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{{Redirect|Gödel|the programming language|Gödel (programming language)|other uses|Godel (disambiguation)}}

{{Redirect|Gödel|the programming language|Gödel (programming language)|other uses|Godel (disambiguation)}}

{{short description|logician and mathematician}}

{{short description|logician and mathematician}}

{{Use mdy dates|date=July 2014}}

{{{简介{逻辑学家和数学家}}

 

{Use mdy dates|date=July 2014}}

 

{{Use mdy dates | date=2014年7月}}

{{Infobox scientist

{{Infobox scientist

| known_for = Gödel's incompleteness theorems<br>Gödel's completeness theorem<br>Gödel's constructible universe<br>Gödel metric (closed timelike curve)<br>Gödel logic<br>Gödel–Dummett logic<br>Gödel's β function<br>Gödel numbering<br>Gödel operation<br>Gödel's speed-up theorem<br>Gödel's ontological proof<br>Gödel–Gentzen translation<br>Von Neumann–Bernays–Gödel set theory<br>ω-consistent theory<br>The consistency of the continuum hypothesis with ZFC<br>Axiom of constructibility<br>Condensation lemma<br>Dialectica interpretation<br>Slingshot argument

| known_for = Gödel's incompleteness theorems<br>Gödel's completeness theorem<br>Gödel's constructible universe<br>Gödel metric (closed timelike curve)<br>Gödel logic<br>Gödel–Dummett logic<br>Gödel's β function<br>Gödel numbering<br>Gödel operation<br>Gödel's speed-up theorem<br>Gödel's ontological proof<br>Gödel–Gentzen translation<br>Von Neumann–Bernays–Gödel set theory<br>ω-consistent theory<br>The consistency of the continuum hypothesis with ZFC<br>Axiom of constructibility<br>Condensation lemma<br>Dialectica interpretation<br>Slingshot argument

哥德尔的不完备性定理哥德尔的完备性定理哥德尔的可构造宇宙哥德尔度量封闭类时曲线哥德尔逻辑哥德尔达米特逻辑哥德尔的 β 函数哥德尔编号哥德尔运算哥德尔加速定理哥德尔的加速定理哥德尔的逻辑证明哥德尔的逻辑证明理论 < br > 连续统假设与 ZFC < br > 构造性公理的一致性 < br > 冷凝引理 < br > 方言解释 < br > 弹弓论点

| known_for =<font color="#ff8000"> 哥德尔不完备性定理<br>哥德尔完备性定理<br>哥德尔可构造宇宙<br>哥德尔度量封闭类时曲线<br>哥德尔逻辑<br>哥德尔达米特逻辑<br>哥德尔的 β 函数<br>哥德尔编号<br>哥德尔运算<br>哥德尔加速定理<br>哥德尔逻辑证明理论<br> 连续统假设与 ZFC <br> 构造性公理的一致性<br> 冷凝引理<br> 方言解释<br>弹弓论点</font>

| prizes = {{Plainlist|

| prizes = {{Plainlist|

* [[Albert Einstein Award]] (1951)

* [[Albert Einstein Award]] (1951)

 

* [[National Medal of Science]] (1974)

* [[National Medal of Science]] (1974)

 

* [[Fellow of the Royal Society|ForMemRS]] (1968)<ref name=frs>{{Cite journal | last1 = Kreisel | first1 = G. | authorlink = Georg Kreisel| doi = 10.1098/rsbm.1980.0005 | title = Kurt Godel. 28 April 1906–14 January 1978 | journal = [[Biographical Memoirs of Fellows of the Royal Society]] | volume = 26 | pages = 148–224| year = 1980 | pmid =  | pmc = | doi-access = free }}</ref>

* [[Fellow of the Royal Society|ForMemRS]] (1968)<ref name=frs>{{Cite journal | last1 = Kreisel | first1 = G. | authorlink = Georg Kreisel| doi = 10.1098/rsbm.1980.0005 | title = Kurt Godel. 28 April 1906–14 January 1978 | journal = [[Biographical Memoirs of Fellows of the Royal Society]] | volume = 26 | pages = 148–224| year = 1980 | pmid =  | pmc = | doi-access = free }}</ref>

}}

}}

Kurt Friedrich Gödel (; ; April 28, 1906&nbsp;– January 14, 1978) was a logician, mathematician, and analytic philosopher. Considered along with Aristotle and Gottlob Frege 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, Alfred North Whitehead, Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging. 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 Protestant. The ancestors of Kurt Gödel were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the  (Men's Choral Union of Brünn).

