Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a
logician,
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, and
philosopher
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
. Considered along with
Aristotle
Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
and
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
,
[For instance, in their "]
Principia Mathematica
' (''Stanford Encyclopedia of Philosophy'' edition). Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...
,
and
David Hilbert were using
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
and
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
to investigate the
foundations of mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
), building on earlier work by Frege,
Richard Dedekind, and
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ; – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
.
Gödel's discoveries in the foundations of mathematics led to the proof of
his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
, and the publication of
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, and cannot prove that it is itself consistent. To prove this, Gödel developed a technique now known as
Gödel numbering, which codes formal expressions as natural numbers.
Gödel also showed that neither the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
nor the
continuum hypothesis can be disproved from the accepted
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to
proof theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
by clarifying the connections between
classical logic
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this c ...
,
intuitionistic logic, and
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
.
Born into a wealthy German-speaking family in
Brno
Brno ( , ; ) is a Statutory city (Czech Republic), city in the South Moravian Region of the Czech Republic. Located at the confluence of the Svitava (river), Svitava and Svratka (river), Svratka rivers, Brno has about 403,000 inhabitants, making ...
, Gödel emigrated to the United States in 1939 to escape the rise of Nazi Germany. Later in life, he suffered from mental illness, which ultimately claimed his life: believing that his food was being poisoned, he refused to eat and starved to death.
Early life and education
Childhood
Gödel was born April 28, 1906, in Brünn,
Austria-Hungary
Austria-Hungary, also referred to as the Austro-Hungarian Empire, the Dual Monarchy or the Habsburg Monarchy, was a multi-national constitutional monarchy in Central Europe#Before World War I, Central Europe between 1867 and 1918. A military ...
(now
Brno
Brno ( , ; ) is a Statutory city (Czech Republic), city in the South Moravian Region of the Czech Republic. Located at the confluence of the Svitava (river), Svitava and Svratka (river), Svratka rivers, Brno has about 403,000 inhabitants, making ...
, Czech Republic), into the German-speaking family of Rudolf Gödel, the managing director and part owner of a major textile firm, and Marianne Gödel (
née
The birth name is the name of the person given upon their birth. The term may be applied to the surname, the given name or to the entire name. Where births are required to be officially registered, the entire name entered onto a births registe ...
Handschuh). At the time of his birth the city had a
German-speaking majority which included his parents. His father was Catholic and his mother was Protestant, and the children were raised as Protestants. Many of Kurt Gödel's ancestors were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer in his time and for some years a member of the (Men's Choral Union of Brünn).
Gödel automatically became a citizen of
Czechoslovakia
Czechoslovakia ( ; Czech language, Czech and , ''Česko-Slovensko'') was a landlocked country in Central Europe, created in 1918, when it declared its independence from Austria-Hungary. In 1938, after the Munich Agreement, the Sudetenland beca ...
at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the
First World War
World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...
. According to his classmate , like many residents of the predominantly German , "Gödel considered himself always Austrian and an exile in Czechoslovakia". In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship. When
Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total popu ...
annexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, after
World War II
World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
, at age 42, he became a U.S. citizen.
In his family, the young Gödel was nicknamed ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven, Kurt suffered from
rheumatic fever; he completely recovered, but remained convinced for the rest of his life that his heart had been permanently damaged. Beginning at age four, Gödel had "frequent episodes of poor health", which continued all his life.
Gödel attended the , a Lutheran school in Brünn, from 1912 to 1916, and was enrolled in the from 1916 to 1924, excelling with honors in all subjects, particularly mathematics, languages, and religion. Although he had first excelled in languages, he became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf left for
Vienna
Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...
, where he attended medical school at the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
. During his teens, Gödel studied
Gabelsberger shorthand, criticism of
Isaac Newton
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
, and the writings of
Immanuel Kant
Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German Philosophy, philosopher and one of the central Age of Enlightenment, Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works ...
.
