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

{{short description|logician and mathematician}}

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

{{Infobox scientist

{{Infobox scientist

{信息盒科学家

| name = Kurt Gödel

| name = Kurt Gödel

| 姓名: 库尔特 · 哥德尔

| image = Kurt gödel.jpg

| image = Kurt gödel.jpg

| image = Kurt gödel.jpg

| image_size =

| image_size =

图片大小 =

| caption =

| caption =

| caption =

| birth_name = Kurt Friedrich Gödel

| birth_name = Kurt Friedrich Gödel

出生名字 = 库尔特·哥德尔

| birth_date = {{birth date|1906|4|28}}

| birth_date =

出生日期

| birth_place = Brünn, [[Austria-Hungary]] {{small|(now [[Brno]], [[Czech Republic]])}}

| birth_place = Brünn, Austria-Hungary 

出生地布鲁恩，Austria-Hungary

| death_date = {{death date and age|1978|1|14|1906|4|28}}

| death_date =

死亡日期

| death_place = [[Princeton, New Jersey|Princeton]], New Jersey, U.S.

| death_place = Princeton, New Jersey, U.S.

死亡地点: 普林斯顿，新泽西州，美国。

| citizenship = {{ubl|Czechoslovak|Austrian|American}}

| citizenship =

公民身份

| field = [[Mathematics]], [[mathematical logic]], [[analytic philosophy]], [[physics]]

| field = Mathematics, mathematical logic, analytic philosophy, physics

数学，数理逻辑，分析哲学，物理

| work_institutions = [[Institute for Advanced Study]]

| work_institutions = Institute for Advanced Study

高等研究院

| alma_mater = [[University of Vienna]]

| alma_mater = University of Vienna

维也纳大学

| thesis_title = Über die Vollständigkeit des Logikkalküls (On the Completeness of the Calculus of Logic)

| thesis_title = Über die Vollständigkeit des Logikkalküls (On the Completeness of the Calculus of Logic)

论文题目: 《逻辑演算的完备性》

| thesis_url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9079526&fileId=S0022481200026633

| thesis_url = http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9079526&fileId=S0022481200026633

Http://journals.cambridge.org/action/displayabstract?frompage=online&aid=9079526&fileid=s0022481200026633

| thesis_year = 1929

| thesis_year = 1929

论文年份 = 1929

| doctoral_advisor = [[Hans Hahn (mathematician)|Hans Hahn]]

| doctoral_advisor = Hans Hahn

博士生导师: Hans Hahn

| doctoral_students =

| doctoral_students =

博士生 =

| influences =

| influences =

影响 =

| influenced =

| influenced =

影响 =

| known_for = [[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 numbering]] [[Gödel operation]] [[Gödel's speed-up theorem]] [[Gödel's ontological proof]] [[Gödel–Gentzen translation]] [[Von Neumann–Bernays–Gödel set theory]] [[ω-consistent theory]] The consistency of the [[continuum hypothesis]] with [[ZFC]] [[Axiom of constructibility]] [[Condensation lemma]] [[Dialectica interpretation]] [[Slingshot argument]]

| known_for = 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 numbering Gödel operation Gödel's speed-up theorem Gödel's ontological proof Gödel–Gentzen translation Von Neumann–Bernays–Gödel set theory ω-consistent theory The consistency of the continuum hypothesis with ZFC Axiom of constructibility Condensation lemma Dialectica interpretation 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 连续统假设与 ZFC 构造性公理的一致性 冷凝引理 方言解释 弹弓论点

| prizes = {{Plainlist|

| prizes = {{Plainlist|

| prizes = {{Plainlist|

* [[Albert Einstein Award]] (1951)

* [[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>

}}

}}

}}

| spouse = {{marriage|Adele Nimbursky|1938}}

| spouse =

配偶 =

| signature = Kurt Gödel signature.svg

| signature = Kurt Gödel signature.svg

| signature = Kurt Gödel signature.svg

}}

}}

}}

'''Kurt Friedrich Gödel''' ({{IPAc-en|ˈ|ɡ|ɜːr|d|əl}};<ref>{{cite Merriam-Webster|Gödel}}</ref> {{IPA-de|ˈkʊɐ̯t ˈɡøːdl̩|lang|Kurt gödel.ogg}}; April 28, 1906 – 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]],<ref name="Stanford&Son">For instance, in their ''[http://plato.stanford.edu/entries/principia-mathematica/ Principia Mathematica]'' (''Stanford Encyclopedia of Philosophy'' edition).</ref> [[Alfred North Whitehead]],<ref name="Stanford&Son"/> and [[David Hilbert]] were analyzing the use of logic and [[set theory]] to understand the [[foundations of mathematics]] pioneered by [[Georg Cantor]].

Kurt Friedrich Gödel (; ; April 28, 1906 – 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).