Kurt Friedrich Gödel (; ; April 28, 1906&nbsp;– January 14, 1978) was a logician, mathematician, and analytic philosopher. Considered along with Aristotle and Gottlob Frege 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, Alfred North Whitehead, Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging. 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 Protestant. The ancestors of Kurt Gödel were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the  (Men's Choral Union of Brünn).

 

Gödel published his two [[Gödel's incompleteness theorems|incompleteness theorems]] in 1931 when he was 25 years old, one year after finishing his doctorate at the [[University of Vienna]]. The first incompleteness theorem states that for any self-consistent [[recursive set|recursive]] [[axiomatic system]] powerful enough to describe the arithmetic of the [[natural number]]s (for example [[Peano arithmetic]]), there are true propositions about the natural numbers that cannot be proved from the [[axioms]]. To prove this theorem, Gödel developed a technique now known as [[Gödel numbering]], which codes formal expressions as natural numbers.

Gödel published his two [[Gödel's incompleteness theorems|incompleteness theorems]] in 1931 when he was 25 years old, one year after finishing his doctorate at the [[University of Vienna]]. The first incompleteness theorem states that for any self-consistent [[recursive set|recursive]] [[axiomatic system]] powerful enough to describe the arithmetic of the [[natural number]]s (for example [[Peano arithmetic]]), there are true propositions about the natural numbers that cannot be proved from the [[axioms]]. To prove this theorem, Gödel developed a technique now known as [[Gödel numbering]], which codes formal expressions as natural numbers.

Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire collapsed, following its defeat in the World War I. (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 annexed Austria in 1938, Gödel automatically became a German citizen at age 32. After World War II (1948), at the age of 42, he became an American citizen.

Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire collapsed, following its defeat in the World War I. (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 annexed Austria in 1938, Gödel automatically became a German citizen at age 32. After World War II (1948), at the age of 42, he became an American citizen.

He also showed that neither the [[axiom of choice]] nor the [[continuum hypothesis]] can be disproved from the accepted [[axiomatic set theory|axioms of set theory]], assuming these 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]].

He also showed that neither the [[axiom of choice]] nor the [[continuum hypothesis]] can be disproved from the accepted [[axiomatic set theory|axioms of set theory]], assuming these 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]].

In his family, young Kurt was known as  ("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.

In his family, young Kurt was known as  ("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.

==Early life and education==

==Early life and education早期教育==

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 Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

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 Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

===Childhood===

===Childhood童年===

Gödel was born April 28, 1906, in Brünn, [[Austria-Hungary]] (now [[Brno]], [[Czech Republic]]) into the German family of Rudolf Gödel (1874–1929), the manager of a textile factory, and Marianne Gödel ([[née]] Handschuh, 1879–1966).<ref>Dawson 1997, pp. 3–4.</ref> Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging.<ref>{{Cite book|url=http://plato.stanford.edu/archives/win2015/entries/johann-herbart/|title=Johann Friedrich Herbart|last=Kim|first=Alan|date=2015-01-01|editor-last=Zalta|editor-first=Edward N.|edition=Winter 2015}}</ref> At the time of his birth the city had a [[German language|German-speaking]] majority which included his parents.<ref>Dawson 1997, p.&nbsp;12</ref> 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 of that time and for some years a member of the {{lang|de|Brünner Männergesangverein}} (Men's Choral Union of Brünn).<ref>Procházka 2008, pp. 30–34.</ref>

Gödel was born April 28, 1906, in Brünn, [[Austria-Hungary]] (now [[Brno]], [[Czech Republic]]) into the German family of Rudolf Gödel (1874–1929), the manager of a textile factory, and Marianne Gödel ([[née]] Handschuh, 1879–1966).<ref>Dawson 1997, pp. 3–4.</ref> Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging.<ref>{{Cite book|url=http://plato.stanford.edu/archives/win2015/entries/johann-herbart/|title=Johann Friedrich Herbart|last=Kim|first=Alan|date=2015-01-01|editor-last=Zalta|editor-first=Edward N.|edition=Winter 2015}}</ref> At the time of his birth the city had a [[German language|German-speaking]] majority which included his parents.<ref>Dawson 1997, p.&nbsp;12</ref> 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 of that time and for some years a member of the {{lang|de|Brünner Männergesangverein}} (Men's Choral Union of Brünn).<ref>Procházka 2008, pp. 30–34.</ref>

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics. Although initially intending to study theoretical physics, 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. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics. Although initially intending to study theoretical physics, 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. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."

