# Godel's Theorem

Edited by Jordan Bohall (University of Illinois, Urbana-Champaign)
 Summary Gödel's Theorems are two of the most critical results in 20th century mathematics and logic. The theorems have had profound implications for logic, philosophy of mathematics, philosophical logic, philosophy of language and more. The two theorems together are a characterization of the far limits of provability within any axiomatized theory T. This is to say that within a consistent formal theory T, there are statements constructible in the language of T that can be neither proved nor disproved (1st Theorem), and T cannot prove that it is itself consistent (2nd Theorem).
 Key works Gödel 1986 Franzén 2005 Raatikainen 2005
 Introductions Enderton 1972  M. Smullyan 1992 Raatikainen 2013 Smorynski 1977
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1. The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty and loss (...)

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2. Defining Gödel Incompleteness Away.P. Olcott - manuscript
We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...)

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3. The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises.

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4. Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

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5. What is the Link between Aristotle’s Philosophy of Mind, the Iterative Conception of Set, Gödel’s Incompleteness Theorems and God? About the Pleasure and the Difficulties of Interpreting Kurt Gödel’s Philosophical Remarks.Eva-Maria Engelen - forthcoming - In Gabriella Crocco & Eva-Maria Engelen (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence.
It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...)
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6. Gödel on Many-Valued Logic.Tim Lethen - 2023 - Review of Symbolic Logic 16 (3):655-671.
This paper collects and presents unpublished notes of Kurt Gödel concerning the field of many-valued logic. In order to get a picture as complete as possible, both formal and philosophical notes, transcribed from the Gabelsberger shorthand system, are included.

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7. The Making of Mathematics: Heuristic Philosophy of Mathematics.Carlo Cellucci - 2022 - Cham, Switzerland: Springer.
Mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century, claims that the philosophy of mathematics cannot concern itself with the making of mathematics, in particular discovery, but only with finished mathematics, namely mathematics presented in finished form. On this basis, mainstream philosophy of mathematics argues that mathematics is theorem proving by the axiomatic method. This, however, is untenable because it is incompatible with Gödel’s incompleteness theorems, and cannot account for many features of mathematics. (...)

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8. Incompleteness and the Halting Problem.Cristian S. Calude - 2021 - Studia Logica 109 (5):1159-1169.
We present an abstract framework in which we give simple proofs for Gödel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’, Chaitin’s and Kritchman-Raz’s Theorems.

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9. Current Research on Gödel’s Incompleteness Theorems.Yong Cheng - 2021 - Bulletin of Symbolic Logic 27 (2):113-167.
We give a survey of current research on Gödel’s incompleteness theorems from the following three aspects: classifications of different proofs of Gödel’s incompleteness theorems, the limit of the applicability of Gödel’s first incompleteness theorem, and the limit of the applicability of Gödel’s second incompleteness theorem.

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10. Czy matematyka jest składnią języka? Kurta Gödla argument przeciwko formalizmowi.Maciej Głowacki - 2021 - Filozofia Nauki 29 (113):43-61.
In this paper, I critically examine Kurt Gödel’s argument against the syntactic interpretation of mathematics. While the main aim is to analyze the argument, I also wish to underscore the relevance of the original elements of Gödel’s philosophical thought. The paper consists of four parts. In the first part, I introduce the reader to Gödel’s philosophy. In the second part, I reconstruct the formalist stance in the philosophy of mathematics, which is the object of Gödel’s criticism. In the third part, (...)

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11. This textbook is a logic manual which includes an elementary course and an advanced course. It covers more than most introductory logic textbooks, while maintaining a comfortable pace that students can follow. The technical exposition is clear, precise and follows a paced increase in complexity, allowing the reader to get comfortable with previous definitions and procedures before facing more difficult material. The book also presents an interesting overall balance between formal and philosophical discussion, making it suitable for both philosophy and (...)

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12. What he could have said (but did not say) about Gödel’s second theorem: A note on Floyd-Putnam’s Wittgenstein.Kaave Lajevardi - 2021 - Wittgenstein-Studien 12 (1):121-129.
In several publications, Juliet Floyd and Hilary Putnam have argued that the so-called ‘notorious paragraph’ of the Remarks on the Foundations of Mathematics contains a valuable philosophical insight about Gödel’s informal proof of the first incompleteness theorem – in a nutshell, the idea they attribute to Wittgenstein is that if the Gödel sentence of a system is refutable, then, because of the resulting ω-inconsistency of the system, we should give up the translation of Gödel’s sentence by the English sentence “I (...)

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13. Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic.Vasil Penchev - 2021 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 13 (7):1-57.
In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n (...)

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14. Remarks on the Gödelian Anti-Mechanist Arguments.Panu Raatikainen - 2020 - Studia Semiotyczne 34 (1):267–278.
Certain selected issues around the Gödelian anti-mechanist arguments which have received less attention are discussed.

