كورت غودل

"Godel" و "Gödel" يـُحوّلا إلى هنا. للاستخدامات الأخرى, انظر Godel (توضيح).

كورت گودل Kurt Gödel
كورت گودل في 1925
وُلِدَ	(1906-04-28)أبريل 28, 1906 برنو، موراڤيا، النمسا-المجر
توفي	يناير 14, 1978(1978-01-14) (aged 71) پرنستون، نيوجرزي، الولايات المتحدة
الجنسية	Austria Czechoslovakia Germany United States
المدرسة الأم	جامعة ڤيينا (PhD, 1930)
اللقب	Gödel's incompleteness theorems Gödel's completeness theorem Gödel's constructible universe Gödel metric (closed timelike curve) Gödel logic Gödel–Dummett logic Gödel's β function Gödel numbering Gödel operation Gödel's speed-up theorem Gödel's ontological proof Gödel–Gentzen translation Gödel–McKinsey–Tarski translation Von Neumann–Bernays–Gödel set theory ω-consistent theory The consistency of the continuum hypothesis with ZFC Axiom of constructibility Compactness theorem Condensation lemma Diagonal lemma Dialectica interpretation Ordinal definable set Slingshot argument
الزوج	Adele Nimbursky ​ (m. 1938)​
الجوائز	جائزة ألبرت أينشتاين (1951) ForMemRS (1968) National Medal of Science (1974)
السيرة العلمية
المجالات	الرياضيات، المنطق الرياضي، analytic philosophy, physics
الهيئات	معهد الدراسات المتقدمة
أطروحة	Über die Vollständigkeit des Logikkalküls (1929)
المشرف على الدكتوراه	هانز هان
طلاب الدكتوراه	لا أحد
التوقيع

كورت گودل (Kurt Gödel ؛ /ˈɡɜːrdəl/ GUR-dəl, ألمانية: [kʊʁt ˈɡøːdl̩]  ( listen)؛ عاش 28 أبريل 1906 - 14 يناير 1978) منطقي، رياضي، فيلسوف. 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, Alfred North Whitehead, 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 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. 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.

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.

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 النشأة والتعليم

 الطفولة

ولد في برون، مورافيا في ما كان يعرف باسم النمسا-المجر، بعد تفكك تلك المملكة أصبح گودل تشيكياً في عمر 12، ثم أصبح نمساويا في عمر 23 وبدخول هتلر إلى النمسا وضمها إلى ألمانيا أصبح غودل مواطنا ألمانيا في عمر 32. بعد انتهاء الحرب العالمية الثانية، سافر گودل إلى الولايات المتحدة حيث أصبح مواطنا أمريكيا في عمر 42.

كورت جودل حل كثيراً من المشكلات الرئيسة في المنطق الرياضي. انضم إلى جامعة ڤيينا في الثلاثينيات، وكان واحداً من أعضاء جماعة ڤيينا إلى جانب فلاسفة وعلماء ورياضيين آخرين أمثال كارناپ Carnap وغيره. وقد شارك زملاءَه في رفضهم للميتافيزيقا وفهمهم الفلسفة على أنها منطق العلم والتحليل المنطقي لمفاهيمه ونظرياته. وهذا التحليل هو البناء المنطقي للغة العلم.

هاجر إلى الولايات المتحدة في عام 1940 بعد ملاحقة الحكومة النمسوية النازية لكل دعاة هذه الجماعة، ومقتل مؤسسها موريتس شليك Moritz Schlick. أصبح عضواً في «معهد العلوم المتقدمة» Institut for Advanced Study في جامعة پرنستون، ودرّس فيه المنطق الحديث والرياضيات. وقد أخضع للفحص جميع أنساق التعريفات الرياضية، وبرهن على عدم اكتمال الأنساق الشكلية axiomatic التي تفترض الصياغة الشكلية لعلم حساب الأعداد الطبيعية. وقد أثبت أن ما من نسق من هذه الأنساق يحتوي في ذاته دليل تماسكه consistency، بمعنى أن هذه الأنساق تحتوي على قضايا داخل إطارها، مما لا يمكن البرهنة عليه، ولا يمكن دحضه على السواء، وقد تسبب الفرض الذي قدمه غودل في قيام البحث في حدود الأنساق الشكلية على أيدي ألونزو تشيرش Alonzo Church وستيفن كول Stephen Kol وألفرد تارسكي Alfred Tarski وموستوفسكي Mostovsky وغيرهم، وهو البحث الذي بلغ ذروته في الاستنباط الفلسفي الأساسي القائل إن الصياغة الشكلية بطريقة كاملة للمعرفة العلمية مستحيلة.

