Kurt Gödel [ This article is about the alphabet officially used in linguistics. ...kurt gøːdl], ( April 28 is the 118th day of the year (119th in leap years) in the Gregorian Calendar, with 247 days remaining. ...April 28, 1906 was a common year starting on Monday (see link for calendar). ...1906 – January 14 is the 14th day of the year in the Gregorian calendar. ...January 14, Events January January 1 _ The Copyright Act of 1976 takes effect, making sweeping changes to United States copyright law. ...1978) was a logician, mathematician, and philosopher of mathematics. He was born in Brünn in Moravia (Czech: Morava) is the eastern part of the Czech Republic. ...Moravia, Austria_Hungary, also known as the Dual monarchy (or: the k. ...Austria_Hungary (now 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. ...Brno in the National motto: Truth prevails ( Czech: Pravda vítězí) Official language Czech Capital Praha ( Prague) President Václav Klaus Prime Minister Stanislav Gross Area  _ Total  _ % water Ranked 114th 78,866 km² 2% Population  _ Total ( 2003)  _ Density Ranked 76th 10. ...Czech Republic), became a 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). ...Czechoslovak citizen at age 12 when the Austro_Hungarian empire was broken up, and an The Republic of Austria ( German: Republik Österreich) is a landlocked country in Central Europe. ...Austrian citizen at age 23. When For other people with the surname Hitler, see Hitler (disambiguation). ...Hitler The general German term Anschluss is part of the specific political incident Anschluss Österreichs referring to the inclusion of Austria in a Greater Germany in 1938. ...annexed Austria, Gödel automatically became a 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. ...German citizen at age 32. After Mushroom cloud from the nuclear explosion over Nagasaki rising 18 km (60,000 ft) into the air. ...World War II, at the age of 42, he obtained The United States of America — also referred to as the United States, the U.S.A., the U.S., America¹, the States, or (archaically) Columbia — is a federal republic of 50 states located primarily in central North America (with the exception of two states: Alaska and Hawaii). ...US citizenship.

He was a deep Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...logician whose most famous work was the In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ...incompleteness theorem stating that any self_consistent 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. ...axiomatic system powerful enough to describe integer arithmetic will allow for propositions about integers that can neither be proven nor disproven from the axioms. He also produced celebrated work on the In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...continuum hypothesis, showing that it cannot be disproven from the accepted Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...set theory axioms, assuming that those axioms are consistent. Gödel made important contributions to Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...proof theory; he clarified the connections between Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ...classical logic, Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...intuitionistic logic and 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. ...modal logic by defining translations between them.

Kurt Gödel was perhaps the greatest logician of the (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...20th century and one of the three greatest logicians of all time with Aristotle (sculpture) Aristotle ( Greek: Αριστοτέλης Aristotelēs) ( 384 BC – March 7, 322 BC) was an ancient Greek philosopher. ...Aristotle and 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. ...Frege. He published his most important result in 1931 is a common year starting on Thursday. ...1931 at age of twenty_five when he worked at The University of Vienna (German: Universität Wien) was founded in 1365 by Rudolph IV and hence named Alma mater Rudolphina. ...Vienna University, Austria.

