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Kurt Gödel

Kurt Gödel (April 28, 1906 – January 14, 1978) was one of the most significant logicians of all time, whose work had an immense impact on 20th century philosophy, logic, and mathematics.

He is best known for his two incompleteness theorems, which he published in 1931 at 25 years of age, a year after finishing his doctorate at the University of Vienna. The theorems and the groundbreaking techniques they employed forced a rethinking of the foundations of mathematics, and initiated the field of metamathematics. The theorems showed that no consistent axiomatic system was capable of capturing all the truths of arithmetic, and this result effectively put an end to nearly a half-century of efforts by mathematicians and logicians, beginning with Gottlob Frege, through Bertrand Russell, Alfred North Whitehead, and David Hilbert, to explain the foundations of mathematics solely in the terms of logic and set theory. Gödel went on in his mathematical work to establish important theorems in set theory and to clarify the connections between classical logic, intuitionistic logic, and modal logic.

Gödel also made direct contributions to philosophy, and was influenced in his thinking by the writings of Immanuel Kant, Edmund Husserl, and, perhaps most strongly, Gottfried Leibniz. He defended mathematical realism and, though he did not make his religious views public, was critical of materialism and described himself as a philosophical theist. Gödel recorded an unpublished revision of Anselm's ontological argument for the existence of God that makes use of modern advances in modal logic, and which has spawned much discussion among logicians and philosophers since it's publication after his death.

Life of Gödel

Kurt Friedrich Gödel was born April 28, 1906, in what was then Brünn, Austria-Hungary (now Brno, Czech Republic), to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh), a well-educated homemaker. At the time of his birth the town had a slight German-speaking majority, and Gödel grew up speaking German. His father was only formally Catholic, while his mother was raised in a Protestant household, and in her childhood attended church regularly. She did not practice her faith as an adult, however, so that Kurt and his older brother Rudolf were not raised religious, though they studied religion in primary and secondary school. Rudolf remained an agnostic his entire life, but Kurt became interested in theology early on and developed unorthodox religious beliefs. In his later private writings he described himself as "theistic rather than pantheistic, following Leibniz rather than Spinoza."

Growing up Kurt was very attached to his mother, to the point that he was known to cry when she would leave the house. His relationship with his father, who was often preoccupied with his work, was less close, though not forbidding. In his family, young Kurt was known as Der Herr Warum ("Mr. Why") because of his insatiable curiosity. At the age of six or seven, Kurt suffered from rheumatic fever; he completely recovered, but after reading about possible complications in a medical book at age eight, he became increasingly preoccupied with his health. He developed hypochondria, and for the rest of his life remained convinced that his heart had suffered permanent damage.

Kurt attended German language primary and secondary school in Brünn and completed them with honors in 1923. Although he 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 left to go to medical school at the University of Vienna. By the time he joined his brother in Vienna and entered the University at age 18, he had already mastered university-level mathematics. Although he initially intended to study theoretical physics at university, Kurt chose to pursue mathematics after being impressed by the lectures of the mathematician P. Furtwängler. He studied primarily under Hans Hahn, who was at the time an active participant in the Vienna Circle, a group of logical positivists organized around another of Gödel's teachers, Moritz Schlick. Gödel attended their meetings frequently, but remained critical of positivist doctrines.

Attending a lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published "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 Hahn's supervision. In it he established the completeness of the first-order predicate calculus (a result 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 any consistent axiomatic system that is powerful enough to describe arithmetic on the natural numbers must be incomplete (i.e. there are arithmetic truths that cannot be proved from those axioms). Among the things that cannot be proved within an axiomatic system is the consistency of the system itself. These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Hilbert's formalism, to find a set of axioms sufficient for all mathematics. They also imply that it is impossible to program a computer to discover all mathematical truths.

Gödel earned his habilitation at the Universtiy of Vienna 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 his teacher Moritz Schlick (whose seminar had aroused Gödel's interest in logic) by a deranged student, which resulted in Gödel's first nervous breakdown.

In 1933, Gödel first traveled to the United States, 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.

