Kurt Friedrich Gödel (ˈkʊʁt ˈɡøːdəl; April 28, 1906 – January 14, 1978) was an
AustriaAustria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n logician,
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, A. N. Whitehead and
David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, were pioneering the use of
logicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
to understand the foundations of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
.
Gödel is best known for his two
incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
, published in 1931 when he was 25 years old, one year after finishing his doctorate at the
University of ViennaThe University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the Germanspeaking world...
. The more famous incompleteness theorem states that for any selfconsistent
recursiveIn computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set....
axiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
powerful enough to describe the arithmetic of the
natural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that neither the
axiom of choice nor the
continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. There are many important results whose only known proofs use the
axiom of choice, which is equivalent to
Zorn's lemmaZorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
. He also made important contributions to
proof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductivelydefined data structures such as plain lists, boxed lists, or trees, which are constructed...
by clarifying the connections between
classical logicClassical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
,
intuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all wellformed statements are assumed to be either true or false, even if we do not have a proof of either...
, and
modal logicModal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
.
Childhood
Gödel was born April 28, 1906, in
BrnoBrno by population and area is the second largest city in the Czech Republic, the largest Moravian city, and the historical capital city of the Margraviate of Moravia. Brno is the administrative centre of the South Moravian Region where it forms a separate district BrnoCity District...
, AustriaHungary into the
ethnic GermanEthnic Germans historically also ), also collectively referred to as the German diaspora, refers to people who are of German ethnicity. Many are not born in Europe or in the modernday state of Germany or hold German citizenship...
family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight
GermanspeakingGerman is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widelyspoken first language in the European Union....
majority, and this was the language of his parents. The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".
Although he spoke very little
CzechCzech is a West Slavic language with about 12 million native speakers; it is the majority language in the Czech Republic and spoken by Czechs worldwide. The language was known as Bohemian in English until the late 19th century...
himself, Gödel automatically became a
CzechoslovakCzechoslovakia or CzechoSlovakia was a sovereign state in Central Europe which existed from October 1918, when it declared its independence from the AustroHungarian Empire, until 1992...
citizen at age 12 when the AustroHungarian empire broke up at the end of World War I. According to his classmate Klepetař, "Gödel considered himself always Austrian and an exile in Czechoslovakia" ("
ein Österreicher im Exil in der Tschechoslowakei") during this time. He chose to become an
AustriaAustria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n citizen at age 23. When
Nazi GermanyNazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...
annexed AustriaThe Anschluss , also known as the ', was the occupation and annexation of Austria into Nazi Germany in 1938....
, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.
In his family, young Kurt was known as
Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from
rheumatic feverRheumatic fever is an inflammatory disease that occurs following a Streptococcus pyogenes infection, such as strep throat or scarlet fever. Believed to be caused by antibody crossreactivity that can involve the heart, joints, skin, and brain, the illness typically develops two to three weeks after...
; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.
Gödel attended the
Evangelische Volksschule, a Lutheran school in Brno from 1912 to 1916, and was enrolled in the
Deutsches StaatsRealgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. 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
ViennaVienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...
to go to medical school at the
University of ViennaThe University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the Germanspeaking world...
. During his teens, Kurt studied
Gabelsberger shorthandGabelsberger shorthand, named for its creator, is a form of shorthand previously common in Germany and Austria. Created circa 1817 by Franz Xaver Gabelsberger, it was first fully described in the 1834 textbook Anleitung zur deutschen Redezeichenkunst oder Stenographie and became rapidly...
,
GoetheJohann Wolfgang von Goethe was a German writer, pictorial artist, biologist, theoretical physicist, and polymath. He is considered the supreme genius of modern German literature. His works span the fields of poetry, drama, prose, philosophy, and science. His Faust has been called the greatest long...
's
Theory of Colours and criticisms of
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, and the writings of
Immanuel KantImmanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
.
