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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. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... The International Phonetic Alphabet is a phonetic alphabet used by linguists to accurately and uniquely represent each of the wide variety of sounds (phones or phonemes) the human vocal apparatus can produce. ... April 28 is the 118th day of the year (119th in leap years) in the Gregorian Calendar, with 247 days remaining. ... 1906 was a common year starting on Monday (see link for calendar). ... January 14 is the 14th day of the year in the Gregorian calendar. ... 1978 was a common year starting on Sunday (the link is to a full 1978 calendar). ... Moravia (Czech: Morava, German: Mähren, Polish: Morawy, Hungarian: Morvaország) is the eastern part of the Czech Republic. ... Austria-Hungary, also known as the Dual monarchy (or: the k. ... Brno  listen? (German: Brünn) is the second-largest city of the Czech Republic, located in the southeast of the country, at the confluence of the Svitava and Svratka rivers. ... Czechoslovakia (Czech: Československo, Slovak: Česko-Slovensko/before 1990 Československo) was a country in Central Europe that existed from 1918 until 1992 (except for the World War II period). ... Adolf Hitler (April 20, 1889–April 30, 1945) was the Führer und Reichskanzler (Leader and Imperial chancellor) of Germany from 1933 to his death. ... March 12, 1938: German troops march into Austria The general German term Anschluss [1] (literally meaning connection, but in this context translated as annexation in the sense of political union) often refers to Anschluss Österreichs — the inclusion of Austria in a Greater Germany in 1938. ... Mushroom cloud from the nuclear explosion over Nagasaki rising 18 km (over 11 miles) into the air. ... Wikiquote has a collection of quotations by or about: United States Wikinews has news related to this article: United States United States government Official website of the United States government - Gateway to governmental sites White House - Official site of the US President Senate. ...


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. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ... In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether a given element belongs to the set or not. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others. ...


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. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... Aristotle (sculpture) Aristotle (Greek: Αριστοτέλης Aristotelēs) (384 BC – March 7, 322 BC) was an ancient Greek philosopher. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 – July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... 1931 is a common year starting on Thursday. ... University of Vienna Main Building The University of Vienna (German: Universität Wien) in Austria was founded in 1365 by Rudolph IV and hence named Alma mater Rudolphina. ...

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. April 28 is the 118th day of the year (119th in leap years) in the Gregorian Calendar, with 247 days remaining. ... 1906 was a common year starting on Monday (see link for calendar). ... Austria-Hungary, also known as the Dual monarchy (or: the k. ... German (called Deutsch in German; in German the term germanisch is equivalent to English Germanic), is a member of the western group of Germanic languages and is one of the worlds major languages. ... Vienna (German: Wien [viːn]) is the capital of Austria, and also one of Austrias nine federal states (Bundesland Wien). ... University of Vienna Main Building The University of Vienna (German: Universität Wien) in Austria was founded in 1365 by Rudolph IV and hence named Alma mater Rudolphina. ... Gabelsberger shorthand, named for its creator Franz Xaver Gabelsberger, is a form of shorthand previously common in Germany. ... Johann Wolfgang von Goethe Johann Wolfgang von Goethe (pronounced [gø tə]) (August 28, 1749 – March 22, 1832) was a German writer, politician, humanist, scientist, and philosopher. ... Color is an important part of the visual arts. ... Sir Isaac Newton in Knellers 1689 portrait Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemist who wrote... A painting of Immanuel Kant in his middle age Immanuel Kant (April 22, 1724 in Königsberg – February 12, 1804) was a German philosopher from Prussia, generally regarded as one of Europes most influential thinkers and the last major philosopher of the Enlightenment. ...


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. Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, and predicting physical phenomena through a physical theory. There are three types of theories in physics: mainstream theories, proposed theories and fringe theories. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense, if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. The various approaches to answering these questions will... The Vienna Circle was a group of philosophers and scientists organized in Vienna under Moritz Schlick. ... Moritz Schlick (April 14, 1882–June 22, 1936) was a German philospher and the founding father of logical positivism and the Vienna Circle. ... Hans Hahn (1879 - 1934) was an Austrian mathematician who made many contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. ... Rudolf Carnap (May 18, 1891 - September 14, 1970) was a German philosopher. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM (18 May 1872–2 February 1970) was an influential mathematician, philosopher, and logician of the modern age, working mostly in the 20th century. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...


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). David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Bologna (from Latin Bononia, Bulaggna in the local dialect) is the capital city of Emilia-Romagna in northern Italy, between the Po River and the Apennines. ... Hans Hahn (1879 - 1934) was an Austrian mathematician who made many contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... Gödels completeness theorem is a fundamental theorem in mathematical logic proved by Kurt Gödel in 1929. ...