وبدءاً من صياغة منطقية لبنية الحساب، وهي ما سماه الميتارياضيات meta- mathematics، أو علم ما وراء الرياضيات، برهن غودل في نظريتين مشهورتين له، على أنه يستحيل إثبات عدم تناقض الحساب؛ لأن هذا الحساب يتضمن منطوقات، «غير بتّية»؛ أي يمتنع إثبات كونها قابلة للبرهان أو للدحض، ولأن هذه النتائج تنطبق على جميع الأنساق الرياضية المتقدمة؛ فإن نظرية غودل ترسم علامة استفهام وشك فيما يتعلق بالأنساق المنطقية والصورية لدى ديفيد هلبرت وبرتراند رسل.

قصر غودل جهوده على المنطق التكويني أو الاستدلالي constructive logic، فضلاً عن علم ما وراء الرياضيات أيضاً، وهو فرع من فروع المنطق الحديث يدرس الحسابات المنطقية، إذ تمكّن، اعتماداً على علم الحساب الخاص به، من تحليل سيرورات معينة ترتبط بالمتغيرات، مثل الاستبدال على سبيل المثال. وقد طمح غودل من ذلك إلى تسهيل أسس المنطق الرياضي، واستبعاد التناقضات. وأسهم أيضاً في تطوير نظرية العوامل الارتدادية. وهذا الإلحاح على النزعة العلمية، واعتماد اللغة جزءاً من الحساب قد أديا به إلى رفض الماورائيات.

له دراسات ومقالات عدة في مجال اللغة والمنطق الرياضي والتحليل المنطقي، تم جمعها في مجلدين صدرا عن جامعة أوكسفورد في عام 1986 وعام 1990. وكان لآراء گودل الفلسفية أثر كبير في تطور الفلسفة الوضعية الجديدة neo-positivism التي أنتجت تيار الفلسفة التحليلية اللغوية.

 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

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.

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.

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).

The idea at the center of the incompleteness theorem is simple. 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.

In his two-page paper Zum intuitionistischen Aussagenkalkül (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof, he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

 Mid-1930s: further work and U.S. visits

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

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. 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 es  (Adele Gödel) (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, 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.

 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.

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".

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.

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.

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. He is said to have given this elaboration to Einstein as a present for his 70th birthday. 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. 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.

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 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. Gödel was elected a resident member of the American Philosophical Society in 1961 and a Foreign Member of the Royal Society (ForMemRS) in 1968. He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.

 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, 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; he weighed 29 kilograms (65 lb) when he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery. Adele died in 1981.

 Religious views

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

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]."

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

 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.

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 مايو 14, 2019 at the Wayback Machine are edited at the Kurt Gödel Research Centre Archived مايو 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 John Dawson published a biography of Gödel, Logical Dilemmas: The Life and Work of Kurt Gödel (A. K. Peters, Wellesley, MA, ISBN 1-56881-256-6). Stephen Budiansky's book about Gödel's life, Journey to the Edge of Reason: The Life of Kurt Gödel (W. W. Norton & Company, New York City, NY, ISBN 978-0-393-35820-9), was a New York Times Critics' Top Book of 2021.

