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Kurt Gödel

Kurt Gödel

Kurt Gödel, pronounced somewhat like "Girdle", (April 28, 1906 - January 14, 1978) wasmathematician bornAustria-Hungary. When Austria-Hungary broke up he became Austrian citizen at age 23later also US citizen at age 42. He wasdeep logician whose most famous work wasIncompleteness Theorem stating that any self-consistent axiomatic system powerful enoughdescribe integer arithmetic will allowpropositions about integers that can neither be proven nor disproven fromaxioms. He also produced celebrated work onContinuum hypothesis, showing thatcannot be disproven fromaccepted set theory axioms, assuming that those axiomsconsistent.

Arguably, Kurt Gödel isgreatest logician of20th centuryone ofthree greatest logiciansall time, withother twothis historical triumvirate being AristotleFrege. He published his most important result1931 at age 25 when he worked at Vienna University, Austria.

Tablecontents
1 Short Biography
2 Important Publications
3 Further Reading
4 External Link

Short Biography

Childhood

Kurt Gödel was born April 28, 1906,Brno, Austria-Hungary (now Czech Republic) asson ofmanager oftextile factory. In his family little Kurt was known as Der Herr Warum (Mr. Why). He attended German-language primarysecondary schoolBrnocompleted themhonors1923. Although Kurt had first excelledlearning languages he later became more fondhistorymathematics. His interestmathematics increased when1920 his older brother Rudolf (born 1902) leftViennagoMedical School atUniversityVienna (UV). Already during his teens Kurt studied Gabelsberger shorthand, Goethe's theorycolorscriticismsIsaac Newton, andwritingsKant.

StudyingVienna

Atage18 Kurt joined his brother RudolfViennaenteredUV. By that time he had already mastered university-level mathematics. Although initially intendingstudy theoretical physics he also attended courses on mathematicsphilosophy. During this time he adopted ideasmathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft,participated inVienna CircleMoritz Schlick, Hans Hahn,Rudolf Carnap. Kurt then studied number theory, but when he took part inseminar run by Moritz Schlick which studied Bertrand Russell's book Introductionmathematical philosophy he became interestedmathematical logic.

While at UV Kurt met his future wife Adele Nimbursky (née Porkert). He startedpublish papers on logicattendedlecture by David HilbertBologna on completenessconsistencymathematical systems. In 1929 Gödel became an Austrian citizenlater that year he completed his doctoral dissertation under Hans Hahn's supervision. In this dissertation he establishedcompleteness offirst-order predicate calculus (also known as Gödel's completeness theorem).

WorkingVienna

In 1930Dr. Philosophy had been grantedGödel. He addedcombinatorial versionhis completeness result, which was published byVienna AcademySciences. In 1931 he published his famous Incompleteness TheoremsÜber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. In this article he proved thatany axiomatic system thatpowerful enoughdescribenatural numbersholds that:

  1. It cannot be both consistentcomplete. (Itthis theorem thatgenerally known as the Incompleteness Theorem.)
  2. Ifsystemconsistent, thenconsistency ofaxioms cannot be proved withinsystem.
These theorems endedhundred yearsattemptsestablishdefinitive setaxiomsputwholemathematics on an axiomatic basis such as inPrincipia MathematicaHilbert's formalism. It also implies thatcomputer can never be programmedanswer all mathematical questions.

In hindsight,basic idea ofincompleteness theoremrather simple. Gödel essentially constructedformula that claims that itunprovable ingiven formal system. Ifwere provablewould be wrong, so one could prove wrong statementsthis system. Otherwise there would be at least one true but unprovable statement.

To make this precise, however, Gödel neededsolve several technical issues, such as encoding proofs andvery conceptprovability within integer numbers. Such formal details aremain reason why his 1931 paperrather longnot so easyread.

Recently (2003)was pointed out that Gödel's self-reference trick can be usedbuild an optimally efficient general problem solver:Gödel machine.

Gödel earned his Habilitation atUV1932in 1933 he becamePrivatdozent (unpaid lecturer) there. When1933 Hitler camepowerGermany this had little effect on Gödel's lifeVienna since he had little interestpolitics. However after Schlick, whose seminar had aroused Gödel's interestlogic, was murdered byNational Socialist student, Gödel was much affectedhad his first nervous breakdown.

VisitingUSA

In this year he took his first trip toUSA, during which he met Albert Einstein who would becomegood friend. He delivered an addressthe annual meeting ofAmerican Mathematical Society. During this year he also developedideascomputabilityrecursive functions topoint where he deliveredlecture on general recursive functions andconcepttruth. This work was developednumber theory, usingconstruction ofGödel numbers.

In 1934 Gödel gaveserieslectures atInstituteAdvanced Study (IAS)Princeton entitled On undecidable propositionsformal mathematical systems. Stephen Kleene who had just completed his Ph.D. at Princeton, took notesthese lectures which have been subsequently published.

Gödel would visitIAS again inautumn1935. The travelling andhard work had exhausted him andnext year he hadrecover fromdepression. He returnedteaching1937during this time he worked onproofconsistency ofContinuum hypothesis; he would go onshow that this hypothesis cannot be disproved fromcommon systemaxiomsset theory. He married Adele on September 20, 1938. Inautumn1938 he visited againIAS. After this he visitedUSA once more inspring1939 atUniversityNotre Dame.

WorkingPrinceton

AfterAnschluss1938 Austria had becomepartNazi Germany. Since Germany had abolishedtitlePrivatdozent Gödel would now havefear conscription intoNazi army. In January 1940 hehis wife left Europe viatrans-Siberian railwaytraveled via RussiaJapan toUSA. WhenarrivedSan Francisco on March 4, 1940, KurtAdele settledPrinceton, where he resumed his membership inIAS. AtInstitute, Gödel's interests turnedphilosophyphysics. He studiedworksGottfried Leibnizdetail and, tolesser extent, thoseKantEdmund Husserl.

Inlate 1940s he demonstratedexistenceparadoxical solutionsAlbert Einstein's field equationsgeneral relativity. These "rotating universes" would allow time travelcaused Einsteinhave doubts about his own theory. He also continuedwork on logicin 1940 he published his work Consistency ofaxiomchoiceofgeneralized continuum-hypothesis withaxiomsset theory which isclassicmodern mathematics. In that work he introducedconstructible universe,modelset theorywhichonly sets which existthose that can be constructed from simpler sets. Gödel showed that bothaxiomchoice andgeneralized continuum hypothesistrue inconstructible universe,therefore must be consistent.

He becamepermanent member ofIAS1946in 1948 he was naturalized as an U.S. citizen. He becamefull professor atinstitute1953an emeritus professor1976.

Inearly seventies, Gödel, who was deeply religious, circulated among his friends an elaboration on Gottfried Leibniz' ontological proofGod's existence. Thisnow known as Gödel's ontological proof.

Gödel wasshywithdrawn person. Towardsendhis life he was extremely concerned about his health; eventually he became convinced that he was being poisoned. To avoid this fate he refusedeatthus starved himselfdeath. He died January 14, 1978,Princeton, New Jersey, USA.

During his life Kurt Gödel received many prestigious awards such as(shared) first Einstein Award1951 andNational MedalScience1974.

Important Publications

Further Reading

External Link


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