Short biography

Childhood

Kurt Gödel was born April 28 is the 118th day of the year (119th in leap years) in the Gregorian Calendar, with 247 days remaining. ...April 28, 1906 was a common year starting on Monday (see link for calendar). ...1906, in Brünn (now Brno), Moravia, Austria_Hungary, also known as the Dual monarchy (or: the k. ...Austria_Hungary (now the National motto: Truth prevails ( Czech: Pravda vítězí) Official language Czech Capital Praha ( Prague) President Václav Klaus Prime Minister Stanislav Gross Area  _ Total  _ % water Ranked 114th 78,866 km² 2% Population  _ Total ( 2003)  _ Density Ranked 76th 10. ...Czech Republic) to Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (née Handschuh). In his 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. ...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 This article is about the city and federal state in Austria. ...Vienna to go to Medical School at the The University of Vienna (German: Universität Wien) was founded in 1365 by Rudolph IV and hence named Alma mater Rudolphina. ...University of Vienna (UV). Already during his teens Kurt studied Gabelsberger shorthand, named for its creator Franz Xaver Gabelsberger, is a form of shorthand previously common in Germany. ...Gabelsberger shorthand, Johann Wolfgang von Goethe Johann Wolfgang von Goethe (pronounced [gø tə]) (August 26, 1749–March 22, 1832) was a German writer, politician, humanist, scientist, and philosopher. ...Goethe's theory of For alternative meanings, see color (disambiguation). ...colors and criticisms of Sir Isaac Newton in Knellers portrait of 1689. ...Isaac Newton, and the writings of Immanuel Kant Immanuel Kant (April 22, 1724 – February 12, 1804) was a Prussian philosopher, generally regarded as one of Europes most influential thinkers and the last major philosopher of the Enlightenment. ...Kant.

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 attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ...theoretical physics he also attended courses on mathematics and philosophy. During this time he adopted ideas of 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...mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the The Vienna Circle was a group of philosophers and scientists organized in Vienna under Moritz Schlick. ...Vienna Circle with Moritz Schlick (April 14, 1882–June 22, 1936) was the founding father of logical positivism and the Vienna Circle. ...Moritz Schlick, 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. ...Hans Hahn, and Rudolf Carnap (May 18, 1891 _ September 14, 1970) was a German philosopher. ...Rudolf Carnap. Kurt then studied Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non_mathematicians. ...number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell ( May 18, 1872 – February 2, 1970) was one of the most influential mathematicians, philosophers, and logicians of the modern age, working mostly in the 20th century. ...Bertrand Russell's book Introduction to mathematical philosophy he became interested in 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. ...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 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. ...David Hilbert in 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. ...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 (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. ...Hans Hahn's supervision. In this dissertation he established the completeness of the First_order predicate calculus or first_order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...first_order predicate calculus (also known as Gödels completeness theorem is a fundamental theorem in mathematical logic proved by Kurt Gödel in 1929. ...Gödel's completeness theorem).

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 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. ...axiomatic system that is powerful enough to describe the 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...natural numbers it holds that:

1. It cannot be both consistent and complete. (It is this theorem that is generally known as the In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. ...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 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. ...Principia Mathematica and 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...Hilbert's formalism. It also implies that not all mathematical questions are In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi_decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if it...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 wrong, so one could prove wrong statements in this system. Otherwise there would be at least one true but unprovable statement.

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. Such formal details are the main reason why his 1931 paper is rather long and not so easy to read.

Gödel earned his 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. ...Habilitation at the UV in 1932 and in 1933 he became a Privatdozent (PD or Priv. ...Privatdozent (unpaid lecturer) there. 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. ...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 A seminar is a form of academic teaching, normally at a university in small groups where students are requested to actively participate during meetings. ...seminar had aroused Gödel's interest in logic, was murdered by a 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). ...National Socialist student, Gödel was much affected and had his first nervous breakdown.

Visiting the USA

In this year he took his first trip to the The United States of America — also referred to as the United States, the U.S.A., the U.S., America¹, the States, or (archaically) Columbia — is a federal republic of 50 states located primarily in central North America (with the exception of two states: Alaska and Hawaii). ...USA, during which he met For other uses of the name Einstein, please see Einstein (disambiguation) Portrait of Albert Einstein taken by Yousuf Karsh on February 11, 1948 Albert Einstein ( March 14, 1879 – April 18, 1955) was a theoretical physicist who is widely regarded as the greatest scientist of the 20th century. ...Albert Einstein who would become a good friend. He delivered an address to the annual meeting of the 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. ...American Mathematical Society. During this year he also developed the ideas of 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. ...computability and 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. ...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 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. ...lectures at the 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. ...Institute for Advanced Study (IAS) in See also: Princeton Township, New Jersey Princeton highlighted in Mercer County. ...Princeton entitled On undecidable propositions of formal mathematical systems. 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. ...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 In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...continuum hypothesis; he would go on to show that this hypothesis cannot be disproved from the common system of axioms of Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...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 Not to be confused with the University of Notre Dame Australia University of Notre Dame du Lac The University of Notre Dame (standard name; full legal name University of Notre Dame du Lac) is a Roman Catholic institution of higher learning located in Notre Dame, Indiana, USA adjacent to the...University of Notre Dame.

