Kurt Gödel

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Kurt Gödel

Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 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 codes 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 one of the most significant logicians of the 20th century in terms of impact, and these works had immense impact upon scientific thinking at a time when many (such as Bertrand Russell) were attempting to codify logic and show its completeness. He published his most important result in 1931 at age of twenty-five when he worked at Vienna University, Austria.

From a certain point of view, Gödel is a father of the postmodern revolution. His fundamental proof that language and logic are by definition not only incomplete, but are unable to be made complete (the problems raised are fundamental to, and inherent in, logic) set the tone for 21st century deconstructive thought trying to sort out what it means to be in a universe where truth is only local, and the meaning of terms is set by context.

1 Short biography
2 Anecdotes
3 Important publications
4 Links and references

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Short biography

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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 Colours and criticisms of Isaac Newton, and the writings of Kant.

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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).

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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:

The system cannot be both consistent and complete. (It is this theorem that is generally known as the incompleteness theorem.)
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.

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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.

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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 took a train to 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.

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.

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 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.

Gödel was awarded (with Julian Schwinger) 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.

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Psychological Disorder

Gödel was a shy, withdrawn and eccentric person, and suffered from paranoid schizophrenia. The great mathematician would wear warm, winter clothing in the middle of summer. In the middle of winter, Gödel would leave all of the windows open in his home, causing it to freeze. He left the windows of his house constantly open because he believed that unknown villains were trying to kill him by pouring poison gas into his house. The great logician was a highly opinionated man, having a strong opinion on just about everything including his diet and his medical prescriptions. He was a somewhat sickly man and was prescribed specific diets and medical regimens by doctors, but being as opinionated as he was, Gödel would often do the opposite of what his doctor would prescribe. All this caused Gödel to suffer further physical illness. Amongst his paranoias was the contention that unknown villains were trying to kill him by poisoning his food. For this reason Gödel would only eat his wife's cooking, refusing to even eat his own cooking for fear of being poisoned; this, in particular, would turn out to be fatal for the great logician. There is an ironically titled biography of the great mathematician called, "Gödel: A Life of Logic."

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Death and honors

As mentioned, Gödel suffered from paranoid psychological disorder. Shortly before Gödel's death, his wife had become extremely ill and was consequently incapacitated in a hospital bed. Not only was this a cause of deep sorrow for Gödel, it also meant that his wife could no longer cook for him. Due to his paranoia this meant that Gödel refused to eat any food at all. Kurt Gödel died of starvation on January 14, 1978, in Princeton, New Jersey, USA. He had no children.

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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Anecdotes

An amusing anecdote about 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. [1][2]

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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)

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