库尔特·哥德尔是一位逻辑学家、数学家和分析哲学家。哥德尔与亚里士多德和 Gottlob Frege 一样被认为是历史上最重要的逻辑学家之一，他对20世纪的科学和哲学思想产生了巨大的影响，在那个时代，其他人，如伯特兰·罗素，阿尔弗雷德·诺思·怀特黑德，在他的一生中，哥德尔都和他的母亲保持着亲密的关系; 他们之间的通信频繁而广泛。在他出生的时候，这个城市有一个讲德语的大多数，包括他的父母。他的父亲是天主教徒，母亲是新教徒，孩子们都是新教徒。库尔特 · 哥德尔的祖先经常活跃在布吕恩的文化生活中。例如，他的祖父约瑟夫 · 哥德尔是当时一位著名的歌唱家，多年来一直是布鲁恩男子合唱团的成员。

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.

哥德尔在12岁时自动成为捷克斯洛伐克公民，因为奥匈帝国在第一次世界大战中失败而垮台。(根据他的同学的说法，像许多德国人占多数的居民一样，“哥德尔认为自己一直是奥地利人，是捷克斯洛伐克的流亡者”。)1929年2月，他被授予捷克斯洛伐克公民身份，并于4月被授予奥地利公民身份。1938年德国吞并奥地利时，哥德尔在32岁时自动成为德国公民。第二次世界大战后(1948年) ，42岁的他成为了美国公民。

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.

在他的家庭里，年轻的库尔特被称为(“为什么先生”) ，因为他贪得无厌的好奇心。据库尔特的哥哥鲁道夫说，库尔特在六七岁的时候得了风湿热，他已经完全康复了，但是在他的余生里，他始终坚信他的心脏受到了永久性的损伤。从四岁开始，哥德尔就患有“频繁发作的健康状况不佳” ，这种状况一直持续到他的一生。

==Early life and education==

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.

哥德尔于1912年至1916年就读于布吕恩的路德教学校，并于1916年至1924年入学，在所有科目上都表现优异，尤其是在数学、语言和宗教方面。尽管库尔特最初擅长语言，但后来他对历史和数学更感兴趣。1920年，他的哥哥鲁道夫(出生于1902年)离开维也纳去维也纳大学医学院学习，这增加了他对数学的兴趣。在他十几岁的时候，库尔特学习了加贝尔伯格的速记，歌德的色彩理论和对艾萨克 · 牛顿的批评，以及伊曼努尔 · 康德的著作。

===Childhood===

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. 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."

18岁时，哥德尔与他的兄弟在维也纳会合，进入维也纳大学学习。到那时，他已经掌握了大学水平的数学。虽然最初打算学习理论物理学，但他也参加了数学和哲学课程。在此期间，他采纳了数学实在论的思想。他阅读康德的著作，并与莫里茨 · 施里克、汉斯 · 哈恩和鲁道夫 · 卡尔纳普一起加入维也纳学派。然后哥德尔学习了数论，但是当他参加了 Moritz Schlick 举办的一个研讨会，研究了伯特兰·罗素的书《数学哲学导论》，他开始对数学逻辑感兴趣。根据哥德尔的说法，数理逻辑是“一门先于所有其他科学的科学，它包含了所有科学的基本思想和原则。”

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. 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?

在博洛尼亚参加大卫 · 希尔伯特关于数学系统的完整性和一致性的讲座，可能为哥德尔的一生奠定了基础。1928年，希尔伯特和威廉·阿克曼出版了《数理逻辑的原理》，在一阶逻辑的导言中提出了完备性的问题: 一个形式系统的公理是否足以推导出所有系统模型中真实的每个陈述？

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.

这个问题成为了哥德尔博士论文的主题。1929年，23岁的他在 Hans Hahn 的指导下完成了他的博士论文。在其中，他建立了关于一阶谓词演算的同名完备性定理。他在1930年获得博士学位，他的论文(附带一些额外的工作)由维也纳科学院出版。

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===

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. 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?''

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.

1930年，哥德尔参加了9月5日至7日在柯尼斯堡举行的第二届精确科学认识论会议。在这里，他发表了他的不完备性定理。

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:

哥德尔在 und verwandter Systeme }(英文名为“论及相关系统的正式不可判定命题”)中发表了他的不完备性定理。在那篇文章中，他证明了任何强大到足以描述自然数算术的可计算公理系统(例如，Peano 公理或 Zermelo-Fraenkel 集合论与选择公理) :

If a (logical or axiomatic formal) system is consistent, it cannot be complete.

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==

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

公理的一致性不能在它们自己的体系中得到证明。

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in and Hilbert's formalism, to find a set of axioms sufficient for all mathematics.

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===

{{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.

