Gödel's discoveries in the foundations of mathematics led to the proof ofhis completeness theorem in 1929 as part of his dissertation to earn a doctorate at theUniversity of Vienna, and the publication ofGödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about thenatural numbers, and cannot prove that it is itself consistent.[4][5] To prove this, Gödel developed a technique now known asGödel numbering, which codes formal expressions as natural numbers.
Born into a wealthy German-speaking family inBrno, Gödel emigrated to the United States in 1939 to escape the rise of Nazi Germany. Later in life, he suffered from mental illness, which ultimately claimed his life: believing that his food was being poisoned, he refused to eat and starved to death.
Gödel was born on 28 April 1906, in Brünn,Austria-Hungary (nowBrno, Czech Republic), into the German-speaking family of Rudolf Gödel,[6] the managing director and part owner of a major textile firm,[a] and Marianne Gödel (née Handschuh).[11] His father was Catholic and his mother was Protestant, and the children were raised as Protestants. Many of Kurt Gödel's ancestors were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer in his time and for some years a member of theBrünner Männergesangverein (Men's Choral Union of Brünn).[12]
Gödel automatically became a citizen ofCzechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in theFirst World War. According to his classmateKlepetař, like many residents of the predominantly GermanSudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[13] In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship.[14] WhenGermanyannexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, afterWorld War II, at age 42, he became a U.S. citizen.[15]
In his family, the young Gödel was nicknamedHerr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven, Kurt suffered fromrheumatic fever; he completely recovered, but remained convinced for the rest of his life that his heart had been permanently damaged. Beginning at age four, Gödel had "frequent episodes of poor health", which continued all his life.[16]
Gödel attended theEvangelische Volksschule, a Lutheran school in Brünn, from 1912 to 1916, and was enrolled in theDeutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all subjects, particularly mathematics, languages, and religion. Although he had first excelled in languages, he became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf left forVienna, where he attended medical school at theUniversity of Vienna. During his teens, Gödel studiedGabelsberger shorthand,[17] criticism ofIsaac Newton, and the writings ofImmanuel Kant.[18]
Attending a lecture byDavid Hilbert inBologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert andWilhelm Ackermann publishedGrundzüge der theoretischen Logik (Principles of Mathematical Logic), an introduction tofirst-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?"[22]
Gödel chose this topic for his doctoral work.[22] In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymouscompleteness theorem regardingfirst-order logic.[22] He was awarded his doctorate in 1930,[22] and his thesis (accompanied by additional work) was published by theVienna Academy of Science.
In 1929 Gödel metAdele Nimbursky [es; ast] (née Porkert), a divorcee living with her parents across the street from him.[23] The two married (in a civil ceremony) a decade later, in September 1938.[24] A trained ballet dancer, Adele was working as a masseuse at the time they met.[23] At one point she worked as a dancer at a downtown nightclub called theNachtfalter ("nocturnal moth").[23] Gödel's parents opposed their relationship because of her background and age (six years older than him).[25] It appears to have been a happy marriage.[26] Adele was an important support to Gödel, whose psychological problems affected their daily lives.[27] The two had no children.
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.
The consistency of axioms cannot be proved within their ownsystem.[30]
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating inPrincipia Mathematica andHilbert's program, to find a non-relatively consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of mathematics).[31]
Gödel constructed a formula that claims it is itself 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 anycomputably 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 not provable in that system. To make this precise, Gödel had to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this by a process known asGödel numbering.[32]
Gödel earned hishabilitation at Vienna in 1932, and in 1933 became aPrivatdozent (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 murdered by one of his former students,Johann Nelböck. This triggered "a severe nervous crisis" in Gödel.[34] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[35]
In 1933, Gödel first traveled to the U.S., where he metAlbert Einstein, who became a good friend.[36] He delivered an address to the annual meeting of theAmerican Mathematical Society. During this year, Gödel also developed the ideas of computability andrecursive 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, usingGödel numbering.
In 1934, Gödel gave a series of lectures at theInstitute for Advanced Study (IAS) inPrinceton, New Jersey, titledOn undecidable propositions of formal mathematical systems.Stephen Kleene, who had just completed his PhD at Princeton, took notes on these lectures that were later published.