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: Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

  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 mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ... For Isaac Newtons 1687 book containing basic laws of physics, see Philosophiae Naturalis Principia Mathematica The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Bertrand Russell and Alfred North Whitehead and published in 1910-1913. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense, if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will be... Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ...


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. Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ...


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. In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ...


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. Habilitation is a term used within the university system in Germany, Austria, and some other European countries such as the German-speaking part of Switzerland, in Poland, the Czech Republic, Slovakia, Hungary and Slovenia. ... Privatdozent (PD or Priv. ... Adolf Hitler Adolf Hitler (April 20, 1889 – April 30, 1945, standard German pronunciation in the IPA) was the Führer (leader) of the National Socialist German Workers Party (Nazi Party) and of Nazi Germany from 1933 to 1945. ... A seminar is a form of academic teaching, normally at a university in small groups where students are requested to actively participate during meetings. ... The Nazi party used a right-facing swastika as their symbol and the red and black colors were said to represent Blut und Boden (blood and soil). ...


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. Albert Einstein, by Yousuf Karsh Albert Einstein (March 14, 1879 – April 18, 1955) was a German theoretical physicist who is widely regarded as the greatest scientist of the 20th century. ... The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. ... Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ... In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are computable in some intuitive sense. ... In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ...


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. A lecture is a talk on a particular subject given in order to teach people about that subject, for example by a university or college teacher. ... Fuld Hall The Institute for Advanced Study is a private institution in Princeton Township, New Jersey, designed to foster pure cutting-edge research by scientists in a variety of fields without the complications of teaching or funding, or the agendas of sponsorship. ... Princeton highlighted in Mercer County. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ...


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. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Not to be confused with the University of Notre Dame Australia University of Notre Dame du Lac The University of Notre Dame is a Roman Catholic institution of higher learning located adjacent to South Bend, Indiana, USA. Notre Dames picturesque campus sits on 1,250 acres (5 km²) containing...


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. March 12, 1938: German troops march into Austria The general German term Anschluss [1] (literally meaning connection, but in this context translated as annexation in the sense of political union) often refers to Anschluss Österreichs — the inclusion of Austria in a Greater Germany in 1938. ... Nazi Germany, or the Third Reich, commonly refers to Germany in the years 1933–1945, when it was under the firm control of the totalitarian and fascist ideology of the Nazi Party, with the Führer Adolf Hitler as dictator. ... The Nazi party used a right-facing swastika as their symbol and the red and black colors were said to represent Blut und Boden (blood and soil). ... Trans-Siberian line in red; Baikal Amur Mainline in green. ... This article is about the city in California. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1 (June 21 O.S.), 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Edmund Husserl Edmund Gustav Albrecht Husserl, (April 8, 1859 - April 26, 1938), philosopher, was born into a Jewish family in Prostějov (Prossnitz), Moravia, Czech Republic (then part of the Austrian Empire). ...


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. Two-dimensional visualisation of space-time distortion. ... Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ...


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 mathematics, the axiom of choice is an axiom of set theory. ... In mathematics, the constructible universe (or Gödels constructible universe) is a particular class of sets which can be described entirely in terms of simpler sets. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the axiom of choice is an axiom of set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ...


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. National Medal of Science The National Medal of Science, also called the Presidential Medal of Science, is an honor given by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social...


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. Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1 (June 21 O.S.), 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... In theology and the philosophy of religion, an ontological argument for the existence of God is an argument that Gods existence can be proved a priori, that is, by intuition and reason alone. ... The term God is used to designate a Supreme Being; however, there are other definitions of God. ... Gödels ontological proof is a formalization of Saint Anselms ontological argument for Gods existence by the mathematician Kurt Gödel. ...


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. In popular culture, the term paranoia is usually used to describe excessive concern about ones own well-being, sometimes suggesting a person holds persecutory beliefs concerning a threat to themselves or their property and is often linked to a belief in conspiracy theories. ... The Scream, the famous painting commonly thought of as depicting the experience of mental illness. ... Princeton highlighted in Mercer County. ...


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)

GEB cover Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book by Douglas Hofstadter, first published in by Basic Books. ...

See also

The Gödel Prize is a prize for outstanding papers in theoretical computer science, named after Kurt Gödel. ... Gödel is a declarative, general-purpose programming language that adheres to the logic programming paradigm. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book by Douglas Hofstadter, first published in by Basic Books. ... This is a list of Austrian scientists Economists Eugen von Böhm-Bawerk Friedrich Hayek, economist and social scientist, Nobel Prize in economics 1974 Leopold Kohr, economist Carl Menger, founder of the Austrian School of economics Ludwig von Mises, free-market economist Oskar Morgenstern, co-founder of game theory Joseph...

External link

  • Biography (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html) at the MacTutor archive

 
 

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