Studies in Vienna

At age 18, Gödel joined his brother at the
University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
. He had already mastered university-level mathematics. Although initially intending to study
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of
mathematical realism. He read Kant's , and participated in the
Vienna Circle with
Moritz Schlick,
Hans Hahn, and
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language 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 theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, but when he took part in a seminar run by
Moritz Schlick that studied
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
's book ''Introduction to Mathematical Philosophy'', he became interested in
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
. 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 Hilbert in
Bologna
Bologna ( , , ; ; ) is the capital and largest city of the Emilia-Romagna region in northern Italy. It is the List of cities in Italy, seventh most populous city in Italy, with about 400,000 inhabitants and 150 different nationalities. Its M ...
on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and
Wilhelm Ackermann published (''
Principles of Mathematical Logic''), an introduction to
first-order logic in which the problem of completeness was posed: "Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?"
[
]
Gödel chose this topic for his doctoral work.
In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous
completeness theorem regarding
first-order logic.
He was awarded his doctorate in 1930,
and his thesis (accompanied by additional work) was published by the
Vienna Academy of Science.
In 1929 Gödel met (née Porkert), a divorcee living with her parents across the street from him.
[Dawson Jr., John W., and Karl Sigmund. “Gödel’s Vienna.” Mathematical Intelligencer, vol. 28, no. 3, Summer 2006, Page 46. EBSCOhost, https://doi.org/10.1007/BF02986884.M] The two married (in a civil ceremony) a decade later, in September 1938. A trained ballet dancer, Adele was working as a masseuse at the time they met.
At one point she worked as a dancer at a downtown nightclub called the ''Nachtfalter'' ("nocturnal moth").
Gödel's parents opposed their relationship because of her background and age (six years older than him). It appears to have been a happy marriage. Adele was an important support to Gödel, whose psychological problems affected their daily lives. The two had no children.
Career
Incompleteness theorems
In 1930 Gödel attended the
Second Conference on the Epistemology of the Exact Sciences, held in
Königsberg
Königsberg (; ; ; ; ; ; , ) is the historic Germany, German and Prussian name of the city now called Kaliningrad, Russia. The city was founded in 1255 on the site of the small Old Prussians, Old Prussian settlement ''Twangste'' by the Teuton ...
on 5–7 September. There, he presented his completeness theorem of first-order logic, and, at the end of the talk, mentioned that this result does not generalise to higher-order logic, thus hinting at his
incompleteness theorems.
Gödel published his incompleteness theorems in (called in English "
On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). In that article, he proved for any
computable axiomatic system powerful enough to describe the arithmetic of the
natural numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
(e.g., the
Peano axioms or
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with the axiom of choice), that:
#If a (logical or axiomatic formal)
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
is
omega-consistent, it cannot be
syntactically complete.
#The consistency of axioms cannot be proved within their own
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
.
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in and
Hilbert's program, to find a non-
relatively
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''.
Philosophy
*Relativism, the concept t ...
consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of mathematics).
Gödel constructed a formula that claims it is itself unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any
computably enumerable set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
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 is true of arithmetic, but not provable in that system. To make this precise, Gödel had to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this by a process known as
Gödel numbering.
In his two-page paper (1932), Gödel refuted the finite-valuedness of
intuitionistic logic. In the proof, he implicitly used what has later become known as
Gödel–Dummett intermediate logic (or
Gödel fuzzy logic).
Mid-1930s: further work and U.S. visits
Gödel earned his
habilitation at Vienna in 1932, and in 1933 became a (unpaid lecturer) there. In 1933,
Adolf Hitler
Adolf Hitler (20 April 1889 – 30 April 1945) was an Austrian-born German politician who was the dictator of Nazi Germany from 1933 until Death of Adolf Hitler, his suicide in 1945. Adolf Hitler's rise to power, He rose to power as the lea ...
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 Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by one of his former students,
Johann Nelböck. This triggered "a severe nervous crisis" in Gödel.
[. From p. 80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgment that Schlick's murder was its trigger, are Rudolf Gödel's. Rudolf knew Kurt well in those years.] 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 Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, who became a good friend. He delivered an address to the annual meeting of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. During this year, Gödel also developed the ideas of computability and
recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using
Gödel numbering.
