Kurt Gödel

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Kurt Gödel
Gödel c. 1926
Born
Kurt Friedrich Gödel

(1906-04-28)April 28, 1906
Brünn, Austria-Hungary (now Brno, Czech Republic)
DiedJanuary 14, 1978(1978-01-14) (aged 71)
Princeton, New Jersey, U.S.
Citizenship
  • Austria
  • Czechoslovakia
  • Germany
  • United States
Alma materUniversity of Vienna (PhD, 1930)
Known for
 
Spouse
Adele Nimbursky
​
(m. 1938)​
Awards
Scientific career
FieldsMathematics, mathematical logic, physics
InstitutionsInstitute for Advanced Study
Thesis Über die Vollständigkeit des Logikkalküls  (1929)
Doctoral advisorHans Hahn
Philosophy career
Era20th-century philosophy
RegionWestern philosophy
SchoolAnalytic philosophy
Main interests
Signature

Kurt Friedrich Gödel (/ˈɡɜːrdəl/ GUR-dəl;[2] German: [kʊʁt ˈɡøːdl̩] ⓘ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell,[3] Alfred North Whitehead,[3] and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor.

Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example, Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms.[4] To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.[5]

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

Childhood

Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic), into the German-speaking family of Rudolf Gödel (1874–1929), the managing director and part owner of a major textile firm, and Marianne Gödel (née Handschuh, 1879–1966).[6] At the time of his birth the city had a German-speaking majority which included his parents.[7] His father was Catholic and his mother was Protestant and the children were raised as Protestants. The ancestors of Kurt Gödel were often 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 the Brünner Männergesangverein (Men's Choral Union of Brünn).[8]

Gödel automatically became a citizen of Czechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the First World War. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[9] In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship.[10] When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, after World War II, at the age of 42, he became an American citizen.[11]

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.[12]

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 Vienna, where he attended medical school at the University of Vienna. During his teens, Gödel studied Gabelsberger shorthand,[13] and criticisms of Isaac Newton, and the writings of Immanuel Kant.[14]

Studies in Vienna

Plaque to Gödel at 43-45 Josefstädter Straße [de], Vienna, where he discovered his incompleteness theorems

At the age of 18, Gödel joined his brother at the University of Vienna. He had already mastered university-level mathematics.[15] Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy.[16] 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 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."[17]

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?"[18]

This problem became the topic that Gödel chose for his doctoral work.[18] In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding first-order logic.[18] He was awarded his doctorate in 1930,[18] and his thesis (accompanied by additional work) was published by the Vienna Academy of Science.

Career

Gödel as a student in 1925

Incompleteness theorems

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[19]

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. There, he presented his completeness theorem of first-order logic, and, at the end of the talk, mentioned that this result does not generalise to higher-order logic, thus hinting at his incompleteness theorems.[20]

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:

  1. If a (logical or axiomatic formal) system is omega-consistent, it cannot be syntactically complete.
  2. The consistency of axioms cannot be proved within their own system.[21]

These theorems ended a half-century of attempts, beginning with the work of Gottlob Frege and culminating in Principia Mathematica and Hilbert'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).[22]

Gödel constructed a formula that claims 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 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 as Gödel numbering.[23]

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).[24]

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.[25] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[26]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.[27] 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, titled 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 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 [es ; ast] (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,[28] 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–Fraenkel axioms for set theory (ZF). This result has had considerable consequences for working mathematicians, as it means they can assume the axiom of choice when proving the Hahn–Banach theorem. 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.

Gödel spent the spring of 1939 at the University of Notre Dame.[29]

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.[30] During this trip, Gödel was supposed to be carrying a secret letter from Viennese physicist Hans Thirring to Albert Einstein to alert Franklin D. Roosevelt of the possibility of Hitler 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.[31] 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.[32]

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 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".[33]

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 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; this has since been dubbed Gödel's Loophole. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. 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 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.[34][35]

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.[36]

During his time 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.[37] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[38] 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 (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.[39] To a lesser extent he studied 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

Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.[40] Gödel was elected a resident member of the American Philosophical Society in 1961 and a Foreign Member of the Royal Society (ForMemRS) in 1968.[41][1] He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.[42]

Later life and death

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

Later in his life, Gödel suffered periods of mental instability and illness. Following the assassination of his close friend Moritz Schlick,[43] Gödel developed an obsessive fear of being poisoned, and would eat only food prepared by his wife Adele. Adele was hospitalized beginning in late 1977, and in her absence Gödel refused to eat;[44] he weighed 29 kilograms (65 lb) when he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.[45] He was buried in Princeton Cemetery. Adele died in 1981.[46]

