Kurt Gödel

== Career ==
[[File:Young Kurt Gödel as a student in 1925.jpg|thumb|Gödel as a student in 1925]]

=== Incompleteness theorems ===
{{blockquote|Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement.|[[John von Neumann]]<ref>{{Cite journal |last=Halmos |first=P.R. |title=The Legend of von Neumann |journal=The American Mathematical Monthly |volume=80 |number=4 |date=April 1973 |pages=382–94|doi=10.1080/00029890.1973.11993293 }}</ref>}}

In 1930 Gödel attended the [[Second Conference on the Epistemology of the Exact Sciences]], held in [[Königsberg]], 5–7 September. Here he delivered his [[Gödel's incompleteness theorems|incompleteness theorems]].<ref name="Stadler">{{cite book |last1=Stadler |first1=Friedrich |title=The Vienna Circle: Studies in the Origins, Development, and Influence of Logical Empiricism |date=2015 |publisher=Springer |isbn=978-3-319-16561-5 |url=https://books.google.com/books?id=2rAlCQAAQBAJ&q=Erkenntnis+1930+Konigsberg&pg=PA161 |language=en}}</ref>

Gödel published his incompleteness theorems in {{lang|de|Über formal unentscheidbare Sätze der {{lang|la|Principia Mathematica}} und verwandter Systeme}} (called in English "[[On Formally Undecidable Propositions of Principia Mathematica and Related Systems|On Formally Undecidable Propositions of {{lang|la|Principia Mathematica|nocat=y}} and Related Systems]]"). In that article, he proved for any [[recursion theory|computable]] [[axiomatic system]] that is powerful enough to describe the arithmetic of the [[natural numbers]] (e.g., the [[Peano axioms]] or [[ZFC|Zermelo–Fraenkel set theory with the axiom of choice]]), that:

# If a (logical or axiomatic formal) [[formal system|system]] is [[omega-consistency|omega-consistent]], it cannot be [[completeness (logic)|syntactically complete]].
# The consistency of [[axiom]]s cannot be proved within their own [[axiomatic system|system]].

These theorems ended a half-century of attempts, beginning with the work of [[Gottlob Frege]] and culminating in {{lang|la|[[Principia Mathematica]]}} and [[Hilbert's Program]], to find a non-[[Relative consistency|relatively]] consistent axiomatization sufficient for number theory (that was to serve as the foundation for other fields of mathematics).

The idea at the center of the incompleteness theorem is simple. Gödel constructed a formula that claims 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]] 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 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 number]]ing.

In his two-page paper {{lang|de|Zum intuitionistischen Aussagenkalkül}} (1932) Gödel refuted the finite-valuedness of [[intuitionistic logic]]. In the proof, he implicitly used what has later become known as [[intermediate logic|Gödel–Dummett intermediate logic]] (or [[t-norm fuzzy logic|Gödel fuzzy logic]]).

=== Mid-1930s: further work and U.S. visits ===
Gödel earned his [[habilitation]] at Vienna in 1932, and in 1933 he became a {{lang|de|[[Privatdozent]]}} (unpaid lecturer) there. In 1933 [[Adolf Hitler]] 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 assassinated by one of his former students, [[Johann Nelböck]]. This triggered "a severe nervous crisis" in Gödel.<ref name=Casti2001>{{Cite book |last1=Casti |first1=John L. |last2=Depauli |first2=Werner |year=2001 |title=Godel: A Life Of Logic, The Mind, And Mathematics |doi= |isbn=978-0-7382-0518-2 |location= Cambridge, Mass. |publisher=Basic Books}}. From p.&nbsp;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.</ref> He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.<ref>Dawson 1997, pp. 110–12</ref>

In 1933, Gödel first traveled to the U.S., where he met [[Albert Einstein]], who became a good friend.<ref>''[[Hutchinson Encyclopedia]]'' (1988), p.&nbsp;518</ref> He delivered an address to the annual meeting of the [[American Mathematical Society]]. During this year, Gödel also developed the ideas of computability and [[Computable function|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]] (IAS) in [[Princeton, New Jersey]], titled ''On undecidable propositions of formal mathematical systems''. [[Stephen Kleene]], 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]] 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.