Studying in Vienna
At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered universitylevel mathematics. Although initially intending to study
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read
KantImmanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
's
Metaphysische Anfangsgründe der NaturwissenschaftImmanuel Kant's Metaphysical Foundations of Natural Science was a basic influence on the rise of science departments of the universities in the Germanspeaking countries in the nineteenth century.Hans Christian Ørsted wrote "Differential and integral...
, and participated in the
Vienna CircleThe Vienna Circle was an association of philosophers gathered around the University of Vienna in 1922, chaired by Moritz Schlick, also known as the Ernst Mach Society in honour of Ernst Mach...
with
Moritz SchlickFriedrich Albert Moritz Schlick was a German philosopher, physicist and the founding father of logical positivism and the Vienna Circle.Early life and works:...
,
Hans HahnHans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.Biography:...
, and
Rudolf CarnapRudolf Carnap was an influential Germanborn philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
. Gödel then studied
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, but when he took part in a seminar run by Moritz Schlick which studied
Bertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
's book
Introduction to Mathematical Philosophy, he became interested in
mathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."
Attending a lecture by
David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
in
BolognaBologna is the capital city of EmiliaRomagna, in the Po Valley of Northern Italy. The city lies between the Po River and the Apennine Mountains, more specifically, between the Reno River and the Savena River. Bologna is a lively and cosmopolitan Italian college city, with spectacular history,...
on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and
Wilhelm AckermannWilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
published
Grundzüge der theoretischen Logik (
Principles of Mathematical Logic), an introduction to
firstorder logicFirstorder logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: firstorder predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 firstorder predicate calculus (
Gödel's completeness theoremGödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in firstorder logic. It was first proved by Kurt Gödel in 1929....
). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.
The Incompleteness Theorem
""Kurt Godel's achievement in modern logic is singular and monumental  indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement." —John von NeumannJohn von Neumann was a HungarianAmerican mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

In 1931 and while still in Vienna, Gödel published his
incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
in
Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "
On Formally Undecidable Propositions of "Principia Mathematica" and Related SystemsÜber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I is a paper in mathematical logic by Kurt Gödel...
"). In that article, he proved for any
computableComputability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
axiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
that is powerful enough to describe the arithmetic of the natural numbers (e.g. the
Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
or Zermelo–Fraenkel set theory with the axiom of choice), that:
 If the system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
is consistentIn logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
, it cannot be completeIn general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
.
 The consistency of the axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s cannot be proven within the systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
.
These theorems ended a halfcentury of attempts, beginning with the work of Frege and culminating in
Principia MathematicaThe Principia Mathematica is a threevolume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
and 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 idea that in a consistent system, provable statements are always true.
Thus there will always be at least one true but unprovable statement.
That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system.
To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as
Gödel numberIn mathematical logic, a Gödel numbering is a function that assigns to each symbol and wellformed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...
ing.
In his twopage paper
Zum intuitionistischen Aussagenkalkül (1932) Gödel refuted the finitevaluedness of
intuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all wellformed statements are assumed to be either true or false, even if we do not have a proof of either...
. In the proof he implicitly used what has later become known as
Gödel–Dummett intermediate logicIn mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic, thus consistent superintuitionistic logics are called intermediate logics .Definition:A superintuitionistic logic is a...
(or Gödel fuzzy logic).
The mid 1930s: further work and visits to the US
Gödel earned his
habilitationHabilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
at Vienna in 1932, and in 1933 he became a
PrivatdozentPrivatdozent or Private lecturer is a title conferred in some European university systems, especially in Germanspeaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...
(unpaid lecturer) there. In 1933
Adolf HitlerAdolf Hitler was an Austrianborn German politician and the leader of the National Socialist German Workers Party , commonly referred to as the Nazi Party). He was Chancellor of Germany from 1933 to 1945, and head of state from 1934 to 1945...
came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians.
In June 1936,
Moritz SchlickFriedrich Albert Moritz Schlick was a German philosopher, physicist and the founding father of logical positivism and the Vienna Circle.Early life and works:...
, whose seminar had aroused Gödel's interest in logic, was assassinated by a proNazi student. This triggered "a severe nervous crisis" in Gödel.
He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.