Gödel visited the IAS again in the autumn of 1935. The traveling and 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 theaxiom of choice and of thecontinuum hypothesis; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
After marrying Adele Nimbursky in 1938, he visited the U.S. again, spending the autumn of 1938 at the IAS and publishingConsistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory,[37] a classic of modern mathematics. In it, he introduced theconstructible universe, a model ofset theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both theaxiom of choice (AC) and thegeneralized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with theZermelo–Fraenkel axioms for set theory (ZF). This result has considerable consequences for working mathematicians, as it means they can assume the axiom of choice when proving theHahn–Banach theorem.Paul Cohen later constructed amodel 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.
After theAnschluss on 12 March 1938, Austria became a part ofNazi Germany. Germany abolished the titlePrivatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially Hahn, weighed against him. The University of Vienna turned his application down.
His predicament worsened 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 forPrinceton. To avoid the difficulty of an Atlantic crossing, the Gödels took theTrans-Siberian Railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then traveled to Princeton by train.[39] During this trip, Gödel was supposed to be carrying a secret letter to Einstein from Viennese physicist Hans Thirring to alert PresidentFranklin D. Roosevelt of the possibility that Hitler was making an atom bomb. Gödel never conveyed that letter to Einstein, although they did meet, because he was not convinced Hitler could achieve this feat.[40] In any case,Leo Szilard had already conveyed the message to Einstein, and Einstein had already warned Roosevelt.
In Princeton, Gödel accepted a position at the Institute for Advanced Study (IAS), which he had visited during 1933–34.[41]
Einstein was also living in Princeton during this time. Gödel and Einstein developed a strong friendship, and were known to take long walks together to and from the IAS. The nature of their conversations was a mystery to the other Institute members. EconomistOskar Morgenstern recounts that toward the end of Einstein's 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".[42]
Gödel and his wife spent the summer of 1942 inBlue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel had a very productive summer of work. UsingHeft 15 [volume 15] of Gödel's still-unpublishedArbeitshefte [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 friendHao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.[43]
On 5 December 1947, Einstein and Morgenstern accompanied Gödel to hisU.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in theU.S. Constitution that could allow the U.S. to become a dictatorship; this has since been dubbedGödel's Loophole. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. The judge turned out to bePhillip 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 theNazi 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.[44][45]
Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.[46]
During his time at the institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involvingclosed timelike curves, toEinstein's field equations ingeneral relativity.[47] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[48] His "rotating universes" would allowtime travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as theGödel metric (an exact solution of theEinstein field equation).
Gödel studied and admired the work ofGottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's work to be suppressed.[49] To a lesser extent he studied Kant andEdmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version ofAnselm of Canterbury'sontological argument for God's existence. This is now known asGödel's ontological proof.
Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery
Gödel married Adele in Vienna in 1938, and they emigrated a year later to the US.
In his later life, Gödel suffered periods ofmental instability and illness. Some scholars have suggestedAsperger syndrome andobsessive-compulsive disorder as diagnoses.[53] After his close friendMoritz Schlick was murdered,[54] Gödel developed anobsessive fear of being poisoned, and ate only food prepared by his wife, Adele. Adele was hospitalized beginning in late 1977, and in her absence Gödel refused to eat;[55] he weighed 29 kilograms (65 lb) when he died of "malnutrition andinanition caused by personality disturbance" inPrinceton Hospital on 14 January 1978.[56] He was buried inPrinceton Cemetery. Adele died in 1981, donating Gödel's papers to the Institute for Advanced Study upon her death.[57]
Gödel believed that God was personal,[58] and called his philosophy "rationalistic, idealistic, optimistic, and theological".[59] He formulated a draft offormal proof of God's existence known asGödel's ontological proof.
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]."[60] He also read widely on otherparanormal topics, including telepathy, reincarnation, and ghosts.[61]
In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief istheistic, not pantheistic, following Leibniz rather than Spinoza."[62] Of religion(s) in general, he said: "Religions are for the most part bad, but not religion itself."[63] According to his wife, Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[64] while ofIslam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."[65]
TheKurt Gödel Society, founded in 1987, is an international organization for the promotion of research in logic, philosophy, and thehistory of mathematics. TheUniversity of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. TheAssociation for Symbolic Logic has held an annualGödel Lecture since 1990. TheGödel Prize is given annually to an outstanding paper in theoretical computer science. Gödel's philosophical notebooks[70] are being edited at the Kurt Gödel Research Centre at theBerlin-Brandenburg Academy of Sciences and Humanities.[71] Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from hisNachlass, and the final two include correspondence.