Gödel automatically became a [[Czechoslovakia|Czechoslovak]] citizen at age 12 when the Austro-Hungarian Empire collapsed, following its defeat in the [[World War I]]. (According to his classmate {{lang|cs|Klepetař|italic=no}}, like many residents of the predominantly German {{lang|de|[[Sudetenland|Sudetenländer]]}}, "Gödel considered himself always Austrian and an exile in Czechoslovakia".)<ref>Dawson 1997, p.&nbsp;15.</ref> In February 1929 he was granted release from his Czechoslovakian citizenship and then, in April, granted Austrian citizenship.<ref>{{Cite book|url=https://books.google.com/books?id=5ya4A0w62skC&pg=PA37|title=Collected works|last=Gödel, Kurt|publisher=|others=Feferman, Solomon|year=1986|isbn=0195039645|location=Oxford|pages=37|oclc=12371326}}</ref> When [[Nazi Germany|Germany]] [[Anschluss|annexed Austria]] in 1938, Gödel automatically became a German citizen at age 32. After [[World War II]] (1948), at the age of 42, he became an American citizen.<ref>{{cite web |last1=Balaguer |first1=Mark |title=Kurt Godel |url=https://school.eb.com/levels/high/article/Kurt-G%C3%B6del/37162 |website=Britannica School High |publisher=Encyclopædia Britannica, Inc. |accessdate=3 June 2019}}</ref>

Gödel automatically became a [[Czechoslovakia|Czechoslovak]] citizen at age 12 when the Austro-Hungarian Empire collapsed, following its defeat in the [[World War I]]. (According to his classmate {{lang|cs|Klepetař|italic=no}}, like many residents of the predominantly German {{lang|de|[[Sudetenland|Sudetenländer]]}}, "Gödel considered himself always Austrian and an exile in Czechoslovakia".)<ref>Dawson 1997, p.&nbsp;15.</ref> In February 1929 he was granted release from his Czechoslovakian citizenship and then, in April, granted Austrian citizenship.<ref>{{Cite book|url=https://books.google.com/books?id=5ya4A0w62skC&pg=PA37|title=Collected works|last=Gödel, Kurt|publisher=|others=Feferman, Solomon|year=1986|isbn=0195039645|location=Oxford|pages=37|oclc=12371326}}</ref> When [[Nazi Germany|Germany]] [[Anschluss|annexed Austria]] in 1938, Gödel automatically became a German citizen at age 32. After [[World War II]] (1948), at the age of 42, he became an American citizen.<ref>{{cite web |last1=Balaguer |first1=Mark |title=Kurt Godel |url=https://school.eb.com/levels/high/article/Kurt-G%C3%B6del/37162 |website=Britannica School High |publisher=Encyclopædia Britannica, Inc. |accessdate=3 June 2019}}</ref>

Attending a lecture by David Hilbert in Bologna on completeness and consistency of 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?

Attending a lecture by David Hilbert in Bologna on completeness and consistency of 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?

In his family, young Kurt was known as {{lang|de|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.<ref>{{Cite book |url=http://plato.stanford.edu/archives/win2015/entries/johann-herbart/ |title=Johann Friedrich Herbart |last=Kim |first=Alan |date=2015-01-01 |editor-last=Zalta |editor-first=Edward N. |edition=Winter 2015 }}</ref>

In his family, young Kurt was known as {{lang|de|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.<ref>{{Cite book |url=http://plato.stanford.edu/archives/win2015/entries/johann-herbart/ |title=Johann Friedrich Herbart |last=Kim |first=Alan |date=2015-01-01 |editor-last=Zalta |editor-first=Edward N. |edition=Winter 2015 }}</ref>

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 regarding the first-order predicate calculus. He was awarded his doctorate in 1930, and his thesis (accompanied by some additional work) was published by the Vienna Academy of Science.

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 regarding the first-order predicate calculus. He was awarded his doctorate in 1930, and his thesis (accompanied by some additional work) was published by the Vienna Academy of Science.

