Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a
logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
,
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, structure, space, models, and change.
History
On ...
, and
philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...
. Considered along with
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
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 phil ...
to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
,
[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 is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...
,
and
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
were using
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
to investigate the
foundations of mathematics
Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
, building on earlier work by the likes of
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
,
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
and Frege.
Gödel published his
first incompleteness theorem
First or 1st is the ordinal form of the number one (#1).
First or 1st may also refer to:
*World record, specifically the first instance of a particular achievement
Arts and media Music
* 1$T, American rapper, singer-songwriter, DJ, and rec ...
in 1931 when he was 25 years old, one year after finishing his
doctorate
A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''l ...
at the
University of Vienna
The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...
. The first incompleteness theorem states that for any
ω-consistent recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
powerful enough to describe the arithmetic of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s (for example
Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms. To prove this, Gödel developed a technique now known as
Gödel numbering
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his ...
, which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.
Gödel also showed that neither the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
nor the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
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 such ...
, 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 Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...
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 class ...
,
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, and
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
.
Early life and education
Childhood
Gödel was born April 28, 1906, in Brünn,
Austria-Hungary
Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise of ...
(now
Brno
Brno ( , ; german: Brünn ) is a city in the South Moravian Region of the Czech Republic. Located at the confluence of the Svitava and Svratka rivers, Brno has about 380,000 inhabitants, making it the second-largest city in the Czech Republic ...
,
Czech Republic
The Czech Republic, or simply Czechia, is a landlocked country in Central Europe. Historically known as Bohemia, it is bordered by Austria to the south, Germany to the west, Poland to the northeast, and Slovakia to the southeast. The ...
) into the German-speaking family of Rudolf Gödel (1874–1929), the managing director and part owner of a major textile firm, and Marianne Gödel (
née
A birth name is the name of a person given upon birth. The term may be applied to the surname, the given name, or the entire name. Where births are required to be officially registered, the entire name entered onto a birth certificate or birth re ...
Handschuh, 1879–1966). At the time of his birth the city had a
German-speaking
German ( ) is a West Germanic language mainly spoken in Central Europe. It is the most widely spoken and official or co-official language in Germany, Austria, Switzerland, Liechtenstein, and the Italian province of South Tyrol. It is also a ...
majority which included his parents. His father was Catholic and his mother was Protestant and the children were raised Protestant. The ancestors of Kurt Gödel were often 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
, rue, Чеськословеньско, , yi, טשעכאסלאוואקיי,
, common_name = Czechoslovakia
, life_span = 1918–19391945–1992
, p1 = Austria-Hungary
, image_p1 ...
at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the
First World War
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
. 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 Czechoslovakian citizenship and then, in April, granted Austrian citizenship. When
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated betwe ...
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, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...
, at the age of 42, he became an American 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
Rheumatic fever (RF) is an inflammatory disease that can involve the heart, joints, skin, and brain. The disease typically develops two to four weeks after a streptococcal throat infection. Signs and symptoms include fever, multiple painful jo ...
; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage. Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire 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 his subjects, particularly in mathematics, languages and religion. Although Gödel had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for
Vienna
en, Viennese
, iso_code = AT-9
, registration_plate = W
, postal_code_type = Postal code
, postal_code =
, timezone = CET
, utc_offset = +1
, timezone_DST ...
, where he attended medical school at the
University of Vienna
The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...
. During his teens, Gödel studied
Gabelsberger shorthand
Gabelsberger shorthand, named for its creator, is a form of shorthand previously common in Germany and Austria. Created c. 1817 by Franz Xaver Gabelsberger, it was first fully described in the 1834 textbook ''Anleitung zur deutschen Redezeichenk ...
,
Goethe
Johann Wolfgang von Goethe (28 August 1749 – 22 March 1832) was a German poet, playwright, novelist, scientist, statesman, theatre director, and critic. His works include plays, poetry, literature, and aesthetic criticism, as well as treat ...
's ''
Theory of Colours
''Theory of Colours'' (german: Zur Farbenlehre, links=no) is a book by Johann Wolfgang von Goethe about the poet's views on the nature of colours and how these are perceived by humans. It was published in German in 1810 and in English in 1840 ...
