Goedel's theorem bilingual英汉双语关于哥德尔不完备定理的一些思考
Some thoughts on G?del's incompleteness theorem (bilingual 英汉双语) 关于哥德尔不完备定理的一些思考
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Let assume that there are two natural numbers A and B. Then A + B must be equal to B + A. Here, the + denotes plus.
However, if we let A, B and + denote something else, say non-numbers, the above statement on A, B and + can be likewise true, provided the + sign changes its meaning from addition to some other operations.
Mixing paint-color example: Let A and B be the yellow and red colors in painting, respectively, and + be the color mixing in painting. Obviously, A + B = B + A, resulting the same purple color C.
Mixing water example: Let A and B be waters in two different but well-insulated glasses at two different temperatures and let + be the mixing operation. Obviously, A + B = B + A, resulting the same mixed water temperature T.
Let G be the assumption or better, the statement, that A + B ≠ B + A, which is, of course, false from humans' point of view but the computer has no idea whether it is true or false even when A and B are natural numbers. G?del's theorems (1931) shows that this natural number statement neither be proved nor not be proved by computaional means. Hence machine-based AI is not necessarily computationable, i.e., not everything in life and thinking is computational.
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The simplest version of G?del's theorem and why it is important
https://www.youtube.com/watch?v=a9eXBUmE-Cc
Chris Niebauer, Ph.D. May 7, 2022
In this video I will show you the simplest way to "get" G?del's theorem. Imagine an all-knowing computer (the limits of the thinking mind) that it can state any truth. Let us call it UTM for universal truth machine. Now if we write out G="UTM will never say G is true" and then ask UTM if G is true, we have put it to its limit. UTM cannot say it is true....which makes it true! This proves the limits of thinking and how there are truths beyond thinking.
Thus, statement G can neither be proved nor disproved.
The incompleteness theorems apply only to formal systems which can prove a sufficient collection of facts about the natural numbers.
The two mixing statements proposed above are outside of the G??del's Incompleteness Theorems.
G is a statement referring only to natural numbers.
However, if the G = C and/or G = T as shown above are digitized into numbers, then after they have been put in computations, they suffer the same conclusions of G in natual numbers - neither proved nor disproved. It is reckless to say AI can replace consiouness by computation.
I am not saying AI is A1, it is not, except that its intelligence is confined to unprovable outcomes. A better term for AI is, therefore, sebome - smart encyclopedia based on machines. [See: https://blog.creaders.net/user_blog_diary.php?did=NTExNjgz
https://www.facebook.com/andrew.colesville
https://x.com/mwsansculotte
https://www.reddit.com/user/Top-Presentation8591/ ]
DeepSeek's opinions on G?del's incompleteness theorems and the dialectic logic:
Okay, so the user is asking if G?del's Incompleteness Theorems are in contradiction from dialectics. Hmm, dialectics usually refers to the method of argument or philosophical process, often associated with Hegel and Marx. The user might be thinking about whether G?del's results conflict with dialectical principles, like the unity of opposites or the idea that contradictions drive development.
First, I need to recall what G?del's theorems actually state. From the search results, the first theorem says that any consistent formal system capable of basic arithmetic will have true statements that can't be proven within the system. The second theorem adds that such a system can't prove its own consistency. These results challenge the idea that mathematics can be fully axiomatized, which was a goal of Hilbert's program.
Now, dialectics often deals with contradictions and their role in progress. For example, in Hegelian dialectics, contradictions are resolved through synthesis. But in formal systems, contradictions are problematic because they lead to explosion (everything becomes provable). G?del showed that avoiding contradictions (consistency) inherently leads to incompleteness.
Looking at the search results, webpage 4 from Philosophy Stack Exchange addresses whether the theorems create a paradox. The answer clarifies that they are not contradictions but highlight limitations. Webpage 5 explains that G?del's theorems use self-referential statements to show incompleteness, which relates to the structure of formal systems.
Dialectics might view the incompleteness as a form of inherent contradiction in formal systems. But according to the theorems, the systems are consistent (no contradictions), just incomplete. So maybe there's a dialectical tension between consistency and completeness. The theorems show that striving for completeness leads to potential inconsistency, which could be seen as a dialectical contradiction.
However, G?del's theorems are mathematical results, not philosophical arguments. They don't directly engage with dialectical materialism or Hegelian ideas. The contradiction here is more about the limits of formal systems rather than a dialectical contradiction in the philosophical sense.
