Kurt Gödel

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Kurt Gödel (1925)

Kurt Gödel signature.svg

Kurt Friedrich Gödel (born April 28, 1906 in Brno , Austria-Hungary , now the Czech Republic ; † January 14, 1978 in Princeton , New Jersey , United States ) was an Austrian and later American mathematician , philosopher and one of the most important logicians of the 20th century. Century. He made significant contributions to predicate logic ( completeness and decision problem in arithmetic and axiomatic set theory ), to the relationships of intuitionist logic both to classical logic and to modal logic, as well as to the theory of relativity in physics.

His philosophical discussions on the fundamentals of mathematics also received wide attention.

Life

Origin and school time

Kurt Gödel came from a wealthy, upper-class family in Brno in Moravia . At the time of Gödel's birth, the city had a German-speaking majority and was part of the Austro-Hungarian monarchy until 1918 . His parents were Marianne (née Handschuh) and Rudolf August Gödel from Brno. The father was a wealthy textile entrepreneur. The mother was Protestant , the father was Catholic , and the family's children were raised Protestants.

Caused by rheumatic fever , Gödel often suffered from poor health in his childhood. Nevertheless, he showed his best academic performance. In 1912 Gödel entered a private elementary and community school , four years later the German-speaking kk Staatsrealgymnasium.

After the First World War , the city of Brno became part of the newly founded Czechoslovak Republic in 1918/1919 . Gödel, who spoke only poorly Czech , felt, according to John D. Dawson, in the newly founded state like an "Austrian exile in Czechoslovakia". In 1923 he took on Austrian citizenship .

Studied in Vienna

In the autumn of 1924, after passing the high school diploma, Gödel moved to Vienna and enrolled at the University of Vienna , initially on the theoretical physics course . In the following year he mainly dealt with physical issues. He also attended the philosophical lectures of Heinrich Gomperz and lectures on number theory by Philipp Furtwängler . These two professors gave Gödel the decisive impetus to deal intensively with the fundamentals of mathematics, which are based on formal logic and set theory.

Memorial plaque in Vienna 8th, Lange Gasse 72, where Gödel lived as a student from July 4, 1928 to November 5, 1929

Shortly after starting his studies, he began to visit the Vienna Circle , an academic circle that Moritz Schlick had set up and dealt with the methodological foundations of thinking and thus the foundations of any philosophy. The conversations with the other members of the group, of whom in particular Hans Hahn , Karl Menger and Olga Taussky were of particular importance to Gödel, also led to the expansion of his mathematical knowledge. Rudolf Carnap's diaries show that he took an active part in meetings of the members of the Vienna Circle in private apartments and cafés. Fascinated by the discussions in the Vienna Circle, Gödel attended Karl Menger's Mathematical Colloquium and became familiar with the basic problems of mathematics and logic of his time. He especially got to know Hilbert's program , which was supposed to prove the consistency of mathematics. For his dissertation entitled On the Completeness of the Logical Calculus (1929) he was awarded the doctorate on February 6, 1930. Hans Hahn was his doctoral supervisor .

The meetings of the circle were also important for his private life, since it was here in 1927 that he met his future wife Adele Nimbursky for the first time. In 1928 Gödel and his brother moved into a new apartment in the 8th district , Florianigasse 42, where there is now a memorial plaque. Coincidentally, it was right across from Adele Nimbursky's apartment, and the two were now dating. Adele, born in 1899 as Adele Porkert, came from a middle-class background, she worked as a cabaret dancer and was poorly educated. She was almost seven years older than Gödel and married to the photographer Nimbursky until 1933 before she divorced him. There was also a denominational difference - she was Catholic and Gödel was Protestant. Godel's parents viewed the relationship as a mesalliance , which led the couple to initially keep it a secret.