Working in Princeton

After the The general German term Anschluss is part of the specific political incident Anschluss Österreichs referring to the inclusion of Austria in a Greater Germany in 1938. ...Anschluss in 1938 Austria had become a part of 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. ...Nazi Germany. Since Germany had abolished the title of Privatdozent Gödel would now have to fear conscription into the 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). ...Nazi army. In January 1940 he and his wife left Europe via the ...trans_Siberian railway and traveled via The Russian Federation ( Russian: Росси́йская Федера́ция, transliteration: Rossiyskaya Federatsiya or Rossijskaja Federacija), or Russia (Russian: Росси́я, transliteration: Rossiya or Rossija), is a country that stretches over a vast expanse of eastern Europe and northern Asia. ...Russia and Official language Japanese Capital Tokyo Largest City Tokyo Emperor Akihito Prime Minister Junichiro Koizumi Area  _ Total  _ % water Ranked 60th 377,835 km² 0. ...Japan to the USA. After they arrived in This article is about the city in California. ...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 Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig _ November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...Gottfried Leibniz in detail and, to a lesser extent, those of Kant and Edmund Husserl Edmund Gustav Albrecht Husserl (April 8, 1859 _ April 26, 1938), philosopher, was born into a Jewish family in Prossnitz, Moravia (Prostejov, Czech Republic), Empire of Austria_Hungary. ...Edmund Husserl.

In the late 1940s he demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ...general relativity. These "rotating universes" would allow 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. ...time travel and caused Einstein to have doubts about his own theory.

He also continued to work on logic and in 1940 he published his work Consistency of the In mathematics, the axiom of choice is an axiom of set theory. ...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 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. ...constructible universe, a model of Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...set theory in which the only sets which exist are those that can be constructed from simpler sets. Gödel showed that both the In mathematics, the axiom of choice is an axiom of set theory. ...axiom of choice and the In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...generalized continuum hypothesis are true in the constructible universe, and therefore must be 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. ...consistent.

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 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...National Medal of Science, in 1974.

In the early seventies, Gödel, who was deeply religious, circulated among his friends an elaboration on Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig _ November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...Gottfried Leibniz' In theology, an ontological argument for the existence of God was first proposed by the medieval philosopher Saint Anselm in Chapter 2 of his Proslogion. ...ontological proof of This article focuses on the monotheistic concept of a singular God. ...God's existence. This is now known as Gödels ontological proof is a formalization of Saint Anselms ontological argument for Gods existence by the mathematician Kurt Gödel. ...Gödel's ontological proof.

Death and honors

Gödel was a shy and withdrawn person, and suffered from Paranoid redirects here. ...paranoid The Scream, the famous painting commonly thought of as depicting the experience of mental illness. ...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 See also: Princeton Township, New Jersey Princeton highlighted in Mercer County. ...Princeton, New Jersey, USA.

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. GEB cover Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book by Douglas Hofstadter, first published in 1979 by Basic Books. ...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)

See also

• Gödel Prize - named after Kurt Gödel
• Gödel programming language - named after Kurt Gödel
• GEB cover Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book by Douglas Hofstadter, first published in 1979 by Basic Books. ...Gödel, Escher, Bach

External link

• Biography (http://www_groups.dcs.st_and.ac.uk/~history/Mathematicians/Godel.html) at the The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...MacTutor archive