In 1933, Gödel first traveled to the U.S., where he met
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, who became a good friend. He delivered an address to the annual meeting of the
American Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
. During this year, Gödel also developed the ideas of computability and
recursive functionsComputable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...
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 StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
(IAS) in
PrincetonPrinceton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, New Jersey, entitled
On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD 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 took a break 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 hypothesisIn mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
; 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, 1899–1981), 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.
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 DameThe University of Notre Dame du Lac is a Catholic research university located in Notre Dame, an unincorporated community north of the city of South Bend, in St. Joseph County, Indiana, United States...
.
Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.
Gödel was not merely vacationing, and had a very productive summer of work. Using
Heft 15 [volume 15] of Gödel's stillunpublished
Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.
Relocation to Princeton, Einstein and US citizenship
After the
AnschlussThe Anschluss , also known as the ', was the occupation and annexation of Austria into Nazi Germany in 1938....
in 1938, Austria had become a part of
Nazi GermanyNazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...
.
Germany abolished the title of
PrivatdozentPrivatdozent or Private lecturer is a title conferred in some European university systems, especially in Germanspeaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...
, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down.
His predicament intensified when the German army found him fit for conscription. World War II started in September 1939.
Before the year was up, Gödel and his wife left Vienna for
PrincetonPrinceton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
. To avoid the difficulty of an Atlantic crossing, the Gödels took the
transSiberian railwayThe TransSiberian Railway is a network of railways connecting Moscow with the Russian Far East and the Sea of Japan. It is the longest railway in the world...
to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the
Institute for Advanced StudyThe Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
(IAS).
Gödel very quickly resumed his mathematical work. In 1940, he published his work
Consistency of the axiom of choice and of the generalized continuumhypothesis with the axioms of set theory which is a classic of modern mathematics. In that work he introduced the
constructible universeIn mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized ContinuumHypothesis"...
, a model of
set theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the
axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF).
Paul CohenPaul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.Early years:Cohen was born in Long Branch, New Jersey, into a...
later constructed a
modelIn universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist
Oskar MorgensternOskar Morgenstern was a Germanborn AustrianSchool economist. He, along with John von Neumann, helped found the mathematical field of game theory ....
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".
On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be
Phillip FormanPhillip Forman was an American lawyer and judge.Early life and career:Forman was born into a Jewish family on November 30, 1895, in New York City. He served in the United States Navy during World War I and received his LL.B...
. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.
Later years and death
Gödel became a permanent member of the Institute of Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.
In 1951, Gödel demonstrated the existence of
paradoxSimilar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
ical solutions to Albert Einstein's field equations in
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. He gave this elaboration to Einstein as a present for his 70th birthday. These "rotating universes" would allow
time travelTime travel is the concept of moving between different points in time in a manner analogous to moving between different points in space. Time travel could hypothetically involve moving backward in time to a moment earlier than the starting point, or forward to the future of that point without the...
and caused Einstein to have doubts about his own theory. His solutions are known as the
Gödel metricThe Gödel metric is an exact solution of the Einstein field equations in which the stressenergy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant...
.
During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of
Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied
Immanuel KantImmanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....
and
Edmund HusserlEdmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of
Anselm of CanterburyAnselm of Canterbury , also called of Aosta for his birthplace, and of Bec for his home monastery, was a Benedictine monk, a philosopher, and a prelate of the church who held the office of Archbishop of Canterbury from 1093 to 1109...
's
ontological proofThe ontological argument for the existence of God is an a priori argument for the existence of God. The ontological argument was first proposed by the eleventhcentury monk Anselm of Canterbury, who defined God as the greatest possible being we can conceive...
of God's existence. This is now known as
Gödel's ontological proofGödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel. It is in a line of development that goes back to Anselm of Canterbury. St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is that for which no...
. Gödel was awarded (with
Julian SchwingerJulian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
) the first
Albert Einstein AwardThe Albert Einstein Award was an award in theoretical physics that was established to recognize high achievement in the natural sciences. It was endowed by the Lewis and Rosa Strauss Memorial Fund in honor of Albert Einstein's 70th birthday...
in 1951, and was also awarded the
National Medal of ScienceThe National Medal of Science is an honor bestowed 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 sciences, biology, chemistry, engineering, mathematics and...