First trips to America

Gödel's pioneering work on completeness and the logic of provability earned him recognition as one of the leading logicians of his time. He was invited by his American colleague Oswald Veblen to Princeton to the newly founded Institute for Advanced Study . In 1933/1934 he traveled to America for the first time. Together with James Alexander , John von Neumann and Oswald Veblen, he became a founding member of the faculty and gave a number of lectures. When Gödel returned to Vienna, which was now governed by dictatorship, in the spring of 1934, he had already received an invitation to continue teaching. Godel was not interested in the political situation in Europe. In July 1934 he got the news of the death of his mentor Hans Hahn. In 1935 he traveled back to Princeton.

Health difficulties

Travel and work exhausted Godel. Now the mental illness, which he probably had latent in him since childhood, made itself felt as depression . In the autumn of 1934 he had to go to a sanatorium for a week. In 1935 he spent several months in a psychiatric clinic. When the philosopher Moritz Schlick , highly respected by Gödel and one of the leading figures in the Vienna Circle, was murdered in June 1936 by his former student Hans Nelböck at the University of Vienna, Gödel suffered a nervous breakdown. He developed hypochondriac obsessions, especially a morbid fear of being poisoned, so that Adele had to prepare and taste all of his dishes in front of his eyes.

Since Godel did not eat properly, his physical health increasingly suffered. His condition worsened over the years. Ever since he suffered from rheumatic fever as a child, he was convinced that he had a weak heart and developed suspicion of the medical profession, who could not find anything like it in him. He avoided doctors, so he nearly died of an untreated duodenal ulcer in the 1940s .

emigration

Vienna 19th, Himmelstrasse 43, in Grinzing , Gödel's apartment November 11, 1937 to November 9, 1939

In March 1938 Austria was annexed to the German Reich . Due to the change in the education system, Gödel lost his Austrian lectureship. He tried to get an adequate academic position in the Nazi education system. The corresponding applications were processed very slowly, however, since Gödel was considered to be a representative of "heavily Jewish mathematics". The paternal inheritance, which Gödel used for his and Adele's maintenance, gradually expired, so that the two no longer had a secure income.

On September 20, 1938, Kurt Gödel and Adele, b. Porkert. After the marriage, Gödel traveled to the United States for a third time. In the fall of 1938 he worked again at the Institute for Advanced Study in Princeton, in the spring of 1939 at the University of Notre Dame in Indiana .

When Gödel returned to Vienna, which was ruled by the National Socialists, he was bullied by people who (wrongly) thought he was a Jew. He was officially classified as fit for use in the war, which is why he finally decided to leave his previous home and emigrate to the USA. Thanks to his supporters there (such as Abraham Flexner and John von Neumann ) and the help of his wife, the two were able to leave the Third Reich in January 1940 on the Trans-Siberian Railway via the Soviet Union and Japan . The United States was not yet actively involved in World War II at the time. At this point in time, the Soviet Union was allied with Nazi Germany through the Hitler-Stalin Pact .

Princeton life

After entering the USA, Gödel continued his work at the Institute for Advanced Study. Paul Arthur Schilpp (1897-1993) invited him to write a contribution to his volume on Bertrand Russell . Gödel was now more concerned with philosophy, especially with Gottfried Wilhelm Leibniz , later also with Edmund Husserl . So he began to grapple more and more with philosophical problems at Princeton and turn away from formal logic.

In 1942 Gödel got to know Albert Einstein better and began to discuss physical problems such as the theory of relativity and philosophical topics with him . A close friendship developed between Einstein and Gödel, which lasted until Einstein's death in 1955. Together they used to walk to the institute and home. Einstein once said that he only came to the institute "to have the privilege of being able to walk home with Gödel". In addition to a few other acquaintances, Gödel became lonely in the 1940s and 1950s due to his progressive mental illness. He suffered from paranoia and continued to fear that food would poison him. As before, Adele had to taste everything for him.