Religious views

Gödel believed that God was personal,[47] and called his philosophy "rationalistic, idealistic, optimistic, and theological".[48] He formulated a formal proof for the existence of God known as Gödel'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]."[49]

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."[50] Of religion(s) in general, he said: "Religions are for the most part bad, but not religion itself."[51] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[52] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."[53]

Legacy

Douglas Hofstadter wrote the 1979 book Gödel, Escher, Bach to celebrate the work and ideas of Gödel, M. C. Escher and Johann Sebastian Bach. It 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. In 2005 John Dawson published a biography, Logical Dilemmas: The Life and Work of Kurt Gödel.[54] Stephen Budiansky's book about Gödel's life, Journey to the Edge of Reason: The Life of Kurt Gödel,[55] was a New York Times Critics' Top Book of 2021.[56] Gödel was one of four mathematicians examined in David Malone's 2008 BBC documentary Dangerous Knowledge.[57]

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 since 1990. The Gödel Prize is given annually to an outstanding paper in theoretical computer science. Gödel's philosophical notebooks[58] are being edited at the Kurt Gödel Research Centre at the Berlin-Brandenburg Academy of Sciences and Humanities.[59] 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.

In the 1994 film I.Q., Lou Jacobi portrays Gödel. In the 2023 movie Oppenheimer, Gödel, played by James Urbaniak, briefly appears walking with Einstein in the gardens of Princeton.

Bibliography

Important publications

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:

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.[60] 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.
  • 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.
  • Philosophische Notizbücher / Philosophical Notebooks: De Gruyter: Berlin/München/Boston. Editor: Eva-Maria Engelen [de].