Webpage 7 mentions that the theorems don't create contradictions but reveal unprovable truths. So the answer is that there's no direct contradiction with dialectics. Instead, the theorems expose limitations in formal systems, which could be interpreted through a dialectical lens as inherent limitations driving the need for evolving systems, but that's an interpretation rather than a contradiction.
I should synthesize this into a structured answer, citing relevant sources. Make sure to clarify that while there's no direct contradiction, there might be philosophical interpretations linking them. Use citations from the webpages that discuss the nature of the theorems and their implications on consistency and completeness.
G?del's Incompleteness Theorems do not directly contradict dialectical principles but instead reveal inherent limitations in formal systems, which can be interpreted through a dialectical lens. Here's a structured analysis:
1. Core Claims of G?del's Theorems
First Incompleteness Theorem: Any consistent formal system capable of expressing basic arithmetic is incomplete—there exist true mathematical statements that cannot be proven within the system 1, 2, 8.
Second Incompleteness Theorem: No such system can prove its own consistency without being inconsistent 1, 6, 8.
These results challenge the Hilbertian ideal of a complete, consistent axiomatic foundation for mathematics.
2. Dialectics and Contradiction
Dialectics (e.g., Hegelian or Marxist) often emphasizes contradictions as drivers of change. However, in formal logic, contradictions (e.g., proving both
P and ?P) lead to logical explosion, rendering systems useless. G?del's theorems highlight a tension between consistency and completeness: avoiding contradictions inherently limits a system's ability to prove all truths. This tension could be framed dialectically as an opposition between two desirable properties 5, 8.
3. Self-Reference and Dialectical Synthesis
G?del's proof relies on self-referential statements (e.g., "This sentence is unprovable"), which create a paradox-like scenario. While dialectics might interpret this as a contradiction to resolve through synthesis (e.g., evolving to a "higher" system, by means of negation of negation), G?del's results show that such incompleteness is inescapable within any sufficiently powerful formal system. Each attempt to "complete" a system by adding axioms only shifts the incompleteness to a new level 3, 8.
4. Philosophical Implications
Limits of Formalization: The theorems undermine the notion of a static, all-encompassing mathematical framework, aligning with dialectical views of knowledge as dynamic and evolving 2, 6.
Human vs. Mechanical Reasoning: Some argue that G?del's results suggest human intuition transcends formal systems—a debated point in philosophy of mind 6, 8. Dialectics might interpret this as a synthesis between human creativity and systemic constraints.
5. No Direct Contradiction
G?del's theorems are mathematical results, not philosophical claims. They do not negate dialectics but expose structural limitations in formal systems. The "contradiction" lies not in dialectical logic but in the inherent trade-off between consistency and completeness within axiomatic frameworks 5, 7.
Conclusion
G?del's theorems do not contradict dialectics but provide a mathematical analogue to dialectical tensions: systems must navigate between consistency and completeness, much like dialectical processes resolve contradictions through synthesis. This interplay enriches both mathematical and philosophical discourse but does not constitute a direct conflict 1, 6, 8.
Webpages:
1. https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
2. https://plato.stanford.edu/entries/goedel-incompleteness/
3. https://www.quantamagazine.org/how-godels-proof-works-20200714/
4. https://www.quantamagazine.org/how-godels-proof-works-20200714/
5. https://philosophy.stackexchange.com/questions/85973/do-godels-incompleteness-theorems-create-a-contradiction-paradox?newreg=b67c19903a54419b9039be57e3117e01
6. https://1000wordphilosophy.com/2024/06/01/godel/
7. https://1000wordphilosophy.com/
8. https://www.spencergreenberg.com/2023/11/understanding-two-of-the-weirdest-theorems-in-math-godels-incompleteness/
[Mark Wain 04/24/2025]
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Godel's 1st Incompleteness Theorem - Proof by Diagonalization
https://www.youtube.com/watch?v=PpSxqde0af4
Stable Sort Jul 7, 2020
"This statement is not true"
Simply reading it out presents a problem, since if we were to assume that the statement is true, it is meaning would suggest the opposite. On the other hand, if the statement is not true, then its implied meaning turns out to be true, hence the paradox.
So, what would it take to construct a similar statement, but express it formally, in mathematical notation?