In 1947 Gödel received US citizenship . The naturalization process required a judicial hearing in which he had to demonstrate knowledge of the country and the constitution. In his preparations for this, Godel discovered that the country's constitution was incomplete in that it would have been possible to establish a dictatorship within the framework of this constitution, despite its individual provisions protecting democracy. Two friends, Albert Einstein and the economist Oskar Morgenstern , accompanied him during the process. Thanks to their help and an enlightened judge, it was possible to avoid Godel getting himself into trouble at the hearing.

It was not until 1953 that he was given a professorship at Princeton, as Hermann Weyl and Carl Ludwig Siegel in particular saw him as unsuitable for his strange behavior. In 1955 he was elected to the National Academy of Sciences , 1957 to the American Academy of Arts and Sciences and 1961 to the American Philosophical Society . In 1972 he became a corresponding member of the British Academy . He stopped giving lectures in the 1960s. His illness left him less and less able to work and participate in social life. Nonetheless, he was still regarded as one of the leading logicians, and was given appropriate academic recognition in the form of awards.

Godel's condition did not improve. In 1970 he tried to publish for the last time. However, the writing had to be withdrawn because he had overlooked many errors due to the effects of psychotropic drugs.

Grave of Adele and Kurt Gödel in Princeton

Last years

Gödel spent the last years of his life at home in Princeton or in various sanatoriums , from which he fled several times. Only the care of his wife, who made sure that he ate at least halfway normally, kept him alive. When Adele Gödel herself was hospitalized in 1977 because of a stroke, she had to watch helplessly as her husband became increasingly emaciated. When she was released six months later - now dependent on a wheelchair - she immediately admitted him to a hospital with a body weight of around 30 kg. Nevertheless, Kurt Gödel died a few weeks later of malnutrition and exhaustion.

Adele Gödel died in 1981. Adele and Kurt Gödel are buried together in Princeton.

Scientific achievements

Research on Hilbert's program

After studying the first edition of the textbook Fundamentals of Theoretical Logic by David Hilbert and Wilhelm Ackermann , Gödel wrote his dissertation on the completeness of the narrower calculus of first-order predicate logic (title of the 1929 dissertation: About the completeness of logic calculus ).

For Gödel, the 1930s were mainly characterized by scientific work, which was initially aimed at the feasibility of the Hilbert program formulated around 1920 . He dealt with the continuum hypothesis and the question of whether arithmetic (the theory of natural numbers ) can be axiomatized completely and without contradictions . These two questions were also the first two of the famous 23 problems , the first ten of which Hilbert had given up in 1900 at the Second International Congress of Mathematicians in Paris at the dawn of the new century.

The continuum hypothesis is the set- theoretical statement that any set that is more powerful than the set of natural numbers is at least as powerful as the set of real numbers , the eponymous continuum. Hilbert was convinced that mathematics - and thus also number theory (arithmetic) and set theory - was complete in the sense that it could ultimately be determined whether a mathematical statement such as the continuum hypothesis was true or not.

The incompleteness sentences

While Gödel's first work could have been regarded as an indication of the feasibility of the project, his most important work, which he published in 1931, was the end of David Hilbert's dream. In the work entitled On formally undecidable theorems of the Principia mathematica and related systems , Gödel proved Gödel's first incompleteness theorem . This means that in a consistent system of axioms , which is rich enough to build up the arithmetic of natural numbers in the usual way, and which is also sufficiently simple, there are always statements that can neither be proven nor refuted from it. Sufficiently simple means that the axiom system is a decidable set. As a second Godel's incompleteness Godel is Corollary referred to the first, after which the consistency of such axiom system is not self-derived from the axiom system.

In particular, all sorts of partial theories of the entire arithmetic - Hilbert wanted to axiomatize the latter completely and consistently - are powerful enough to represent their own syntax and their inference rules. Corresponding axiomatizations are therefore either

  1. not sufficiently simple or
  2. not complete or
  3. not free of contradictions.

