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He also showed that neither the [[axiom of choice]] nor the [[continuum hypothesis]] can be disproved from the accepted [[axiomatic set theory|axioms of set theory]], assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to [[proof theory]] by clarifying the connections between [[classical logic]], [[intuitionistic logic]], and [[modal logic]].
He also showed that neither the [[axiom of choice]] nor the [[continuum hypothesis]] can be disproved from the accepted [[axiomatic set theory|axioms of set theory]], assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to [[proof theory]] by clarifying the connections between [[classical logic]], [[intuitionistic logic]], and [[modal logic]].
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。
In his family, young Kurt was known as ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage. Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire life.
In his family, young Kurt was known as ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage. Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire life.
在他的家庭里，年轻的库尔特因为他贪得无厌的好奇心被称为(“为什么先生”) 。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。
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>
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>
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|捷克斯洛伐克]]公民。（根据他的同学{lang| cs | klepata|italic=no}的说法，和德国人占主导地位的{lang de |[[Sudetenland| sudeten|nder]]}}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. 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]][[Anschluss |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>
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|捷克斯洛伐克]]公民。（根据他的同学{lang| cs | klepata|italic=no}的说法，和德国人占主导地位的{lang de |[[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. 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]][[Anschluss |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>
Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?
Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?
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？
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命题？
In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.
In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems.
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>。
Gödel published his incompleteness theorems in und verwandter Systeme}} (called in English "On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
Gödel published his incompleteness theorems in und verwandter Systeme}} (called in English "On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
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) :
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<font color="#ff8000"> 不完备性定理</font>。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) :
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.
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努力，从弗雷格的工作开始，到希尔伯特的形式主义，试图找到一套足以适用于所有数学的公理。
===Incompleteness theorem不完全性定理===
===Incompleteness theorem不完全性定理===
In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false.
In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false.
事后看来，<font color="#ff8000"> 不完备性定理</font>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。
In his two-page paper (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof, he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).
In his two-page paper (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof, he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).
在他1932年的两页论文中，哥德尔反驳了直觉主义逻辑的有限价值。在证明中，他隐含地使用了后来被称为的<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)。
# The consistency of [[axiom]]s cannot be proved within their own [[axiomatic system|system]].
# The consistency of [[axiom]]s cannot be proved within their own [[axiomatic system|system]].
Thus there will always be at least one true but unprovable statement.
Thus there will always be at least one true but unprovable statement.
事后看来，不完备性定理的核心思想相当简单。哥德尔基本上构造了一个公式，证明它在给定的形式系统中是不可证明的。如果这是可以证明的，那就错了。
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。
In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.
In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.
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。
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，使用了哥德尔编码。
That is, for any [[computably enumerable]] set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system.
That is, for any [[computably enumerable]] set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system.
In his two-page paper {{lang|de|Zum intuitionistischen Aussagenkalkül}} (1932) Gödel refuted the finite-valuedness of [[intuitionistic logic]]. In the proof, he implicitly used what has later become known as [[intermediate logic|Gödel–Dummett intermediate logic]] (or [[t-norm fuzzy logic|Gödel fuzzy logic]]).
In his two-page paper {{lang|de|Zum intuitionistischen Aussagenkalkül}} (1932) Gödel refuted the finite-valuedness of [[intuitionistic logic]]. In the proof, he implicitly used what has later become known as [[intermediate logic|Gödel–Dummett intermediate logic]] (or [[t-norm fuzzy logic|Gödel fuzzy logic]]).
哥德尔在两页纸的论文{lang| de | Zum直觉主义者ischen Aussagenkalkül}（1932）中驳斥了[[直觉逻辑]]的有限值性。在证明中，他含蓄地使用了后来被称为[[中间逻辑|哥德尔-达米特中间逻辑]]（或[[t-范数模糊逻辑|哥德尔模糊逻辑]]）。
哥德尔在两页纸的论文{lang| de | Zum直觉主义者ischen Aussagenkalkül}（1932）中驳斥了[[直觉逻辑]]的有限值性。在证明中，他隐含地使用了后来被称为的[[中间逻辑|哥德尔-达米特中间逻辑]]（或[[t-范数模糊逻辑|哥德尔模糊逻辑]]）。
Gödel visited the IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.
Gödel visited the IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.