There are two immediate problems to handle. First, the word "true," somewhat surprisingly, is not defined axiomatically. Numbers and formulas do not have a property of "true." We think of axioms as being defined to be true and any theorems that are derived from them would therefore also be true. But the concept of truth would then still be outside of the system itself.
Tarski's indefinability theorem:
https://en.wikipedia.org/wiki/Tarski%...
The other problematic word is "this." It refers to the statement itself, without the statement having been defined yet. It is like saying "This video is exactly 16 minutes and 9 seconds long" without first having made the video.
To get around this problem, G?del established a way to convert statements, such as theorems and even proofs of those theorems, into natural numbers as a type of encoding. Then, having listed encodings of specific kinds of formulas, he showed that there is a natural number in that list that corresponds to a formula that states its own unprovability.
G?del Encoding establishes a one-to-one correspondence between a unique sequence of characters and some unique number. This allowed G?del to convert any mathematical statement, as well as a proof of that statement into two unique numbers.
A Simple Proof of G?del's Incompleteness Theorems
https://mat.iitm.ac.in/home/asingh/pu...
Examples of unprovable statements
https://en.wikipedia.org/wiki/Paris%E...
Further reading on G?del's Incompleteness Theorems:
https://plato.stanford.edu/entries/go...
Written and narrated by Andre Violentyev
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G?del's incompleteness theorems
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2020)
Raatikainen, Panu (2020). "G?del's Incompleteness Theorems". Stanford Encyclopedia of Philosophy. Retrieved November 7, 2022.
The unprovable statement GF referred to by the theorem is often referred to as "the G?del sentence" for the system F. The proof constructs a particular G?del sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the G?del sentence and any logically valid sentence.
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Mathematician explains G?del's Incompleteness Theorem | Edward Frenkel and Lex Fridman
https://www.youtube.com/watch?v=u3GYrEOoKGk
Lex Clips
1.48M subscribers Apr 13, 2023
@arsartium108
11 months ago
Here are brief statements of the theorems for those interested:
G?del's First Incompleteness Theorem states that "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable within that theory."
G?del's Second Incompleteness Theorem states that "For any effectively generated formal theory T including basic arithmetical truths and certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent."
Just prior to publication of his incompleteness results in 1931, G?del already had proved the completeness of the First Order logical calculus; but a number-theoretic system consists of both logic plus number-theoretic axioms, so the completeness of PM and the goal of Hilbert's Programme (Die Grundlagen der Mathematik) remained open questions. G?del proved (1) If the logic is complete, but the whole is incomplete, then the number-theoretic axioms must be incomplete; and (2) It is impossible to prove the consistency of any number-theoretic system within that system. In the context of Mr. Dean's discussion, G?del's Incompleteness results show that any formal system obtained by combining Peano's axioms for the natural numbers with the logic of PM is incomplete, and that no consistent system so constructed can prove its own consistency.
What led G?del to his Incompleteness theorems is fascinating. G?del was a mathematical realist (Platonist) who regarded the axioms of set theory as obvious in that they "force themselves upon us as being true." During his study of Hilbert's problem to prove the consistency of Analysis by finitist means, G?del attempted to "divide the difficulties" by proving the consistency of Number Theory using finitist means, and to then prove the consistency of Analysis by Number Theory, assuming not only the consistency but also the truth of Number Theory.
According to Wang (1981):
"[G?del] represented real numbers by formulas...of number theory and found he had to use the concept of truth for sentences in number theory in order to verify the comprehension axiom for analysis. He quickly ran into the paradoxes (in particular, the Liar and Richard's) connected with truth and definability. He realized that truth in number theory cannot be defined in number theory, and therefore his plan...did not work."
[Hao Wang (1921-1995) was a Chinese-American logician, philosopher, mathematician, and commentator on Kurt G?del. One of Wang's most important contributions was the Wang tile.He showed that any Turing machine can be turned into a set of Wang tiles. A philosopher in his own right, Wang also developed a penetrating interpretation of Ludwig Wittgenstein's later philosophy of mathematics, which he called "anthropologism." He chronicled Kurt G?del's philosophical ideas and authored several books on the subject, thereby providing contemporary scholars with many insights elucidating G?del's later philosophical thought. See https://digitalcommons.rockefeller.edu ? faculty-members]
As a mathematical realist, G?del already doubted the underlying premise of Hilbert's Formalism, and after discovering that truth could not be defined within number theory using finitist means, G?del realized the existence of undecidable propositions within sufficiently strong systems. Thereafter, he took great pains to remove the concept of truth from his 1931 results in order to expose the flaw in the Formalist project using only methods to which the Formalist could not object.