In the 1994 filmI.Q.,Lou Jacobi portrays Gödel. In the 2023 movieOppenheimer, Gödel, played byJames Urbaniak, briefly appears walking with Einstein in the gardens of Princeton.
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 byRichard Braithwaite. Dover reprint of the 1962 Basic Books edition.
Kurt Gödel, 2000.[72]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).
^The factory was involved in wool trade - originally founded by Friedrich Redlich (1828, Brno – 1893/4, Brno) 1850 forworsted yarn imported from England, France and Belgium - had aweaving mill andfinishing shop, nospinning mill.[7][8][9][6][10]
^abcdIn the Scope of Logic, Methodology and Philosophy of Science. 11th International Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999. Vol. 1. 2002. p. 291.
^abcDawson Jr., John W., and Karl Sigmund. “Gödel’s Vienna.” Mathematical Intelligencer, vol. 28, no. 3, Summer 2006, Page 46. EBSCOhost,https://doi.org/10.1007/BF02986884.M
^Dawson Jr., John W., and Karl Sigmund. “Gödel’s Vienna.” Mathematical Intelligencer, vol. 28, no. 3, Summer 2006, Page 52. EBSCOhost,https://doi.org/10.1007/BF02986884.M
^Troelstra, A. S. (1988).Constructivism in Mathematics: An Introduction. Vol. 1. North-Holland. pp. 64–66.
^Casti, John L.; Depauli, Werner (2001).Godel: A Life Of Logic, The Mind, And Mathematics. Cambridge, Mass.: Basic Books.ISBN978-0-7382-0518-2.. From p. 80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgment that Schlick's murder was its trigger, are Rudolf Gödel's. Rudolf knew Kurt well in those years.
^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
^Dawson 1997, pp. 179–80. The story of Gödel's citizenship hearing has 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.
^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 fromthe original(PDF) on December 28, 2013. RetrievedDecember 4, 2017.
^Brewer, William D. "Kurt Gödel: The Genius of Metamathematics". Springer Nature. 2022. Pages 209-210
^Toates, Frederick; Olga Coschug Toates (2002).Obsessive Compulsive Disorder: Practical Tried-and-Tested Strategies to Overcome OCD. Class Publishing. p. 221.ISBN978-1-85959-069-0.
^Tucker McElroy (2005).A to Z of Mathematicians. Infobase Publishing. p. 118.ISBN978-0-8160-5338-4.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 remained a theist (a believer in a personal God) throughout his life.
^Feldman, Burton.112 Mercer Street: Einstein, Russell, Gödel, Pauli, and the End of Innocence in Science. Arcade Publishing. 2007. Page 7.
^Gödel's answer to a special questionnaire sent him by the sociologist Burke Grandjean. This answer is quoted directly inWang 1987, p. 18, and indirectly inWang 1996, p. 112. It is also quoted directly inDawson 1997, p. 6, who citesWang 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 did not return it. "Theistic" is italicized in bothWang 1987 andWang 1996. It is possible that this italicization is Wang's and not Gödel's. The quote followsWang 1987, with two corrections taken fromWang 1996.Wang 1987 reads "Baptist Lutheran" whereWang 1996 has "baptized Lutheran".Wang 1987 has "rel. cong.", which inWang 1996 is expanded to "religious congregation".
^Gödel, Kurt (2003)."Marianne Gödel". InFeferman, Solomon; Dawson, John W. (eds.).Kurt Gödel: Collected Works: Volume IV. OUP Oxford. p. 425.doi:10.1093/oso/9780198500735.003.0018.ISBN978-0-19-968961-3.Godel was not unmoved by religious concerns. On the contrary, his library included many books and tracts devoted to various religious sects; among his notebooks are two devoted to theology; and in a shorthand manuscript found in hisNachlaß he wrote, "Die Religionen sind zum größten Teil schlecht, aber nicht die Religion." ("Religions are for the most part bad, but not religion itself.")
^Wang 1996, p. 148, 4.4.3. It is one of Gödel's observations, made between 16 November and 7 December 1975, that Wang found hard to classify under the main topics considered elsewhere in the book.