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15. 이유의 외부 한계'에 대한 검토 (The Outer Limits of Reason) Noson Yanofsky 403p (2013).Michael Richard Starks - 2020 - In 지구상의 지옥에 오신 것을 환영합니다 : 아기, 기후 변화, 비트 코인, 카르텔, 중국, 민주주의, 다양성, 역학, 평등, 해커, 인권, 이슬람, 자유주의, 번영, 웹, 혼돈, 기아, 질병, 폭력, 인공 지능, 전쟁. Las Vegas, NV USA: Reality Press. pp. 210-227.
나는 비텐슈타인과 진화 심리학의 통일 된 관점에서 노슨 야노프스키에 의해 '이성의 외부 한계'에 대한 자세한 리뷰를 제공합니다. 나는 언어와 수학의 역설, 불완전성, 부정성, 계산성, 컴퓨터와 같은 뇌와 우주와 같은 문제의 어려움은 모두 적절한 맥락에서 언어의 사용을 주의 깊게 바라보지 못하고 따라서 언어가 작동하는 방법의 문제에서 과학적 사실의 문제를 분리하지 못하는 데서 발생한다는 것을 나타냅니다. 나는 불완전성, 파라불일치 및 부정성에 대한 비트겐슈타인의 견해와 계산의 한계에 대한 울퍼트의 작품에 대해 논의한다. 요약하자면: 브루클린---좋은 과학에 따르면 우주, 그리 좋은 철학. 현대 의 두 systems보기에서인간의 (...)

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16. Último sermão da Igreja do naturalismo fundamentalista pelo pastor Hofstadter. Como o seu muito mais famoso (ou infame por seus erros filosóficos implacáveis) Godel, Escher, Bach, ele tem uma plausibilidade superficial, mas se se compreende que este é um cientificismo desenfreado que mistura questões científicas reais com os filosóficos (ou seja, o somente as edições reais são que jogos da língua nós devemos jogar) então quase todo seu interesse desaparece. Eu forneci um quadro para análise baseada na psicologia evolutiva e (...)

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17. 私は奇妙なループです」のレビュー(I am a Strange Loop) by Douglas Hofstadter (2007) (レビュー改訂2019).Michael Richard Starks - 2020 - In 地獄へようこそ 赤ちゃん、気候変動、ビットコイン、カルテル、中国、民主主義、多様性、ディスジェニックス、平等、ハッカー、人権、イスラム教、自由主義、繁栄、ウェブ、カオス、飢餓、病気、暴力、人工知能、戦争. Las Vegas, NV, USA: Reality Press. pp. 102-118.
ホフスタッター牧師による原理主義自然主義教会からの最新の説教。彼のはるかに有名な(または容赦ない哲学的誤りで悪名高い)作品ゴーデル、エッシャー、バッハのように、それは表面的な妥当性を持っていますが、こ れが哲学的なものと実際の科学的問題を混ぜ合わせた横行するサイエンティズムであることを理解すれば(つまり、唯一の本当の問題は、私たちがプレイすべき言語ゲームです)、その後、ほとんどすべての関心が消えます 。進化心理学とヴィトゲンシュタインの仕事に基づく分析のフレームワークを提供しています(最近の著作で更新されて以来)。 現代の2つのシス・エムスの見解から人間の行動のための包括的な最新の枠組みを望む人は、私の著書「ルートヴィヒ・ヴィトゲンシュタインとジョン・サールの第2回(2019)における哲学、心理学、ミンと言語の論 理的構造」を参照することができます。私の著作の多くにご興味がある人は、運命の惑星における「話す猿--哲学、心理学、科学、宗教、政治―記事とレビュー2006-2019 第3回(2019)」と21世紀4日(2019年)の自殺ユートピア妄想st Century 4th ed (2019)などを見ることができます .

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18. Ich gebe einen ausführlichen Überblick über 'The Outer Limits of Reason' von Noson Yanofsky aus einer einheitlichen Perspektive von Wittgenstein und Evolutionspsychologie. Ich weise darauf hin, dass die Schwierigkeit bei Themen wie Paradoxon in Sprache und Mathematik, Unvollständigkeit, Unbedenklichkeit, Berechenbarkeit, Gehirn und Universum als Computer usw. allesamt auf das Versäumnis zurückzuführen ist, unseren Sprachgebrauch im geeigneten Kontext sorgfältig zu prüfen, und daher das Versäumnis, Fragen der wissenschaftlichen Tatsache von Fragen der Funktionsweise von Sprache zu trennen. Ich bespreche Wittgensteins Ansichten über (...)

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19. В «Godel's Way» три видных ученых обсуждают такие вопросы, как неплатежеспособность, неполнота, случайность, вычислительность и последовательность. Я подхожу к этим вопросам с точки зрения Витгенштейна, что есть две основные проблемы, которые имеют совершенно разные решения. Есть научные или эмпирические вопросы, которые являются факты о мире, которые должны быть исследованы наблюдений и философские вопросы о том, как язык может быть использован внятно (которые включают в себя определенные вопросы в математике и логике), которые должны быть решены, глядят, как мы на самом деле (...)

