Kurt Gödel . He was an Austrian-American logician, mathematician, and philosopher. Recognized as one of the most important logicians of all time. Gödel’s work has had an immense impact on 20th century scientific and philosophical thought .
Gödel, like other thinkers such as Bertrand Russell , AN Whitehead, and David Hilbert tried to use logic and set theory to understand the foundations of mathematics. Gödel is best known for his two incompleteness theorems , published in 1931 at the age of 25, one year after completing his doctorate at the University of Vienna .
The most famous of his Incompleteness Theorems states that for every self-consistent recursive axiomatic system powerful enough to describe the arithmetic of natural numbers (Peano’s arithmetic), there are true propositions about naturals that cannot be proved from of the axioms. To prove this theorem he developed a technique now called Gödel numbering, which encodes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be refuted from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by elucidating the connections between classical logic, intuitionist logic, and modal logic.
Summary
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- 1 Biographical synthesis
- 1 Studies in Vienna
- 2 Work in Vienna
- 3 Visits to the United States
- 4 Work at Princeton
- 5 Friendship with Einstein
- 6 Death
- 2 Legacy and distinctions
- 3 Gödel Award
- 4 Impact on the cinema
- 5 Bibliography
- 6 Sources
Biographical synthesis
Born on 28 of April of 1906 the capital, in Brunn Moravia Austro – Hungarian (now Brno, Czech Republic ) in an ethnic-German family wealthy, composed by Rudolf August Gödel, businessman and manager of a textile factory, and Marianne Gödel (born Handschuh), an educated and cultured woman who remained close to Gödel throughout her life (as can be seen in the extensive correspondence between the two).
At the time of his birth, his city had the majority of the German-speaking population and this was the language of his parents. Gödel who spoke very little Czech automatically became Czechoslovak at the age of 12 after the fall of the Austro-Hungarian Empire at the end of the First World War . He later told his biographer John W. Dawson that during that time he felt like an “Austrian exile in Czechoslovakia ” (“ein Österreicher im Exil in der Tschechoslowakei”).
He decided to become an Austrian citizen at the age of 23. When Nazi Germany annexed Austria , Gödel automatically became a German citizen at the age of 32. After World War II , at the age of 42, he became an American citizen.
In his family, young Kurt was called Herr Warum (Mr. Why) because of his insatiable curiosity. The only exception to an uneventful childhood was the one that from the age of four Kurt suffered poor health and rheumatic fevers, from which he fully recovered, but remained convinced for the rest of his life that his heart had suffered damage. permanent.
He attended German language primary and secondary school in Brno from which he graduated with honors in 1923 and excelled in mathematics, languages and religion. In the course of his adolescence, Kurt studied, among other subjects, Goethe’s Theory of Colors , critiques of Isaac Newton, and the work of Immanuel Kant .
Studies in Vienna
Kurt Gödel while he was a student in Vienna
At the age of 18 Kurt met his older brother Rudolf (born 1902 ) and entered the University of Vienna. By then he had mastered mathematics at the university level and although at first he intended to study theoretical physics, he also attended courses in philosophy taught by Heinrich Gomperz and mathematics .
During this period he adopted ideas of mathematical realism, read Kant’s Metaphysische Anfangsgründe der Naturwissenschaft (Metaphysical Foundations of Natural Science), and although he himself was not a “logical positivist” he participated in meetings of the Vienna Circle with Moritz Schlick , Hans Hahn and Rudolf Carnap , the latter two being from whom he learned logic. Later he also studied number theory, and it was by attending a seminar directed by Schlick, in which he was studying Bertrand Russell’s Introduction to Mathematical Logic , that motivated him to become interested in mathematical logic.
Attending a Hilbert lecture on the completeness and consistency of mathematical systems might have been what decided the course of his life.
In 1928 Hilbert and Wilhelm Ackermann published the 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 each of the true propositions in all the models of the system? ” This was the subject chosen by Gödel for his doctoral dissertation.
In 1929 , at the age of 23, he completed his dissertation under the supervision of Hans Hahn, in which Gödel established the completeness of the calculus of first-order predicates (this result is now known as Gödel’s completeness theorem). The title of Dr. Phil. It was awarded to him in 1930 and his thesis, along with additional work, was published by the Vienna Academy of Sciences.