'' and criticisms of
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
, and the writings of
Immanuel Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
.
Studies in Vienna
At the age of 18, Gödel joined his brother at the
University of Vienna
The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...
. By that time, 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 natural phenomena. This is in contrast to experimental physics, which uses experim ...
, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of
mathematical realism
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
. He read
Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...
's , and participated in the
Vienna Circle
The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, cha ...
with
Moritz Schlick
Friedrich Albert Moritz Schlick (; ; 14 April 1882 – 22 June 1936) was a German philosopher, physicist, and the founding father of logical positivism and the Vienna Circle.
Early life and works
Schlick was born in Berlin to a wealthy Prussian f ...
,
Hans Hahn, and
Rudolf Carnap. Gödel then studied
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, but when he took part in a seminar run by
Moritz Schlick
Friedrich Albert Moritz Schlick (; ; 14 April 1882 – 22 June 1936) was a German philosopher, physicist, and the founding father of logical positivism and the Vienna Circle.
Early life and works
Schlick was born in Berlin to a wealthy Prussian f ...
which studied
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
's book ''Introduction to Mathematical Philosophy'', he became interested in
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
. 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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
in
Bologna
Bologna (, , ; egl, label= Emilian, Bulåggna ; lat, Bononia) is the capital and largest city of the Emilia-Romagna region in Northern Italy. It is the seventh most populous city in Italy with about 400,000 inhabitants and 150 different nat ...
on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and
Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation.
Biography
...
published (''
Principles of Mathematical Logic''), an introduction to
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
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 problem became the topic that Gödel chose for his doctoral work. In 1929, at the age of 23, he completed his doctoral
dissertation under Hans Hahn's supervision. In it, he established his eponymous
completeness theorem
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
regarding the
first-order predicate calculus
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quan ...
. He was awarded his doctorate in 1930, and his thesis (accompanied by some additional work) was published by the
Vienna Academy of Science.
Career
Incompleteness theorems
In 1930 Gödel attended the
Second Conference on the Epistemology of the Exact Sciences The Second Conference on the Epistemology of the Exact Sciences (german: 2. Tagung für Erkenntnislehre der exakten Wissenschaften in Königsberg) was held on 5–7 September 1930 in Königsberg, then located in East Prussia. It was at this conferen ...
, held in
Königsberg
Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was named ...
, 5–7 September. Here he delivered his
incompleteness theorems
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
.
Gödel published his incompleteness theorems in (called in English "
On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any
computable
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is close ...
axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
that is powerful enough to describe the arithmetic of the
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
(e.g., the
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
or
Zermelo–Fraenkel set theory with the axiom of choice), that:
# If a (logical or axiomatic formal)
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
is
omega-consistent, it cannot be
syntactically complete.
# The consistency of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s cannot be proved within their own
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
.
These theorems ended a half-century of attempts, beginning with the work of
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 phil ...
and culminating in and
Hilbert's Program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathema ...
, to find a non-
relatively consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of mathematics).
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.
Thus there will always be at least one true but unprovable statement.
That is, for any
computably enumerable
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 ...
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 is true of arithmetic, but which is not provable in that system.
To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this using a process known as
Gödel numbering.
In his two-page paper (1932) Gödel refuted the finite-valuedness of
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
. In the proof, he implicitly used what has later become known as
Gödel–Dummett intermediate logic (or
Gödel fuzzy logic
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
).
Mid-1930s: further work and U.S. visits
Gödel earned his
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
at Vienna in 1932, and in 1933 he became a (unpaid lecturer) there. In 1933
Adolf Hitler
Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Nazi Germany, Germany from 1933 until Death of Adolf Hitler, his death in 1945. Adolf Hitler's rise to power, He rose to power as the le ...
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
Friedrich Albert Moritz Schlick (; ; 14 April 1882 – 22 June 1936) was a German philosopher, physicist, and the founding father of logical positivism and the Vienna Circle.
Early life and works
Schlick was born in Berlin to a wealthy Prussian f ...