In particular, complete and consistent arithmetic is then not sufficiently simple. Hilbert last (around 1930) actually tried an ω-rule, according to which (roughly) correct universal statements should be axioms, apparently in order to save his conviction that it was completely axiomatizable. But as it is, the set of axioms is no longer sufficiently simple.

The proof of the incompleteness theorems is based on a formalization of antinomies of the form I am not speaking the truth now. He formulated this paradox mathematically precisely by looking at the mathematical statements for natural numbers and found that each of these statements can be written as a natural number yourself. This is called Gödel number , and its calculation is then called Gödelization. However, if statements about natural numbers can themselves be understood as natural numbers, then one can formulate self-referential statements of the kind mentioned. This is a variant of Cantor's diagonal method . More precisely, he constructed a provability predicate as a number-theoretic formula Bew ( x ), which becomes true if and only if one replaces the variable x everywhere with a formal representation of the Godel number of a provable theorem of the theory under study. He showed that there is a natural number n with a formal representation N such that n is the Gödel number of the negation of Bew (N). The associated negated formula ¬Bew (N) thus expresses its own unprovability and is neither provable nor refutable in the theory under study if it is free of contradictions.

The second theorem of incompleteness is usually interpreted in such a way that Hilbert's program to prove the consistency of mathematics or at least arithmetic is not feasible and the second problem from Hilbert's list of 23 mathematical problems is unsolvable. However, this conclusion refers to Gödel's natural arithmetic representation of provability, the provability predicate Bew ( x ). With certain artificial modifications of Gödel's provability predicate, the second incompleteness clause no longer applies. Such a modification was first proposed by John Barkley Rosser soon after Godel's publication; in the meantime specialists are trying to clarify what the difference between natural and artificial actually consists of.

Intuitionist logic, demonstrability logic

Hilbert's program was part of general attempts of his time to clarify the fundamentals of mathematics . Hilbert's approach to this, known as formalism , was contrasted by Luitzen Egbertus Jan Brouwer's intuitionism . The philosophical approach of intuitionism found expression as intuitionist logic in the field of mathematical logic , created by Arend Heyting . For Gödel, the intuitionist approach was hardly less interesting than Hilbert's program. Above all, the relationship between intuitionist logic and classical logic has been a captivating subject of investigation for Gödel and for other logicians since then - regardless of whether one viewed oneself philosophically as an intuitionist.

There is a problem of understanding between such mathematicians and those with a classical background. From a classical point of view, mathematicians with an intuitionist orientation use the same words as classically influenced mathematicians - only with a completely different, puzzling meaning. From a classical point of view, an intuitionist only seems to mean A is provable when he says A. However, especially after Gödel's discoveries, truth does not mean provability. Intuitionist mathematicians therefore subject their assertions to stricter requirements than classically shaped ones. An intuitionist does not believe everything that a classically trained mathematician believes.

In 1933, Gödel refuted this limited notion of Heyting's arithmetic in a certain way. Heyting's arithmetic is superficially inferior to classical arithmetic theories, but this appearance disappears when one pays attention to the special role of negation. Gödel gave an interpretation of classical arithmetic in Heyting arithmetic, which assumed that every atomic formula was transformed into its double negation. Gödel showed that the initial formula can be derived in classical Peano arithmetic (limited to one type of variable) if its translation can be derived in Heyting arithmetic. Modulo of this translation one can derive all theorems of classical arithmetic also in Heyting arithmetic.

Godel also supported the idea that an A uttered by an intuitionist can only be interpreted by classical logicians as being provable - in a modified way. Proof can be formally represented by adding modal operators to predicate logic systems , as they are otherwise used for the logic of necessary and possible . Only recently has the idea been thoroughly pursued to examine provability as a variety of necessity (provability logic). Gödel made an early contribution to this line of research by comparing the modal logics for different provability predicates. In this context he gave an interpretation of the intuitionist propositional logic in the modal logic S4 . The translation essentially takes place by inserting the necessity operator in front of each real partial formula. Theorems of intuitionist propositional logic are thus translated into theorems of S4 . In 1948, McKinsey and Tarski confirmed Gödel's mere conjecture that beyond that only theorems of intuitionist propositional logic are translated into S4 theorems .

