库尔特·哥德尔 Kurt Gödel

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"Gödel" redirects here. For the programming language, see Gödel (programming language). For other uses, see Godel (disambiguation).

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{{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 = (1906-模板:MONTHNUMBER-28)28, 1906

| birth_date =

出生日期

| birth_place = Brünn, Austria-Hungary
模板:Small

| birth_place = Brünn, Austria-Hungary

出生地布鲁恩，Austria-Hungary

| death_date = January 14, 1978(1978-01-14) (aged 71)

| death_date =

死亡日期

| death_place = 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 = 模板:Ubl

| 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

| 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 < 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

| prizes =

| spouse = 模板:Marriage

| spouse =

配偶 =

| signature = Kurt Gödel signature.svg

| signature = Kurt Gödel signature.svg

| signature = Kurt Gödel signature.svg

}}

}}

}}

Kurt Friedrich Gödel (模板:IPAc-en;[1] German: [ˈkʊɐ̯t ˈɡøːdl̩]模板:IPA audio link; 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,[2] Alfred North Whitehead,[2] 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 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 axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the natural numbers that 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 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).[3] Throughout his life, Gödel would remain close to his mother; their correspondence was frequent and wide-ranging.[4] At the time of his birth the city had a German-speaking majority which included his parents.[5] 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 脚本错误：没有“lang”这个模块。 (Men's Choral Union of Brünn).[6]

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 Czechoslovak citizen at age 12 when the Austro-Hungarian Empire collapsed, following its defeat in the World War I. (According to his classmate 脚本错误：没有“lang”这个模块。, like many residents of the predominantly German 脚本错误：没有“lang”这个模块。, "Gödel considered himself always Austrian and an exile in Czechoslovakia".)[7] In February 1929 he was granted release from his Czechoslovakian citizenship and then, in April, granted Austrian citizenship.[8] 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.[9]

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 脚本错误：没有“lang”这个模块。 ("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.[10]

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 脚本错误：没有“lang”这个模块。, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the 脚本错误：没有“lang”这个模块。 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.


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.[11] 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 脚本错误：没有“lang”这个模块。, 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."[12]


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 脚本错误：没有“lang”这个模块。 (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 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

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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 incompleteness theorems.[13]

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 脚本错误：没有“lang”这个模块。 (called in English "On Formally Undecidable Propositions of 脚本错误：没有“lang”这个模块。 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:

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

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

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in 脚本错误：没有“lang”这个模块。 and 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 numbering.

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 脚本错误：没有“lang”这个模块。 (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).

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 脚本错误：没有“lang”这个模块。 (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.[14] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[15]


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.[16] 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.


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, 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 模板:Ill (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,[17] 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.

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.[18]

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 脚本错误：没有“lang”这个模块。, 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. 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.[19]


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".[20]


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 脚本错误：没有“lang”这个模块。 [volume 15] of Gödel's still-unpublished 脚本错误：没有“lang”这个模块。 [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.


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.[21][22]

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.[23]

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 curves, to Einstein's field equations in general relativity.[24] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[25] 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.[26] 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.


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.[27] Gödel was elected a Foreign Member of the Royal Society (ForMemRS) in 1968.[28] He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts.[29] 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。

文件:Kurt godel tomb 2004.jpg
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 instability and illness. Following the assassination of his close friend Moritz Schlick,[30] 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.[31] He weighed 模板:Convert 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.[32] He was buried in Princeton Cemetery. Adele's death followed in 1981.[33]

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.[34] 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]."[35]


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."[36] Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not".[37] According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning",[38] while of Islam, he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded."[39]


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.

Gödel's Philosophical Notebooks are edited at the Kurt Gödel Research Centre which is situated at the 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 脚本错误：没有“lang”这个模块。, and the final two include correspondence.


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.[40]


In English translation:

英文翻译:

Douglas Hofstadter wrote a popular book in 1979 called 脚本错误：没有“lang”这个模块。 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.


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 Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
  • Kurt Gödel, 2000.[41] 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.
  • Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
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