Work in Vienna
In 1931 Gödel published his famous incompleteness theorems in “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On formally undecidable propositions of Principia Mathematica and related systems”). In this article he showed that for any computable axiomatic system that is powerful enough to describe the arithmetic of natural numbers (eg Peano’s axioms (or ZFC), then: 1. If the system is consistent, it cannot be complete. ( This is generally known as the incompleteness theorem .) 2. The consistency of the axioms cannot be proved within the system).
These theorems ended half a century of academic attempts (beginning with Frege’s work and culminating in the Principia Mathematica and Hilbert’s formalism) to find a sufficient set of axioms for all of Mathematics . The incompleteness theorem also implies that not all mathematics is computable.
The basic idea of the incompleteness theorem is rather simple. Essentially Gödel constructed a formula that claims to be unprovable for a certain formal system. If it were provable, it would be false, which contradicts the fact that in a consistent system provable propositions are always true. So there will always be at least one true but unprovable proposition.
That is, for every set of axioms of arithmetic that can be constructed by man there is a formula which is obtained from arithmetic but is unprovable in that system. However, to make this precise, Gödel needed to resolve several technical questions, such as coding propositions and the very concept of provability in natural number theory. The latter was done through a process called Gödel numbering.
In his two-page essay “Zum intuitionistischen Aussagenkalkül” ( 1932 ) Gödel refuted the finite “valuability” of intuitionist logic. In the proof he implicitly employed what later became known as Gödel – Dummett intermediate logic (or Gödel fuzzy logic).
Gödel received his habilitation from the University of Vienna in 1932 , and in 1933 he became a Privatdozent (unpaid professor). Hitler’s rise in Germany in 1933 affected Gödel little in Vienna, as he had little interest in politics. However, he was deeply affected by the murder of Moritz Schlick (whose seminar had piqued his interest in logic) at the hands of a disturbed student, an incident that resulted in his first nervous breakdown.
Visits to the United States
In 1933 Gödel traveled for the first time to the United States where he met Albert Einstein , with whom he strengthened ties of friendship. He presented a lecture at the annual meeting of the American Mathematical Society . In the course of that year Gödel also developed ideas about computability and the recursive function to the point that he presented a lecture on these functions and on the concept of truth. Later, this work was developed in number theory, using Gödel’s numbering.
In 1934 Gödel presented a series of lectures at the Institute for Advanced Study (IAS) in Princeton, entitled On the Undecidable Propositions of Formal Mathematical Systems. Stephen Kleene , who had just completed his doctorate at Princeton, took notes from this lecture, which were subsequently published.
Gödel would visit the IEA again in the fall of 1935 , but travel and hard work had exhausted him, and the following year he convalesced from a depression, and did not return to teaching until 1937 . During that time he devoted himself to the consistency test of the axiom of choice and to the continuum hypothesis in whose work he continued to show that these hypotheses cannot be refuted from the common system of axioms of set theory.
Married on 20 September as as 1938 with Adele Nimbursky (born Porkert , 1899 – 1981 ), whom he had known for 10 years. Gödel’s parents objected to the relationship on the grounds that she was a divorced dancer and six years older than him. They never had children.
Later he made another visit to the United States , where he spent the fall of 1938 at the IEA and the spring of 1939 at the University of Notre Dame. During their vacation from the IEA, Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine. However Gödel was not merely vacationing as he had a very productive working summer. John W. Dawson conjectures that during these vacations Gödel, using volume 15 of his still unpublished work Arbeitshefte (working notebooks), discovered a proof of the independence of the axiom of choice from finite type theory, a weakened form of set theory. Hao wangGödel’s close friend supports this conjecture, noting that Blue Hill’s notebooks contain his most extensive treatment of the problem.
I work at Princeton
After the Anschluss in 1938 , Austria became part of Nazi Germany . Germany abolished the title of Privatdozent, so Gödel had to run for a different office in the new order. However, his previous ties to Jewish members of the Vienna Circle , especially Hahn, weighed against him. His situation was precipitated when he was found fit for military service, being at risk of being called up to the ranks of the German army, which is why he emigrated to the United States to assume a teaching position at the IEA.
He quickly resumed his work in mathematics and in 1940 published his work Consistency of the axiom of choice and the hypothesis of the generalized continuum with the axioms of set theory, which is a classic of modern mathematics.
In this work he introduced the constructable universe, a model of set theory in which the only sets that exist are those that can be built from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (HCG) are true in the constructable universe and therefore must be consistent with the Zermelo-Fraenkel axioms for set theory (ZF). Later Paul Cohen built a model of ZF in which AC and HCG are fake; Together these proofs mean that AC and HCG are independent of the ZF axioms for set theory.