, whose seminar had aroused Gödel's interest in logic, was assassinated by one of his former students,
Johann Nelböck
Johann "Hans" Nelböck (May 12, 1903 – February 3, 1954) was an Austrian former student and murderer of Moritz Schlick, the founder of the group of philosophers and scientists known as the Vienna Circle.
After attending the gymnasium in Wels, ...
. 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 judgement that the Schlick assassination was its trigger, are from the Rudolf Gödel quote. 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 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
, 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 mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was developed by Kurt Gödel for the proof of his ...
.
In 1934, Gödel gave a series of lectures at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
(IAS) in
Princeton, New Jersey
Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...
, titled ''On undecidable propositions of formal mathematical systems''.
Stephen Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.
Gödel visited the IAS again in the autumn of 1935. The travelling and the 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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and of the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
He married (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Gödel's parents had opposed their relationship because she was a divorced dancer, six years older than he was.
Subsequently, he left for another visit to the United States, 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 that work he introduced the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
, a model of
set theory
Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly conce ...
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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(AC) and the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
(GCH) are true in the constructible universe, and therefore must be consistent with the
Zermelo–Fraenkel axioms for set theory (ZF). This result has had considerable consequences for working mathematicians, as it means they can assume the axiom of choice when proving the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
.
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
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 ''modulus'', a measure.
Models c ...
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 ( ) or ND, is a private Catholic research university in Notre Dame, Indiana, outside the city of South Bend. French priest Edward Sorin founded the school in 1842. The main campu ...
.
Princeton, Einstein, U.S. citizenship
After the
Anschluss
The (, or , ), also known as the (, en, Annexation of Austria), was the annexation of the Federal State of Austria into the German Reich on 13 March 1938.
The idea of an (a united Austria and Germany that would form a " Greater Germany ...
on 12 March 1938, Austria had become a part of
Nazi Germany
Nazi Germany (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...
.
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 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
Princeton
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
. To avoid the difficulty of an Atlantic crossing, the Gödels took the
Trans-Siberian Railway
The Trans-Siberian Railway (TSR; , , ) connects European Russia to the Russian Far East. Spanning a length of over , it is the longest railway line in the world. It runs from the city of Moscow in the west to the city of Vladivostok in the ea ...
to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the US by train to Princeton. There Gödel accepted a position at the Institute for Advanced Study (IAS), which he had previously visited during 1933–34.
Albert Einstein was also living at Princeton during this time. Gödel and Einstein 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 Morgenstern
Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decis ...
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".
Gödel and his wife, Adele, spent the summer of 1942 in
Blue Hill, Maine
Blue Hill is a town in Hancock County, Maine, United States. The population was 2,792 at the 2020 census. It is home to the Blue Hill Public Library, Blue Hill Memorial Hospital, George Stevens Academy, the Blue Hill Harbor School, The Bay Sch ...
, at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but 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
Citizenship of the United States is a legal status that entails Americans with specific rights, duties, protections, and benefits in the United States. It serves as a foundation of fundamental rights derived from and protected by the Constituti ...
exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the
U.S. Constitution
The Constitution of the United States is the supreme law of the United States of America. It superseded the Articles of Confederation, the nation's first constitution, in 1789. Originally comprising seven articles, it delineates the natio ...
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 (lit. "National Socialist State"), ' (lit. "Nazi State") for short; also ' (lit. "National Socialist Germany") (officially known as the German Reich from 1933 until 1943, and the Greater German Reich from 1943 to 1945) was ...
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 curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van St ...
s, to
Einstein's field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. He is said to have given this elaboration to Einstein as a present for his 70th birthday. His "rotating universes" would allow
time travel
Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...
to the past and caused Einstein to have doubts about his own theory. His solutions are known as the
Gödel metric
The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous d ...
(an exact solution of the
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
).
He studied and admired the works of
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
, 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 Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
and
Edmund Husserl
, thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations)
, thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view
, thesis1_year = 1883
, thesis2_title ...
. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of
Anselm of Canterbury
Anselm of Canterbury, OSB (; 1033/4–1109), also called ( it, Anselmo d'Aosta, link=no) after his birthplace and (french: Anselme du Bec, link=no) after his monastery, was an Italian Benedictine monk, abbot, philosopher and theologian of th ...