In 1934, Gödel gave a series of lectures at the
Institute for Advanced Study
The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
(IAS) in
Princeton, New Jersey
The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Pri ...
, titled ''On undecidable propositions of formal mathematical systems''.
Stephen Kleene, who had just completed his PhD at Princeton, took notes on these lectures that were later published.
Gödel visited the IAS again in the autumn of 1935. The traveling and hard work had exhausted him and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
and of the
continuum hypothesis; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
After marrying Adele Nimbursky in 1938, he visited the U.S. again, spending the autumn of 1938 at the IAS and publishing ''Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory'', a classic of modern mathematics. In it, he introduced the
constructible universe, a model of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
(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). This result has considerable consequences for working mathematicians, as it means they can assume the axiom of choice when proving the
Hahn–Banach theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
.
Paul Cohen later constructed a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
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.
Gödel spent the spring of 1939 at the
University of Notre Dame
The University of Notre Dame du Lac (known simply as Notre Dame; ; ND) is a Private university, private Catholic research university in Notre Dame, Indiana, United States. Founded in 1842 by members of the Congregation of Holy Cross, a Cathol ...
.
Princeton, Einstein, U.S. citizenship
After the
Anschluss
The (, or , ), also known as the (, ), was the annexation of the Federal State of Austria into Nazi Germany on 12 March 1938.
The idea of an (a united Austria and Germany that would form a "German Question, Greater Germany") arose after t ...
on 12 March 1938, Austria became a part of
Nazi Germany
Nazi Germany, officially known as the German Reich and later the Greater German Reich, was the German Reich, German state between 1933 and 1945, when Adolf Hitler and the Nazi Party controlled the country, transforming it into a Totalit ...
. Germany abolished the title , so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially Hahn, weighed against him. The University of Vienna turned his application down.
His predicament worsened 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
Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the
Trans-Siberian Railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then traveled to Princeton by train.
During this trip, Gödel was supposed to be carrying a secret letter to Einstein from Viennese physicist Hans Thirring to alert President
Franklin D. Roosevelt
Franklin Delano Roosevelt (January 30, 1882April 12, 1945), also known as FDR, was the 32nd president of the United States, serving from 1933 until his death in 1945. He is the longest-serving U.S. president, and the only one to have served ...
of the possibility that Hitler was making an atom bomb. Gödel never conveyed that letter to Einstein, although they did meet, because he was not convinced Hitler could achieve this feat.
In any case,
Leo Szilard had already conveyed the message to Einstein, and Einstein had already warned Roosevelt.
In Princeton, Gödel accepted a position at the Institute for Advanced Study (IAS), which he had visited during 1933–34.
Einstein was also living in Princeton during this time. Gödel and Einstein developed a strong friendship, and were known to take long walks together to and from the IAS. The nature of their conversations was a mystery to the other Institute members. Economist
Oskar Morgenstern
Oskar Morgenstern (; January 24, 1902 – July 26, 1977) was a German-born economist. In collaboration with mathematician John von Neumann, he is credited with founding the field of game theory and its application to social sciences and strategic ...
recounts that toward the end of Einstein's 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".
Gödel and his wife spent the summer of 1942 in
Blue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel had a very productive summer of work. Using
olume 15of Gödel's still-unpublished
orking 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.
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 that could allow the U.S. to become a dictatorship; this has since been dubbed
Gödel's Loophole. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. The judge turned out to be
Phillip Forman, who 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
Nazi Germany, officially known as the German Reich and later the Greater German Reich, was the German state between 1933 and 1945, when Adolf Hitler and the Nazi Party controlled the country, transforming it into a totalitarian dictat ...
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.
Gödel became a permanent member of the Institute for 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.
During his time at the institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving
closed timelike curves, to
Einstein's field equations in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. He is said to have given this elaboration to Einstein as a present for his 70th birthday. His "rotating universes" would allow
time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the
Gödel metric (an exact solution of the
Einstein field equation).
Gödel studied and admired the work of
Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in ad ...
