Anne Treisman 2014: Robert Axelrod 2015: Albert Bandura

Biological sciences


1963: C. B. van Niel 1964: Marshall W. Nirenberg 1965: Francis P. Rous George G. Simpson Donald D. Van Slyke 1966: Edward F. Knipling Fritz Albert Lipmann William C. Rose Sewall Wright 1967: Kenneth S. Cole Harry F. Harlow Michael Heidelberger Alfred H. Sturtevant 1968: Horace Barker Bernard B. Brodie Detlev W. Bronk Jay Lush Burrhus Frederic Skinner 1969: Robert Huebner Ernst Mayr


1970: Barbara McClintock Albert B. Sabin 1973: Daniel I. Arnon Earl W. Sutherland Jr. 1974: Britton Chance Erwin Chargaff James V. Neel James Augustine Shannon 1975: Hallowell Davis Paul Gyorgy Sterling B. Hendricks Orville Alvin Vogel 1976: Roger Guillemin Keith Roberts Porter Efraim Racker E. O. Wilson 1979: Robert H. Burris Elizabeth C. Crosby Arthur Kornberg Severo Ochoa Earl Reece Stadtman George Ledyard Stebbins Paul Alfred Weiss


1981: Philip Handler 1982: Seymour Benzer Glenn W. Burton Mildred Cohn 1983: Howard L. Bachrach Paul Berg Wendell L. Roelofs Berta Scharrer 1986: Stanley Cohen Donald A. Henderson Vernon B. Mountcastle George Emil Palade Joan A. Steitz 1987: Michael E. DeBakey Theodor O. Diener Harry Eagle Har Gobind Khorana Rita Levi-Montalcini 1988: Michael S. Brown Stanley Norman Cohen Joseph L. Goldstein Maurice R. Hilleman Eric R. Kandel Rosalyn Sussman Yalow 1989: Katherine Esau Viktor Hamburger Philip Leder Joshua Lederberg Roger W. Sperry Harland G. Wood


1990: Baruj Benacerraf Herbert W. Boyer Daniel E. Koshland Jr. Edward B. Lewis David G. Nathan E. Donnall Thomas 1991: Mary Ellen Avery G. Evelyn Hutchinson Elvin A. Kabat Salvador Luria Paul A. Marks Folke K. Skoog Paul C. Zamecnik 1992: Maxine Singer Howard Martin Temin 1993: Daniel Nathans Salome G. Waelsch 1994: Thomas Eisner Elizabeth F. Neufeld 1995: Alexander Rich 1996: Ruth Patrick 1997: James Watson Robert A. Weinberg 1998: Bruce Ames Janet Rowley 1999: David Baltimore Jared Diamond Lynn Margulis


2000: Nancy C. Andreasen Peter H. Raven Carl Woese 2001: Francisco J. Ayala Mario R. Capecchi Ann Graybiel Gene E. Likens Victor A. McKusick Harold Varmus 2002: James E. Darnell Evelyn M. Witkin 2003: J. Michael Bishop Solomon H. Snyder Charles Yanofsky 2004: Norman E. Borlaug Phillip A. Sharp Thomas E. Starzl 2005: Anthony S. Fauci Torsten N. Wiesel 2006: Rita R. Colwell Nina Fedoroff Lubert Stryer 2007: Robert J. Lefkowitz Bert W. O'Malley 2008: Francis S. Collins Elaine Fuchs J. Craig Venter 2009: Susan L. Lindquist Stanley B. Prusiner


2010: Ralph L. Brinster Shu Chien Rudolf Jaenisch 2011: Lucy Shapiro Leroy Hood Sallie Chisholm 2014: May Berenbaum Bruce Alberts 2015: Stanley Falkow Rakesh K. Jain Mary-Claire King Simon Levin



1982: F. Albert Cotton Gilbert Stork 1983: Roald Hoffmann George C. Pimentel Richard N. Zare 1986: Harry B. Gray Yuan Tseh Lee Carl S. Marvel Frank H. Westheimer 1987: William S. Johnson Walter H. Stockmayer Max Tishler 1988: William O. Baker Konrad E. Bloch Elias J. Corey 1989: Richard B. Bernstein Melvin Calvin Rudolph A. Marcus Harden M. McConnell


1990: Elkan Blout Karl Folkers John D. Roberts 1991: Ronald Breslow Gertrude B. Elion Dudley R. Herschbach Glenn T. Seaborg 1992: Howard E. Simmons Jr. 1993: Donald J. Cram Norman Hackerman 1994: George S. Hammond 1995: Thomas Cech Isabella L. Karle 1996: Norman Davidson 1997: Darleane C. Hoffman Harold S. Johnston 1998: John W. Cahn George M. Whitesides 1999: Stuart A. Rice John Ross Susan Solomon


2000: John D. Baldeschwieler Ralph F. Hirschmann 2001: Ernest R. Davidson Gábor A. Somorjai 2002: John I. Brauman 2004: Stephen J. Lippard 2006: Marvin H. Caruthers Peter B. Dervan 2007: Mostafa A. El-Sayed 2008: Joanna Fowler JoAnne Stubbe 2009: Stephen J. Benkovic Marye Anne Fox


2010: Jacqueline K. Barton Peter J. Stang 2011: Allen J. Bard M. Frederick Hawthorne 2014: Judith P. Klinman Jerrold Meinwald 2015: A. Paul Alivisatos Geraldine L. Richmond

Engineering sciences


1962: Theodore von Kármán 1963: Vannevar Bush John Robinson Pierce 1964: Charles S. Draper 1965: Hugh L. Dryden Clarence L. Johnson Warren K. Lewis 1966: Claude E. Shannon 1967: Edwin H. Land Igor I. Sikorsky 1968: J. Presper Eckert Nathan M. Newmark 1969: Jack St. Clair Kilby


1970: George E. Mueller 1973: Harold E. Edgerton Richard T. Whitcomb 1974: Rudolf Kompfner Ralph Brazelton Peck Abel Wolman 1975: Manson Benedict William Hayward Pickering Frederick E. Terman Wernher von Braun 1976: Morris Cohen Peter C. Goldmark Erwin Wilhelm Müller 1979: Emmett N. Leith Raymond D. Mindlin Robert N. Noyce Earl R. Parker Simon Ramo


1982: Edward H. Heinemann Donald L. Katz 1983: William Redington Hewlett George M. Low John G. Trump 1986: Hans Wolfgang Liepmann T. Y. Lin Bernard M. Oliver 1987: R. Byron Bird H. Bolton Seed Ernst Weber 1988: Daniel C. Drucker Willis M. Hawkins George W. Housner 1989: Harry George Drickamer Herbert E. Grier


1990: Mildred Dresselhaus Nick Holonyak Jr. 1991: George H. Heilmeier Luna B. Leopold H. Guyford Stever 1992: Calvin F. Quate John Roy Whinnery 1993: Alfred Y. Cho 1994: Ray W. Clough 1995: Hermann A. Haus 1996: James L. Flanagan C. Kumar N. Patel 1998: Eli Ruckenstein 1999: Kenneth N. Stevens


2000: Yuan-Cheng B. Fung 2001: Andreas Acrivos 2002: Leo Beranek 2003: John M. Prausnitz 2004: Edwin N. Lightfoot 2005: Jan D. Achenbach Tobin J. Marks 2006: Robert S. Langer 2007: David J. Wineland 2008: Rudolf E. Kálmán 2009: Amnon Yariv


2010: Shu Chien 2011: John B. Goodenough 2014: Thomas Kailath

Mathematical, statistical, and computer sciences


1963: Norbert Wiener 1964: Solomon Lefschetz H. Marston Morse 1965: Oscar Zariski 1966: John Milnor 1967: Paul Cohen 1968: Jerzy Neyman 1969: William Feller


1970: Richard Brauer 1973: John Tukey 1974: Kurt Gödel 1975: John W. Backus Shiing-Shen Chern George Dantzig 1976: Kurt Otto Friedrichs Hassler Whitney 1979: Joseph L. Doob Donald E. Knuth