Gödel attended the {{lang|de|Evangelische Volksschule}}, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the {{lang|de|Deutsches Staats-Realgymnasium}} from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for [[Vienna]] to go to medical school at the [[University of Vienna]]. During his teens, Kurt studied [[Gabelsberger shorthand]], [[Johann Wolfgang von Goethe|Goethe]]'s ''[[Theory of Colours (book)|Theory of Colours]]'' and criticisms of [[Isaac Newton]], and the writings of [[Immanuel Kant]].

Gödel attended the {{lang|de|Evangelische Volksschule}}, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the {{lang|de|Deutsches Staats-Realgymnasium}} from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for [[Vienna]] to go to medical school at the [[University of Vienna]]. During his teens, Kurt studied [[Gabelsberger shorthand]], [[Johann Wolfgang von Goethe|Goethe]]'s ''[[Theory of Colours (book)|Theory of Colours]]'' and criticisms of [[Isaac Newton]], and the writings of [[Immanuel Kant]].

 

===Studying in Vienna在维也纳的学习===

===Studying in Vienna===

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.<ref>Dawson 1997, p.&nbsp;24.</ref> Although initially intending to study [[theoretical physics]], he also attended courses on mathematics and philosophy. During this time, he adopted ideas of [[mathematical realism]]. He read [[Immanuel Kant|Kant]]'s {{lang|de|[[Metaphysical Foundations of Natural Science|Metaphysische Anfangsgründe der Naturwissenschaft]]|italic=yes}}, and participated in the [[Vienna Circle]] with [[Moritz Schlick]], [[Hans Hahn (mathematician)|Hans Hahn]], and [[Rudolf Carnap]]. Gödel then studied [[number theory]], but when he took part in a seminar run by [[Moritz Schlick]] which studied [[Bertrand Russell]]'s book ''Introduction to Mathematical Philosophy'', he became interested in [[mathematical logic]]. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."<ref>Gleick, J. (2011) ''[[The Information: A History, a Theory, a Flood]],'' London, Fourth Estate, p. 181.</ref>

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.<ref>Dawson 1997, p.&nbsp;24.</ref> Although initially intending to study [[theoretical physics]], he also attended courses on mathematics and philosophy. During this time, he adopted ideas of [[mathematical realism]]. He read [[Immanuel Kant|Kant]]'s {{lang|de|[[Metaphysical Foundations of Natural Science|Metaphysische Anfangsgründe der Naturwissenschaft]]|italic=yes}}, and participated in the [[Vienna Circle]] with [[Moritz Schlick]], [[Hans Hahn (mathematician)|Hans Hahn]], and [[Rudolf Carnap]]. Gödel then studied [[number theory]], but when he took part in a seminar run by [[Moritz Schlick]] which studied [[Bertrand Russell]]'s book ''Introduction to Mathematical Philosophy'', he became interested in [[mathematical logic]]. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."<ref>Gleick, J. (2011) ''[[The Information: A History, a Theory, a Flood]],'' London, Fourth Estate, p. 181.</ref>

Attending a lecture by [[David Hilbert]] in [[Bologna]] on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and [[Wilhelm Ackermann]] published {{lang|de|Grundzüge der theoretischen Logik|italic=yes}} (''[[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?''

参加[[davidhilbert]]在[[Bologna]]举办的一场关于数学系统的完整性和一致性的讲座，可能已经为哥德尔的人生道路定下了伏笔。1928年，希尔伯特和[[威廉·阿克曼]]出版了{lang | de | Grundzüge der theoretischen Logik | italic=yes}}（'[[数学逻辑原理]]'，[[一阶逻辑]]的导论，其中提出了完备性问题：“一个形式系统的公理是否足以导出系统所有模型中的每一个正确的陈述？”

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.

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 [[Gödel's completeness theorem|completeness theorem]] regarding the [[first-order predicate calculus]]. He was awarded his doctorate in 1930, and his thesis (accompanied by some additional work) was published by the [[Vienna Academy of Science]].

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 [[Gödel's completeness theorem|completeness theorem]] regarding the [[first-order predicate calculus]]. He was awarded his doctorate in 1930, and his thesis (accompanied by some additional work) was published by the [[Vienna Academy of Science]].

