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[[Image:Goel.jpg|thumb|right|Kurt Godel at Institute for Advanced Study]]
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[[Image:Goel.jpg|thumb|right|Kurt Gödel at Institute for Advanced Study]]
'''Kurt Gödel''' ([[International Phonetic Alphabet|IPA]]: {{IPA|[kuɹt gøːdl]}}) ([[April 28]], [[1906]] [[Brno|Brünn]], [[Austria-Hungary]] (now Brno, Czech Republic) – [[January 14]], [[1978]] [[Princeton, New Jersey]]) was an [[Austria]][[United States|American]] [[mathematician]] and [[philosopher]].
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'''Kurt Gödel''' (1906-1978) was an Austrian-born mathematician who is considered perhaps the greatest logician since [[Aristotle]].  His Incompleteness Theorem was a stunning proof of limitations on logic, at a time when leading mathematicians were working diligently to establish its completeness.  He immigrated to the [[United States]] and worked at the Institute for Advanced Study at Princeton, New Jersey, and is buried in that town with his wife.  Gödel was a devout [[Christian]] who believed in an afterlife and read the [[Bible]] regularly.<ref>https://www.quora.com/What-led-Kurt-G%C3%B6del-to-become-a-Christian</ref>  He completely rejected the [[atheism]] of the [[Vienna Circle]] to which he was introduced.
  
One of the most significant logicians of all time, Gödel's work has had immense impact upon scientific and philosophical thinking in the [[20th century]], a time when many, such as [[Bertrand Russell]], [[A. N. Whitehead]] and [[David Hilbert]], were attempting to use [[logic]] and [[set theory]] to understand the foundations of [[mathematics]].
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Gödel was ostracized from academia despite his immense intellectual achievements, and was never offered a tenure-track position of professor.<ref>Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.</ref>  He worked on developing a logical proof of [[God]] that build on the ontological proof by [[Saint Anselm]]. Politically, he supported Republican President [[Dwight Eisenhower]] and was critical of the [[Democrat]] [[Harry Truman]].<ref>https://www.usna.edu/Users/math/meh/godel.html</ref>
  
Gödel is best known for his two [[Gödel's incompleteness theorems|incompleteness theorems]], published in 1931 when he was 25 years of age, one year after finishing his doctorate at the [[University of Vienna]]. The more famous incompleteness theorem states that for any self-consistent [[Recursive set|recursive]] [[axiomatic system]] powerful enough to describe the arithmetic of the [[natural number]]s ([[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 number]]ing, which codes formal expressions as natural numbers.
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Gödel published his remarkable proof of the Incompleteness Theorem in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete.  Moreover, the consistency of the axioms cannot be proven within the system.
  
He also showed that the [[continuum hypothesis]] cannot be disproved from the accepted [[axiomatic set theory|axioms of set theory]], if those axioms are consistent. He made important contributions to [[proof theory]] by clarifying the connections between [[classical logic]], [[intuitionistic logic]], and [[modal logic]].
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Gödel's work abruptly 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 first-order axioms for all of mathematics that is both provably consistent as well as complete. [[Bertrand Russell]] had already published, in ''Principia Mathematica'' (1910–13), a massive attempt to axiomatize mathematics in a consistent way.  Gödel's proof also showed that the formalist approach of [[David Hilbert]] was bound to fail to prove consistency.
  
== Life ==
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Gödel took [[set theory]], as originally developed by [[Georg Cantor]], to new heights, as explained by Boston University Professors Juliet Floyd and Akihiro Kanamori in "How Gödel Transformed Set Theory":
=== Childhood ===
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{{cquote|How Gödel transformed set theory can be broadly cast as follows: On the larger stage, from the time of Cantor, sets began making their way into topology, algebra, and analysis so that by the time of Gödel, they were fairly entrenched in the structure and language of mathematics. But how were sets viewed among set theorists, those investigating sets as such? Before Gödel, the main concerns were what sets are and how sets and their axioms can serve as a reductive basis for mathematics. Even today, those preoccupied with ontology, questions of mathematical existence, focus mostly upon the set theory of the early period. After Gödel, the main concerns became what sets do and how set theory is to advance as an autonomous field of mathematics.<ref>www.ams.org/notices/200604/fea-kanamori.pdf
Kurt Friedrich Gödel was born [[April 28]], [[1906]], in [[Brno]] ([[German language|German]]: ''Brünn''), [[Moravia]], [[Austria-Hungary]] (now the [[Czech Republic]]) into the [[ethnic German]] family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight [[German language|German-speaking]] majority and this was the language of his parents.  
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</ref>}}
  