1982: Marshall Harvey Stone 1983: Herman Goldstine Isadore Singer 1986: Peter Lax Antoni Zygmund 1987: Raoul Bott Michael Freedman 1988: Ralph E. Gomory Joseph B. Keller 1989: Samuel Karlin Saunders Mac Lane Donald C. Spencer


1990: George F. Carrier Stephen Cole Kleene John McCarthy 1991: Alberto Calderón 1992: Allen Newell 1993: Martin David Kruskal 1994: John Cocke 1995: Louis Nirenberg 1996: Richard Karp Stephen Smale 1997: Shing-Tung Yau 1998: Cathleen Synge Morawetz 1999: Felix Browder Ronald R. Coifman


2000: John Griggs Thompson Karen K. Uhlenbeck 2001: Calyampudi R. Rao Elias M. Stein 2002: James G. Glimm 2003: Carl R. de Boor 2004: Dennis P. Sullivan 2005: Bradley Efron 2006: Hyman Bass 2007: Leonard Kleinrock Andrew J. Viterbi 2009: David B. Mumford


2010: Richard A. Tapia S. R. Srinivasa Varadhan 2011: Solomon W. Golomb Barry Mazur 2014: Alexandre Chorin David Blackwell 2015: Michael Artin

Physical sciences


1963: Luis W. Alvarez 1964: Julian Schwinger Harold Clayton Urey Robert Burns Woodward 1965: John Bardeen Peter Debye Leon M. Lederman William Rubey 1966: Jacob Bjerknes Subrahmanyan Chandrasekhar Henry Eyring John H. Van Vleck Vladimir K. Zworykin 1967: Jesse Beams Francis Birch Gregory Breit Louis Hammett George Kistiakowsky 1968: Paul Bartlett Herbert Friedman Lars Onsager Eugene Wigner 1969: Herbert C. Brown Wolfgang Panofsky


1970: Robert H. Dicke Allan R. Sandage John C. Slater John A. Wheeler Saul Winstein 1973: Carl Djerassi Maurice Ewing Arie Jan Haagen-Smit Vladimir Haensel Frederick Seitz Robert Rathbun Wilson 1974: Nicolaas Bloembergen Paul Flory William Alfred Fowler Linus Carl Pauling Kenneth Sanborn Pitzer 1975: Hans A. Bethe Joseph O. Hirschfelder Lewis Sarett Edgar Bright Wilson Chien-Shiung Wu 1976: Samuel Goudsmit Herbert S. Gutowsky Frederick Rossini Verner Suomi Henry Taube George Uhlenbeck 1979: Richard P. Feynman Herman Mark Edward M. Purcell John Sinfelt Lyman Spitzer Victor F. Weisskopf


1982: Philip W. Anderson Yoichiro Nambu Edward Teller Charles H. Townes 1983: E. Margaret Burbidge Maurice Goldhaber Helmut Landsberg Walter Munk Frederick Reines Bruno B. Rossi J. Robert Schrieffer 1986: Solomon J. Buchsbaum H. Richard Crane Herman Feshbach Robert Hofstadter Chen-Ning Yang 1987: Philip Abelson Walter Elsasser Paul C. Lauterbur George Pake James A. Van Allen 1988: D. Allan Bromley Paul Ching-Wu Chu Walter Kohn Norman F. Ramsey Jack Steinberger 1989: Arnold O. Beckman Eugene Parker Robert Sharp Henry Stommel


1990: Allan M. Cormack Edwin M. McMillan Robert Pound Roger Revelle 1991: Arthur L. Schawlow Ed Stone Steven Weinberg 1992: Eugene M. Shoemaker 1993: Val Fitch Vera Rubin 1994: Albert Overhauser Frank Press 1995: Hans Dehmelt Peter Goldreich 1996: Wallace S. Broecker 1997: Marshall Rosenbluth Martin Schwarzschild George Wetherill 1998: Don L. Anderson John N. Bahcall 1999: James Cronin Leo Kadanoff


