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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.

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。

事后看来，<font color="#ff8000"> 不完全性定理</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。

To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this using a process known as Gödel numbering.

To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this using a process known as Gödel numbering.

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

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<font color="#ff8000"> 哥德尔编码Gödel numbering</font>。

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:

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)。

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)。

# 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.

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，使用了哥德尔编码。

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<font color="#ff8000"> 哥德尔编码</font>。

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.

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.

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。

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。

1934年，哥德尔在新泽西州的[[普林斯顿，新泽西州|普林斯顿]]的[[高等研究所]]（IAS）做了一系列讲座，题目是“关于形式数学系统的不可判定命题”。刚刚在普林斯顿完成博士学位的[[Stephen Kleene]]记下了随后出版的这些讲座。

1934年，哥德尔在新泽西州的[[普林斯顿，新泽西州|普林斯顿]]的[[高等研究所]]（IAS）做了一系列讲座，题目是“关于形式数学系统的不可判定命题”。刚刚在普林斯顿完成博士学位的[[Stephen Kleene]]记下了随后出版的这些讲座。

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

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

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

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

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===

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