Gödel published his incompleteness theorems in  und verwandter Systeme}} (called in English "On Formally Undecidable Propositions of  and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

Gödel published his incompleteness theorems in  und verwandter Systeme}} (called in English "On Formally Undecidable Propositions of  and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

如果一个(逻辑或公理化的正式)系统是一致的，那么它就不能是完整的。

如果一个(逻辑或公理化的正式)系统是一致的，那么它就不能是完整的。

==Career==

==Career职业生涯==

  The consistency of axioms cannot be proved within their own system.

  The consistency of axioms cannot be proved within their own system.

这些定理结束了半个世纪的尝试，开始于弗雷格的工作，最终在希尔伯特的形式主义，找到一套公理足以为所有数学。

这些定理结束了半个世纪的尝试，开始于弗雷格的工作，最终在希尔伯特的形式主义，找到一套公理足以为所有数学。

===Incompleteness theorem===

===Incompleteness theorem不完全性定理===

{{quote|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>}}

{{quote|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 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.

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.

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=9783319165615 |url=https://books.google.com/books?id=2rAlCQAAQBAJ&q=Erkenntnis+1930+Konigsberg&pg=PA161 |language=en}}</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=9783319165615 |url=https://books.google.com/books?id=2rAlCQAAQBAJ&q=Erkenntnis+1930+Konigsberg&pg=PA161 |language=en}}</ref>

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 which is not provable in that system.

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 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.

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.

然而，要做到这一点，哥德尔需要产生一种方法来编码(自然数)的陈述，证明，和可证明的概念; 他这样做使用的过程称为哥德尔编号。

然而，要做到这一点，哥德尔需要产生一种方法来编码(自然数)的陈述，证明，和可证明的概念; 他这样做使用的过程称为哥德尔编码。

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:

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 [[consistency proof|consistent]], it cannot be [[completeness (logic)|complete]].

# If a (logical or axiomatic formal) [[formal system|system]] is [[consistency proof|consistent]], it cannot be [[completeness (logic)|complete]].

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).

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).

在他1932年的两页论文中，哥德尔反驳了直觉主义逻辑的有限价值。在证明中，他隐含地使用了后来被称为哥德尔-达米特中间逻辑逻辑(或哥德尔模糊逻辑)。

在他1932年的两页论文中，哥德尔反驳了直觉主义逻辑的有限价值。在证明中，他隐含地使用了后来被称为的哥德尔-达米特中间逻辑(或哥德尔模糊逻辑)。

# The consistency of [[axiom]]s cannot be proved within their own [[axiomatic system|system]].

# 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 [[Frege]] and culminating in {{lang|la|[[Principia Mathematica]]}} and [[philosophy of mathematics#Formalism|Hilbert's formalism]], to find a set of axioms sufficient for all mathematics.

These theorems ended a half-century of attempts, beginning with the work of [[Frege]] and culminating in {{lang|la|[[Principia Mathematica]]}} and [[philosophy of mathematics#Formalism|Hilbert's formalism]], to find a set of axioms sufficient for all mathematics.

 

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a  (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. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a  (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. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.

1932年，哥德尔在维也纳获得了学位，1933年，他在那里成为一名(无薪讲师)。1933年，阿道夫 · 希特勒在德国掌权，随后几年，纳粹在奥地利和维也纳的数学家中的影响力不断上升。1936年6月，莫里茨 · 施里克的研讨会引起了哥德尔对逻辑学的兴趣，却被他以前的学生约翰 · 内尔博克暗杀。这在哥德尔引发了“一场严重的神经危机”。他出现了偏执症状，包括害怕中毒，并因神经疾病在疗养院度过了几个月。

1932年，哥德尔在维也纳获得了学位，1933年，他在那里成为一名(无薪讲师)。1933年，阿道夫 · 希特勒在德国掌权，随后几年，纳粹在奥地利和维也纳的数学家中的影响力不断上升。1936年6月，莫里茨 · 施里克的研讨会引起了哥德尔对逻辑学的兴趣，却被他以前的学生约翰 · 内尔博克暗杀。这对哥德尔引发了“一场严重的神经危机”。他出现了偏执症状，包括害怕中毒，并因神经疾病在疗养院度过了几个月。

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.

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.

Thus there will always be at least one true but unprovable statement.

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. 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 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. 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.

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 number]]ing.

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 number]]ing.

In 1934, Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

In 1934, Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

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]]).

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]]).

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.

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.