Towards the end of 1940 Gödel demonstrated the existence of paradoxical solutions to Albert Einstein’s field equations of general relativity . These “rotating universes” would allow time travel and raised doubts in Einstein about his own theory. Its solutions are known as Gödel’s metric (or Gödel’s Universe).
During his many years at the Institute, Gödel’s interests turned to Philosophy and [Physics]]. He studied and admired the works of Gottfried Leibniz, but came to the conclusion (without evidence) that most of Leibniz’s work had been suppressed. To a lesser extent he also studied Kant and Edmund Husserl . In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz’s ontological proof of the existence of God, which is now known as Gödel’s ontological proof.
In 1946 Gödel became a permanent member of the IEA. Around this period he stopped publishing, although he continued working. He became a full professor at the Institute in 1955 and a professor emeritus in 1976 .
In 1951 Gödel was recognized (along with Julian Schwinger) with the first Albert Einstein Prize, and he was also awarded the National Medal of Science in 1974 .
Friendship with Einstein
Albert Einstein and Gödel struck up a legendary friendship, shared on the walks they took together at the IEA. The nature of their conversations remained a mystery to the other members of the Institute. The economist Oskar Morgenstern recalls that towards the end of his life Einstein confided to him that “his own work no longer mattered much, that he came to the Institute only to have the privilege of walking home with Gödel.”
Einstein and Morgenstern counseled Gödel for his US citizenship test, concerned that their friend’s unpredictable behavior would jeopardize his chance. When the Nazi regime was briefly mentioned, Gödel informed the presiding judge that he had discovered a way in which a dictatorship could be legally established in the US, through a logical contradiction in the Constitution. Neither the judge nor Einstein or Morgenstern allowed Gödel to finish the elaboration of his thought and citizenship was handed over to Gödel in popular culture.
Death
Gödel’s grave in Princeton Cemetery, NJ
In his later years Gödel suffered from periods of instability and mental illness. He had obsessive fears of being poisoned, and would not eat unless his wife Adele tasted the food before him.
At the end of 1977 Adele was hospitalized for six months and could not continue to taste Gödel’s food. In his absence he refused to eat, to the point of starving himself. At the time of his death he weighed 65 pounds (approximately 30 kg). The death certificate at Princeton Hospital, the 14 of January of 1978 , reportedly died of “malnutrition and starvation caused by disturbances in the personality.”
Legacy and distinctions
The Kurt Gödel Society , founded in 1987 , was named in his honor. It is an international organization dedicated to the promotion of research in Logic, Philosophy and the history of mathematics . He was named an honorary doctor of literature by Yale University in 1951 .
He also received an honorary doctorate of science from Harvard University in 1952 with a mention declaring him “the discoverer of the most significant mathematical truth of the century.” He was elected as a member of the National Academy of Sciences in 1955 and of the American Academy of Arts and Sciences in 1957 .
In 1961 he joined the American Philosophical Society and in 1967 was elected an honorary member of the Mathematical Society of London . Finally, in 1975 President Gerald Ford presented him with the National Medal of Science.
Gödel Award
In honor of Kurt Gödel, since 1993 the Gödel Prize has been awarded , it is given to authors of articles related to the theory of computation. The award is awarded annually by the European Association for Theoretical Computer Science (EATCS) and the Special Interest Group on Algorithms and Computation Theory of the Association for Computing Machinery (ACM).
The award ceremonies are made to coincide with relevant meetings of specialists in these matters, either at the STOC (ACM Symposium on Theory of Computing), one of the main North American conferences in the area of theoretical computing, or at the ICALP (International Colloquium on Automata, Languages, and Programming), one of the main European conferences in the same area. Aside from the distinction, the award includes a $ 5,000 bonus.
As a requirement to receive the award, the winning article must have been published in a scientific journal no more than 14 years ago (previously it was only 7 years).
Impact on the cinema
In the 1994 romantic comedy IQ directed by Fred Schepisi , Gödel was dramatized as a supporting character played by actor Lou Jacobi ; in the film he appears without his paranoia and fully enjoying his retirement. In 2007 students from the Nederlandse Filmacademie (Dutch) (Dutch Film Academy) graduated with a 25-minute short, directed by Igor Kramer with Austrian actor Robert Stuc in the lead role; a retired Gödel realizes that his surroundings are a film set, which fuels his paranoia.
