Mathematics

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**Kurt Gödel** (IPA: [kurt gřːdl]) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was an Austrian logician, mathematician, and philosopher of mathematics

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.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, and only one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (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 the continuum hypothesis cannot be disproved from the accepted 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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Kurt Friedrich Gödel was born April 28, 1906, in Brno (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-speaking majority and this was the language of his parents.

He automatically became a 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 was never able to speak Czech and refused to learn it at school. He became an Austrian citizen by choice at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became a naturalized American citizen.

In his family, young Kurt was known as *Der 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.

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, Goethe's *Theory of Colours* and criticisms of Isaac Newton, and the writings of Immanuel Kant.

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 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.

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 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?* This was the topic chosen by Gödel for his doctorate work. 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.

In 1931, Gödel published his famous incompleteness theorems in "Über formal unentscheidbare Sätze der *Principia Mathematica* und verwandter Systeme." In that 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), then:

- The system cannot be both consistent and complete. (This is generally known as
*the*incompleteness theorem.) - The consistency of the axioms cannot be proved within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and

In 1933, Gödel first traveled to the USA, 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 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 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 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 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.

He married Adele Nimbursky (née Porkert), 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. They had no children.

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.

After the Anschluss in 1938, Austria had become a part of Nazi Germany. 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. His predicament precipitated when he was found fit for military service and was now at risk of being conscripted into the German army. World War II started in September 1939. 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 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 IAS in Princeton, New Jersey.

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 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.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the words 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 proof of God's existence. This is now known as Gödel's ontological proof.

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.

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.

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

Gödel was shy, withdrawn and eccentric. He would wear warm, winter clothing in the middle of summer. In the middle of winter, he would leave all of the windows open in his home because he believed that conspirators were trying to assassinate him with poison gas. He was a somewhat sickly man and was prescribed specific diets and medical regimens by doctors, but Gödel often ignored their advice, or even would do the opposite of what his prescription indicated. This caused him to suffer further illness. In the 1940s he suffered from a bleeding ulcer, but his distrust of doctors led him to delay treatment; he risked death and was saved only by emergency blood transfusion.

Amongst his delusions was the belief 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.

Late in 1977, Adele became incapacitated due to illness and so could no longer cook for Gödel. Due to his paranoia, he refused to eat any food at all and thus died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He weighed 65 pounds.

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.

Gödel had a most distinguished coach for his citizenship exam: Albert Einstein, who had earlier earned his own citizenship, but knowing of Gödel's unpredictable behavior, was concerned that his friend might somehow behave erratically during the exam. 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 [1][2].)

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)

In German:

- 1931, "Über formal unentscheidbare Sätze der
*Principia Mathematica*und verwandter Systeme,"*Monatshefte für Mathematik und Physik 38*: 173-98.

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.

In English translation:

- 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. - Jean van Heijenoort, 1967.
*A Source Book in Mathematical Logic, 1879-1931*. Harvard Univ. Press.- 1930. "The completeness of the axioms of the functional calculus of logic," 582-91.
- 1930. "Some metamathematical results on completeness and consistency," 595-96. Abstract to (1931).
- 1931. "On formally undecidable propositions of
*Principia Mathematica*and related systems," 596-616. - 1931a. "On completeness and consistency," 616-17.

- Collected Works : Volume I: Publications 1929-1936 ISBN 0195039645, Volume II: Publications 1938-1974 ISBN 0195039726, Volume III: Unpublished Essays and Lectures ISBN 0195072553, Volume IV: Correspondence, A-G ISBN 0198500734. Publisher: Oxford University Press, USA

- [3] Dawson, John W., 1997.
*Logical dilemmas: The life and work of Kurt Gödel*. Wellesley MA: A K Peters. - Depauli-Schimanovich, Werner, and Casti, John L., 19nn.
*Gödel: A life of logic.*Perseus. - [4]Franzén, Torkel, 2005.
*Gödel's Theorem: An Incomplete Guide to Its Use and Abuse*. Wellesley, MA: A K Peters. - Goldstein, Rebecca, 2005.
*Incompleteness: The Proof and Paradox of Kurt Godel (Great Discoveries)*. W. W. Norton. - 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. - Ernst Nagel and Newman, James R., 1958.
*Gödel's Proof.*New York Univ. Press. - Raymond Smullyan, 1992.
*Godel's Incompleteness Theorems*. Oxford University Press. - Hao Wang, 1987.
*Reflections on Kurt Gödel.*MIT Press. - Wang, Hao. 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.

- Gödel's incompleteness theorem
- Gödel dust, an exact solution of the Einstein field equation
- Gödel Prize
- Gödel programming language
*Gödel, Escher, Bach*- Gödel's Slingshot
- List of Austrian scientists

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