G?del writes:
“I may add that my objectivist conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my work in logic. How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to consist of meaningless symbols which acquire some substitute of meaning only through metamathematics...It should be noted that the heuristic principle of my construction of undecidable number theoretical propositions in the formal systems of mathematics is the highly transfinite concept of 'objective mathematical truth' as opposed to that of demonstrability...” Wang (1974)
In an unpublished letter to a graduate student, G?del writes:
“However, in consequence of the philosophical prejudices of our times, 1. nobody was looking for a relative consistency proof because [it] was considered that a consistency proof must be finitary in order to make sense, 2. a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless.”
Clearly, despite G?del's ontological commitment to mathematical truth, he justifiably feared rejection by the formalist establishment dominated by Hilbert's perspective of any results that assumed foundationalist concepts. In so doing, he was led to a result even he did not anticipate - his second Incompleteness theorem -- which established that no sufficiently strong formal system can demonstrate its own consistency.
See also,
G?del, Kurt "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" Jean van Heijenoort (trans.), From Frege to G?del: A Sourcebook in Mathematical Logic, 1879-1931 (Harvard 1931)
汉语译文
关于哥德尔不完备定理的一些思考
https://blog.creaders.net/user_blog_diary.php?did=NTE0MTk5
https://www.facebook.com/andrew.colesville
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假设有两个自然数 A 和 B。那么 A + B 必定等于 B + A。这里，+ 表示加。
但是，如果我们让 A、B 和 + 表示其他东西，比如非数，那么上述关于 A、B 和 + 的陈述同样成立，前提是 + 符号的含义从加法变为其他运算。
混合颜料示例：设 A 和 B 分别为绘画中的黄色和红色，+ 表示绘画中的颜色混合。显然，A + B = B + A，结果相同，为紫色 C。
混合水示例：设 A 和 B 分别是盛在两个不同温度、但隔热良好的玻璃杯中的水，+ 表示混合运算。显然，A + B = B + A，混合水温相同，均为 T。
设 G 为假设，或者更确切地说，为 A + B ≠ B + A 的命题。从人类的角度来看，这当然是错误的，但即使 A 和 B 是自然数，计算机也不知道它是真还是假。哥德尔定理（1931 年）表明，这个自然数命题既不能用计算方法证明其是，也不能用计算方法证明其非。因此，基于机器的人工智能不一定是可计算的，也就是说，生活和思维中的所有事物并非都是可计算的。
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哥德尔定理的最简版本及其重要性
https://www.youtube.com/watch?v=a9eXBUmE-Cc
Chris Niebauer 博士，2022 年 5 月 7 日
在本视频中，我将向您展示“理解”哥德尔定理的最简单方法。想象一台无所不知的计算机（思维的极限），它可以陈述任何真理。我们称之为通用真理机器（UTM）。现在，如果我们写出 G="UTM 永远不会说 G 为真"，然后询问 UTM G 是否为真，我们就把它推到了极限。UTM 不能说它是真的……这使得它为真！这证明了思维的极限，以及存在超越思维的真理。
因此，命题 G 既不能被证明也不能被证伪。
不完备性定理仅适用于能够证明关于自然数的充分事实集合的形式系统。
上面提出的两个混合命题不在哥德尔不完备性定理的范围内。
G 是一个仅指自然数的命题。
然而，如果将上面所示的 G = C 和/或 G = T 数字化为数字，那么在它们被计算之后，它们会得到与自然数中的 G 相同的结论——既不能被证明也不能被证伪。说人工智能可以用计算取代意识是鲁莽的。
我并不是说人工智能是A1，它不是，只是它的智能仅限于无法证明的结果。因此，对人工智能更好的定义是“sebome”【或赛博梅】——基于机器的进步得到的聪明百科全书。[参见：https://blog.creaders.net/user_blog_diary.php?did=NTExNjgz
https://www.facebook.com/andrew.colesville
https://x.com/mwsansculotte
https://www.reddit.com/user/Top-Presentation8591/]
DeepSeek 对哥德尔不完备定理和辩证逻辑的看法：
好的，所以这位用户想问的是哥德尔不完备定理是否与辩证法相矛盾。嗯，辩证法通常指的是论证方法或哲学过程，经常与黑格尔和马克思联系在一起。用户可能会思考哥德尔的结论是否与辩证法原理相冲突，例如对立统一或矛盾推动发展的理念。
首先，我需要回顾一下哥德尔定理的实际含义。从搜索结果来看，第一个定理指出，任何能够进行基本算术运算的一致形式系统都会包含无法在系统内部证明的真陈述。第二个定理补充说，这样的系统无法证明其自身的一致性。这些结果挑战了数学可以完全公理化的观点，而这正是希尔伯特纲领的目标之一。
辩证法经常处理矛盾及其在进步中的作用。例如，在黑格尔辩证法中，矛盾通过综合来解决。但在形式系统中，矛盾是有问题的，因为它们会导致爆炸（一切都变得可证）。哥德尔表明，避免矛盾（即一致性）本质上会导致不完备性。
查看搜索结果，Philosophy Stack Exchange 的第 4 个网页探讨了这些定理是否构成了悖论。答案澄清了它们并非矛盾，而是强调了局限性。第 5 个网页解释说，哥德尔定理使用自指陈述来表示不完备性，这与形式系统的结构有关。
辩证法可能将不完备性视为形式系统中固有矛盾的一种形式。但根据这些定理，这些系统是一致的（没有矛盾），只是不完备。因此，一致性和完备性之间可能存在辩证张力。这些定理表明，追求完备性会导致潜在的不一致性，这可以被视为一种辩证矛盾。
然而，哥德尔定理是数学结果，而非哲学论证。它们并不直接涉及与辩证唯物主义或黑格尔思想相悖。这里的矛盾更多的是关于形式系统的局限性，而不是哲学意义上的辩证矛盾。
网页7提到，这些定理不会制造矛盾，而是揭示无法证明的真理。因此，答案是，它们与辩证法没有直接矛盾。相反，这些定理揭示了形式系统的局限性，这些局限性可以通过辩证法的视角来解释，即驱动系统不断发展的内在局限性，但这只是一种解释，而不是矛盾。
我应该将其综合成一个结构化的答案，并引用相关来源。请务必澄清，虽然没有直接矛盾，但可能存在一些哲学解释将它们联系起来。请使用讨论这些定理的性质及其对一致性和完整性影响的网页中的引用。
哥德尔不完备定理并不直接与辩证法原理相矛盾，而是揭示了形式系统的固有局限性，这些局限性可以通过辩证法的视角来解读。以下是结构化分析：
1. 哥德尔定理的核心主张
第一不完备定理：任何能够表达基本算术的一致形式系统都是不完备的——存在无法在系统见1、2、8内证明的真数学陈述。
第二不完备定理：任何这样的系统都无法在不矛盾见1、6、8的情况下证明其自身的一致性。
这些结果挑战了希尔伯特关于数学完整、一致的公理基础的理想。
2. 辩证法与矛盾论
辩证法（例如黑格尔辩证法或马克思辩证法）经常强调矛盾是变革的驱动力。然而，在形式逻辑中，矛盾（例如同时证明 P 和 ?P）会导致逻辑爆炸，使系统变得毫无用处。哥德尔定理强调了一致性和完备性之间的矛盾：避免矛盾本质上会限制系统证明所有真理的能力。这种矛盾可以用辩证法来表述为两个理想属性 5、8 之间的对立。
3. 自指与辩证综合
哥德尔的证明依赖于自指性陈述（例如，“这句话无法证明”），这创造了一种类似悖论的场景。虽然辩证法可能将其解释为需要通过综合来解决的矛盾（例如，通过否定之否定演化到“更高级”的系统），但哥德尔的结果表明，在任何足够强大的形式系统中，这种不完备性都是不可避免的。每次试图通过添加公理来“完善”一个系统，只会将不完整性提升到一个新的层次 3, 8。
4. 哲学含义
形式化的局限性：这些定理破坏了静态、包罗万象的数学框架的概念，这与辩证法关于知识是动态和不断发展的观点相一致 2, 6。
人类推理与机械推理：有人认为，哥德尔的结果表明人类直觉超越了形式系统——这是心灵哲学中一个备受争议的观点 6, 8。辩证法可能将其解释为人类创造力与系统约束之间的综合。
5. 无直接矛盾
哥德尔定理是数学结果，而非哲学主张。它们并非否定辩证法，而是揭示了形式系统的结构局限性。 “矛盾”并非在于辩证逻辑，而在于公理框架内一致性与完备性之间的内在权衡 5, 7。
结论
哥德尔定理并不与辩证法相矛盾，而是为辩证法的张力提供了一个数学类比：系统必须在一致性与完备性之间寻找平衡，就像辩证法过程通过综合来解决矛盾一样。这种相互作用丰富了数学和哲学的论述，但并不构成直接冲突 1, 6, 8。
网页：
1. https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
2. https://plato.stanford.edu/entries/goedel-incompleteness/
3. https://www.quantamagazine.org/how-godels-proof-works-20200714/
4. https://www.quantamagazine.org/how-godels-proof-works-20200714/
5. https://philosophy.stackexchange.com/questions/85973/do-godels-incompleteness-theorems-create-a-contradiction-paradox?newreg=b67c19903a54419b9039be57e3117e01
6. https://1000wordphilosophy.com/2024/06/01/godel/
7. https://1000wordphilosophy.com/
8. https://www.spencergreenberg.com/2023/11/understanding-two-of-the-weirdest-theorems-in-math-godels-incompleteness/
[Mark Wain 2025年4月24日]
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哥德尔第一不完备定理 - 证明对角化
https://www.youtube.com/watch?v=PpSxqde0af4
稳定排序 2020年7月7日
“这个陈述不正确”
简单地读出来会存在问题，因为如果我们假设这个陈述是正确的，那么它的含义就会反过来。另一方面，如果这个陈述不正确，那么它的含义就会是正确的，因此就会出现悖论。
那么，构造一个类似的陈述需要什么呢？用数学符号正式地表达它？
有两个亟待解决的问题。首先，“真”这个词，有点令人惊讶，并没有公理化的定义。数字和公式不具备“真”的属性。我们认为公理被定义为真，因此由此推导出的任何定理也为真。但这样一来，真的概念仍然存在于系统本身之外。
塔斯基的不可定义性定理：
https://en.wikipedia.org/wiki/Tarski%...
另一个有问题的词是“this”。它指的是陈述本身，但该陈述尚未被定义。这就像在没有制作视频的情况下说“这段视频正好16分9秒”。
为了解决这个问题，哥德尔建立了一种方法，将陈述（例如定理，甚至是这些定理的证明）转换为自然数，作为一种编码。然后，在列出了特定类型公式的编码后，他证明了列表中存在一个自然数，它对应于一个陈述其自身不可证明性的公式。
哥德尔编码在唯一的字符序列和某个唯一的数字之间建立了一一对应关系。这使得哥德尔能够将任何数学陈述及其证明转换成两个唯一的数字。
哥德尔不完备性定理的简单证明
https://mat.iitm.ac.in/home/asingh/pu...
无法证明的语句示例
https://en.wikipedia.org/wiki/Paris%E...
更多关于哥德尔不完备性定理的阅读材料：
https://plato.stanford.edu/entries/go...