, in 1974.
In later life, Gödel suffered periods of mental instability and illness. He had an
obsessive fear of being poisonedPersecutory delusions are a delusional condition in which the affected person believes they are being persecuted...
; he would eat only food that his wife, Adele, prepared for him. Late in 1977, Adele was hospitalized for six months and could no longer prepare Gödel's food. In her absence, he refused to eat, eventually starving to death. He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.
Religious views
Gödel was a convinced theist. He rejected the notion of others like his friend
Albert EinsteinAlbert Einstein was a Germanborn theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
that God was impersonal.
He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world [Welt] is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."
In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is
theisticTheism, in the broadest sense, is the belief that at least one deity exists.In a more specific sense, theism refers to a doctrine concerning the nature of a monotheistic God and God's relationship to the universe....
, not
pantheisticPantheism is the view that the Universe and God are identical. Pantheists thus do not believe in a personal, anthropomorphic or creator god. The word derives from the Greek meaning "all" and the Greek meaning "God". As such, Pantheism denotes the idea that "God" is best seen as a process of...
, following Leibniz rather than Spinoza."
Legacy
The
Kurt Gödel SocietyThe Kurt Gödel Society was founded in Vienna, Austria in 1987. It is an international organization aimed at promoting research primarily on logic, philosophy and the history of mathematics, with special attention to connections with Kurt Gödel, in whose honour it was named.The group also organizes...
, 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. The
University of ViennaThe University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the Germanspeaking world...
hosts the Kurt Gödel Research Center for Mathematical Logic. The Association of Symbolic Logic has invited an annual Kurt Gödel lecture each year since 1990.
Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's
Nachlass, and the final two include correspondence.
A biography of Gödel was published by
John DawsonJohn W. Dawson, Jr. is Professor of Mathematics, Emeritus at Pennsylvania State University at York. Born in Wichita, Kansas, he attended M.I.T. as a National Merit Scholar before earning a doctorate in mathematical logic from the University of Michigan in 1972...
in 2005. Gödel was also one of four mathematicians examined in the 2008
BBCThe British Broadcasting Corporation is a British public service broadcaster. Its headquarters is at Broadcasting House in the City of Westminster, London. It is the largest broadcaster in the world, with about 23,000 staff...
documentary entitled "Dangerous Knowledge".
Douglas HofstadterDouglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogymaking, artistic creation, literary translation, and discovery in mathematics and physics...
wrote a popular book in 1979 called
Gödel, Escher, Bach: An Eternal Golden Braid to celebrate the work and ideas of Gödel, along with those of artist
M. C. EscherMaurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
and composer
Johann Sebastian BachJohann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turingcomplete computational system, which may include the human brain.
Important publications
In German:
 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Mathematik und Physik 37: 349–60.
 1931, "Über formal unentscheidbare Sätze der Principia Mathematica
The Principia Mathematica is a threevolume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173–98.
 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.
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
Paul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....
and Hilary PutnamHilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470–85.
 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 1: 175–81
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
Richard Braithwaite or Brathwait was an English poet.He was born near Kendal, and educated at Oxford. He is believed to have served with the Royalist army in the Civil War...
. Dover reprint of the 1962 Basic Books edition.
 Kurt Godel, 2000. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
 Jean van Heijenoort
Jean Louis Maxime van Heijenoort was a pioneer historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and from then until 1947, an American Trotskyist activist.Life:Van Heijenoort was born in Creil, France...
, 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.
 "My philosophical viewpoint", c. 1960, unpublished.
 "The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.
 Collected Works: Oxford University Press: New York. Editorinchief: Solomon Feferman
Solomon Feferman is an American philosopher and mathematician with major works in mathematical logic.He was born in New York City, New York, and received his Ph.D. in 1957 from the University of California, Berkeley under Alfred Tarski...
.
