Kurt Gödel

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"Gödel" redirects here. For the programming language, see Gödel (programming language). For other uses, see Godel (disambiguation).
Kurt Gödel
Kurt gödel.jpg
Born Kurt Friedrich Gödel
(1906-04-28)April 28, 1906
Brünn, Austria-Hungary
(now Brno, Czech Republic)
Died Script error: The function "death_date_and_age" does not exist.
Princeton, New Jersey, U.S.
Residence United States
Citizenship Austria, United States
Fields Mathematics, Mathematical logic
Institutions Institute for Advanced Study
Alma mater University of Vienna
Thesis Über die Vollständigkeit des Logikkalküls
Doctoral advisor Hans Hahn
Known for Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel metric, Gödel's ontological proof
Notable awards <templatestyles src="Plainlist/styles.css"/>
Signature

Kurt Friedrich Gödel (/ˈkɜːrt ɡɜːrdəl/;[2] German: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[3] A. N. Whitehead,[3] and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these 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 ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh).[4] At the time of his birth the city had a German-speaking majority[5] which included his parents.[6] The marriage of his parents was confessionally mixed, his father being a Catholic and his mother a Protestant. The children were raised in the Protestant confession. The ancestors of Kurt Gödel were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".[7]

Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[8] He chose to become an Austrian citizen at age 23[citation needed]. When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as 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.

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 Kurt 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 to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.[9] Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. 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."[10]

Attending a lecture by David Hilbert in Bologna on completeness and consistency of 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 became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

Career

The Incompleteness Theorem

"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[11]

In 1931 and while still in Vienna, 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 the system is consistent, it cannot be complete.
  2. The consistency of the axioms cannot be proven within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all 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, which contradicts the idea that in a consistent system, provable statements are always true. 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 intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

The mid-1930s: further work and visits to the US

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 Johann Nelböck, a former student of Moritz Schlick. This triggered "a severe nervous crisis" in Gödel.[12] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[13]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.[14] 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, entitled 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 traveling 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 (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he was.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Relocation to Princeton, Einstein and U.S. citizenship

After the Anschluss in 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 U.S. by train to Princeton, where Gödel accepted a position at the Institute for Advanced Study (IAS).

Gödel very quickly resumed his mathematical work. In 1940, he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, which is a classic of modern mathematics.[citation needed] 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 that 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.

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".[15]

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. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. Fortunately, 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.[16][17]

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.[18]

During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein's field equations in general relativity.[19] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[20] 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.[21] 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.[citation needed] Gödel was elected a Foreign Member of the Royal Society (ForMemRS) in 1968.[1]

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

Later life and death

Later in his life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.[22] He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.[23] Adele's death followed in 1981.

Personal life

Religious views

Gödel was a convinced theist, in the Christian tradition.[24] He held the notion that God was personal, which differed from the religious views of his friend Albert Einstein.

He believed firmly in an afterlife, stating: "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]."[25]

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."[26] Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not".[27] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[28] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded".[29]

Legacy

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of 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 invited an annual Kurt Gödel lecturer each year since 1990.

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

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled Dangerous Knowledge by David Malone.[30]

Douglas Hofstadter wrote a popular book in 1979 called Gödel, Escher, Bach to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book 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.

Gödel is the main character of Daniel Kehlmann's play Ghosts in Princeton (2011).

The character Basyli in Quentin Canterel's "The Jolly Coroner" (2015) is largely based on Gödel.

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, 1: 175–81

In English translation:

See also

Notes

  1. ↑ 1.0 1.1 Lua error in package.lua at line 80: module 'strict' not found.
  2. ↑ "Gödel". Collins English Dictionary.
  3. ↑ 3.0 3.1 For instance, in their Principia Mathematica (Stanford Encyclopedia of Philosophy edition).
  4. ↑ Dawson 1997, pp. 3–4
  5. ↑  Lua error in package.lua at line 80: module 'strict' not found.
  6. ↑ Dawson 1997, p. 12
  7. ↑ Procházka 2008, pp. 30–34.
  8. ↑ Dawson 1997, p. 15.
  9. ↑ Dawson 1997, p. 24.
  10. ↑ Gleick, J. (2011) The Information: A History, a Theory, a Flood, London, Fourth Estate, p181.
  11. ↑ Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
  12. ↑ Lua error in package.lua at line 80: module 'strict' not found.. 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.
  13. ↑ Dawson 1997, pp. 110–112
  14. ↑ Hutchinson Encyclopedia (1988), p.518
  15. ↑ Goldstein (2005), p. 33.
  16. ↑ Dawson 1997, pp. 179–180. 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.
  17. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  18. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  19. ↑ An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation, Rev. Mod. Phys. 21, 447, published July 1, 1949 [1].
  20. ↑ Das Genie & der Wahnsinn, Der Tagesspiegel, January 13, 2008 (in German).
  21. ↑ John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. A K Peters, Ltd., 2005. P. 166.
  22. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  23. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  24. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  25. ↑ Hao Wang, "A Logical Journey: From Gödel to Philosophy", 1996, pp. 104–105.
  26. ↑ 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".
  27. ↑ Wang 1996 p. 316
  28. ↑ Wang 1996, p. 51.
  29. ↑ 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.
  30. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  31. ↑ Lua error in package.lua at line 80: module 'strict' not found.

References

  • Dawson, John W., 1997. Logical dilemmas: The life and work of Kurt Gödel. Wellesley MA: A K Peters.
  • 1911 Encyclopædia Britannica/Brünn. (September 19, 2007). In Wikisource, The Free Library. Retrieved 10 pm EST March 13, 2008.
  • Rebecca Goldstein, 2005. Incompleteness: The Proof and Paradox of Kurt Gödel. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.

Further reading

  • John L. Casti and Werner DePauli, 2000. Gödel: A Life of Logic, Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.
  • John W. Dawson, Jr. Logical Dilemmas: The Life and Work of Kurt Gödel. AK Peters, Ltd., 1996.
  • John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", Scientific American, vol. 280 num. 6, pp. 76–81
  • Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
  • Jaakko Hintikka, 2000. On Gödel. Wadsworth.
  • Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage.
  • Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint ca. 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.
  • Procházka, Jiří, 2006, 2006, 2008, 2008, 2010. Kurt Gödel: 1906–1978: Genealogie. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl. Volume V,Brno,Princeton 2010, ISBN 80-903476-9-X. In Germ., Engl.
  • Procházka, Jiří, 2012. "Kurt Gödel: 1906–1978: Historie". ITEM,Brno, Wien, Princeton. Volume I. ISBN 978-80-903476-2-5. In Ger., Engl.
  • 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.
  • Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press.
  • Hao Wang, 1996. A Logical Journey: From Godel to Philosophy. MIT Press.
  • 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. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), pp. 861–868):

External links

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