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Kurt Gödel

Kurt Gödel [kurt gøːdl], (April 28, 1906January 14, 1978) was a logician, mathematician, and philosopher of mathematics. He was born in Brünn in Moravia, Austria-Hungary (now Brno in the Czech Republic), became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire was broken up, and an Austrian citizen at age 23. When Hitler annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he obtained US citizenship.

Gödel's most famous works were his incompleteness theorems, the most famous of which states that any self-consistent recursive axiomatic system powerful enough to describe integer arithmetic will allow for "true" propositions about integers that can not be proven from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which maps formal expressions into arithmetic. He also produced celebrated work on the continuum hypothesis, showing that it cannot be disproven from the accepted set theory axioms, assuming that those axioms are consistent. Gödel made important contributions to proof theory; he clarified the connections between classical logic, intuitionistic logic and modal logic by defining translations between them.

Kurt Gödel was perhaps the greatest logician of the 20th century and one of the three greatest logicians of all time with Aristotle and Frege. He published his most important result in 1931 at age of twenty-five when he worked at Vienna University, Austria.

Contents

Short biography

Childhood

Kurt Gödel was born April 28, 1906, in Brünn (now Brno), Moravia, Austria-Hungary (now the Czech Republic) to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh). In his German-speaking family young Kurt was known as Der Herr Warum (Mr Why). 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). Already during his teens Kurt studied Gabelsberger shorthand, Goethe's theory of colors and criticisms of Isaac Newton, and the writings of Kant.

Studying in Vienna

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 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. 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 he became interested in mathematical logic.

While at UV Kurt met his future wife Adele Nimbursky (née Porkert). He started to publish papers on logic and attended a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems. In 1929 Gödel became an Austrian citizen and later that year he completed his doctoral dissertation under Hans Hahn's supervision. In this dissertation he established the completeness of the first-order predicate calculus (also known as Gödel's completeness theorem).

Working in Vienna

In 1930 a doctorate in Philosophy was granted to Gödel. He added a combinatorial version to his completeness result, which was published by the Vienna Academy of Sciences. In 1931 he published his famous incompleteness theorems in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. In this article he proved that for any computable axiomatic system that is powerful enough to describe arithmetic on the natural numbers (e.g. the Peano axioms or ZFC) it holds that:

  1. The system cannot be both consistent and complete. (It is this theorem that is generally known as the incompleteness theorem.)
  2. If the system is consistent, then the consistency of the axioms cannot be proved within the system.

These theorems ended a hundred years of attempts to establish a definitive set of axioms to put the whole of mathematics on an axiomatic basis such as in the Principia Mathematica and Hilbert's formalism. It also implies that not all mathematical questions are computable.

In hindsight, the basic idea 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. Thus there will always be at least one true but unprovable statement. That is, a formula which obtains in arithmetic, but which is not provable from any humanly constructible set of axioms for arithmetic.

To make this precise, however, Gödel needed to solve several technical issues, such as encoding proofs and the very concept of provability within integer numbers. He did this using a process known as Gödel numbering.

Gödel earned his Habilitation at the UV in 1932 and in 1933 he became a Privatdozent (unpaid lecturer) there. Hitler's rise to power in 1933, in Germany had little effect on Gödel's life in Vienna since he did not have much interest in politics. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a National Socialist student, Gödel was much affected and had his first nervous breakdown.

Visiting the USA

In this year he took his first trip to the USA, during which he met Albert Einstein who would become a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year he also developed the ideas of computability and recursive functions 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 the construction of the Gödel numbers.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton 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.

Gödel would visit the IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he had to recover from a depression. He returned to teaching in 1937 and during this time he worked on the proof of consistency of the continuum hypothesis; he would go on to show that this hypothesis cannot be disproved from the common system of axioms of set theory. He married Adele on September 20, 1938. In the autumn of 1938 he visited the IAS again. After this he visited the USA once more in the spring of 1939 at the University of Notre Dame.

Working in Princeton

After the Anschluss in 1938 Austria had become a part of Nazi Germany. Since Germany had abolished the title of Privatdozent Gödel would now have to fear conscription into the Nazi army. In January 1940 he and his wife left Europe via the trans-Siberian railway and traveled via Russia and Japan to the USA. After they arrived in San Francisco on March 4, 1940, Kurt and Adele settled in Princeton, where he resumed his membership in the IAS. At the Institute, Gödel's interests turned to philosophy and physics. He studied the works of Gottfried Leibniz in detail and, to a lesser extent, those of Kant and Edmund Husserl.

In the late 1940s he 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.

He also continued to work on logic and 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 which exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice and the generalized continuum hypothesis are true in the constructible universe, and therefore must be consistent.

He became a permanent member of the IAS in 1946 and in 1948 he was naturalized as an U.S. citizen. He became a full professor at the institute in 1953 and an emeritus professor in 1976.

An amusing anecdote relating to Gödel relates that he apparently informed the presiding judge at his citizenship hearing, against the pleadings of Einstein, that he had discovered a way in which a dictatorship could be legally installed in the United States. Despite this minor fiasco, the judge, who was apparently a very patient person, still awarded Gödel his citizenship.

Gödel was awarded (with another nominee) the first Einstein Award, in 1951, and was also awarded the National Medal of Science, in 1974.

In the early seventies, Gödel, who was deeply religious, circulated among his friends an elaboration on Gottfried Leibniz' ontological proof of God's existence. This is now known as Gödel's ontological proof.

Death and honors

Gödel was a shy and withdrawn person, and suffered from paranoid psychological disorder. Towards the end of his life, he grew extremely obsessed with his health; eventually becoming convinced that he was being poisoned. To avoid this fate he refused to eat and thus starved himself to death. He died January 14, 1978, in Princeton, New Jersey, USA.

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.

Important publications

  • Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931). (Available in English at http://home.ddc.net/ygg/etext/godel/ )
  • The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press, Princeton, NJ. (1940)

Links and references

Further reading

  • Dawson, John W. Logical dilemmas: The life and work of Kurt Gödel. A K Peters. (ISBN 1568810253)
  • Depauli-Schimanovich, Werner, & Casti, John L. Gödel: A life of logic. Perseus. (ISBN 0738205184)
  • Goldstein, Rebecca (2005). Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries). W. W. Norton & Company. (ISBN 0393051692)
  • Hintikka, Jaakko (2000). On Gödel. Wadsworth. (ISBN 0534575951)
  • Hofstadter, Douglas. Gödel, Escher, Bach (ISBN 0465026567)
  • Nagel, Ernst, & Newman, James R..Gödel's Proof. New York University Press. (ISBN 0-8147-5816-9)
  • Wang, Hao (1996). A logical journey: From Gödel to philosophy. Cambridge, MA: MIT Press.
  • Yourgrau, Palle (2004). A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books. (ISBN 0465092934)
  • Yourgrau, Palle (1999). Gödel Meets Einstein: Time Travel in the Gödel Universe. Open Court. (ISBN 0812694082)

See also

External link

es:Kurt Gödel eo:Kurt GÖDEL fr:Kurt Gödel io:Kurt Godel it:Kurt Gödel he:קורט גדל nl:Kurt Gödel ja:クルト・ゲーデル pl:Kurt Gödel pt:Kurt Gödel ru:Гёдель, Курт sk:Kurt Gödel sl:Kurt Gödel sv:Kurt Gödel tr:Kurt Gödel uk:Гедель Курт zh:库尔特·哥德尔

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