 Volume I: Publications 1929–1936 ISBN 9780195039641 / Paperback:ISBN 9780195147209,
 Volume II: Publications 1938–1974 ISBN 9780195039726 / Paperback:ISBN 9780195147216,
 Volume III: Unpublished Essays and Lectures ISBN 9780195072556 / Paperback:ISBN 9780195147223,
 Volume IV: Correspondence, A–G ISBN 9780198500735,
 Volume V: Correspondence, H–Z ISBN 9780198500759.
Further reading
 John L. Casti and Werner DePauli, 2000. Gödel: A Life of Logic, Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0738205184.
 John W. Dawson, Jr
John W. Dawson, Jr. is Professor of Mathematics, Emeritus at Pennsylvania State University at York. Born in Wichita, Kansas, he attended M.I.T. as a National Merit Scholar before earning a doctorate in mathematical logic from the University of Michigan in 1972...
. Logical Dilemmas: The Life and Work of Kurt Gödel. AK Peters, Ltd., 1996.
 John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", Scientific American, vol. 280 num. 6, pp. 76–81
 Torkel Franzén
Torkel Franzén was a Swedish academic. He worked at the Department of Computer Science and Electrical Engineering at Luleå University of Technology, Sweden, in the fields of mathematical logic and computer science. He was known for his work on Gödel's incompleteness theorems and for his...
, 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, MA: A K Peters.
 Ivor GrattanGuinness
Ivor GrattanGuinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
 Jaakko Hintikka
Kaarlo Jaakko Juhani Hintikka is a Finnish philosopher and logician.Hintikka was born in Vantaa. After teaching for a number of years at Florida State University, Stanford, University of Helsinki, and the Academy of Finland, he is currently Professor of Philosophy at Boston University...
, 2000. On Gödel. Wadsworth.
 Douglas Hofstadter
Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogymaking, artistic creation, literary translation, and discovery in mathematics and physics...
, 1980. Gödel, Escher, BachGödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....
. Vintage.
 Stephen Kleene, 1967. Mathematical Logic. Dover paperback reprint ca. 2001.
 Stephen Kleene, 1980. Introduction to Metamathematics. North Holland ISBN 0720421039 (Ishi Press paperback. 2009. ISBN 9780923891572)
 J.R. Lucas, 1970. The Freedom of the Will. Clarendon Press, Oxford.
 Ernest Nagel
Ernest Nagel was a CzechAmerican philosopher of science. Along with Rudolf Carnap, Hans Reichenbach, and Carl Hempel, he is sometimes seen as one of the major figures of the logical positivist movement....
and Newman, James R., 1958. Gödel's Proof. New York Univ. Press.
 Procházka, Jiří, 2006, 2006, 2008, 2008, 2O1O. Kurt Gödel: 1906–1978: Genealogie. ITEM, Brno. Volume I. Brno 2006, ISBN 8090229794. In Ger., Engl. Volume II. Brno 2006, ISBN 8090347606. In Germ., Engl. Volume III. Brno 2008, ISBN 8090347649. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 9788090347656. In Germ., Engl.Volume V,Brno,Princeton 2O1O, ISBN 8O9O34769X.In Germ.,Engl.
 Ed Regis
Ed Regis is an American author and educator. He specializes in books and articles about science, philosophy and intelligence. His topics have included nanotechnology, transhumanism and biological warfare...
, 1987. Who Got Einstein's Office? AddisonWesley Publishing Company, Inc.
 Raymond Smullyan
Raymond Merrill Smullyan is an American mathematician, concert pianist, logician, Taoist philosopher, and magician.Born in Far Rockaway, New York, his first career was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959...
, 1992. Godel's Incompleteness Theorems. Oxford University Press.
 Olga TausskyTodd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.
 Hao Wang, 1987. Reflections on Kurt Gödel. MIT Press.
 Hao Wang, 1996. A Logical Journey: From Godel to Philosophy. MIT Press.
 Yourgrau, Palle, 1999. Gödel Meets Einstein: Time Travel in the Gödel Universe. Chicago: Open Court.
 Yourgrau, Palle, 2004. A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), p 861–868):
External links