事后看来，不完备性定理的核心基本思想相当简单。哥德尔实质上构造了一个公式，声称它在给定的形式系统中是不可证明的。如果可以证明，那就是错误的。

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

因此，总会有至少一个真实但无法证明的陈述。

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

也就是说，对于任何可计算枚举的算术公理集(也就是说，原则上可以由拥有无限资源的理想计算机打印出来的公理集) ，有一个算术公式是正确的，但在该系统中无法证明。

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。

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]].

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)。

# 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.

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。

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.

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.

1933年，哥德尔第一次来到美国，在那里他遇到了阿尔伯特 · 爱因斯坦，爱因斯坦成了他的好朋友。他在美国数学学会的年会上发表了演讲。在这一年里，哥德尔还发展了可计算性和递归函数的概念，以至于他能够提出一个关于一般递归函数和真理概念的演讲。这项工作是在数论中发展起来的，使用了哥德尔编号。

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 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.

1934年，哥德尔在新泽西州普林斯顿的高级研究所(IAS)做了一系列演讲，题目是关于正式数学系统的不可判定命题。斯蒂芬 · 克莱恩(Stephen Kleene)刚刚在普林斯顿大学完成了他的博士学位，他记下了这些讲座的笔记，这些讲座随后被出版。

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.

哥德尔在1935年秋天再次参观了国际会计准则。旅行和艰苦的工作使他筋疲力尽，第二年他休息一下，从抑郁症中恢复过来。他于1937年重返教学岗位。在此期间，他致力于证明选择公理和连续统假设公理的一致性; 他继续表明，这些假设不能从集合论公理系统的共同体系中被推翻。

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

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.

他于1938年9月20日与认识了10多年的波克特结婚。哥德尔的父母反对他们的关系，因为她是一个离异的舞蹈演员，比哥德尔大六岁。

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 : 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. 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.

随后，他又前往美国，在国际会计准则所度过了1938年秋天，并出版了选择公理和广义连续统一体假设与集合论公理的一致性，集合论是现代数学的经典。在这部著作中，他提出了可构造的宇宙，这是一个集合论模型，在这个模型中，只有那些可以由简单集合构造出来的集合存在。哥德尔指出，选择公理(AC)和广义连续统假设公理(GCH)在可构造的宇宙中都是正确的，因此必须与集合论的 Zermelo-Fraenkel 公理(ZF)一致。这个结果对从事数学工作的人来说有相当大的影响，因为这意味着他们在证明哈恩-巴纳赫定理时可以假定选择公理。保罗 · 科恩后来构造了一个 ZF 模型，其中 AC 和 GCH 是假的; 这些证明一起意味着 AC 和 GCH 是独立于集合论的 ZF 公理的。

In 1933, Gödel first traveled to the U.S., where he met [[Albert Einstein]], who became a good friend.<ref>''[[Hutchinson Encyclopedia]]'' (1988), p. 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.

1939年春天，哥德尔在圣母大学度过。

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.

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

1938年3月12日德国合并后，奥地利成为纳粹德国的一部分。

Germany abolished the title , so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down.

德国废除了这个头衔，因此哥德尔不得不在新的秩序下申请一个不同的职位。他以前与维也纳学派的犹太成员，特别是与哈恩的关系对他不利。维也纳大学拒绝了他的申请。

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.

当德国军队发现他适合征兵时，他的困境加剧了。第二次世界大战开始于1939年9月。

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.

年底前，哥德尔和他的妻子离开维也纳去了普林斯顿。为了避免横渡大西洋的困难，哥德尔夫妇乘坐西伯利亚铁路航行到太平洋，从日本到旧金山(他们在1940年3月4日到达) ，然后乘火车横渡美国到达普林斯顿。在那里，哥德尔接受了高级研究学院(IAS)的一个职位，他在1933-34年期间曾到过这个学院。

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".

阿尔伯特 · 爱因斯坦在这段时间也住在普林斯顿。哥德尔和爱因斯坦建立了深厚的友谊，他们一起在高等研究院进行长距离的散步。他们谈话的性质对研究所的其他成员来说是个谜。经济学家约翰 · 奥斯卡·摩根斯腾回忆说，在他生命的最后时刻，他曾坦言自己的工作不再意味着什么，他来到这个研究所仅仅是为了... ... 享受和哥德尔一起走回家的特权。

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

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.

1942年的夏天，哥德尔和他的妻子阿黛尔在缅因州的蓝山海湾顶端的蓝山旅馆度过。哥德尔不仅仅是在度假，而且还有一个非常富有成效的夏季工作。小约翰 · w · 道森(John w. Dawson jr.)利用哥德尔尚未出版的[工作笔记本][卷15]推测，哥德尔在1942年《布鲁希尔》(Blue Hill)一书中发现了选择公理独立于集合论弱化形式——有限型理论的证明。哥德尔的密友王支持这一猜想，指出哥德尔的蓝山笔记本包含了他对这一问题最广泛的论述。

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.

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.