===Mid-1930s: further work and U.S. visits===

===Mid-1930s: further work and U.S. visits20世纪30年代中期：进一步的工作和美国访问===

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.

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.

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=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}. 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>

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=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}. 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>

 

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, 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 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.

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, 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 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.

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 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]].

 

Gödel spent the spring of 1939 at the University of Notre Dame.

Gödel spent the spring of 1939 at the University of Notre Dame.

In 1934, Gödel gave a series of lectures at the [[Institute for Advanced Study]] (IAS) in [[Princeton, New Jersey|Princeton]], New Jersey, entitled ''On undecidable propositions of formal mathematical systems''. [[Stephen Kleene]], who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

In 1934, Gödel gave a series of lectures at the [[Institute for Advanced Study]] (IAS) in [[Princeton, New Jersey|Princeton]], New Jersey, entitled ''On undecidable propositions of formal mathematical systems''. [[Stephen Kleene]], who had just completed his PhD at Princeton, took notes of these lectures 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.

1935年秋，哥德尔再次访问了国际会计学院。旅行和艰苦的工作使他筋疲力尽，第二年他休息一下，从抑郁中恢复过来。他于1937年重返教书岗位。在这段时间里，他致力于证明[[选择公理]]和[[连续统假设]]的一致性；他接着指出，这些假设不能从集合论公理的共同体系中得到反驳。

After the Anschluss on 12 March 1938, Austria had become a part of Nazi Germany.

After the Anschluss on 12 March 1938, Austria had become a part of Nazi Germany.

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.

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.

 

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939.

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939.

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 }}</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.

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 }}</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.

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 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.

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 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.

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 |date= |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>

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 |date= |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>

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".

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".

===Princeton, Einstein, U.S. citizenship===

===Princeton, Einstein, U.S. citizenship普林斯顿，爱因斯坦，美国公民===

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  [volume 15] of Gödel's still-unpublished  [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

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  [volume 15] of Gödel's still-unpublished  [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

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.

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.

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. 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.

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. 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.

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939.

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939.

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.

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.

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}}</ref>

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}}</ref>

 

During his many years 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. 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).

During his many years 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. 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).

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>Goldstein (2005), p.&nbsp;33.</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>Goldstein (2005), p.&nbsp;33.</ref>

 

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. 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 proof of God's existence. This is now known as Gödel's ontological proof.

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. 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 proof of God's existence. This is now known as Gödel's ontological proof.

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.

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.

Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974. Gödel was elected a Foreign Member of the Royal Society (ForMemRS) in 1968. The Gödel Prize, an annual prize for outstanding papers in the area of theoretical computer science, is named after him.

Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974. Gödel was elected a Foreign Member of the Royal Society (ForMemRS) in 1968. The Gödel Prize, an annual prize for outstanding papers in the area of theoretical computer science, is named after him.

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 |accessdate=December 1, 2015 }}</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 |accessdate=December 1, 2015 }}</ref>

Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery

Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery

During his many years 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]]).

During his many years 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]]).

 

Later in his life, Gödel suffered periods of mental instability and illness. Following the assassination of his close friend Moritz Schlick, Gödel had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could subsequently no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death. He weighed  when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery. Adele's death followed in 1981.

Later in his life, Gödel suffered periods of mental instability and illness. Following the assassination of his close friend Moritz Schlick, Gödel had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could subsequently no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death. He weighed  when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He was buried in Princeton Cemetery. Adele's death followed in 1981.

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=9781568812564 }}</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]].

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=9781568812564 }}</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]].

 

==Awards and honours奖励与荣誉==

==Awards and honours==

Gödel was awarded (with [[Julian Schwinger]]) the first [[Albert Einstein Award]] in 1951, and was also awarded the [[National Medal of Science]], in 1974.<ref>{{cite web|url=https://www.nsf.gov/od/nms/recip_details.jsp?recip_id=138|title=The President's National Medal of Science: Recipient Details {{!}} NSF – National Science Foundation|website=www.nsf.gov|access-date=2016-09-17}}</ref> Gödel was elected a [[List of Fellows of the Royal Society elected in 1968|Foreign Member of the Royal Society (ForMemRS) in 1968]].<ref name=frs/> He was a Plenary Speaker of the [[International Congress of Mathematicians|ICM]] in 1950 in Cambridge, Massachusetts.<ref>{{cite book|author=Gödel, Kurt|chapter=Rotating universes in general relativity theory|title=''In:'' Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, August 30–September 6, 1950|volume=vol. 1|pages=175–81|year=1950|chapter-url=http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0175.0181.ocr.pdf}}</ref> The [[Gödel Prize]], an annual prize for outstanding papers in the area of theoretical computer science, is named after him.