's
ontological proof of God's existence. This is now known as
Gödel's ontological proof
Gödel's ontological proof is a formal proof, formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development that goes back to Anselm of Canterbury (1033–1109). St. Anselm's ontol ...
.
Awards and honours
Gödel was awarded (with
Julian Schwinger
Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
) the first
Albert Einstein Award
The Albert Einstein Award (sometimes mistakenly called the ''Albert Einstein Medal'' because it was accompanied with a gold medal) was an award in theoretical physics, given periodically from 1951 to 1979, that was established to recognize high ac ...
in 1951, and was also awarded 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 and social scienc ...
, in 1974. Gödel was elected a resident member of the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
in 1961 and a
Foreign Member of the Royal Society (ForMemRS) in 1968.
[ He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts. The ]Gödel Prize
The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interes ...
, an annual prize for outstanding papers in the area of theoretical computer science, is named after him.
Later life and death
Later in his life, Gödel suffered periods of mental instability and illness. Following the assassination of his close friend Moritz Schlick
Friedrich Albert Moritz Schlick (; ; 14 April 1882 – 22 June 1936) was a German philosopher, physicist, and the founding father of logical positivism and the Vienna Circle.
Early life and works
Schlick was born in Berlin to a wealthy Prussian f ...
, Gödel developed an obsessive fear of being poisoned, and would eat 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
Starvation is a severe deficiency in caloric energy intake, below the level needed to maintain an organism's life. It is the most extreme form of malnutrition. In humans, prolonged starvation can cause permanent organ damage and eventually, deat ...
caused by personality disturbance" in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery
Princeton Cemetery is located in Princeton, New Jersey, United States. It is owned by the Nassau Presbyterian Church. John F. Hageman in his 1878 history of Princeton, New Jersey refers to the cemetery as "The Westminster Abbey of the United Stat ...
. Adele died in 1981.
Religious views
Gödel believed that God was personal, and called his philosophy "rationalistic, idealistic, optimistic, and theological".
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"
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
Theism is broadly defined as the belief in the existence of a supreme being or deities. In common parlance, or when contrasted with ''deism'', the term often describes the classical conception of God that is found in monotheism (also referred t ...
'', not pantheistic
Pantheism is the belief that reality, the universe and the cosmos are identical with divinity and a supreme supernatural being or entity, pointing to the universe as being an immanent creator deity still expanding and creating, which has ...
, following Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
rather than Spinoza
Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, b ...
." Of religion(s) in general, he said: "Religions are, for the most part, bad—but religion is not". 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 (; ar, ۘالِإسلَام, , ) is an Abrahamic religions, Abrahamic Monotheism#Islam, monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God in Islam, God (or ...
, 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'', or in Irela ...
idea of religion and open-minded."
Legacy
Douglas Hofstadter wrote the 1979 book to celebrate the work and ideas of Gödel, M. C. Escher
Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints.
Despite wide popular interest, Escher was for most of his life neglected in t ...
and Johann Sebastian Bach
Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard w ...
. It partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Tur ...
computational system, which may include the human brain
The human brain is the central organ of the human nervous system, and with the spinal cord makes up the central nervous system. The brain consists of the cerebrum, the brainstem and the cerebellum. It controls most of the activities of the ...
.
The Kurt Gödel Society The 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, ...
, founded in 1987, was named in his honor. It 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 mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
. The University of Vienna
The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...
hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic
The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Alonzo Church. The current president of the ASL is ...
has invited an annual Kurt Gödel lecturer each year since 1990
Gödel's Philosophical Notebooks
are edited at th
Kurt Gödel Research Centre
which is situated at th
Berlin-Brandenburg Academy of Sciences and Humanities
in Germany.
Lou Jacobi
Lou Jacobi (born Louis Harold Jacobovitch; December 28, 1913October 23, 2009) was a Canadian character actor.
Life and early career
Jacobi was born Louis Harold Jacobovitch in Toronto, Canada, to Joseph and Fay Jacobovitch.
Jacobi began acting ...
plays Gödel in the 1994 film ''I.Q.