1947年12月5日，爱因斯坦和摩根斯坦陪同哥德尔参加了他的美国公民身份考试，他们在考试中充当了见证人。哥德尔曾向他们透露，他发现美国宪法存在不一致之处，这可能使美国成为一个独裁国家。爱因斯坦和摩根斯坦担心他们的朋友不可预测的行为可能会危及他的申请。法官原来是菲利普 · 福曼，他认识爱因斯坦，并在爱因斯坦自己的公民听证会上主持了宣誓。一切都很顺利，直到 Forman 碰巧问哥德尔，他是否认为像纳粹政权一样的独裁会在美国发生，然后哥德尔开始向 Forman 解释他的发现。福尔曼明白发生了什么事，打断了哥德尔的话，把听证会转移到其他问题和例行结论上。

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.

哥德尔于1946年成为普林斯顿高等研究院的常任成员。大约在这个时候，他停止了出版，尽管他继续工作。1953年，他成为该研究所的全职教授，1976年成为名誉教授的全职教授。

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).

哥德尔在研究所的多年时间里，他的兴趣转向了哲学和物理学。1949年，他证明了包含封闭时间型曲线的解的存在性，这些解是爱因斯坦在《广义相对论的场方程。据说他把这个精心设计作为爱因斯坦70岁生日的礼物送给了他。他的“旋转宇宙”将允许时间旅行回到过去，并使爱因斯坦对自己的理论产生怀疑。他的解被称为哥德尔度量(爱因斯坦场方程的精确解)。

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. 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.

他研究和欣赏莱布尼茨的作品，但是他开始相信是一个敌对的阴谋导致了莱布尼茨的一些作品被压制。在一定程度上，他研究了康德和胡塞尔。在20世纪70年代早期，哥德尔在他的朋友中间传阅了莱布尼茨版本的关于上帝存在的本体论证明的安瑟伦。这就是现在众所周知的哥德尔的本体论证明。

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. 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 |accessdate=April 16, 2019 }}</ref>

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.

哥德尔于1951年被授予(与朱利安·施温格一起)第一个阿尔伯特·爱因斯坦奖勋章，并于1974年获得美国国家科学奖章勋章。1968年，哥德尔被选为英国皇家学会的外籍会员。哥德尔奖是一年一度的理论计算机科学领域杰出论文奖，以他的名字命名。

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

新泽西州普林斯顿公墓的库尔特和阿黛尔 · 哥德尔墓碑

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.

后来，哥德尔经历了一段精神不稳定和疾病的时期。在他的密友 Moritz Schlick 被暗杀后，哥德尔对中毒有着强迫性的恐惧; 他只吃他的妻子阿黛尔为他准备的食物。1977年底，她住院6个月，随后不能再为丈夫准备食物。在她不在的时候，他拒绝进食，最终饿死。他死的时候称了体重。他的死亡证明显示他于1978年1月14日在普林斯顿医院死于“人格障碍引起的营养不良和缺乏意识”。他被埋葬在普林斯顿公墓。阿黛尔于1981年去世。

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==

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.

哥德尔是一个坚定的有神论者，在基督教传统中。他认为上帝是个人的。

[[File:Kurt godel tomb 2004.jpg|right|thumb|200px|Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery]]

He believed firmly in an afterlife, stating: "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]."

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 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

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." Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not". According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning", while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."

在一份未经邮寄的问卷调查中，哥德尔将他的宗教描述为“受洗的路德教徒(但不是任何宗教集会的成员)。我的信仰是有神论的，而不是泛神论的，遵循的是莱布尼茨而不是斯宾诺莎。”在描述宗教时，哥德尔说: “宗教在很大程度上是坏的ーー但宗教不是。”。据他的妻子阿黛尔说，“哥德尔虽然没有去教堂，但他是个虔诚的教徒，每个星期天早上都在床上读《圣经》。”而对于伊斯兰教，他说，“我喜欢伊斯兰教: 它是一个持续的(或者说是重要的)宗教观念，思想开放。”

| last = Toates

| first = Frederick

|author2=Olga Coschug Toates

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of 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 invited an annual Kurt Gödel lecturer each year since 1990.

1987年成立的库尔特 · 哥德尔协会就是以他的名字命名的。它是一个国际组织，促进研究领域的逻辑，哲学和数学史。维也纳大学主办了库尔特 · 哥德尔数理逻辑研究中心。自1990年以来，符号逻辑协会每年都会邀请一位库尔特 · 哥德尔的年度讲师。

| title = Obsessive Compulsive Disorder: Practical Tried-and-Tested Strategies to Overcome OCD

[http://www.bbaw.de/en/research/goedel Gödel's Philosophical Notebooks] are edited at the [http://www.bbaw.de/en/research/goedel Kurt Gödel Research Centre] which is situated at the [http://www.bbaw.de/en/academy Berlin-Brandenburg Academy of Sciences and Humanities] in Germany.

