Kurt Gödel

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"Godel" and "Gödel" redirect here. For other uses, see Godel (disambiguation).

Kurt Gödel
PhD
, 1930)
Known for
 
Spouse
Adele Nimbursky
​
(m. 1938)​
Awards
Scientific career
FieldsMathematics, mathematical logic, analytic philosophy, physics
InstitutionsInstitute for Advanced Study
ThesisÜber die Vollständigkeit des Logikkalküls (1929)
Doctoral advisorHans Hahn
Signature

Kurt Friedrich Gödel (

philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect 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 using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Gottlob Frege
.

Gödel's discoveries in the foundations of mathematics led to the proof of

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,

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

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 Czechoslovakian 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

Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.[citation needed
]

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

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."[15]

Attending a lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: "Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?"

This problem became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral

first-order predicate calculus. He was awarded his doctorate in 1930, and his thesis (accompanied by some 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[16]

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.[17]

Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (called in English "

Zermelo–Fraenkel set theory with the axiom of choice
), that:

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

These theorems ended a half-century of attempts, beginning with the work of

relatively
consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of 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

Gödel fuzzy logic
).

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.[18] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[19]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.[20] 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

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,

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.[22]

Princeton, Einstein, U.S. citizenship

After the Anschluss on 12 March 1938, Austria had become a part of Nazi Germany. Germany abolished the title Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down.

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the Trans-Siberian Railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the US by train to Princeton. There Gödel accepted a position at the Institute for Advanced Study (IAS), which he had previously visited during 1933–34.[23]

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".[24]

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

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.[25][26]

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.[27]

During his time at the institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving

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.[30] 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

Foreign Member of the Royal Society (ForMemRS) in 1968.[32][1] He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.[33]

Later life and death

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

Later in his life, Gödel suffered periods of

Princeton Hospital on January 14, 1978.[36] He was buried in Princeton Cemetery. Adele died in 1981.[37]

Religious views

Gödel believed that God[38] was personal, and called his philosophy "rationalistic, idealistic, optimistic, and theological".[39]

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]."[40]

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is

Spinoza."[41] Of religion(s) in general, he said: "Religions are, for the most part, bad—but religion is not".[42] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[43] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."[44]

Legacy

Turing-complete computational system, which may include the human brain
.

The Kurt Gödel Society, founded in 1987, is an international organization for the promotion of research in logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has held an annual Gödel Lecture each year since 1990. Gödel's Philosophical Notebooks Archived May 14, 2019, at the Wayback Machine are edited at the Kurt Gödel Research Centre Archived May 14, 2019, at the Wayback Machine which is situated at the Berlin-Brandenburg Academy of Sciences and Humanities in Germany.

Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from his Nachlass, and the final two include correspondence.

In 2005

ISBN 978-0-393-35820-9), was a New York Times Critics' Top Book of 2021.[45]

Gödel was also one of four mathematicians examined in David Malone's 2008 BBC documentary Dangerous Knowledge.[46]

The Gödel Prize is given annually for an outstanding paper in theoretical computer science.

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:

See also

Notes

  1. ^
    S2CID 120119270
    .
  2. ^ "Gödel". Merriam-Webster.com Dictionary.
  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. 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. ^ Dawson 1997, p. 24.
  14. ISBN 978-3-8348-0173-9
    .
  15. ^ Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p. 181.
  16. doi:10.1080/00029890.1973.11993293
    .
  17. ISBN 978-3-319-16561-5
    .
  18. 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.
  19. ^ Dawson 1997, pp. 110–12
  20. ^ Hutchinson Encyclopedia (1988), p. 518
  21. PMID 16577857
    .
  22. ^ 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
  23. ^ "Kurt Gödel". Institute for Advanced Study. December 9, 2019.
  24. ^ Goldstein 2005, p. 33
  25. ^ 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.
  26. ^ Oskar Morgenstern (September 13, 1971). "History of the Naturalization of Kurt Gödel" (PDF). Retrieved April 16, 2019.
  27. ^ "Kurt Gödel – Institute for Advanced Study". Retrieved December 1, 2015.
  28. doi:10.1103/RevModPhys.21.447
    .
  29. ^ "Das Genie & der Wahnsinn". Der Tagesspiegel (in German). January 13, 2008.
  30. ISBN 978-1-56881-256-4
    .
  31. ^ "The President's National Medal of Science: Recipient Details | NSF – National Science Foundation". www.nsf.gov. Retrieved September 17, 2016.
  32. ^ "APS Member History". search.amphilsoc.org. Retrieved January 28, 2021.
  33. ^ 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.
  34. ^ "Tragic deaths in science: Kurt Gödel - looking over the edge of reason - Paperpile".
  35. doi:10.1038/435019a
    .
  36. ISBN 978-1-85959-069-0
    .
  37. ^ Dawson, John W. (June 1, 2006). "Gödel and the limits of logic". Plus. University of Cambridge. Retrieved November 1, 2020.
  38. 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.
  39. ^ Wang 1996, p. 8.
  40. ^ Wang 1996, p. 104-105.
  41. ^ 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".
  42. ^ Wang 1996, p. 316.
  43. ^ Wang 1996, p. 51.
  44. ^ 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.
  45. ^ "Times Critics' Top Books of 2021". The New York Times. December 15, 2021. Retrieved July 5, 2022.
  46. ^ "Dangerous Knowledge". BBC. June 11, 2008. Retrieved October 6, 2009.
  47. S2CID 197663120
    .

References

Further reading

External links

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