An intelligence quotient (IQ) is a total score derived from a set of standardized tests or subtests designed to assess human intelligence. The abbreviation "IQ" was coined by the psychologist William Stern for the German term ''Intelligenzq ...
''
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 2005 John Dawson published a biography of Gödel, ''Logical Dilemmas: The Life and Work of Kurt Gödel'' ( A. K. Peters, Wellesley, MA, ). Stephen Budiansky's book about Gödel's life, ''Journey to the Edge of Reason: The Life of Kurt Gödel'' ( W. W. Norton & Company, New York City, NY, ), was a ''New York Times'' Critics' Top Book of 2021.
Gödel was also one of four mathematicians examined in David Malone's 2008 BBC #REDIRECT BBC #REDIRECT BBC
Here i going to introduce about the best teacher of my life b BALAJI sir. He is the precious gift that I got befor 2yrs . How has helped and thought all the concept and made my success in the 10th board exam. ...
...
documentary ''Dangerous Knowledge''.
In German:
* 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." ''Monatshefte für Mathematik und Physik'' 37: 349–60.
* 1931, "Über formal unentscheidbare Sätze der ''
'' 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
, 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
. Dover reprint of the 1962 Basic Books edition.
* Kurt Gödel, 2000.
''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel
*
, 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.
c. 1960, unpublished.
1961, unpublished.
* ''Collected Works'': Oxford University Press: New York. Editor-in-chief:
.
** Volume I: Publications 1929–1936 / Paperback: ,
** Volume II: Publications 1938–1974 / Paperback: ,
** Volume III: Unpublished Essays and Lectures / Paperback: ,
** Volume IV: Correspondence, A–G ,
** Volume V: Correspondence, H–Z .
* ''Philosophische Notizbücher / Philosophical Notebooks'': De Gruyter: Berlin/München/Boston. Editor: .
** Volume 1: Philosophie I Maximen 0 / Philosophy I Maxims 0 .
** Volume 2: Zeiteinteilung (Maximen) I und II / Time Management (Maxims) I and II .
** Volume 3: Maximen III / Maxims III
, 2021. ''Journey to the Edge of Reason: The Life of Kurt Gödel''. W.W. Norton & Company.
* .
* .
* .
* .
*
, 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.
*
*
, 2000. ''On Gödel''. Wadsworth.
*
, 1980. ''
''. Vintage.
*
, 1967. ''Mathematical Logic''. Dover paperback reprint c. 2001.
* Stephen Kleene, 1980. ''Introduction to Metamathematics''. North Holland (Ishi Press paperback. 2009. )
*
, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford.
*
and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.
* Procházka, Jiří, 2006, 2006, 2008, 2008, 2010. ''Kurt Gödel: 1906–1978: Genealogie''. ITEM, Brno. Volume I. Brno 2006, . In German, English. Volume II. Brno 2006, . In German, English. Volume III. Brno 2008, . In German, English. Volume IV. Brno, Princeton 2008, . In German, English Volume V, Brno, Princeton 2010, . In German, English.
* Procházka, Jiří, 2012. "Kurt Gödel: 1906–1978: Historie". ITEM, Brno, Wien, Princeton. Volume I. . In German, English.
*
, 1987. ''Who Got Einstein's Office?'' Addison-Wesley Publishing Company, Inc.
*
, 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.
*
Engineering & Science, Winter 1988.
* Gödel, Alois, 2006. Brünn 1679–1684. ITEM, Brno 2006, edited by Jiří Procházka,
* Procházka, Jiří 2017. "Kurt Gödel: 1906–1978: Curriculum vitae". ITEM, Brno, Wien, Princeton 2017. Volume I. (). In German, English.
* Procházka, Jiří 2019. "Kurt Gödel 1906-1978: Curriculum vitae". ITEM, Brno, Wien, Princeton 2019. Volume II. (). In German, English.
* Procházka, Jiří 2O2O. "Kurt Gödel: 19O6-1978. Curriculum vitae". ITEM, Brno, Wien, Princeton 2020. Volume III. (). In German, English. 223 Pages.
* 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)
.
Edge: A Talk with Rebecca Goldstein on Kurt Gödel.