Some thoughts on Gödel's incompleteness theorem (bilingual 英漢雙語) 關於哥德爾不完備定理的一些思考
https://blog.creaders.net/user_blog_diary.php?did=NTE0MTk5 https://www.facebook.com/andrew.colesville https://x.com/mwsansculotte
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. Gödel'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. * The simplest version of Gödel'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" Gödel'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 Gödel'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 Gödel's incompleteness theorems and the dialectic logic: Okay, so the user is asking if Gödel'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 Gödel's results conflict with dialectical principles, like the unity of opposites or the idea that contradictions drive development. First, I need to recall what Gödel'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). Gödel 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 Gödel'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, Gödel'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. Gödel'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 Gödel'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. Gödel'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 Gödel'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), Gödel'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 Gödel'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 Gödel'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 Gödel'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] * 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, Gödel 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. Gödel Encoding establishes a one-to-one correspondence between a unique sequence of characters and some unique number. This allowed Gödel to convert any mathematical statement, as well as a proof of that statement into two unique numbers. A Simple Proof of Gödel'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 Gödel's Incompleteness Theorems: https://plato.stanford.edu/entries/go... Written and narrated by Andre Violentyev * Gödel'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). "Gödel'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 Gödel sentence" for the system F. The proof constructs a particular Gödel 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 Gödel sentence and any logically valid sentence. * Mathematician explains Gödel'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: Gödel'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." Gödel'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, Gödel 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. Gödel 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, Gödel'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 Gödel to his Incompleteness theorems is fascinating. Gödel 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, Gödel 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): "[Gödel] 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 Gödel. 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 Gödel's philosophical ideas and authored several books on the subject, thereby providing contemporary scholars with many insights elucidating Gödel's later philosophical thought. See https://digitalcommons.rockefeller.edu › faculty-members] As a mathematical realist, Gödel already doubted the underlying premise of Hilbert's Formalism, and after discovering that truth could not be defined within number theory using finitist means, Gödel 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. Gödel 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, Gödel 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 Gödel'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, Gödel, Kurt "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" Jean van Heijenoort (trans.), From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931 (Harvard 1931) 漢語譯文 關於哥德爾不完備定理的一些思考
https://blog.creaders.net/user_blog_diary.php?did=NTE0MTk5 https://www.facebook.com/andrew.colesville https://x.com/mwsansculotte
假設有兩個自然數 A 和 B。那麼 A + B 必定等於 B + A。這裡，+ 表示加。 但是，如果我們讓 A、B 和 + 表示其他東西，比如非數，那麼上述關於 A、B 和 + 的陳述同樣成立，前提是 + 符號的含義從加法變為其他運算。 混合顏料示例：設 A 和 B 分別為繪畫中的黃色和紅色，+ 表示繪畫中的顏色混合。顯然，A + B = B + A，結果相同，為紫色 C。 混合水示例：設 A 和 B 分別是盛在兩個不同溫度、但隔熱良好的玻璃杯中的水，+ 表示混合運算。顯然，A + B = B + A，混合水溫相同，均為 T。 設 G 為假設，或者更確切地說，為 A + B ≠ B + A 的命題。從人類的角度來看，這當然是錯誤的，但即使 A 和 B 是自然數，計算機也不知道它是真還是假。哥德爾定理（1931 年）表明，這個自然數命題既不能用計算方法證明其是，也不能用計算方法證明其非。因此，基於機器的人工智能不一定是可計算的，也就是說，生活和思維中的所有事物並非都是可計算的。 * 哥德爾定理的最簡版本及其重要性 https://www.youtube.com/watch?v=a9eXBUmE-Cc Chris Niebauer 博士，2022 年 5 月 7 日 在本視頻中，我將向您展示“理解”哥德爾定理的最簡單方法。想象一台無所不知的計算機（思維的極限），它可以陳述任何真理。我們稱之為通用真理機器（UTM）。現在，如果我們寫出 G="UTM 永遠不會說 G 為真"，然後詢問 UTM G 是否為真，我們就把它推到了極限。UTM 不能說它是真的……這使得它為真！這證明了思維的極限，以及存在超越思維的真理。 因此，命題 G 既不能被證明也不能被證偽。 不完備性定理僅適用於能夠證明關於自然數的充分事實集合的形式系統。 上面提出的兩個混合命題不在哥德爾不完備性定理的範圍內。 G 是一個僅指自然數的命題。 然而，如果將上面所示的 G = C 和/或 G = T 數字化為數字，那麼在它們被計算之後，它們會得到與自然數中的 G 相同的結論——既不能被證明也不能被證偽。