Kurt Gödel
Kurt Friedrich Gödel (/ˈɡɜːrdəl/ GUR-dəl,[2] German: [kʊʁt ˈɡøːdl̩] ⓘ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege 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,[3] Alfred North Whitehead,[3] and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Richard Dedekind, Georg Cantor and Gottlob Frege.
"Godel" and "Gödel" redirect here. For other uses, see Godel (disambiguation).
Kurt Gödel
January 14, 1978 (aged 71)
- Austria
- Czechoslovakia
- Germany
- United States
University of Vienna (PhD, 1930)
- Gödel's incompleteness theorems
- Gödel's completeness theorem
- Gödel's constructible universe
- Gödel metric (closed timelike curve)
- Gödel logic
- Gödel–Dummett logic
- Gödel's β function
- Gödel's Loophole
- Gödel numbering
- Gödel operation
- Gödel's speed-up theorem
- Gödel's ontological proof
- Gödel–Gentzen translation
- Gödel–McKinsey–Tarski translation
- Von Neumann–Bernays–Gödel set theory
- ω-consistent theory
- The consistency of the continuum hypothesis with ZFC
- Axiom of constructibility
- Compactness theorem
- Condensation lemma
- Diagonal lemma
- Dialectica interpretation
- Ordinal definable set
- Slingshot argument
- Albert Einstein Award (1951)
- ForMemRS (1968)[1]
- National Medal of Science (1974)
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, and the publication of Gödel's incompleteness theorems two years later, in 1931. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example, Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms.[4] To prove this, Gödel developed a technique now known as Gödel numbering, 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.[5]
Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, 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 by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
Early life and education[edit]
Childhood[edit]
Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic), 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 Handschuh, 1879–1966).[6] At the time of his birth the city had a German-speaking majority which included his parents.[7] His father was Catholic and his mother was Protestant and the children were raised as Protestants. 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 Brünner Männergesangverein (Men's Choral Union of Brünn).[8]
Gödel automatically became a citizen of Czechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the First World War. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[9] In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship.[10] When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, after World War II, at the age of 42, he became an American citizen.[11]
In his family, the young Gödel was nicknamed Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven, Kurt suffered from rheumatic fever; 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.[12]
Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium 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, where he attended medical school at the University of Vienna. During his teens, Gödel studied Gabelsberger shorthand,[13] and criticisms of Isaac Newton, and the writings of Immanuel Kant.[14]
Awards and honours[edit]
Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.[34] Gödel was elected a resident member of the American Philosophical Society in 1961 and a Foreign Member of the Royal Society (ForMemRS) in 1968.[35][1] He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.[36]
Religious views[edit]
Gödel believed that God was personal,[41] and called his philosophy "rationalistic, idealistic, optimistic, and theological".[42]
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 [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."[43]
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."[44] Of religion(s) in general, he said: "Religions are, for the most part, bad—but religion is not".[45] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[46] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."[47]
Legacy[edit]
Douglas Hofstadter wrote the 1979 book Gödel, Escher, Bach to celebrate the work and ideas of Gödel, M. C. Escher and Johann Sebastian Bach. 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 Kurt Gödel Society, founded in 1987, is an international organization for the promotion of research in logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has held an annual Gödel Lecture each year since 1990. Gödel's Philosophical Notebooks Archived May 14, 2019, at the Wayback Machine are edited at the Kurt Gödel Research Centre Archived May 14, 2019, at the Wayback Machine which is situated at the Berlin-Brandenburg Academy of Sciences and Humanities in Germany.
Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from his Nachlass, 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, ISBN 1-56881-256-6). 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, ISBN 978-0-393-35820-9), was a New York Times Critics' Top Book of 2021.[48]
Gödel was also one of four mathematicians examined in David Malone's 2008 BBC documentary Dangerous Knowledge.[49]
The Gödel Prize is given annually for an outstanding paper in theoretical computer science.
In the 2023 movie Oppenheimer, Gödel, played by James Urbaniak, briefly appears walking with Einstein in the gardens of Princeton.
In English:
In English translation: