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'''库尔特·弗里德里希·哥德尔 Kurt Friedrich Gödel'''（1906年4月28日至1978年1月14日）是一位逻辑学家、数学家和分析哲学家。与[[亚里士多德 Aristotle]]和[[哥特洛布·弗雷格 Gottlob Frege]]一起被认为是历史上最重要的逻辑学家之一。当时其他人，如伯特兰·罗素 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>阿尔弗雷德·诺斯·怀特黑德 Alfred North Whitehead<ref name="Stanford&Son"/> 和大卫·希尔伯特 David Hilbert都在分析逻辑和[[集合论]]的使用，以理解乔治·坎托 Georg Cantor开创的数学基础。

'''库尔特·弗里德里希·哥德尔 Kurt Friedrich Gödel'''（1906年4月28日至1978年1月14日）是一位逻辑学家、数学家和分析哲学家。与[[亚里士多德 Aristotle]]和[[哥特洛布·弗雷格 Gottlob Frege]]一起被认为是历史上最重要的逻辑学家之一。当时其他人，如伯特兰·罗素 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>阿尔弗雷德·诺斯·怀特黑德 Alfred North Whitehead<ref name="Stanford&Son"/> 和大卫·希尔伯特 David Hilbert都在分析逻辑和[[集合论]]的使用，以理解乔治·坎托 Georg Cantor开创的数学基础。

1931年，25岁的哥德尔在[[维也纳大学]完成博士学位一年后，发表了两篇[[哥德尔不完全性定理|不完全性定理]]。第一个不完全性定理指出，对于任何强大到足以描述[[自然数]]的算术的自洽[[递归集|递归]][[公理系统]]，都存在无法从[[公理]]证明的关于自然数的真实命题。为了证明这个定理，哥德尔发展了一种技术，现在称为[[哥德尔编码]]，它将形式表达式编码为自然数。

1931年，25岁的哥德尔在[[维也纳大学]完成博士学位一年后，发表了两篇[[哥德尔不完全性定理|不完全性定理]]。第一个不完全性定理指出，对于任何强大到足以描述[[自然数]]的算术的自洽[[递归集|递归]][[公理系统]]，都存在无法从[[公理]]证明的关于自然数的真实命题。为了证明这个定理，哥德尔发展了一种技术，现在称为[[哥德尔编码]]，它将形式表达式编码为自然数。

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。

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。

他还指出，如若这些公理一致，[[选择公理]]和[[连续性假设]]都不能从公认的[[公理集理论|集合论公理]]中证伪。前一个结果为数学家在证明中假设选择公理打开了大门。他还通过澄清[[经典逻辑]]、[[直觉逻辑]]和[[模态逻辑]]之间的联系，对[[证明理论]]作出了重要贡献。

他还指出，如若这些公理一致，[[选择公理]]和[[连续性假设]]都不能从公认的[[公理集理论|集合论公理]]中证伪。前一个结果为数学家在证明中假设选择公理打开了大门。他还通过澄清[[经典逻辑]]、[[直觉逻辑]]和[[模态逻辑]]之间的联系，对[[证明理论]]作出了重要贡献。

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.

奥匈帝国在一战中战败，12岁时，哥德尔自动成为了捷克斯洛伐克公民。（根据他的同学的说法，和德国人占主导地位的捷克苏台德区的许多居民一样，“哥德尔一直认为自己是奥地利人，是捷克斯洛伐克的流亡者。”<ref>Dawson 1997, p.&nbsp;15.</ref>1929年2月，他被授予捷克斯洛伐克国籍，4月获得奥地利国籍。<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> 1938年纳粹德国吞并奥地利时，哥德尔在32岁时自动成为德国公民。第二次世界大战（1948年）之后，42岁的他成为美国公民。<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>

===Childhood童年===

===在维也纳的学习===

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

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Moritz Schlick、Hans Hahn和Rudolf Carnap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<ref>Gleick, J. (2011) ''[[The Information: A History, a Theory, a Flood]],'' London, Fourth Estate, p. 181.</ref>讨会，研究了Bertrand Russell的书《数学哲学导论 Introduction to Mathematical Philosophy》 ，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。”

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."

在博洛尼亚参加David Hilbert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，Hilbert和Wilhelm Ackermann出版了《数理逻辑的原理 Principles of Mathematical Logic》 ，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命题？”

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

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<font color="#ff8000"> 不完备性定理</font>。<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>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) :

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?

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

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.

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 Albert Einstein]]，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，使用了哥德尔编码 Gödel numbering。

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证伪。

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

===不完全性定理===

 

 

==Career职业生涯==

 

 

 

 

 

 

 

===Incompleteness theorem不完全性定理===

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

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

{{引述|库尔特·哥德尔在现代逻辑方面的成就是独一无二的和具有纪念意义的——事实上它不仅仅是一座纪念碑，它是一座里程碑，它将在遥远的时空中保持可见。。。随着哥德尔的成就，逻辑学的学科无疑已经完全改变了它的性质和可能性[[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>}}

{{引述|库尔特·哥德尔在现代逻辑方面的成就是独一无二的和具有纪念意义的——事实上它不仅仅是一座纪念碑，它是一座里程碑，它将在遥远的时空中保持可见。。。随着哥德尔的成就，逻辑学的学科无疑已经完全改变了它的性质和可能性[[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>}}

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

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

哥德尔把他的不完全性定理发表在{{lang|de|Über formal unentscheidbare Sätze der {{lang|la|Principia Mathematica}} und verwandter Systeme}} （英文名为“[[关于数学原理和相关系统的形式不可判定命题{lang | la | Principia Mathematica | nocat=y}}和相关系统的形式不可判定命题]”）。在这篇文章中，他证明了任何强大到足以描述[[自然数]]算术的[[递归理论|可计算]][[公理系统]（例如，[[Peano公理]]或[[ZFC | Zermelo–Fraenkel集理论与选择公理]]）：

哥德尔把他的不完全性定理发表在{{lang|de|Über formal unentscheidbare Sätze der {{lang|la|Principia Mathematica}} und verwandter Systeme}} （英文名为“[[关于数学原理和相关系统的形式不可判定命题{lang | la | Principia Mathematica | nocat=y}}和相关系统的形式不可判定命题]”）。在这篇文章中，他证明了任何强大到足以描述[[自然数]]算术的[[递归理论|可计算]][[公理系统]（例如，[[Peano公理]]或[[ZFC | Zermelo–Fraenkel集理论与选择公理]]）：

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

1. 如果一个（逻辑或公理形式）形式系统是一致性的，它的逻辑就不可能是完整的。

#如果一个（逻辑或公理形式）[[形式系统|系统]]是[[证明一致性|一致性]]的，它就不可能是[[完整（逻辑）|完整性]]的。

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值性。在证明中，他隐含地使用了后来被称为的<font color="#ff8000"> 哥德尔-达米特中间逻辑 Gödel–Dummett intermediate logic</font>（或哥德尔模糊逻辑）。

 

 

 

 

 

因此，总会有至少一个真实但无法证明的陈述。

因此，总会有至少一个真实但无法证明的陈述。

也就是说，对于任何[[可计算可枚举]]的算术公理集（也就是说，一个原则上可以由一台具有无限资源的理想计算机打印出来的集合），都有一个公式是正确的，但在该系统中是不可证明的。

也就是说，对于任何[[可计算可枚举]]的算术公理集（也就是说，一个原则上可以由一台具有无限资源的理想计算机打印出来的集合），都有一个公式是正确的，但在该系统中是不可证明的。

 

 

Gödel earned his [[habilitation]] at Vienna in 1932, and in 1933 he became a {{lang|de|[[Privatdozent]]}} (unpaid lecturer) there. In 1933 [[Adolf Hitler]] came to power in Germany, and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, [[Moritz Schlick]], whose seminar had aroused Gödel's interest in logic, was assassinated by one of his former students, [[Johann Nelböck]].  This triggered "a severe nervous crisis" in Gödel.<ref name=Casti2001>{{Cite book |last1=Casti |first1=John L. |last2=Depauli |first2=Werner |year=2001 |title=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}. From p.&nbsp;80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgement that the Schlick assassination was its trigger, are from the Rudolf Gödel quote. Rudolf knew Kurt well in those years.</ref> He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.<ref>Dawson 1997, pp. 110–12</ref>

Gödel earned his [[habilitation]] at Vienna in 1932, and in 1933 he became a {{lang|de|[[Privatdozent]]}} (unpaid lecturer) there. In 1933 [[Adolf Hitler]] came to power in Germany, and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, [[Moritz Schlick]], whose seminar had aroused Gödel's interest in logic, was assassinated by one of his former students, [[Johann Nelböck]].  This triggered "a severe nervous crisis" in Gödel.<ref name=Casti2001>{{Cite book |last1=Casti |first1=John L. |last2=Depauli |first2=Werner |year=2001 |title=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}. From p.&nbsp;80, which quotes Rudolf Gödel, Kurt's brother and a medical doctor. The words "a severe nervous crisis", and the judgement that the Schlick assassination was its trigger, are from the Rudolf Gödel quote. Rudolf knew Kurt well in those years.</ref> He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.<ref>Dawson 1997, pp. 110–12</ref>

1932年，哥德尔在维也纳为人熟知（获得了[[习惯化]]），1933年他成为了{lang | de |[[Privatdozent]]}}（无薪讲师）。1933年[[阿道夫希特勒]在德国掌权，在接下来的几年里，纳粹在奥地利和维也纳数学家中的影响力不断上升。1936年6月，[[Moritz Schlick]]的研讨会引起了哥德尔对逻辑学的兴趣，他被他的一个前学生[[Johann Nelböck]]暗杀。这在哥德尔引发了“严重的神经危机”。<ref name="Casti2001">{{Cite book |last1=Casti |first1=John L. |last2=Depauli |first2=Werner |year=2001 |title=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}摘自第80页，其中引用了库尔特的哥哥、医生鲁道夫·哥德尔的话。这句话引述了索尔夫的话：“这是一次严重的刺杀，这是鲁德的一次严重的刺杀。”。鲁道夫在那几年很了解库尔特。</ref>他出现了偏执症状，包括害怕中毒，并在一所治疗神经疾病的疗养院呆了几个月。<ref>Dawson 1997, pp. 110–12</ref>

1932年，哥德尔在维也纳为人熟知（获得了[[习惯化]]），1933年他成为了{lang | de |[[Privatdozent]]}}（无薪讲师）。1933年[[阿道夫希特勒]在德国掌权，在接下来的几年里，纳粹在奥地利和维也纳数学家中的影响力不断上升。1936年6月，[[Moritz Schlick]]的研讨会引起了哥德尔对逻辑学的兴趣，他被他的一个前学生[[Johann Nelböck]]暗杀。这在哥德尔引发了“严重的神经危机”。<ref name="Casti2001">{{Cite book |last1=Casti |first1=John L. |last2=Depauli |first2=Werner |year=2001 |title=Gödel&thinsp;: a life of logic |doi=10.1287/moor.1050.0169 |isbn=978-0-7382-0518-2 |location=Cambridge, Mass. |publisher=Basic Books |journal=Mathematics of Operations Research |volume=31 |page=147 |last3=Koppe |first3=Matthias |last4=Weismantel |first4=Robert |arxiv=math/0410111 |s2cid=9054486 }}摘自第80页，其中引用了库尔特的哥哥、医生鲁道夫·哥德尔的话。这句话引述了索尔夫的话：“这是一次严重的刺杀，这是鲁德的一次严重的刺杀。”。鲁道夫在那几年很了解库尔特。</ref>他出现了偏执症状，包括害怕中毒，并在一所治疗神经疾病的疗养院呆了几个月。<ref>Dawson 1997, pp. 110–12</ref>

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。

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。

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

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

1938年3月12日德国合并后，奥地利成为纳粹德国的一部分。

1938年3月12日德国合并后，奥地利成为纳粹德国的一部分。

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。

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。

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

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

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。

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普林斯顿，爱因斯坦，美国公民===

===普林斯顿，爱因斯坦，美国公民===

Gödel and his wife, Adele, spent the summer of 1942 in Blue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but had a very productive summer of work. Using  [volume 15] of Gödel's still-unpublished  [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

Gödel and his wife, Adele, spent the summer of 1942 in Blue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but had a very productive summer of work. Using  [volume 15] of Gödel's still-unpublished  [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

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。”

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

<ref>{{cite book| last = Toates| first = Frederick|author2=Olga Coschug 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.

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.

===Important publications重要出版物===

===重要出版物===

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

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

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

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

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

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

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

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

英文翻译:

 

 

 

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, 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.

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

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

==See also请参阅==

==另见==

 

* [[Gödel machine]]

* [[Gödel machine]]

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

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

*[[哥德尔机器]]

*[[哥德尔机器]]

* [[哥德尔完备性定理的原始证明]]

* [[哥德尔完备性定理的原始证明]]

==Notes注释==

==参考文献==

{{reflist|colwidth=30em}}

{{reflist|colwidth=30em}}

==References参考文献==

==其他文献==

* {{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 | 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 }}.

* {{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拓展阅读==

==进一步阅读==

* {{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 | 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}}.

* 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.&nbsp;861–68): <!-- Comment<ref>http://www.ams.org/notices/200707/tx070700861p.pdf</ref>-->

* 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.&nbsp;861–68): <!-- Comment<ref>http://www.ams.org/notices/200707/tx070700861p.pdf</ref>-->

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

==外部链接==

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

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