Http://www.bbaw.de/en/Research/goedel · 哥德尔的哲学笔记》由 http://www.bbaw.de/en/Research/goedel · 库尔特 · 哥德尔研究中心编辑，该中心位于德国柏林-勃兰登堡科学与人文 http://www.bbaw.de/en/Academy。

| publisher=Class Publishing

| year = 2002

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's , and the final two include correspondence.

哥德尔的五部作品集已经出版。前两个包括哥德尔的出版物; 第三个包括哥德尔未出版的手稿，最后两个包括信件。

| page = 221

| isbn = 978-1-85959-069-0}}</ref> He was buried in [[Princeton Cemetery]]. Adele's death followed in 1981.<ref>{{cite web |last1=Dawson |first1=John W. |authorlink1=John W. Dawson Jr. |title=Gödel and the limits of logic |url=https://plus.maths.org/content/goumldel-and-limits-logic |website=Plus |publisher=University of Cambridge |accessdate=November 1, 2020 |language=en |date=June 1, 2006}}</ref>

A biography of Gödel was published by John Dawson in 2005: Logical Dilemmas: The Life and Work of Kurt Gödel (A. K. Peters, Wellesley, MA, ). Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled Dangerous Knowledge by David Malone.

约翰 · 道森于2005年出版了哥德尔的传记: 《逻辑困境: 科特 · 哥德尔的生活和工作》(a · k · 彼得斯，韦尔斯利，MA)。哥德尔也是2008年 BBC 纪录片《危险的知识》中的四位数学家之一。

==Personal life==

Douglas Hofstadter wrote a popular book in 1979 called to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book 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.

侯世达在1979年写了一本很受欢迎的书，名字叫做《哥德尔的作品和思想》，同时出版的还有艺术家 m. c. Escher 和作曲家约翰·塞巴斯蒂安·巴赫。这本书在一定程度上探索了哥德尔的不完备性定理可以应用于任何图灵完备计算系统(可能包括人脑)的结果。

===Religious views===

Gödel is played by Lou Jacobi in the 1994 film I.Q.

哥德尔由卢 · 雅各比在1994年的电影《智商》中扮演。

Gödel was a convinced [[theist]], in the Christian tradition.<ref>{{cite book|title=A to Z of Mathematicians|year=2005|publisher=Infobase Publishing|isbn=978-0-8160-5338-4|author=Tucker McElroy|page=[https://archive.org/details/tozofmathematici0000mcel/page/118 118]|quote=Gödel had a happy childhood, and was called "Mr. Why" by his family, due to his numerous questions. He was baptized as a Lutheran, and re-mained a theist (a believer in a personal God) throughout his life.|url=https://archive.org/details/tozofmathematici0000mcel/page/118}}</ref> He held the notion that God was personal.

He believed firmly in an afterlife, stating: "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]."<ref>Hao Wang, "A Logical Journey: From Gödel to Philosophy", 1996, pp. 104–05.</ref>

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is ''[[Theism|theistic]]'', not [[Pantheism|pantheistic]], following [[Gottfried Wilhelm Leibniz|Leibniz]] rather than [[Spinoza]]."<ref>Gödel's answer to a special questionnaire sent him by the sociologist Burke Grandjean. This answer is quoted directly in Wang 1987, p. 18, and indirectly in Wang 1996, p. 112. It's also quoted directly in Dawson 1997, p. 6, who cites Wang 1987. The Grandjean questionnaire is perhaps the most extended autobiographical item in Gödel's papers. Gödel filled it out in pencil and wrote a cover letter, but he never returned it. "Theistic" is italicized in both Wang 1987 and Wang 1996. It is possible that this italicization is Wang's and not Gödel's. The quote follows Wang 1987, with two corrections taken from Wang 1996. Wang 1987 reads "Baptist Lutheran" where Wang 1996 has "baptized Lutheran". Wang 1987 has "rel. cong.", which in Wang 1996 is expanded to "religious congregation".</ref> Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not".<ref>Wang 1996 p. 316</ref> According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",<ref>Wang 1996, p. 51.</ref> while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."<ref>Wang 1996, p. 148, 4.4.3. It is one of Gödel's observations, made between

In German:

德语:

16 November and 7 December 1975, which Wang found hard to classify under the main topics considered elsewhere in the book.</ref>

==Legacy==

The [[Kurt Gödel Society]], founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of 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 invited an annual Kurt Gödel lecturer each year since 1990.

[http://www.bbaw.de/en/research/goedel Gödel's Philosophical Notebooks] are edited at the [http://www.bbaw.de/en/research/goedel Kurt Gödel Research Centre] which is situated at the [http://www.bbaw.de/en/academy Berlin-Brandenburg Academy of Sciences and Humanities] in Germany.

In English:

英语:

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's {{lang|de|Nachlass}}, and the final two include correspondence.