, but came to believe that a hostile conspiracy had caused some of Leibniz's work to be suppressed. To a lesser extent he studied Kant and
Edmund Husserl
Edmund Gustav Albrecht Husserl (; 8 April 1859 – 27 April 1938) was an Austrian-German philosopher and mathematician who established the school of Phenomenology (philosophy), phenomenology.
In his early work, he elaborated critiques of histori ...
. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of
Anselm of Canterbury's
ontological argument for God's existence. This is now known as
Gödel's ontological proof.
Awards and honours
Gödel was awarded (with
Julian Schwinger) the first
Albert Einstein Award in 1951 and the
National Medal of Science
The 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 science, behavior ...
in 1974. Gödel was elected a resident member of the
American Philosophical Society
The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
in 1961 and a
Foreign Member of the Royal Society (ForMemRS) in 1968.
[ He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts.
]
Later life and death
Later in life, Gödel suffered periods of mental instability and illness. Some scholars have suggested Asperger syndrome and obsessive-compulsive disorder as diagnoses. After his close friend Moritz Schlick was murdered, Gödel developed an obsessive fear of being poisoned, and ate only food prepared by his wife, Adele. Adele was hospitalized beginning in late 1977, and in her absence Gödel refused to eat; he weighed when he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery. Adele died in 1981, donating Gödel's papers to the Institute for Advanced Study upon her death.
Religious views
Gödel believed that God was personal, and called his philosophy "rationalistic, idealistic, optimistic, and theological". He formulated a formal proof
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the s ...
of God's existence known as Gödel's ontological proof.
Gödel believed in an afterlife, saying, "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 he 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 is rationally constructed and has meaning, then there must be such a thing s an afterlife" He also read widely on other paranormal topics, including telepathy, reincarnation, and ghosts.
In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is ''theistic'', not pantheistic, following Leibniz rather than Spinoza." Of religion(s) in general, he said: "Religions are for the most part bad, but not religion itself." According to his wife, Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning", while of Islam
Islam is an Abrahamic religions, Abrahamic monotheistic religion based on the Quran, and the teachings of Muhammad. Adherents of Islam are called Muslims, who are estimated to number Islam by country, 2 billion worldwide and are the world ...
, he said, "I like Islam: it is a consistent r consequential
R, or r, is the eighteenth letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ar'' (pronounced ), plural ''ars''.
The lette ...
idea of religion and open-minded."
Legacy
Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange ...
's 1979 book interweaves the work and ideas of Gödel, M. C. Escher, and Johann Sebastian Bach
Johann Sebastian Bach (German: Help:IPA/Standard German, �joːhan zeˈbasti̯an baχ ( – 28 July 1750) was a German composer and musician of the late Baroque music, Baroque period. He is known for his prolific output across a variety ...
. It partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain
The human brain is the central organ (anatomy), organ of the nervous system, and with the spinal cord, comprises the central nervous system. It consists of the cerebrum, the brainstem and the cerebellum. The brain controls most of the activi ...
. In 2005, John W. Dawson Jr. published a biography, ''Logical Dilemmas: The Life and Work of Kurt Gödel''. That year, Rebecca Goldstein published ''Incompleteness: The Proof and Paradox of Kurt Gödel'' as part of the Great Discoveries series. Stephen Budiansky's Gödel's biography, ''Journey to the Edge of Reason: The Life of Kurt Gödel'', was a ''New York Times'' Critics' Top Book of 2021. Gödel was one of four mathematicians examined in David Malone's 2008 BBC
The British Broadcasting Corporation (BBC) is a British public service broadcaster headquartered at Broadcasting House in London, England. Originally established in 1922 as the British Broadcasting Company, it evolved into its current sta ...
documentary ''Dangerous Knowledge''.
The Kurt Gödel Society, founded in 1987, is an international organization for the promotion of research in logic, philosophy, and the history of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
. The University of Vienna
The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has held an annual Gödel Lecture since 1990. The Gödel Prize is given annually to an outstanding paper in theoretical computer science. Gödel's philosophical notebooks are being edited at the Kurt Gödel Research Centre at the Berlin-Brandenburg Academy of Sciences and Humanities. Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from his , and the final two include correspondence.