These two results were published in 1933, two others on intuitionist logic in 1932 and 1958. Stal Anderaa found a gap in Gödel's work on the decision problem of the logical function calculus from 1933 in the mid-1960s and Warren Goldfarb , who proved in 1984 that the corresponding theory (with two all quantifiers followed by any number of existential quantifiers and identity) against Gödel's statement was even undecidable.

Continuum hypothesis

In his study of the continuum hypothesis , Gödel worked with John von Neumann , among others . He tried to prove the independence of the continuum hypothesis from the other axioms of set theory. For this purpose, he worked out an axiomatic set theory with classes, the original version of the Neumann-Bernays-Gödel set theory , which corresponds to the Zermelo-Fraenkel set theory (ZFC) in the set area.

On this basis he proved in 1938 the theorem published in 1940 that one cannot prove the negation of the continuum hypothesis with the ZFC axioms if they are free of contradictions (see constructability axiom ). In 1963 the American Paul Cohen completed this theorem of independence and showed that the continuum hypothesis itself cannot be proven if ZFC is consistent.

This became the first mathematically or fundamentally theoretically relevant example of a statement independent of ZFC (supposedly all of mathematics), the existence of which Gödel had proven with his First Incompleteness Theorem under more general conditions (Gödel's own undecidable arithmetic statement was mathematically uninteresting). This also showed that Hilbert's first problem of 1900 is unsolvable.

Ontological proof of God and philosophy

In his late work in 1941, Gödel undertook a reconstruction of the ontological proof of God using modal logic . Gödel's concern "consisted [...] in the proof that an ontological proof of God could be carried out in a way that does justice to modern logical standards". The evidence, which was only published in 1970 as part of the Collected Works , was checked for formal correctness with the aid of a computer in 2013. Its validity, however, depends on accepting the axioms and definitions used by Gödel as evidence.

Although he was a member of the Vienna Circle, which represented a logical empiricism, he also dealt intensively with philosophy and especially metaphysics , whereby he represented a rational metaphysics with which he wanted to establish different scientific disciplines. It should be interdisciplinary and one of the subjects it was supposed to deal with included theology. According to Eva-Maria Engelen , who publishes his Philosophical Notebooks, he rejected the rigorous logical empiricism of Rudolf Carnap , for example , because for him mathematics is not a syntax of language and theoretical thinking is limited by it. He was a platonist in mathematics because he saw in their constructions a reality independent of the human mind. However, he dealt little with Plato himself, but all the more with Gottfried Wilhelm Leibniz and Thomas Aquinas , but also with Ludwig Wittgenstein , for example , because the entries in his notebook show, according to Engelen, that some of Wittgenstein's later English students were in philosophy Investigations into fixed ideas were probably already discussed in the Vienna Circle in the 1930s. Gödel's philosophical notebooks were kept from 1934 to 1955, but will not be published by De Gruyter until 2019 , since - according to Engelen - they were outside the main mathematical and logical interests of the editors of his works. They served as a reading plan for his philosophical studies, but were also preparation for philosophical work, whereby Gödel found it difficult to publish. There are also other notebooks on mathematics, quantum physics, and general topics from the 1960s, in which he processed his newspaper reading.

In the philosophical notebooks he rarely commented on the events of the time, but there are considerations as to whether it would be morally justifiable to wear a swastika (which he initially considers justifiable if it were a small swastika) and in the Nazi lecturers' association to enter (which he did not). There are indications that Gödel could have known Max Horkheimer's criticism of the consequences of the ethical position of the Vienna Circle . Although he dealt with theology, the Engelen notebooks do not reveal any pronounced spiritual inclinations, so far it cannot even be said whether he was a believer, even if he was thinking about converting to Catholicism.