A biography of Gödel was published by [[John W. Dawson, Jr|John Dawson]] in 2005: ''Logical Dilemmas: The Life and Work of Kurt Gödel'' ([[A. K. Peters]], Wellesley, MA, {{isbn|1-56881-256-6}}). Gödel was also one of four mathematicians examined in the 2008 [[BBC]] documentary entitled ''Dangerous Knowledge'' by [[David Malone (independent filmmaker)|David Malone]].<ref>{{cite web|url=https://www.bbc.co.uk/bbcfour/documentaries/features/dangerous-knowledge.shtml|title=Dangerous Knowledge|work=BBC |date=June 11, 2008|accessdate=October 6, 2009}}</ref>

In English translation:

英文翻译:

[[Douglas Hofstadter]] wrote a popular book in 1979 called {{lang|de|[[Gödel, Escher, Bach]]|italic=yes}} to celebrate the work and ideas of Gödel, along with those of artist [[M. C. Escher]] and composer [[Johann Sebastian Bach]]. The book 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]].

Gödel is played by [[Lou Jacobi]] in the 1994 film ''[[I.Q. (film)|I.Q.]]''

==Bibliography==

===Important publications===

In German:

* 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." ''Monatshefte für Mathematik und Physik'' '''37''': 349–60.

* 1931, "Über formal unentscheidbare Sätze der ''[[Principia Mathematica]]'' und verwandter Systeme, I." ''Monatshefte für Mathematik und Physik'' '''38''': 173–98.

* 1932, "Zum intuitionistischen Aussagenkalkül", ''Anzeiger Akademie der Wissenschaften Wien'' '''69''': 65–66.

In English:

* 1940. ''The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory.'' Princeton University Press.

* 1947. "What is Cantor's continuum problem?" ''The American Mathematical Monthly 54'': 515–25. Revised version in [[Paul Benacerraf]] and [[Hilary Putnam]], eds., 1984 (1964). ''Philosophy of Mathematics: Selected Readings''. Cambridge Univ. Press: 470–85.

* 1950, "Rotating Universes in General Relativity Theory." ''Proceedings of the international Congress of Mathematicians in Cambridge,'' '''1''': 175–81

In English translation:

* Kurt Gödel, 1992. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. B. Meltzer, with a comprehensive introduction by [[R. B. Braithwaite|Richard Braithwaite]]. Dover reprint of the 1962 Basic Books edition.

* Kurt Gödel, 2000.<ref>{{cite journal|doi=10.1007/BF01700692|author=Kurt Godel |year=1931|url=http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf|title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I|trans-title=On formally undecidable propositions of Principia Mathematica and related systems I|journal=Monatshefte für Mathematik und Physik|volume= 38|pages= 173–98|s2cid=197663120 }}</ref> ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel

* [[Jean van Heijenoort]], 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press.

** 1930. "The completeness of the axioms of the functional calculus of logic," 582–91.

** 1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).

** 1931. "On formally undecidable propositions of ''Principia Mathematica'' and related systems," 596–616.

** 1931a. "On completeness and consistency," 616–17.

* [https://web.archive.org/web/20160304074030/http://www.geocities.ws/kandathil/phil_view.html "My philosophical viewpoint"], c. 1960, unpublished.

* [https://web.archive.org/web/20160304080342/http://www.geocities.ws/kandathil/foundations_mathematics.html "The modern development of the foundations of mathematics in the light of philosophy"], 1961, unpublished.

* ''Collected Works'': Oxford University Press: New York. Editor-in-chief: [[Solomon Feferman]].

** Volume I: Publications 1929–1936 {{isbn|978-0-19-503964-1}} / Paperback: {{isbn|978-0-19-514720-9}},

** Volume II: Publications 1938–1974 {{isbn|978-0-19-503972-6}} / Paperback: {{isbn|978-0-19-514721-6}},

** Volume III: Unpublished Essays and Lectures {{isbn|978-0-19-507255-6}} / Paperback: {{isbn|978-0-19-514722-3}},

** Volume IV: Correspondence, A–G {{isbn|978-0-19-850073-5}},

** Volume V: Correspondence, H–Z {{isbn|978-0-19-850075-9}}.

* ''Philosophische Notizbücher / Philosophical Notebooks'': De Gruyter: Berlin/München/Boston. Editor: [[:de:Eva-Maria_Engelen|Eva-Maria Engelen]].

** Volume 1: Philosophie I Maximen 0 / Philosophy I Maxims 0 {{isbn|978-3-11-058374-8}}.

==See also==

{{Portal|Biography|Philosophy}}

* [[Gödel machine]]

* [[Gödel Prize]]

* [[List of Austrian scientists]]

* [[List of pioneers in computer science]]

* [[Mathematical Platonism]]

* [[Original proof of Gödel's completeness theorem]]

==Notes==

{{reflist|colwidth=30em}}

==References==

* {{Citation | last = Dawson | first = John W | year = 1997 | title = Logical dilemmas: The life and work of Kurt Gödel | place = Wellesley, MA | publisher = AK Peters}}.