He married {{ill|Adele Gödel|lt=Adele Nimbursky|es || ast}} (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,''<ref>{{Cite journal |last=Gödel |first=Kurt |date=November 9, 1938 |title=The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=24 |issue=12 |pages=556–57 |issn=0027-8424 |pmc=1077160 |pmid=16577857 |bibcode=1938PNAS...24..556G |doi=10.1073/pnas.24.12.556 |doi-access=free }}</ref> a classic of modern mathematics. In that work he introduced the [[constructible universe]], a model of [[set theory]] in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the [[axiom of choice]] (AC) and the [[generalized continuum hypothesis]] (GCH) are true in the constructible universe, and therefore must be consistent with the [[Zermelo–Fraenkel axioms]] for set theory (ZF). 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]]. [[Paul Cohen]] later constructed a [[structure (mathematical logic)|model]] 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]].<ref>{{cite web |url=https://math.nd.edu/assets/13975/logicatndweb.pdf |title=Kurt Gödel at Notre Dame |last=Dawson |first=John W. Jr |page=4 |quote=the Mathematics department at the University of Notre Dame was host ... for a single semester in the spring of 1939 [to] Kurt Gödel }}</ref>

=== Princeton, Einstein, U.S. citizenship ===
After the [[Anschluss]] on 12 March 1938, Austria had become a part of [[Nazi Germany]].
Germany abolished the title {{lang|de|[[Privatdozent]]}}, 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, New Jersey|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 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.<ref>{{Cite web|url=https://www.ias.edu/scholars/godel|title=Kurt Gödel|website=Institute for Advanced Study|date=December 9, 2019}}</ref>

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]] 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".<ref>{{Harvnb|Goldstein|2005|p=[https://books.google.com/books?id=tXk2AAAAQBAJ&pg=PA33 33]}}</ref>

Gödel and his wife, Adele, spent the summer of 1942 in [[Blue Hill, Maine]], 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 {{lang|de|Heft 15}} [volume 15] of Gödel's still-unpublished {{lang|de|Arbeitshefte}} [working notebooks], [[John W. Dawson Jr.]] conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend [[Hao Wang (academic)|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]] 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.<ref>Dawson 1997, pp. 179–80. The story of Gödel's citizenship hearing is repeated in many versions. Dawson's account is the most carefully researched, but was written before the rediscovery of Morgenstern's written account. Most other accounts appear to be based on Dawson, hearsay or speculation.</ref><ref>{{cite web |url=https://robert.accettura.com/wp-content/uploads/2010/10/Morgenstern_onGoedelcitizenship.pdf |title=History of the Naturalization of Kurt Gödel |date=September 13, 1971 |author=Oskar Morgenstern |access-date=April 16, 2019 }}</ref>

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.<ref>{{cite web |url=https://www.ias.edu/people/godel |title=Kurt Gödel – Institute for Advanced Study |access-date=December 1, 2015 }}</ref>

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]]s, to [[Einstein's field equations]] in [[general relativity]].<ref>{{cite journal |last=Gödel |first=Kurt |title=An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation |journal=[[Rev. Mod. Phys.]] |volume=21 |issue=447 |pages=447–450 |date=July 1, 1949 |doi=10.1103/RevModPhys.21.447 |bibcode=1949RvMP...21..447G |doi-access=free }}</ref> He is said to have given this elaboration to Einstein as a present for his 70th birthday.<ref>{{cite news |url=http://www.tagesspiegel.de/magazin/wissen/Albert-Einstein-Kurt-Goedel;art304,2454513 |title=Das Genie & der Wahnsinn |work=[[Der Tagesspiegel]] |date=January 13, 2008 |language=de }}</ref> 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]]).

He studied and admired the works of [[Gottfried Leibniz]], but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed.<ref>{{cite book | first=John W. Jr. |last=Dawson |url=https://books.google.com/books?id=gA8SucCU1AYC&q=godel+leibniz&pg=PA166 |title=Logical Dilemmas: The Life and Work of Kurt Gödel. |publisher=A K Peters |year=2005 |page=166 |isbn=978-1-56881-256-4 }}</ref> To a lesser extent he studied [[Immanuel Kant]] and [[Edmund Husserl]]. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of [[Anselm of Canterbury]]'s [[ontological argument|ontological proof]] of God's existence. This is now known as [[Gödel's ontological proof]].