Princeton

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, weighted 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, Kurt and Adele left Europe via the trans-Siberian railway and Japan to the USA. Arriving in San Francisco on March 4, 1940, they crossed the USA by train so that Kurt could take up a position at the IAS in Princeton.

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 the works of Gottfried Leibniz, whom he came to admire, in detail and, to a lesser extent, those of 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. 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.

Psychological disorder

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.

Death

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.

Legacy

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's friendship with Einstein

File:Godel Einstein 1950.jpeg

Gödel and Einstein at the IAS (1950)

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. 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)

Einstein often worried about his friend—and sometimes found his behavior utterly exasperating. One November day in 1952, a colleague encountered Einstein on the street, and noting his unusually perturbed expression, inquired what was wrong.

"Gödel has gone completely crazy!" was the reply.

"Why, what has he done now?"

Einstein explained: "He voted for Eisenhower!" ^{[citation needed]}

Important publications

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

Links and references

External links

John J. O'Connor and Edmund F. Robertson. Kurt Gödel at the MacTutor archive
Kurt Gödel at the Mathematics Genealogy Project
Time Bandits - an article about the relationship between Gödel and Einstein by Jim Holt
International Journal of Computers, Communications & Control, ISSN 1841-9844 (online), ISSN 1841-9836 (print), Vol. I (2006), No.1, pp. 69-71 / "Logic will never be the same again"- Kurt Gödel Centenary, by Gabriel Ciobanu
Eric W. Weisstein, Gödel, Kurt (1906-1978) at ScienceWorld.
"Gödel and the limits of logic" by John W Dawson Jr. (June 2006)
Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue

Persondata
NAME	Gödel, Kurt
ALTERNATIVE NAMES
SHORT DESCRIPTION	logician, mathematician, and philosopher of mathematics
DATE OF BIRTH	April 28, 1906
PLACE OF BIRTH	Brno
DATE OF DEATH	January 14, 1978
PLACE OF DEATH	Princeton, New Jersey

fo:Kurt Gödel

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@@ Line 3: / Line 3: @@
 '''Kurt Gödel''' (April 28, 1906 &ndash; January 14, 1978) was one of the most significant logicians of all time, whose work had an immense impact on 20th century philosophy, logic, and mathematics.
-He is best known for his two incompleteness theorems, which he published in 1931 at 25 years of age, a year after finishing his doctorate at the University of Vienna. The theorems and the groundbreaking techniques they employed forced a rethinking of the foundations of mathematics, and initiated the field of metamathematics. The theorems showed that no consistent axiomatic system was capable of capturing all the truths of arithmetic, and this result effectively put an end to nearly a half-century of efforts by mathematicians and logicians, beginning with [[Gottlob Frege]], through [[Bertrand Russell]], [[Alfred North Whitehead]], and [[David Hilbert]], to explain the foundations of mathematics solely in the terms of formal logic. Gödel went on in his mathematical work to establish important theorems in set theory and to clarify the connections between [[classical logic]], [[intuitionistic logic]], and [[modal logic]].
+He is best known for his two incompleteness theorems, which he published in 1931 at 25 years of age, a year after finishing his doctorate at the University of Vienna. The theorems and the groundbreaking techniques they employed forced a rethinking of the foundations of mathematics, and initiated the field of metamathematics. The theorems showed that no consistent axiomatic system was capable of capturing all the truths of arithmetic, and this result effectively put an end to nearly a half-century of efforts by mathematicians and logicians, beginning with [[Gottlob Frege]], through [[Bertrand Russell]], [[Alfred North Whitehead]], and [[David Hilbert]], to explain the foundations of mathematics solely in the terms of logic and set theory. Gödel went on in his mathematical work to establish important theorems in set theory and to clarify the connections between [[classical logic]], [[intuitionistic logic]], and [[modal logic]].
-Gödel also made direct contributions to philosophy, and was influenced in his thinking by the writings of [[Immanuel Kant]], [[Edmund Husserl]], and, perhaps most strongly, [[Gottfried Leibniz]]. He defended mathematical realism, and in his private writings revealed himself to be an opponent of [[materialism]] and a philosophical theist. In his notebooks Gödel recorded a revision of Anselm's [[ontological argument]] for the existence of God that makes use of modern advances in modal logic, and which has spawned much discussion among leading logicians and philosophers since it's publication after his death.
+Gödel also made direct contributions to philosophy, and was influenced in his thinking by the writings of [[Immanuel Kant]], [[Edmund Husserl]], and, perhaps most strongly, [[Gottfried Leibniz]]. He defended mathematical realism and, though he did not make his religious views public, was critical of [[materialism]] and described himself as a philosophical theist. Gödel recorded an unpublished revision of Anselm's [[ontological argument]] for the existence of God that makes use of modern advances in modal logic, and which has spawned much discussion among logicians and philosophers since it's publication after his death.
 == Life of Gödel ==
@@ Line 12: / Line 12: @@
 Growing up Kurt was very attached to his mother, to the point that he was known to cry when she would leave the house. His relationship with his father, who was often preoccupied with his work, was less close, though not forbidding. In his family, young Kurt was known as ''Der Herr Warum'' ("Mr. Why") because of his insatiable curiosity. At the age of six or seven, Kurt suffered from rheumatic fever; he completely recovered, but after reading about possible complications in a medical book at age eight, he became increasingly preoccupied with his health. He developed hypochondria, and for the rest of his life remained convinced that his heart had suffered permanent damage.
-Kurt attended German language primary and secondary school in Brünn and completed them with honors in 1923. Although he 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 left to go to medical school at the University of Vienna. By the time he joined his brother in Vienna and entered the University at age 18, he had already mastered university-level mathematics. Although initially he intended to study theoretical physics at university, Kurt chose to pursue mathematics after being impressed by the lectures of the mathematician [[P. Furtwängler]]. He read Kant's ''Metaphysische Anfangsgründe der Naturwissenschaft'', and participated in the [[Vienna Circle]] with [[Moritz Schlick]], [[Hans Hahn]], and [[Rudolf Carnap]]. After Kurt took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book ''Introduction to Mathematical Philosophy'', he also became interested in mathematical logic.
+Kurt attended German language primary and secondary school in Brünn and completed them with honors in 1923. Although he 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 left to go to medical school at the University of Vienna. By the time he joined his brother in Vienna and entered the University at age 18, he had already mastered university-level mathematics. Although he initially intended to study theoretical physics at university, Kurt chose to pursue mathematics after being impressed by the lectures of the mathematician [[P. Furtwängler]]. He studied primarily under [[Hans Hahn]], who was at the time an active participant in the [[Vienna Circle]], a group of [[logical positivism|logical positivists]] organized around another of Gödel's teachers, [[Moritz Schlick]]. Gödel attended their meetings frequently, but remained critical of positivist doctrines.
-Attending a lecture by David Hilbert in Bologna on completeness and consistency in 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 he established the completeness of the first-order predicate calculus (a result 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.
+Attending a lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and [[Wilhelm Ackermann]] published "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 Hahn's supervision. In it he established the completeness of the first-order predicate calculus (a result 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.
-=== Working in Vienna ===
+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 any consistent axiomatic system that is powerful enough to describe arithmetic on the natural numbers must be incomplete (i.e. there are arithmetic truths that cannot be proved from those axioms). Among the things that cannot be proved within an axiomatic system is the consistency of the system itself. These theorems ended a half-century of attempts, beginning with the work of [[Frege]] and culminating in Hilbert's[[philosophy of mathematics#formalism| formalism]], to find a set of axioms sufficient for all mathematics. They also imply that it is impossible to program a computer to discover all mathematical truths.
-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 [[Recursion theory|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'' [[Gödel's incompleteness theorem|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  [[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.
-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.
+Gödel earned his habilitation at the Universtiy of Vienna 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 his teacher Moritz Schlick (whose seminar had aroused Gödel's interest in logic) by a deranged student, which resulted in Gödel's first nervous breakdown.
-Thus there will always be at least one true but unprovable statement.
-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.
-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.
-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]].
+In 1933, Gödel first traveled to the United States, 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.
-=== Visits to the USA ===
-In 1933, Gödel first traveled to the [[United States|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 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.