* {{Citation | first = Rebecca | last = Goldstein | authorlink = Rebecca Goldstein | year = 2005 | title = Incompleteness: The Proof and Paradox of Kurt Gödel | publisher = W.W. Norton & Co | place = New York | isbn = 978-0-393-32760-1 }}.

==Further reading==

* {{Citation | first1 = John L | last1 = Casti | first2 = Werner | last2 = DePauli | year = 2000 | title = Gödel: A Life of Logic | publisher = Basic Books (Perseus Books Group) | place = Cambridge, MA | isbn = 978-0-7382-0518-2}}.

* {{Citation | first = John W | last = Dawson, Jr | authorlink =John W. Dawson, Jr | title = Logical Dilemmas: The Life and Work of Kurt Gödel | publisher = AK Peters | year = 1996}}.

* {{Citation | first = John W | last = Dawson, Jr | year = 1999 | title = Gödel and the Limits of Logic | journal = Scientific American | volume = 280 | number = 6 | pages = 76–81| pmid = 10048234 | bibcode = 1999SciAm.280f..76D | doi = 10.1038/scientificamerican0699-76 }}.

* {{Citation | first = Torkel | last = Franzén | authorlink = Torkel Franzén | year = 2005 | title = Gödel's Theorem: An Incomplete Guide to Its Use and Abuse | place = Wellesley, MA | publisher = AK Peters}}.

* [[Ivor Grattan-Guinness]], 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.

* {{cite book | author=Hämeen-Anttila, Maria | title=Gödel on Intuitionism and Constructive Foundations of Mathematics | type=Ph.D. thesis | location=Helsinki | publisher=University of Helsinki | year=2020 | isbn=978-951-51-5922-9 | url=http://urn.fi/URN:ISBN:978-951-51-5923-6 }}

* [[Jaakko Hintikka]], 2000. ''On Gödel''. Wadsworth.

* [[Douglas Hofstadter]], 1980. ''[[Gödel, Escher, Bach]]''. Vintage.

* [[Stephen Kleene]], 1967. ''Mathematical Logic''. Dover paperback reprint c. 2001.

* Stephen Kleene, 1980. ''Introduction to Metamathematics''. North Holland {{isbn|0-7204-2103-9}} (Ishi Press paperback. 2009. {{isbn|978-0-923891-57-2}})

* [[J.R. Lucas]], 1970. ''The Freedom of the Will''. Clarendon Press, Oxford.

* [[Ernest Nagel]] and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.

* Procházka, Jiří, 2006, 2006, 2008, 2008, 2010. ''Kurt Gödel: 1906–1978: Genealogie''. ITEM, Brno. Volume I. Brno 2006, {{isbn|80-902297-9-4}}. In German, English. Volume II. Brno 2006, {{isbn|80-903476-0-6}}. In German, English. Volume III. Brno 2008, {{isbn|80-903476-4-9}}. In German, English. Volume IV. Brno, Princeton 2008, {{isbn|978-80-903476-5-6}}. In German, English Volume V, Brno, Princeton 2010, {{isbn|80-903476-9-X}}. In German, English.

* Procházka, Jiří, 2012. "Kurt Gödel: 1906–1978: Historie". ITEM, Brno, Wien, Princeton. Volume I. {{isbn|978-80-903476-2-5}}. In German, English.

* [[Ed Regis (author)|Ed Regis]], 1987. ''Who Got Einstein's Office?'' Addison-Wesley Publishing Company, Inc.

* [[Raymond Smullyan]], 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.

* [[Olga Taussky-Todd]], 1983. [http://calteches.library.caltech.edu/605/02/Todd.pdf Remembrances of Kurt Gödel]. Engineering & Science, Winter 1988.

* Gödel, Alois, 2OO6. Brünn 1679–1684. ITEM, Brno 2OO6, edited by Jiří Procházka, {{ISBN|80-902297-8-6}}

* Procházka, Jiří 2017. "Kurt Gödel: 1906–1978: Curriculum vitae". ITEM, Brno, Wien, Princeton 2017. Volume I. ({{ISBN|978-80-903476-9-4}}). In German, English.

* Procházka, Jiří 2O19. "Kurt Gödel 19O6-1978: Curriculum vitae". ITEM, Brno, Wien, Princeton 2O19. Volume II. ({{ISBN|978-80-903476-1-8}}). In German, English.

* [[Hao Wang (academic)|Hao Wang]], 1987. ''Reflections on Kurt Gödel.'' MIT Press.

* Hao Wang, 1996. ''A Logical Journey: From Godel to Philosophy''. MIT Press.

* Yourgrau, Palle, 1999. ''Gödel Meets Einstein: Time Travel in the Gödel Universe.'' Chicago: Open Court.