He automatically became a [[Czechoslovakia|Czechoslovak]] citizen at age 12 when the Austro-Hungarian empire broke up at the end of [[World War I]]. He later told his biographer John D. Dawson that he felt like an "exiled Austrian in Czechoslovakia" ("''ein österreichischer Verbannter in Tschechoslowakien''") during this time. He spoke very little [[Czech language|Czech]]. He became an [[Austria]]n citizen by choice at age 23. When [[Nazi Germany]] [[Anschluss|annexed Austria]], Gödel automatically became a [[Germany|German]] citizen at age 32. After [[World War II]], at the age of 42, he became an [[United States|American]] citizen.
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The [[Gödel's incompleteness theorems|incompleteness theorems]] also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.  
  
In his family, young Kurt was known as ''Der Herr Warum'' ("Mr. Why") because of his insatiable curiosity.
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Gödel's proof was a landmark for mathematics, and demonstrated that it can never be a finished project.
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.
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He attended German language primary and secondary school in Brno and completed them with honors in [[1923]]. 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]] (UV). During his teens, Kurt studied [[Gabelsberger shorthand]], [[Johann Wolfgang von Goethe|Goethe]]'s ''[[Theory of Colours]]'' and criticisms of [[Isaac Newton]], and the writings of [[Immanuel Kant]].
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Among Gödel's other remarkable achievements:  the first to discover a solution to the equation for general relativity in which there are closed, time-like curves. This means it is mathematically possible for there to be universes in which one can go back in time (provided one has enough fuel and time—something probably not physically possible).  Gödel was also the first to recognize the significance of the P=NP problem, in a letter he wrote to [[John von Neumann]] in 1956.<ref>http://rjlipton.wordpress.com/the-gdel-letter/</ref>
  
=== Studying in Vienna ===
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A profoundly religious man who reportedly read the [[Bible]] every morning, Gödel is also noted for giving Gödel's Ontological Proof,<ref>[http://www.stats.uwaterloo.ca/~cgsmall/Godel.final.revision.PDF A paper about Godel's ontological proof of the existence of God].  Godel did not publish his proof of the existence of God until 1971.</ref> an attempt to make [[Saint Anselm|Anselm's]] [[ontological argument]] into a completely logically rigorous argument.  This had the useful property of making very explicit and precise the assumptions necessary for one to accept the ontological argument.
At the age of 18, Kurt joined his brother Rudolf in Vienna and entered the UV. By that time, he had already mastered university-level mathematics. Although initially intending to study [[theoretical physics]], Kurt also attended courses on mathematics and philosophy. During this time, he adopted ideas of [[mathematical realism]].  He read [[Immanuel Kant|Kant]]'s ''Metaphysische Anfangsgründe der Naturwissenschaft'', and participated in the [[Vienna Circle]] with [[Moritz Schlick]], [[Hans Hahn]], and [[Rudolf Carnap]]. Kurt 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'', Kurt became interested in [[mathematical logic]].
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In this proof, Gödel demonstrated that if one accepts only five modest and seemingly obvious axioms, it is necessary to conclude that God exists.
  
Attending a lecture by [[David Hilbert]] in [[Bologna]] on completeness and consistency of mathematical systems may have set Gödel's life course.
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== Sources ==
In [[1928]], Hilbert and [[Wilhelm Ackermann]] published ''Grundzüge der theoretischen Logik'' ([[Principles of Theoretical 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?''
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This was the topic chosen by Gödel for his doctorate work.
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In 1929, at the age of 23, he completed his doctoral [[dissertation]] under [[Hans Hahn]]'s supervision. In it, Gödel established the completeness of the [[first-order predicate calculus]] (this result is known as [[Gödel's completeness theorem]]). He was awarded the doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.
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=== Working in Vienna ===
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*[http://www.answers.com/topic/kurt-g-del Biography from answers.com]
In 1931, Gödel published his famous 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 that for any [[Recursion theory|computable]] [[axiomatic system]] that is powerful enough to describe the arithmetic of the [[natural numbers]] (e.g. the [[Peano axioms]] or [[ZFC]]), then:
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*[http://www.usna.edu/Users/math/meh/godel.html Insightful biography from the U.S. Naval Academy]
# If the system is consistent, it cannot be complete. (This is generally known as ''the'' [[Gödel's incompleteness theorems|incompleteness theorem]].)
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# The consistency of the axioms cannot be proved within the system.
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These theorems ended a half-century of attempts, beginning with the work of [[Frege]] and culminating in [[Principia Mathematica]] and  [[philosophy of mathematics#formalism|Hilbert's formalism]], to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.
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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 fact that provable statements are always true.
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== References ==
Thus there will always be at least one true but unprovable statement.
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That is, for any [[recursion theory|humanly constructible]] set of axioms for arithmetic, there is a formula which obtains in arithmetic, but which is not provable in that system.
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To make this precise, however, Gödel needed to solve several technical issues, such as encoding statements, proofs, and the very concept of provability into the natural numbers. He did this using a process known as [[Gödel number]]ing.
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Gödel earned his [[habilitation]] at the UV in [[1932]], and in 1933 he became a ''[[Privatdozent]]'' (unpaid lecturer) there. Hitler's 1933 ascension in Germany had little effect on Gödel in Vienna, as he took little interest in politics. He was, however, much affected by the 1936 murder of [[Moritz Schlick]] (whose seminar had aroused Gödel's interest in logic) by a deranged student, which resulted in Gödel's first [[nervous breakdown]].
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<references/>
  