Godel universe

In 1949 Gödel gave the first solution to the general theory of relativity with closed time-like world lines, which shows that time travel is possible in this theory (see Gödel universe ). His example of a rotating universe wasn't very realistic, however. Nevertheless, the search for a chronology protection mechanism in physics was opened. In 1950 he gave a plenary lecture at the International Congress of Mathematicians in Cambridge (Massachusetts) on his cosmology with the title Rotating universes in general relativity theory .

P-NP problem

In the 1980s it became known that Gödel had already formulated the P-NP problem in a letter to John von Neumann in 1956 and highlighted its great importance.

reception

Godel as namesake

In mathematics and physics, the following are named after Gödel:

The following are also named after Gödel:

In 2016, a planned Gödelgasse near Triester Strasse south of Raxstrasse was officially named in Vienna- Favoriten (10th district) . A Gödelgasse near Vienna Central Station was planned for 2009–2015 ; but this traffic area did not materialize.

Reception in literature

The cognitive scientist Douglas R. Hofstadter achieved an award-winning bestseller with Gödel, Escher, Bach (first published in English in 1979). In the book, Hofstadter combines Kurt Gödel's mathematics with the artistic graphics of MC Escher and the music of Johann Sebastian Bach .

In 1997 John W. Dawson published an authoritative biography of Gödel (German: Kurt Gödel. Life and work , see secondary literature ).

Hans Magnus Enzensberger (poet, writer, editor, translator and editor) published the poem Hommage à Gödel in 1971 , in which he depicts Gödel's incompleteness sentence. Well-known mathematicians confirmed that Hans Magnus Enzensberger's presentation is correct.

In 2011, the Austrian author Daniel Kehlmann brought out his play Geister in Princeton in Salzburg and Graz , which deals with Kurt Gödel (world premiere on September 24, 2011 at the Schauspielhaus Graz , director: Anna Badora ). Kehlmann was awarded the Nestroy Theater Prize in Vienna in 2012 as the author of the best play.

Gödel plays an important role in As far as we know (Berlin 2017), the debut novel by Zia Haider Rahman.

literature

Fonts (selection)

  • On the completeness of the axioms of the logical function calculus . Dissertation, 1929. In: Monthly notebooks for mathematics and physics. Akademische Verlagsgesellschaft, Leipzig 37.1930, 2, pp. 349–360. (Also in: Erg. 3.1932, pp. 12–13)
  • About formally undecidable sentences of the Principia Mathematica and related systems I. In: Monthly books for mathematics and physics. Akademische Verlagsgesellschaft, Leipzig 38.1931, pp. 173–198.
  • Discussion on the foundation of mathematics, knowledge 2. In: Monthly books for mathematics and physics. Akademische Verlagsgesellschaft, Leipzig 39, 1931-32, pp. 147–148.
  • The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. (= Annals of Mathematical Studies. Volume 3). Princeton University Press, Princeton, NJ 1940.
  • Russell's mathematical logic. In: Alfred North Whitehead, Bertrand Russell: Principia Mathematica. Preface, pp. V – XXXIV. Suhrkamp 1986, ISBN 3-518-28193-3 .
  • Solomon Feferman et al. (Ed.): Kurt Gödel. Collected Works . Clarendon Press, Oxford. (The complete collection of all published and unpublished writings by Gödel in German and English)
  • Eva-Maria Engelen (Ed.): Kurt Gödel. Philosophical Notebooks / Philosophical Notebooks: Philosophy I Maximen 0 / Philosophy I Maxims . De Gruyter, Berlin / Munich / Boston. (Bilingual edition in German and English, the volumes appear annually)

The Kurt Gödel Research Center of the Berlin-Brandenburg Academy of Sciences publishes 15 Gödel's Philosophical Notebooks.