Gödel was awarded (with [[Julian Schwinger]]) the first [[Albert Einstein Award]] in 1951, and was also awarded the [[National Medal of Science]], in 1974.<ref>{{cite web|url=https://www.nsf.gov/od/nms/recip_details.jsp?recip_id=138|title=The President's National Medal of Science: Recipient Details {{!}} NSF – National Science Foundation|website=www.nsf.gov|access-date=2016-09-17}}</ref> Gödel was elected a [[List of Fellows of the Royal Society elected in 1968|Foreign Member of the Royal Society (ForMemRS) in 1968]].<ref name=frs/> He was a Plenary Speaker of the [[International Congress of Mathematicians|ICM]] in 1950 in Cambridge, Massachusetts.<ref>{{cite book|author=Gödel, Kurt|chapter=Rotating universes in general relativity theory|title=''In:'' Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, August 30–September 6, 1950|volume=vol. 1|pages=175–81|year=1950|chapter-url=http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0175.0181.ocr.pdf}}</ref> The [[Gödel Prize]], an annual prize for outstanding papers in the area of theoretical computer science, is named after him.

 

Gödel was a convinced theist, in the Christian tradition. He held the notion that God was personal.

Gödel was a convinced theist, in the Christian tradition. He held the notion that God was personal.

他坚定地相信来世，他说: “当然，这是假设有许多关系，今天的科学和公认的智慧没有任何暗示。但我相信这个(来世) ，与任何神学都无关。”“今天，通过纯粹的推理，我们有可能认识到” ，它“与已知的事实完全一致”“如果世界是合理构建的，并且有意义，那么就一定存在(来世)。”

他坚定地相信来世，他说: “当然，这是假设有许多关系，今天的科学和公认的智慧没有任何暗示。但我相信这个(来世) ，与任何神学都无关。”“今天，通过纯粹的推理，我们有可能认识到” ，它“与已知的事实完全一致”“如果世界是合理构建的，并且有意义，那么就一定存在(来世)。”

==Later life and death==

==Later life and death晚年生活与死亡==

Later in his life, Gödel suffered periods of [[mental disorder|mental instability]] and illness. Following the assassination of his close friend [[Moritz Schlick]],<ref name="pape_Trag">{{Cite web | title = Tragic deaths in science: Kurt Gödel - looking over the edge of reason - Paperpile | url = https://paperpile.com/blog/kurt-goedel/}}</ref> Gödel had an [[persecutory delusion|obsessive fear of being poisoned]]; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could subsequently no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.<ref>{{cite journal|title=Gödel's universe|author=Davis, Martin|journal=Nature|date=May 4, 2005|volume=435|issue=7038|doi=10.1038/435019a|pages=19–20|bibcode=2005Natur.435...19D|doi-access=free}}</ref> He weighed {{convert|65|lbs|kg|order=flip}} when he died. His death certificate reported that he died of "malnutrition and [[inanition]] caused by personality disturbance" in [[Princeton Hospital]] on January 14, 1978.<ref>{{cite book

Later in his life, Gödel suffered periods of [[mental disorder|mental instability]] and illness. Following the assassination of his close friend [[Moritz Schlick]],<ref name="pape_Trag">{{Cite web | title = Tragic deaths in science: Kurt Gödel - looking over the edge of reason - Paperpile | url = https://paperpile.com/blog/kurt-goedel/}}</ref> Gödel had an [[persecutory delusion|obsessive fear of being poisoned]]; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could subsequently no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.<ref>{{cite journal|title=Gödel's universe|author=Davis, Martin|journal=Nature|date=May 4, 2005|volume=435|issue=7038|doi=10.1038/435019a|pages=19–20|bibcode=2005Natur.435...19D|doi-access=free}}</ref> He weighed {{convert|65|lbs|kg|order=flip}} when he died. His death certificate reported that he died of "malnutrition and [[inanition]] caused by personality disturbance" in [[Princeton Hospital]] on January 14, 1978.<ref>{{cite book