作者：Andre Violentyev
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哥德尔不完备性定理
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
第一不完备性定理：“任何一致的形式系统 F，只要能进行一定量的基本算术运算，都是不完备的；也就是说，F 语言中存在一些语句，既不能被证明，也不能被证伪。” (Raatikainen 2020)
Raatikainen, Panu (2020). “哥德尔不完备性定理”。斯坦福哲学百科全书。检索日期：2022年11月7日。
该定理所指的不可证明的语句GF通常被称为系统F的“哥德尔语句”。该证明为系统F构建了一个特定的哥德尔语句，但该系统语言中存在无数个具有相同属性的语句，例如哥德尔语句与任何逻辑上有效的语句的合取。
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数学家解释哥德尔不完备性定理 | Edward Frenkel 和 Lex Fridman
https://www.youtube.com/watch?v=u3GYrEOoKGk
Lex Clips
148万订阅者 2023年4月13日
@arsartium108
11个月前
以下是这些定理的简要说明，供感兴趣的读者参考：
哥德尔第一不完备定理指出：“任何能够有效表达初等算术的理论，都不可能同时具有一致性和完备性。具体而言，对于任何能够有效生成并证明某些基本算术真理的形式理论，都存在一个为真但在该理论内无法证明的算术陈述。”
哥德尔第二不完备定理指出：“对于任何有效生成并包含基本算术真理和某些关于形式可证性的真理的形式理论T，当且仅当T不一致时，T才包含其自身一致性的陈述。”
就在1931年发表其不完备性结果之前，哥德尔已经证明了一阶逻辑演算的完备性；但数论系统由逻辑和数论公理组成，因此PM的完备性和希尔伯特纲领（数学基础）的目标仍然是悬而未决的问题。哥德尔证明了：(1) 如果逻辑完备，但整体不完备，则数论公理必定不完备；(2) 任何数论系统都不可能在该系统内证明其一致性。在迪恩先生的讨论中，哥德尔的不完备性结果表明，任何将皮亚诺的自然数公理与PM逻辑相结合而得到的形式系统都是不完备的，并且任何以此方式构建的一致系统都无法证明其自身的一致性。
哥德尔得出不完备性定理的原因令人着迷。哥德尔是一位数学实在论者（柏拉图主义者），他认为集合论的公理显而易见，因为它们“强加于我们为真”。在研究希尔伯特问题以有限主义方法证明分析的一致性时，哥德尔试图“化解难题”，先用有限主义方法证明数论的一致性，然后再用数论证明分析的一致性，不仅假设数论具有一致性，还假设数论为真。
据王（1981）所述：“[哥德尔]用数论公式表示实数……并发现他必须使用数论中语句的真值概念才能验证其全面性。当他对分析性的理解公理进行分析的时候，他很快就陷入了与真理和可定义性相关的悖论（尤其是说谎者悖论和理查德悖论）。他意识到数论中的真理无法在数论中定义，因此他的计划……没有奏效。
[王浩（1921-1995）是一位美籍华裔逻辑学家、哲学家、数学家和库尔特·哥德尔的评论家。王浩最重要的贡献之一是王氏拼贴。他证明了任何图灵机都可以转化为一组王氏拼贴。作为一位杰出的哲学家，王浩还对路德维希·维特根斯坦后期的数学哲学进行了深刻的阐释，他称之为“人类学主义”。他记录了库尔特·哥德尔的哲学思想，并撰写了多本关于该主题的著作，从而为当代学者提供了许多阐明哥德尔后期哲学思想的洞见。参见 https://digitalcommons.rockefeller.edu ? 教职员工]
作为一名数学实在论者，哥德尔早已对希尔伯特形式主义的根本前提产生了怀疑。在发现真理无法用有限主义方法在数论中定义之后，哥德尔意识到在足够强的系统中存在不可判定的命题。此后，他煞费苦心地从1931年的成果中移除了真理的概念，以便仅使用形式主义者无法反对的方法来揭示形式主义方案的缺陷。
哥德尔写道：
“我可以补充一点，我对数学和元数学，尤其是超限推理的客观主义观念，对我的逻辑研究也至关重要。如果数学系统本身被认为是由无意义的符号组成，只有通过元数学才能获得某种意义的替代，那么我们又如何能想到用数学系统本身来表达元数学呢？……需要注意的是，我在数学形式系统中构建不可判定数论命题的启发式原则，是高度超限的‘客观数学真理’概念，而不是可证明性概念……” 王 (1974)
在一封未发表的致研究生的信中，哥德尔写道：
“然而，由于我们时代的哲学偏见，1. 没有人寻求相对一致性的证明，因为人们认为一致性证明必须是有限的才能有意义；2. 与可证明性相对的客观数学真理概念，被认为是……哥德尔的“逻辑”遭到了极大的怀疑，并被广泛地认为毫无意义而遭到拒绝。
显然，尽管哥德尔在本体论上致力于数学真理，但他有理由担心，任何假设基础主义概念的结果，都会被希尔伯特观点主导的形式主义建制派所排斥。如此一来，他便得出了一个连他自己都未曾预料到的结果——他的第二不完备定理——该定理确立了任何足够强的形式系统都无法证明其自身的一致性。
另见：
库尔特·哥德尔，《论《数学原理》及其相关系统的形式不可判定命题 I》，让·范·海耶诺特（译），《从弗雷格到哥德尔：数理逻辑资料集，1879-1931》（哈佛大学，1931年）