Secondary literature

Web links

Commons : Kurt Gödel  - Collection of images, videos and audio files
Wikiquote: Kurt Gödel  - Quotes

English

Newspaper article (German)

Individual evidence

  1. ↑ 64% of the 109,000 inhabitants were German-speaking and 36% Czech. Source: AL Hickmann's geographic-statistical pocket atlas of Austria-Hungary. 3. Edition. G. Freytag & Berndt, Vienna / Leipzig 1909.
  2. ^ John D. Dawson: Logical Dilemmas. Springer Verlag, 1997, p. 15. Dawson quotes a letter from Harry Klepetař (Gödel's classmate) to Dawson from 1983, in which Klepetař also reported that he had never heard Gödel speak a word of Czech. Dawson adds, however, that Godel could probably speak Czech.
  3. ↑ Interview with Eva-Maria Engelen, Der neue Aristoteles, Frankfurter Allgemeine Sonntagszeitung, January 5, 2020, p. 56 https://www.faz.net/aktuell/wissen/computer-mathematik/die-philosophischen-notizen-kurt- goedels-16565151.html? printPagedArticle = true # pageIndex_2
  4. ↑ Kurt Gödel in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  5. ↑ Guerrerio: Kurt Gödel. P. 34.
  6. ↑ Guerrerio: Kurt Gödel. P. 72.
  7. ↑ Guerrerio: Kurt Gödel. P. 71.
  8. ↑ Guerrerio: Kurt Gödel. P. 74.
  9. ↑ a b on this Yourgrau 2005.
  10. ↑ Goldstein 2006.
  11. ↑ Tagesspiegel: Das Genie und der Wahnsinn , accessed on July 9, 2017
  12. ^ Morgenstern on Goedel citizenship.pdf. (PDF) Retrieved March 10, 2019 .
  13. ↑ Jaakko Hintikka: On Gödel. 2000, p. 9.
  14. ↑ Member History: Kurt Gödel. American Philosophical Society, accessed August 23, 2018 .
  15. ^ Deceased Fellows. British Academy, accessed June 1, 2020 .
  16. ↑ The genius & the madness In: Der Tagesspiegel. January 13, 2008.
  17. ↑ The foundation of the elementary theory of numbers. In: Mathematical Annals. 104, pp. 485-494; Proof of the Tertium non datur , Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1931, pp. 120–125.
  18. ↑ Cf. e.g. Karl-Georg Niebergall: on the metamathematics of non-axiomatizable theories. CIS University of Munich, Munich 1996, ISBN 3-930859-04-1 .
  19. ↑ The following presentation is based on that in the Stanford Encyclopedia of Philosophy .
  20. ^ George Boolos : The Logic of Provability. Cambridge University Press, Cambridge (England) 1993, ISBN 0-521-43342-8 .
  21. ↑ monthly Math. Phys. 40, 1933, pp. 433-443.
  22. ↑ Warren D. Goldfarb: The Godel class with identity is unsolvable . In: Bulletin of the American Mathematical Society . tape 10 , no. 1 , 1984, p. 113–115 , doi : 10.1090 / S0273-0979-1984-15207-8 (commentary by Goldfarb in Gödel's collected works).
  23. ↑ J. Floyd, A. Kanamori: How Gödel Transformed Set Theory. In: Notices of the AMS. 53 (2006), p. 424 ( ams.org PDF).
  24. ↑ Modern representation, for example in K. Kunen: Set Theory. North-Holland, Amsterdam 1980, Chapter VI, ISBN 0-444-85401-0 .
  25. ↑ See Kurt Gödel: Ontological proof. In: Kurt Gödel: Collected Works Vol. 3: Unpublished Essays and Letters . Oxford University Press, 1970. and Kurt Gödel, Appendix A. Notes in Kurt Godel's Hand, in: JH Sobel: Logic and Theism: Arguments for and Against Beliefs in God. Cambridge University Press, 2004, pp. 144-145.