=== Visits to the USA ===
 
In 1933, Gödel first traveled to the [[United States|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 function]]s to the point where he delivered 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|Princeton]], [[New Jersey]], entitled ''On undecidable propositions of formal mathematical systems''. [[Stephen Kleene]], who had just completed his Ph.D. at Princeton, took notes of these lectures which have been subsequently published.
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[[Category:Mathematicians|Godel, Kurt]]
 
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Gödel would visit the IAS again in the autumn of [[1935]]. The traveling and the hard work had exhausted him and the next year he had to recover from a depression. 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 would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
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He married [[Adele Nimbursky]] (née Porkert), whom he had known for over 10 years, on September 20, [[1938]].
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Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he.
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They had no children.
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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]].
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=== Princeton ===
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After the [[Anschluss]] in 1938, Austria had become a part of [[Nazi Germany]].
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Germany abolished the title of ''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.
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His predicament precipitated when he was found fit for military service and was now at risk of being conscripted into the German army.
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[[World War II]] started in September 1939.
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In January [[1940]], Gödel and his wife left Europe.  Due to the difficulty of an Atlantic crossing, they took the [[trans-Siberian railway]] and passed through [[Japan]] en route to the [[United States|U.S.]]. Arriving in [[San Francisco, California]] on March 4, 1940, they crossed the U.S. by train so that Gödel could take up a position at the [[Institute for Advanced Study|IAS]] in [[Princeton, New Jersey]].
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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. 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 [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] (GCH) are true in the constructible universe, and therefore must be consistent with the [[Zermelo-Frankel axioms]] for set theory (ZF). [[Paul Cohen (mathematician)|Paul Cohen]] later constructed a [[structure (mathematical logic)|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.
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During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of [[Gottfried Leibniz]], but came around to the (unsupported) belief that most of Leibniz's works had been suppressed.  To a lesser extent he studied [[Kant]] and [[Edmund Husserl]]. In the early 1970s, Gödel circulated among his friends an elaboration of [[Leibniz]]'s [[ontological argument|ontological proof]] of [[God]]'s existence. This is now known as [[Gödel's ontological proof]].
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In the late [[1940s]], Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in [[general relativity]]. These "rotating universes" would allow [[time travel]] and caused Einstein to have doubts about his own theory.
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Gödel became a permanent member of the IAS 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]].
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Gödel was awarded (with [[Julian Schwinger]]) the first [[Albert Einstein Award]], in [[1951]], and was also awarded the [[National Medal of Science]], in [[1974]].
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=== Death ===
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Late in 1977, Adele became incapacitated due to illness and could no longer cook for Gödel. Due to his paranoia, he refused to eat, and thus died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on [[January 14]], [[1978]].  He weighed 65 pounds.
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== Legacy ==
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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.
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== Gödel's friendship with Einstein ==
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Gödel had a most distinguished coach for his citizenship exam: [[Albert Einstein]], who had earlier earned his own citizenship and who was concerned that his friend's unpredictable behavior might jeopardize his chances.  Einstein accompanied Gödel to the hearing.  To everyone's consternation, Gödel suddenly informed the presiding judge that he had discovered a way in which a [[dictatorship]] could be legally installed in the United States, through a logical contradiction in the [[U.S. Constitution]]. Fortunately, the judge, who was apparently a very patient person, took this in good part and awarded Gödel his citizenship. (See [http://web.archive.org/web/20060413083135/http://www.newyorker.com/critics/atlarge/?050228crat_atlarge][http://linguafranca.mirror.theinfo.org/9802/hyp.html].)
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Einstein and Gödel had a legendary friendship, shared in the walks they took together to and from the Institute for Advanced Studies. 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". (Rebecca Goldstein, ISBN 0393051692, p 33)
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{{-}}
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== Gödel in popular culture ==
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In the 1994 romantic comedy "[[I.Q. (film)]]" directed by Fred Schepisi, Gödel was dramatized as a secondary character portrayed by actor [[Lou Jacobi]].  The film portrays Gödel without his paranoia and fully enjoying his retirement.
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In 2007 students from the [http://www.filmacademie.nl Nederlandse Filmacademie (Dutch)] (Dutch Film Academy) graduated with a 25-minute short "[http://us.imdb.com/title/tt1100094/ Gödel]". It was directed by Igor Kramer with Austrian actor Robert Stuc in the title role. In this short a retired Gödel realizes his surroundings are a filmset, feeding his paranoia.
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== Important publications ==
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In German:
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*1931, "Über formal unentscheidbare Sätze der ''[[Principia Mathematica]]'' und verwandter Systeme," ''Monatshefte für Mathematik und Physik 38'': 173-98.
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In English:
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*1940. ''The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory.'' Princeton University Press.
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*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.
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In English translation:
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* Kurt Godel, 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.
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* Kurt Godel, 2000. http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel
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*[[Jean van Heijenoort]], 1967. ''A Source Book in Mathematical Logic, 1879-1931''. Harvard Univ. Press.
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**1930. "The completeness of the axioms of the functional calculus of logic," 582-91.
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**1930. "Some metamathematical results on completeness and consistency," 595-96. Abstract to (1931).
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**1931. "On formally undecidable propositions of ''Principia Mathematica'' and related systems," 596-616.
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**1931a. "On completeness and consistency," 616-17.
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*''Collected Works'': Oxford University Press: New York.  Editor-in-chief: [[Solomon Feferman]].
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**Volume I: Publications 1929-1936 ISBN 0195039645,
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**Volume II: Publications 1938-1974 ISBN 0195039726,
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**Volume III: Unpublished Essays and Lectures ISBN 0195072553,
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**Volume IV: Correspondence, A-G ISBN 0198500734.
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== Further reading and references ==
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* [http://www.akpeters.com/product.asp?ProdCode=2566] Dawson, John W., 1997. ''Logical dilemmas: The life and work of Kurt Gödel''. Wellesley MA: A K Peters.
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* Depauli-Schimanovich, Werner, and Casti, John L., 19nn. ''Gödel: A life of logic.'' Perseus.
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* [http://www.akpeters.com/product.asp?ProdCode=2388]Franzén, Torkel, 2005. ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse''. Wellesley, MA: A K Peters.
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* [[Rebecca Goldstein|Goldstein, Rebecca]], 2005. ''Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries)''. W. W. Norton.
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* [[Ivor Grattan-Guinness]], 2000. ''The Search for Mathematical Roots 1870&ndash;1940''. Princeton Univ. Press.
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* [[Jaakko Hintikka]], 2000. ''On Gödel''.  Wadsworth.
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* [[Douglas Hofstadter]], 1980. ''[[Gödel, Escher, Bach]]''. Vintage.
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* [[Stephen Kleene]], 1967. ''Mathematical Logic''.  Dover paperback reprint ca. 2001.
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* [[Ernst Nagel]] and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.
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* [[Ed Regis]], 1987. ''Who Got Einstein's Office''. Addison-Wesley Publishing Company, Inc.
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* [[Raymond Smullyan]], 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.
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* [[Hao Wang]], 1987. ''Reflections on Kurt Gödel.'' MIT Press.
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* Wang, Hao. 1996. A Logical Journey: From Godel to Philosophy. MIT Press.
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* Yourgrau, Palle, 1999. ''Gödel Meets Einstein: Time Travel in the Gödel Universe.'' Chicago: Open Court.
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* Yourgrau, Palle, 2004. ''A World Without Time: The Forgotten Legacy of Gödel and Einstein.'' Basic Books.
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== See also ==
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*[[Gödel's incompleteness theorem]]
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*[[Gödel metric|Gödel dust]], an [[exact solution]] of the [[Einstein field equation]]
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*[[Gödel Prize]]
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*[[Gödel programming language]]
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*''[[Gödel, Escher, Bach]]''
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*[[Gödel's Slingshot]]
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== External links ==
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* Kennedy, Juliette. [http://plato.stanford.edu/entries/goedel "Kurt Gödel."] In Stanford Encyclopedia of Philosophy.
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* [http://www.newyorker.com/critics/atlarge/?050228crat_atlarge Time Bandits] - an article about the relationship between Gödel and Einstein by Jim Holt
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* [http://plus.maths.org/issue39/features/dawson/ "Gödel and the limits of logic"] by John W Dawson Jr. (June 2006)
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* [http://www.ams.org/notices/200604/200604-toc.html Notices of the AMS, April 2006, Volume 53, Number 4] Kurt Gödel Centenary Issue
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* [http://www.abc.net.au/rn/scienceshow/stories/2006/1807626.htm Paul Davies and Freeman Dyson discuss Kurt Godel]
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* [http://www.edge.org/3rd_culture/goldstein05/goldstein05_index.html "Gödel and the Nature of Mathematical Truth"] Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
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* [http://video.google.com/videoplay?docid=-3503877302082311448&hl=en Dangerous Knowledge] Google Video of a BBC documentary featuring Kurt Gödel and other revolutionary mathematical thinkers.
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* [http://www.whoisgodel.com Who is Gödel?] The official website for 2007 studentfilm "Gödel" (dutch)
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[[Category:Mathematicians|Gödel, Kurt]]
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Revision as of 00:28, June 23, 2019