* Yourgrau, Palle, 2004. ''A World Without Time: The Forgotten Legacy of Gödel and Einstein.'' Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society ('''54''' (7), pp. 861–68): 

==External links==

{{Commons category|Kurt Gödel}}

{{Wikiquote}}

* {{ScienceWorldBiography | urlname=Goedel | title=Gödel, Kurt (1906–1978)}}

* {{cite SEP |url-id=goedel |title=Kurt Gödel |last=Kennedy |first=Juliette}}

* [http://www.newyorker.com/archive/2005/02/28/050228crat_atlarge Time Bandits]: an article about the relationship between Gödel and Einstein by Jim Holt

Category:1906 births

类别: 1906出生

* [http://www.ams.org/notices/200604/200604-toc.html Notices of the AMS, April 2006, Volume 53, Number 4] Kurt Gödel Centenary Issue

Category:1978 deaths

分类: 1978年死亡

* [http://www.abc.net.au/rn/scienceshow/stories/2006/1807626.htm Paul Davies and Freeman Dyson discuss Kurt Godel]

Category:20th-century American mathematicians

范畴: 20世纪美国数学家

* [http://www.edge.org/3rd_culture/goldstein05/goldstein05_index.html "Gödel and the Nature of Mathematical Truth"] Edge: A Talk with Rebecca Goldstein on Kurt Gödel.

Category:20th-century American philosophers

范畴: 20世纪美国哲学家

* [https://web.archive.org/web/20091106003330/http://simplycharly.com/godel/gregory_chaitin_interview.htm It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.]

Category:20th-century Austrian mathematicians

范畴: 20世纪奥地利的数学家

* [https://web.archive.org/web/20090301015757/http://www.univie.ac.at/bvi/photo-gallery/photo_gallery.htm Gödel photo gallery.]

Category:American Protestants

类别: 美国新教徒

* [http://www-history.mcs.st-andrews.ac.uk/Biographies/Godel.html Kurt Gödel]

Category:American logicians

类别: 美国逻辑学家

* [http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/gdel-kurt.pdf National Academy of Sciences Biographical Memoir]

Category:American people of Moravian-German descent

类别: 具有摩拉维亚-德国血统的美国人

Category:Analytic philosophers

范畴: 分析哲学家

{{Set theory}}

Category:Austrian emigrants to the United States

类别: 移居美国的奥地利移民

{{Winners of the National Medal of Science|math-stat-comp}}

Category:Austrian logicians

类别: 奥地利逻辑学家

{{relativity}}

Category:Austrian mathematicians

类别: 奥地利数学家

{{Analytic philosophy}}

Category:Austrian people of Moravian-German descent

类别: 有摩拉维亚-德国血统的奥地利人

{{Platonists}}

Category:Austrian philosophers

范畴: 奥地利哲学家

Category:Austrian Protestants

类别: 奥地利新教徒

{{Authority control}}

Category:Austro-Hungarian mathematicians

类别: 奥匈数学家

Category:Burials at Princeton Cemetery

类别: 普林斯顿公墓的葬礼

{{DEFAULTSORT:Godel, Kurt}}

Category:Deaths by starvation

分类: 死于饥饿

[[Category:1906 births]]

Category:Foreign Members of the Royal Society

类别: 皇家学会的外国成员

[[Category:1978 deaths]]

Category:Institute for Advanced Study faculty

类别: 高级研究学院

[[Category:20th-century American mathematicians]]

Category:National Medal of Science laureates

类别: 美国国家科学奖章

[[Category:20th-century American philosophers]]

Category:Ontologists

范畴: 本体论者

[[Category:20th-century Austrian mathematicians]]

Category:People from Brno

分类: 来自布尔诺的人

[[Category:American Protestants]]

Category:People from the Margraviate of Moravia

类别: 来自摩拉维亚伯国的人

[[Category:American logicians]]

Category:People with acquired American citizenship

类别: 获得美国国籍的人

[[Category:American people of Moravian-German descent]]

Category:People with paranoid personality disorder

类别: 患有妄想型人格违常的人

[[Category:Analytic philosophers]]

Category:Platonists

类别: 柏拉图主义者

[[Category:Austrian emigrants to the United States]]

Category:Princeton University faculty

类别: 普林斯顿大学

[[Category:Austrian logicians]]

Category:Protestant philosophers

类别: 新教哲学家

[[Category:Austrian mathematicians]]

Category:Set theorists

范畴: 集合理论家

[[Category:Austrian people of Moravian-German descent]]

Category:Vienna Circle

类别: 维也纳圈

[[Category:Austrian philosophers]]

Category:University of Notre Dame faculty

类别: 圣母大学

[[Category:Austrian Protestants]]

Category:University of Vienna alumni

类别: 维也纳大学校友

<noinclude>

This page was moved from [[wikipedia:en:Kurt Gödel]]. Its edit history can be viewed at [[哥德尔/edithistory]]</noinclude>

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2020年11月16日 (一) 21:39的版本