  26. ^ André Fuhrmann: Logic in Philosophy . Ed .: W. Spohn. Synchron, Heidelberg 2005, Existence and Necessity - Kurt Gödel's axiomatic theology, p. 349-374 ( online [PDF]). Logic in Philosophy ( Memento of the original from May 18, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  @1@ 2Template: Webachiv / IABot / dl.dropboxusercontent.com
  27. ↑ Joachim Bromand: Evidence of God from Anselm to Gödel (=  Suhrkamp Taschenbuch Wissenschaft ). 1st edition. Suhrkamp, ​​Berlin 2011, ISBN 978-3-518-29546-5 , pp. 393 .
  28. ↑ Christoph Benzmüller, Bruno Woltzenlogel Paleo: Formalization, Mechanization and Automation of Gödel's Proof of God's Existence . In: IOS Press Ebooks - Automating Gödel's Ontological Proof of God's Existence with Higher-order Automated Theorem Provers (=  Frontiers in Artificial Intelligence and Applications ). 2013, p. 93-98 , doi : 10.3233 / 978-1-61499-419-0-93 ( arxiv.org [PDF]). ; Th. Gawlick: What are and what should mathematical proofs of God be? (with autograph of Gödel's Ontological Proof, PDF; 520 kB).
  29. ↑ Interview with Eva-Maria Engelen in Frankfurter Allgemeine Sonntagzeitung, January 5, 2020, p. 56, Der neue Aristoteles
  30. ↑ This can be found in the announced second volume of the philosophical notebooks, Engelen, FAS, January 5, 2020, p. 56
  31. ^ Reviews of Modern Physics. Volume 21, 1949, 447, as well as in Schilpp (Ed.) Albert Einstein. 1955. Gödel proved that in this model the energy expenditure necessary for time travel was unrealistically high, but the possibility of communication remained open and was for Gödel a possible explanation for ghostly apparitions (Kreisel 1980, p. 155);
    Ellis on Gödel's work on cosmology, in Petr Hajek (ed.): Gödel 96 , 1996 projecteuclid.org ( memento of the original from January 10, 2017 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. . @1@ 2Template: Webachiv / IABot / projecteuclid.org
  32. ↑ John Dawson: Kurt Gödel - Life and Work. Springer Verlag, 1997, p. 177, there the letter is quoted.
  33. ^ The Gödel Letter, Lipton's blog , with English translation.
  34. ↑ K. Gödel: An example of a new type of cosmological solution of Einstein's field equations of gravitation . In: Rev. Mod. Phys. tape 21 , 1949, pp. 447-450 , doi : 10.1103 / RevModPhys.21.447 .
  35. ↑ 3366 Gödel 3366 Godel (1985 SD1) JPL Small-Body Database Browser (accessed April 23, 2010).
  36. ^ Hans Magnus Enzensberger: Poems. 1955–1970 , Suhrkamp Taschenbuch, Frankfurt am Main 1971, ISBN 3-518-06504-1 , pp. 168f .; online: sternenfall.de: Enzensberger - Homage to Gödel
  37. ^ Reason for the award on the website of the award
  38. ↑ Kurt Gödel Research Center: The "Philosophical Comments Kurt Gödels" , 2019
  39. ^ Website of the Kurt Gödel Research Center in English. Retrieved May 22, 2019 .
This version was added to the list of articles worth reading on December 22, 2005 .
personal data
SURNAME Godel, Kurt
ALTERNATIVE NAMES Gödel, Kurt Friedrich (full name)
BRIEF DESCRIPTION Austrian-American mathematician and logician
DATE OF BIRTH April 28, 1906
PLACE OF BIRTH Brno , Austria-Hungary , today Brno , Czech Republic
DATE OF DEATH January 14, 1978
Place of death Princeton , New Jersey , United States