Kurt Gödel at Institute for Advanced Study

Kurt Gödel (1906-1978) was an Austrian-born mathematician who is considered perhaps the greatest logician since Aristotle. His Incompleteness Theorem was a stunning proof of limitations on logic, at a time when leading mathematicians were working diligently to establish its completeness. He immigrated to the United States and worked at the Institute for Advanced Study at Princeton, New Jersey, and is buried in that town with his wife. Gödel was a devout Christian who believed in an afterlife and read the Bible regularly.[1] He completely rejected the atheism of the Vienna Circle to which he was introduced.

Gödel was ostracized from academia despite his immense intellectual achievements, and was never offered a tenure-track position of professor.[2] He worked on developing a logical proof of God that build on the ontological proof by Saint Anselm. Politically, he supported Republican President Dwight Eisenhower and was critical of the Democrat Harry Truman.[3]

Gödel published his remarkable proof of the Incompleteness Theorem in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete. Moreover, the consistency of the axioms cannot be proven within the system.

Gödel's work abruptly 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 first-order axioms for all of mathematics that is both provably consistent as well as complete. Bertrand Russell had already published, in Principia Mathematica (1910–13), a massive attempt to axiomatize mathematics in a consistent way. Gödel's proof also showed that the formalist approach of David Hilbert was bound to fail to prove consistency.

Gödel took set theory, as originally developed by Georg Cantor, to new heights, as explained by Boston University Professors Juliet Floyd and Akihiro Kanamori in "How Gödel Transformed Set Theory":

How Gödel transformed set theory can be broadly cast as follows: On the larger stage, from the time of Cantor, sets began making their way into topology, algebra, and analysis so that by the time of Gödel, they were fairly entrenched in the structure and language of mathematics. But how were sets viewed among set theorists, those investigating sets as such? Before Gödel, the main concerns were what sets are and how sets and their axioms can serve as a reductive basis for mathematics. Even today, those preoccupied with ontology, questions of mathematical existence, focus mostly upon the set theory of the early period. After Gödel, the main concerns became what sets do and how set theory is to advance as an autonomous field of mathematics.[4]

The incompleteness theorems also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.

Gödel's proof was a landmark for mathematics, and demonstrated that it can never be a finished project.

Among Gödel's other remarkable achievements: the first to discover a solution to the equation for general relativity in which there are closed, time-like curves. This means it is mathematically possible for there to be universes in which one can go back in time (provided one has enough fuel and time—something probably not physically possible). Gödel was also the first to recognize the significance of the P=NP problem, in a letter he wrote to John von Neumann in 1956.[5]

A profoundly religious man who reportedly read the Bible every morning, Gödel is also noted for giving Gödel's Ontological Proof,[6] an attempt to make Anselm's ontological argument into a completely logically rigorous argument. This had the useful property of making very explicit and precise the assumptions necessary for one to accept the ontological argument. In this proof, Gödel demonstrated that if one accepts only five modest and seemingly obvious axioms, it is necessary to conclude that God exists.

Sources

References

  1. https://www.quora.com/What-led-Kurt-G%C3%B6del-to-become-a-Christian
  2. Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.
  3. https://www.usna.edu/Users/math/meh/godel.html
  4. www.ams.org/notices/200604/fea-kanamori.pdf
  5. http://rjlipton.wordpress.com/the-gdel-letter/
  6. A paper about Godel's ontological proof of the existence of God. Godel did not publish his proof of the existence of God until 1971.