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

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:

• 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
• 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470–85.
• 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 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 by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
• Kurt Gödel, 2000. 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).
  • 1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596–616.
  • 1931a. "On completeness and consistency," 616–17.
• 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: de (Eva-Maria Engelen).
  • Volume 1: Philosophie I Maximen 0 / Philosophy I Maxims 0 ISBN 978-3-11-058374-8.
  • Volume 2: Zeiteinteilung (Maximen) I und II / Time Management (Maxims) I and II ISBN 978-3-11-067409-5.
  • Volume 3: Maximen III / Maxims III ISBN 978-3-11-075325-7

 See also

• Original proof of Gödel's completeness theorem
• Gödel fuzzy logic
• Gödel–Löb logic
• Gödel Prize
• Gödel's ontological proof
• Infinite-valued logic
• List of Austrian scientists
• List of pioneers in computer science
• Mathematical Platonism
• Primitive recursive functional
• Strange loop
• Tarski's undefinability theorem
• World Logic Day
• Gödel machine

 Notes

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 References

• Dawson, John W (1997), Logical dilemmas: The life and work of Kurt Gödel, Wellesley, MA: AK Peters, https://archive.org/details/logicaldilemmasl0000daws .
• Goldstein, Rebecca (2005), Incompleteness: The Proof and Paradox of Kurt Gödel, New York: W.W. Norton & Co, ISBN 978-0-393-32760-1, https://books.google.com/books?id=tXk2AAAAQBAJ .
• Wang, Hao (1987), Reflections on Kurt Gödel, Cambridge: MIT Press, ISBN 0-262-73087-1, https://books.google.com/books?id=wLLePwhDOMYC 
• Wang, Hao (1996), A Logical Journey: From Gödel to Philosophy, Cambridge: MIT Press, ISBN 0-262-23189-1, https://books.google.com/books?id=pckvCy6L_ocC 

 Further reading

• Stephen Budiansky, 2021. Journey to the Edge of Reason: The Life of Kurt Gödel. W.W. Norton & Company.
• Casti, John L; DePauli, Werner (2000), Gödel: A Life of Logic, Cambridge, MA: Basic Books (Perseus Books Group), ISBN 978-0-7382-0518-2 .
• Dawson, John W Jr (1996), Logical Dilemmas: The Life and Work of Kurt Gödel, AK Peters .
• Dawson, John W Jr (1999), "Gödel and the Limits of Logic", Scientific American 280 (6): 76–81, doi:10.1038/scientificamerican0699-76, PMID 10048234, Bibcode: 1999SciAm.280f..76D .
• Franzén, Torkel (2005), Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, Wellesley, MA: AK Peters .
• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
• Hämeen-Anttila, Maria (2020). Gödel on Intuitionism and Constructive Foundations of Mathematics (Ph.D. thesis). Helsinki: University of Helsinki. ISBN 978-951-51-5922-9.
• Jaakko Hintikka, 2000. On Gödel. Wadsworth.
• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
• Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint c. 2001.
• Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. ISBN 978-0-923891-57-2)
• J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
• Ernest Nagel and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
• Ed Regis, 1987. Who Got Einstein's Office? Addison-Wesley Publishing Company, Inc.
• Raymond Smullyan, 1992. Godel's Incompleteness Theorems. Oxford University Press.
• Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.
• Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
• Yourgrau, Palle, 2004. A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books. ISBN 978-0-465-09293-2. (Reviewed by John Stachel in the Notices of the American Mathematical Society (54 (7), pp. 861–68).

 وصلات خارجية

• Weisstein, Eric W., Gödel, Kurt (1906–1978) at ScienceWorld.
• قالب:Cite SEP
• Time Bandits: an article about the relationship between Gödel and Einstein by Jim Holt
• Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue
• Paul Davies and Freeman Dyson discuss Kurt Godel (transcript)
• "Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
• It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.
• Gödel photo gallery. (archived)
• Kurt Gödel MacTutor History of Mathematics archive page
• National Academy of Sciences Biographical Memoir

 انظر أيضاً

بوابة المنطق

• گودل، إشر، باخ