In the 1994 film '' I.Q.'', Lou Jacobi portrays Gödel. In the 2023 movie '' Oppenheimer'', Gödel, played by James Urbaniak, briefly appears walking with Einstein in the gardens of Princeton.
Bibliography
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'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'' 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 and Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
, 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,'' Vol. 1, pp. 175–81.
In English translation:
* Kurt Gödel, 1992. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
* Kurt Gödel, 2000. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel
* Jean van Heijenoort, 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press.
** 1930. "The completeness of the axioms of the functional calculus of logic," 582–91.
** 1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).
** 1931. "On formally undecidable propositions of ''Principia Mathematica'' and related systems," 596–616.
** 1931a. "On completeness and consistency," 616–17.
* ''Collected Works'': Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
*
: / Paperback:
: / Paperback:
: / Paperback:
:
:
* ''Philosophische Notizbücher / Philosophical Notebooks'': De Gruyter: Berlin/München/Boston. Editor: .
** Volume 1: Philosophie I Maximen 0 / Philosophy I Maxims 0 / Paperback: .
** Volume 2: Zeiteinteilung (Maximen) I und II / Time Management (Maxims) I and II .
** Volume 3: Maximen III / Maxims III .
** Volume 4: Maximen IV / Maxims IV .
** Volume 5: Maximen V / Maxims V .
** Volume 6: Maximen VI / Maxims VI .
See also
* Original proof of Gödel's completeness theorem
* Gödel fuzzy logic
* Gödel–Löb logic
* Gödel Prize
* Gödel's ontological proof
* Infinite-valued logic
* List of Austrian scientists
* List of pioneers in computer science
* Mathematical Platonism
* Primitive recursive functional
* Strange loop
* Tarski's undefinability theorem
* World Logic Day
* Gödel machine
Notes
References
* .
* .
*
*
Further reading
* Stephen Budiansky, 2021. ''Journey to the Edge of Reason: The Life of Kurt Gödel''. W.W. Norton & Company.
* .
* .
* .
* Ivor Grattan-Guinness, 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.
*
* Jaakko Hintikka, 2000. '' On Gödel''. Wadsworth.
* Douglas Hofstadter
Douglas Richard Hofstadter (born 15 February 1945) is an American cognitive and computer scientist whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, Strange loop, strange ...
, 1980. ''Gödel, Escher, Bach
''Gödel, Escher, Bach: an Eternal Golden Braid'' (abbreviated as ''GEB'') is a 1979 nonfiction book by American cognitive scientist Douglas Hofstadter.
By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Esc ...
''. Vintage.
* Stephen Kleene, 1967. ''Mathematical Logic''. Dover paperback reprint c. 2001.
* Stephen Kleene, 1980. ''Introduction to Metamathematics''. North Holland (Ishi Press paperback. 2009. )
* J.R. Lucas, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford.
* Ernest Nagel and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.
* Ed Regis, 1987. ''Who Got Einstein's Office?'' Addison-Wesley Publishing Company, Inc.
* Raymond Smullyan, 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.
* Olga Taussky-Todd
Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and later Czech Americans, Czech-American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and Matrix (mathematics), ...
, 1983
Remembrances of Kurt Gödel
Engineering & Science, Winter 1988.
* 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. . (Reviewed by John Stachel in the '' Notices of the American Mathematical Society'' (54 (7)
pp. 861–68
.
External links
*
*
Time Bandits
an article about the relationship between Gödel and Einstein by Jim Holt
Kurt Gödel Centenary Issue
Paul Davies and Freeman Dyson discuss Kurt Godel
(transcript)
Edge: A Talk with Rebecca Goldstein on Kurt Gödel.
* ttps://web.archive.org/web/20090301015757/http://www.univie.ac.at/bvi/photo-gallery/photo_gallery.htm Gödel photo gallery.(archived)
Kurt Gödel
MacTutor History of Mathematics archive page
National Academy of Sciences Biographical Memoir
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