說人工智能可以用計算取代意識是魯莽的。 我並不是說人工智能是A1，它不是，只是它的智能僅限於無法證明的結果。因此，對人工智能更好的定義是“sebome”【或賽博梅】——基於機器的進步得到的聰明百科全書。[參見：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 對哥德爾不完備定理和辯證邏輯的看法： 好的，所以這位用戶想問的是哥德爾不完備定理是否與辯證法相矛盾。嗯，辯證法通常指的是論證方法或哲學過程，經常與黑格爾和馬克思聯繫在一起。用戶可能會思考哥德爾的結論是否與辯證法原理相衝突，例如對立統一或矛盾推動發展的理念。 首先，我需要回顧一下哥德爾定理的實際含義。從搜索結果來看，第一個定理指出，任何能夠進行基本算術運算的一致形式系統都會包含無法在系統內部證明的真陳述。第二個定理補充說，這樣的系統無法證明其自身的一致性。這些結果挑戰了數學可以完全公理化的觀點，而這正是希爾伯特綱領的目標之一。 辯證法經常處理矛盾及其在進步中的作用。例如，在黑格爾辯證法中，矛盾通過綜合來解決。但在形式系統中，矛盾是有問題的，因為它們會導致爆炸（一切都變得可證）。哥德爾表明，避免矛盾（即一致性）本質上會導致不完備性。 查看搜索結果，Philosophy Stack Exchange 的第 4 個網頁探討了這些定理是否構成了悖論。答案澄清了它們並非矛盾，而是強調了局限性。第 5 個網頁解釋說，哥德爾定理使用自指陳述來表示不完備性，這與形式系統的結構有關。 辯證法可能將不完備性視為形式系統中固有矛盾的一種形式。但根據這些定理，這些系統是一致的（沒有矛盾），只是不完備。因此，一致性和完備性之間可能存在辯證張力。這些定理表明，追求完備性會導致潛在的不一致性，這可以被視為一種辯證矛盾。 然而，哥德爾定理是數學結果，而非哲學論證。它們並不直接涉及與辯證唯物主義或黑格爾思想相悖。這裡的矛盾更多的是關於形式系統的局限性，而不是哲學意義上的辯證矛盾。 網頁7提到，這些定理不會製造矛盾，而是揭示無法證明的真理。因此，答案是，它們與辯證法沒有直接矛盾。相反，這些定理揭示了形式系統的局限性，這些局限性可以通過辯證法的視角來解釋，即驅動系統不斷發展的內在局限性，但這只是一種解釋，而不是矛盾。 我應該將其綜合成一個結構化的答案，並引用相關來源。請務必澄清，雖然沒有直接矛盾，但可能存在一些哲學解釋將它們聯繫起來。請使用討論這些定理的性質及其對一致性和完整性影響的網頁中的引用。 哥德爾不完備定理並不直接與辯證法原理相矛盾，而是揭示了形式系統的固有局限性，這些局限性可以通過辯證法的視角來解讀。以下是結構化分析： 1. 哥德爾定理的核心主張 第一不完備定理：任何能夠表達基本算術的一致形式系統都是不完備的——存在無法在系統見1、2、8內證明的真數學陳述。 第二不完備定理：任何這樣的系統都無法在不矛盾見1、6、8的情況下證明其自身的一致性。 這些結果挑戰了希爾伯特關於數學完整、一致的公理基礎的理想。 2. 辯證法與矛盾論 辯證法（例如黑格爾辯證法或馬克思辯證法）經常強調矛盾是變革的驅動力。然而，在形式邏輯中，矛盾（例如同時證明 P 和 ¬P）會導致邏輯爆炸，使系統變得毫無用處。哥德爾定理強調了一致性和完備性之間的矛盾：避免矛盾本質上會限制系統證明所有真理的能力。這種矛盾可以用辯證法來表述為兩個理想屬性 5、8 之間的對立。 3. 自指與辯證綜合 哥德爾的證明依賴於自指性陳述（例如，“這句話無法證明”），這創造了一種類似悖論的場景。雖然辯證法可能將其解釋為需要通過綜合來解決的矛盾（例如，通過否定之否定演化到“更高級”的系統），但哥德爾的結果表明，在任何足夠強大的形式系統中，這種不完備性都是不可避免的。每次試圖通過添加公理來“完善”一個系統，只會將不完整性提升到一個新的層次 3, 8。 4. 哲學含義 形式化的局限性：這些定理破壞了靜態、包羅萬象的數學框架的概念，這與辯證法關於知識是動態和不斷發展的觀點相一致 2, 6。 人類推理與機械推理：有人認為，哥德爾的結果表明人類直覺超越了形式系統——這是心靈哲學中一個備受爭議的觀點 6, 8。辯證法可能將其解釋為人類創造力與系統約束之間的綜合。 5. 無直接矛盾 哥德爾定理是數學結果，而非哲學主張。它們並非否定辯證法，而是揭示了形式系統的結構局限性。 “矛盾”並非在於辯證邏輯，而在於公理框架內一致性與完備性之間的內在權衡 5, 7。 結論 哥德爾定理並不與辯證法相矛盾，而是為辯證法的張力提供了一個數學類比：系統必須在一致性與完備性之間尋找平衡，就像辯證法過程通過綜合來解決矛盾一樣。這種相互作用豐富了數學和哲學的論述，但並不構成直接衝突 1, 6, 8。 網頁： 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] * 哥德爾第一不完備定理 - 證明對角化 https://www.youtube.com/watch?v=PpSxqde0af4 穩定排序 2020年7月7日 “這個陳述不正確” 簡單地讀出來會存在問題，因為如果我們假設這個陳述是正確的，那麼它的含義就會反過來。另一方面，如果這個陳述不正確，那麼它的含義就會是正確的，因此就會出現悖論。 那麼，構造一個類似的陳述需要什麼呢？用數學符號正式地表達它？ 有兩個亟待解決的問題。首先，“真”這個詞，有點令人驚訝，並沒有公理化的定義。數字和公式不具備“真”的屬性。我們認為公理被定義為真，因此由此推導出的任何定理也為真。但這樣一來，真的概念仍然存在於系統本身之外。 塔斯基的不可定義性定理： https://en.wikipedia.org/wiki/Tarski%... 另一個有問題的詞是“this”。它指的是陳述本身，但該陳述尚未被定義。這就像在沒有製作視頻的情況下說“這段視頻正好16分9秒”。 為了解決這個問題，哥德爾建立了一種方法，將陳述（例如定理，甚至是這些定理的證明）轉換為自然數，作為一種編碼。然後，在列出了特定類型公式的編碼後，他證明了列表中存在一個自然數，它對應於一個陳述其自身不可證明性的公式。 哥德爾編碼在唯一的字符序列和某個唯一的數字之間建立了一一對應關係。這使得哥德爾能夠將任何數學陳述及其證明轉換成兩個唯一的數字。 哥德爾不完備性定理的簡單證明 https://mat.iitm.ac.in/home/asingh/pu... 無法證明的語句示例 https://en.wikipedia.org/wiki/Paris%E... 更多關於哥德爾不完備性定理的閱讀材料： https://plato.stanford.edu/entries/go... 作者：Andre Violentyev * 哥德爾不完備性定理 https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems 第一不完備性定理：“任何一致的形式系統 F，只要能進行一定量的基本算術運算，都是不完備的；也就是說，F 語言中存在一些語句，既不能被證明，也不能被證偽。” (Raatikainen 2020) Raatikainen, Panu (2020). “哥德爾不完備性定理”。斯坦福哲學百科全書。檢索日期：2022年11月7日。 該定理所指的不可證明的語句GF通常被稱為系統F的“哥德爾語句”。該證明為系統F構建了一個特定的哥德爾語句，但該系統語言中存在無數個具有相同屬性的語句，例如哥德爾語句與任何邏輯上有效的語句的合取。 * 數學家解釋哥德爾不完備性定理 | Edward Frenkel 和 Lex Fridman https://www.youtube.com/watch?v=u3GYrEOoKGk Lex Clips 148萬訂閱者 2023年4月13日 @arsartium108 11個月前 以下是這些定理的簡要說明，供感興趣的讀者參考： 哥德爾第一不完備定理指出：“任何能夠有效表達初等算術的理論，都不可能同時具有一致性和完備性。具體而言，對於任何能夠有效生成並證明某些基本算術真理的形式理論，都存在一個為真但在該理論內無法證明的算術陳述。” 哥德爾第二不完備定理指出：“對於任何有效生成並包含基本算術真理和某些關於形式可證性的真理的形式理論T，當且僅當T不一致時，T才包含其自身一致性的陳述。” 就在1931年發表其不完備性結果之前，哥德爾已經證明了一階邏輯演算的完備性；但數論系統由邏輯和數論公理組成，因此PM的完備性和希爾伯特綱領（數學基礎）的目標仍然是懸而未決的問題。哥德爾證明了：(1) 如果邏輯完備，但整體不完備，則數論公理必定不完備；(2) 任何數論系統都不可能在該系統內證明其一致性。在迪恩先生的討論中，哥德爾的不完備性結果表明，任何將皮亞諾的自然數公理與PM邏輯相結合而得到的形式系統都是不完備的，並且任何以此方式構建的一致系統都無法證明其自身的一致性。 哥德爾得出不完備性定理的原因令人着迷。哥德爾是一位數學實在論者（柏拉圖主義者），他認為集合論的公理顯而易見，因為它們“強加於我們為真”。在研究希爾伯特問題以有限主義方法證明分析的一致性時，哥德爾試圖“化解難題”，先用有限主義方法證明數論的一致性，然後再用數論證明分析的一致性，不僅假設數論具有一致性，還假設數論為真。 據王（1981）所述：“[哥德爾]用數論公式表示實數……並發現他必須使用數論中語句的真值概念才能驗證其全面性。當他對分析性的理解公理進行分析的時候，他很快就陷入了與真理和可定義性相關的悖論（尤其是說謊者悖論和理查德悖論）。他意識到數論中的真理無法在數論中定義，因此他的計劃……沒有奏效。 [王浩（1921-1995）是一位美籍華裔邏輯學家、哲學家、數學家和庫爾特·哥德爾的評論家。王浩最重要的貢獻之一是王氏拼貼。他證明了任何圖靈機都可以轉化為一組王氏拼貼。作為一位傑出的哲學家，王浩還對路德維希·維特根斯坦後期的數學哲學進行了深刻的闡釋，他稱之為“人類學主義”。他記錄了庫爾特·哥德爾的哲學思想，並撰寫了多本關於該主題的著作，從而為當代學者提供了許多闡明哥德爾後期哲學思想的洞見。參見 https://digitalcommons.rockefeller.edu › 教職員工] 作為一名數學實在論者，哥德爾早已對希爾伯特形式主義的根本前提產生了懷疑。在發現真理無法用有限主義方法在數論中定義之後，哥德爾意識到在足夠強的系統中存在不可判定的命題。此後，他煞費苦心地從1931年的成果中移除了真理的概念，以便僅使用形式主義者無法反對的方法來揭示形式主義方案的缺陷。 哥德爾寫道： “我可以補充一點，我對數學和元數學，尤其是超限推理的客觀主義觀念，對我的邏輯研究也至關重要。如果數學系統本身被認為是由無意義的符號組成，只有通過元數學才能獲得某種意義的替代，那麼我們又如何能想到用數學系統本身來表達元數學呢？……需要注意的是，我在數學形式系統中構建不可判定數論命題的啟發式原則，是高度超限的‘客觀數學真理’概念，而不是可證明性概念……” 王 (1974) 在一封未發表的致研究生的信中，哥德爾寫道： “然而，由於我們時代的哲學偏見，1. 沒有人尋求相對一致性的證明，因為人們認為一致性證明必須是有限的才能有意義；2. 與可證明性相對的客觀數學真理概念，被認為是……哥德爾的“邏輯”遭到了極大的懷疑，並被廣泛地認為毫無意義而遭到拒絕。 顯然，儘管哥德爾在本體論上致力於數學真理，但他有理由擔心，任何假設基礎主義概念的結果，都會被希爾伯特觀點主導的形式主義建制派所排斥。如此一來，他便得出了一個連他自己都未曾預料到的結果——他的第二不完備定理——該定理確立了任何足夠強的形式系統都無法證明其自身的一致性。 另見： 庫爾特·哥德爾，《論《數學原理》及其相關系統的形式不可判定命題 I》，讓·范·海耶諾特（譯），《從弗雷格到哥德爾：數理邏輯資料集，1879-1931》（哈佛大學，1931年）
|