See also

Notes

  1. ^ 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. S2CID 120119270.
  2. ^ "Gödel". Merriam-Webster.com Dictionary. Merriam-Webster.
  3. ^ a b For instance, in their "Principia Mathematica " (Stanford Encyclopedia of Philosophy edition).
  4. ^ Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. V.
  5. ^ Smullyan, R. M. (1992). Gödel's Incompleteness Theorems. New York, Oxford: Oxford University Press, ch. IX.
  6. ^ Dawson 1997, pp. 3–4.
  7. ^ Dawson 1997, p. 12
  8. ^ Procházka 2008, pp. 30–34.
  9. ^ Dawson 1997, p. 15.
  10. ^ Gödel, Kurt (1986). Collected works. Feferman, Solomon. Oxford. p. 37. ISBN 0-19-503964-5. OCLC 12371326.{{cite book}}: CS1 maint: location missing publisher (link)
  11. ^ Balaguer, Mark. "Kurt Godel". Britannica School High. Encyclopædia Britannica, Inc. Retrieved June 3, 2019.
  12. ^ Kim, Alan (January 1, 2015). Zalta, Edward N. (ed.). Johann Friedrich Herbart (Winter 2015 ed.). Metaphysics Research Lab, Stanford University.
  13. ^ "Gabelsberger stenography | Gödel Enigma | University of Helsinki". www.helsinki.fi.
  14. ^ Parsons, Charles (2010). "Gödel and philosophical idealism". Philosophia Mathematica. Series III. 18 (2): 166–192. doi:10.1093/philmat/nkq001. MR 2669137.
  15. ^ Dawson 1997, p. 24.
  16. ^ At the University of Vienna, Gödel attended mathematics and philosophy courses side by side with Hermann Broch, who was in his early forties. See: Sigmund, Karl; Dawson Jr., John W.; Mühlberger, Kurt (2007). Kurt Kurt Gödel: Das Album. Springer-Verlag. p. 27. ISBN 978-3-8348-0173-9.
  17. ^ Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p. 181.
  18. ^ a b c d In 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.
  19. ^ Halmos, P.R. (April 1973). "The Legend of von Neumann". The American Mathematical Monthly. 80 (4): 382–94. doi:10.1080/00029890.1973.11993293.
  20. ^ Stadler, Friedrich (2015). The Vienna Circle: Studies in the Origins, Development, and Influence of Logical Empiricism. Springer. ISBN 978-3-319-16561-5.
  21. ^ Dawson 1997, pp. 61–63.
  22. ^ Nagel, Ernest (2001). Gödel's Proof. New York University Press. pp. 85–87.
  23. ^ Raatikainen, Panu (2015). Gödel's Incompleteness Theorems. Stanford Encyclopedia of Philosophy.
  24. ^ Troelstra, A. S. (1988). Constructivism in Mathematics: An Introduction. Vol. 1. North-Holland. pp. 64–66.
  25. ^ Casti, John L.; Depauli, Werner (2001). Godel: A Life Of Logic, The Mind, And Mathematics. Cambridge, Mass.: Basic Books. ISBN 978-0-7382-0518-2.. From p. 80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgement that the Schlick assassination was its trigger, are from the Rudolf Gödel quote. Rudolf knew Kurt well in those years.
  26. ^ Dawson 1997, pp. 110–12
  27. ^ Hutchinson Encyclopedia (1988), p. 518
  28. ^ Gödel, Kurt (November 9, 1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12): 556–57. Bibcode:1938PNAS...24..556G. doi:10.1073/pnas.24.12.556. ISSN 0027-8424. PMC 1077160. PMID 16577857.
  29. ^ 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
  30. ^ Dawson Jr, John W (October 2002). "Max Dehn, Kurt Gödel, and the Trans-Siberian Escape Route" (PDF). Notices of the American Mathematical Society. 49 (9): 1068–1075.
  31. ^ Sigmund K (March 2024). "The spy who flunked it: Kurt Gödel's forgotten part in the atom-bomb story". Nature. 627 (8002): 26–28. Bibcode:2024Natur.627...26S. doi:10.1038/d41586-024-00644-1. PMID 38438543.
  32. ^ "Kurt Gödel". Institute for Advanced Study. December 9, 2019.
  33. ^ Goldstein 2005, p. 33
  34. ^ 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.
  35. ^ Oskar Morgenstern (September 13, 1971). "History of the Naturalization of Kurt Gödel" (PDF). Retrieved April 16, 2019.
  36. ^ "Kurt Gödel – Institute for Advanced Study". Retrieved December 1, 2015.
  37. ^ Gödel, Kurt (July 1, 1949). "An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation". Rev. Mod. Phys. 21 (447): 447–450. Bibcode:1949RvMP...21..447G. doi:10.1103/RevModPhys.21.447.
  38. ^ "Das Genie & der Wahnsinn". Der Tagesspiegel (in German). January 13, 2008.
  39. ^ Dawson, John W. Jr. (2005). Logical Dilemmas: The Life and Work of Kurt Gödel. A K Peters. p. 166. ISBN 978-1-56881-256-4.
  40. ^ "The President's National Medal of Science: Recipient Details | NSF – National Science Foundation". www.nsf.gov. Retrieved September 17, 2016.
  41. ^ "APS Member History". search.amphilsoc.org. Retrieved January 28, 2021.
  42. ^ 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.
  43. ^ "Tragic deaths in science: Kurt Gödel - looking over the edge of reason - Paperpile".
  44. ^ Davis, Martin (May 4, 2005). "Gödel's universe". Nature. 435 (7038): 19–20. Bibcode:2005Natur.435...19D. doi:10.1038/435019a.
  45. ^ 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.
  46. ^ Dawson, John W. (June 1, 2006). "Gödel and the limits of logic". Plus. University of Cambridge. Retrieved November 1, 2020.
  47. ^ Tucker McElroy (2005). A to Z of Mathematicians. Infobase Publishing. p. 118. ISBN 978-0-8160-5338-4. 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 re-mained a theist (a believer in a personal God) throughout his life.
  48. ^ Wang 1996, p. 8.
  49. ^ Wang 1996, p. 104-105.
  50. ^ 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".
  51. ^ Gödel, Kurt (2003). "Marianne Gödel". In Feferman, Solomon; Dawson, John W. (eds.). Kurt Gödel: Collected Works: Volume IV. OUP Oxford. p. 425. doi:10.1093/oso/9780198500735.003.0018. ISBN 978-0-19-968961-3. 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 his Nachlaß he remarked that "Die Religionen sind zum größten Teil schlecht , aber nicht die Religion." ("Religions are for the most part bad, but not religion itself.")
  52. ^ Wang 1996, p. 51.
  53. ^ 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.
  54. ^ A. K. Peters, Wellesley, MA, ISBN 1-56881-256-6
  55. ^ W. W. Norton & Company, New York City, ISBN 978-0-393-35820-9
  56. ^ "Times Critics' Top Books of 2021". The New York Times. December 15, 2021. Retrieved July 5, 2022.
  57. ^ "Dangerous Knowledge". BBC. June 11, 2008. Retrieved October 6, 2009.
  58. ^ "Kurt-Gödel-Forschungsstelle: die "Philosophischen Bemerkungen" Kurt Gödels (Kurt Gödel Research Centre: The 'Philosophical Remarks' of Kurt Gödel) – Berlin-Brandenburg Academy of Sciences and Humanities". www.bbaw.de.
  59. ^ "The Academy – Berlin-Brandenburg Academy of Sciences and Humanities". www.bbaw.de.
  60. ^ Kurt Godel (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I" [On formally undecidable propositions of Principia Mathematica and related systems I] (PDF). Monatshefte für Mathematik und Physik. 38: 173–98. doi:10.1007/BF01700692. S2CID 197663120.

References

Further reading

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