2000: Willis E. Lamb Jeremiah P. Ostriker Gilbert F. White 2001: Marvin L. Cohen Raymond Davis Jr. Charles Keeling 2002: Richard Garwin W. Jason Morgan Edward Witten 2003: G. Brent Dalrymple Riccardo Giacconi 2004: Robert N. Clayton 2005: Ralph A. Alpher Lonnie Thompson 2006: Daniel Kleppner 2007: Fay Ajzenberg-Selove Charles P. Slichter 2008: Berni Alder James E. Gunn 2009: Yakir Aharonov Esther M. Conwell Warren M. Washington


2011: Sidney Drell Sandra Faber Sylvester James Gates 2014: Burton Richter Sean C. Solomon 2015: Shirley Ann Jackson

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Special relativity


Principle of relativity Special
relativity Doubly special relativity


Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations


Galilean relativity Galilean transformation Lorentz transformation


Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks


Light cone World line Spacetime
diagram Biquaternions Minkowski space

General relativity


Introduction Mathematical formulation

Fundamental concepts

relativity Equivalence principle World line Riemannian geometry Minkowski diagram Penrose diagram


Black hole Event horizon Frame-dragging Geodetic effect Lenses Singularity Waves Ladder paradox Twin paradox Two-body problem BKL singularity


ADM formalism BSSN formalism Einstein field equations Geodesic equation Friedmann equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation Tolman–Oppenheimer–Volkoff equation

Advanced theories

Brans–Dicke theory Kaluza–Klein Mach's principle Quantum gravity


Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Friedmann–Lemaître–Robertson–Walker Taub–NUT Milne pp-wave van Stockum dust Weyl−Lewis−Papapetrou


Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedmann Milne Zwicky Lemaître Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Taylor Hulse Stockum Taub Newman Yau Thorne Weiss Bondi Misner others

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Analytic philosophy


J. L. Austin A. J. Ayer G. E. M. Anscombe Nick Bostrom Robert Brandom C. D. Broad Patricia Churchland David Chalmers Noam Chomsky James F. Conant Alice Crary Donald Davidson Daniel Dennett Cora Diamond Michael Dummett Paul Feyerabend Antony Flew Bas van Fraassen Gottlob Frege Jerry Fodor Philippa Foot Peter Geach Paul Grice Ian Hacking R. M. Hare Carl Gustav Hempel Peter van Inwagen Christine Korsgaard Saul Kripke Thomas Kuhn David Lewis Alasdair MacIntyre J. L. Mackie Norman Malcolm John McDowell G. E. Moore Ernest Nagel Thomas Nagel Robert Nozick Derek Parfit Alvin Plantinga Karl Popper Hilary Putnam W. V. O. Quine John Rawls Hans Reichenbach Richard Rorty Bertrand Russell Gilbert Ryle Moritz Schlick John Searle Wilfrid Sellars Peter Singer Richard Swinburne Charles Taylor Michael Walzer Bernard Williams Timothy Williamson Ludwig Wittgenstein


Actualism Analytical feminism Analytical Marxism Anti-realism Berlin Circle Descriptivist theory of names Emotivism Functional contextualism Linguistic turn Logical positivism Modal realism Model-dependent realism Neopragmatism Neurophilosophy Ordinary language philosophy Postanalytic philosophy Pragmatic theory of truth Verificationism Vienna


Analysis Analytic–synthetic distinction Causal / Deductive / epistemic closure Concept Counterfactual Denotation / reference Definite description Factive Family resemblance Intuition Meaning (Proposition) Modality Natural kind / projectability Necessary–sufficient conditions Paradox of analysis Possible world Reduction Reflective equilibrium Rigid–flaccid designators Sense data Supervenience Thought experiment Truth function Truthmaker Truth-bearer Type–token distinction

Related articles

Aretaic turn Australian realism Communitarianism Ordinary language philosophy Philosophical logic Philosophy of language Philosophy of science Postanalytic philosophy

Index Category

Authority control

WorldCat Identities VIAF: 97851774 LCCN: n79007770 ISNI: 0000 0001 1031 567X GND: 11869569X SELIBR: 237197 SUDOC: 028039734 BNF: cb12133987b (data) MGP: 19539 NDL: 00549746 NKC: jn20000602196 ICCU: ITICCUMILV24928 BNE: XX988655 SNAC: w6254