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20. Wolpert, Chaitin et Wittgenstein sur l’impossibilité, l’incomplétude, le paradoxe menteur, le théisme, les limites du calcul, un principe d’incertitude mécanique non quantique et l’univers comme ordinateur, le théorème ultime dans Turing Machine Theory (révisé 2019).Michael Richard Starks - 2020 - In Bienvenue en Enfer sur Terre : Bébés, Changement climatique, Bitcoin, Cartels, Chine, Démocratie, Diversité, Dysgénique, Égalité, Pirates informatiques, Droits de l'homme, Islam, Libéralisme, Prospérité, Le Web, Chaos, Famine, Maladie, Violence, Intellige. Las Vegas, NV , USA: Reality Press. pp. 185-189.
J’ai lu de nombreuses discussions récentes sur les limites du calcul et de l’univers en tant qu’ordinateur, dans l’espoir de trouver quelques commentaires sur le travail étonnant du physicien polymathe et théoricien de la décision David Wolpert, mais n’ont pas trouvé une seule citation et je présente donc ce résumé très bref. Wolpert s’est avéré quelques théoricaux d’impossibilité ou d’incomplétude renversants (1992 à 2008-voir arxiv dot org) sur les limites de l’inférence (computation) qui sont si généraux qu’ils sont indépendants de (...)

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21. Último sermón de la iglesia del naturalismo fundamentalista por el pastor Hofstadter. Al igual que su mucho más famoso (o infame por sus incesantemente errores filosóficos) trabajo Godel, Escher, Bach, tiene una plausibilidad superficial, pero si se entiende que se trata de un científico rampante que mezcla problemas científicos reales con los filosóficos (es decir, el sólo los problemas reales son los juegos de idiomas que debemos jugar) entonces casi todo su interés desaparece. Proporciono un marco para el análisis basado (...)

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22. 'गोडेल के रास्ते' में तीन प्रख्यात वैज्ञानिकों ने अनिर्णय, अपूर्णता, यादृच्छिकता, गणनाऔरता और परासंगति जैसे मुद्दों पर चर्चा की। मैं Wittgensteinian दृष्टिकोण से इन मुद्दों दृष्टिकोण है कि वहाँ दो बुनियादी मुद्दों जो पूरी तरह से अलग समाधान है. वहाँ वैज्ञानिक या अनुभवजन्य मुद्दों, जो दुनिया के बारे में तथ्य है कि अवलोकन और दार्शनिक मुद्दों की जांच की जरूरत है के रूप में कैसे भाषा intelligibly इस्तेमाल किया जा सकता है (जो गणित और तर्क में कुछ सवाल शामिल हैं), (...)

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23. In "Godel es Way" diskutieren drei namhafte Wissenschaftler Themen wie Unentschlossenheit, Unvollständigkeit, Zufälligkeit, Berechenbarkeit und Parakonsistenz. Ich gehe diese Fragen aus Wittgensteiner Sicht an, dass es zwei grundlegende Fragen gibt, die völlig unterschiedliche Lösungen haben. Es gibt die wissenschaftlichen oder empirischen Fragen, die Fakten über die Welt sind, die beobachtungs- und philosophische Fragen untersuchen müssen, wie Sprache verständlich verwendet werden kann (die bestimmte Fragen in Mathematik und Logik beinhalten), die entschieden werden müssen, indem man sich anschaut,wie wir Wörter in bestimmten (...)

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24. 《我是一个奇怪的循环》的回顾由道格拉斯·霍夫施塔特 (2007)(Review of I Am a Strange Loop by Douglas Hofstadter (2007)) (审查修订 2019).Michael Richard Starks - 2020 - In 欢迎来到地球上的地狱: 婴儿，气候变化，比特币，卡特尔，中国，民主，多样性，养成基因，平等，黑客，人权，伊斯兰教，自由主义，繁荣，网络，混乱。饥饿，疾病，暴力，人工智能，战争. Las Vegas, NV USA: Reality Press. pp. 105-120.
霍夫施塔特牧师从原教旨主义自然主义教会的最新讲道。像他更出名(或臭名昭著的无情的哲学错误)的工作戈德尔,埃舍尔,巴赫,它有一个肤浅的合理性,但如果人们明白,这是猖獗的科学主义,混合真正的科学问题与哲学 问题(即,只有真正的问题是我们应该玩什么语言游戏),然后几乎所有的兴趣消失。我提供了一个基于进化心理学和维特根斯坦工作的分析框架(自从我最近的著作中更新)。 那些希望从现代两个系统的观点来看为人类行为建立一个全面的最新框架的人,可以查阅我的书《路德维希的哲学、心理学、心神 (Mind) 和语言的逻辑结构》维特根斯坦和约翰·西尔的《第二部》(2019年)。那些对我更多的作品感兴趣的人可能会看到《会说话的猴子——一个末日星球上的哲学、心理学、科学、宗教和政治——文章和评论2006-201 9年第3次(2019年)和自杀乌托邦幻想21篇世纪4日 (2019).

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25. Я читал много недавних дискуссий о границах вычислений и Вселенной, как компьютер, надеясь найти некоторые комментарии по удивительной работы физика полимата и теоретик решений Дэвид Вольперт, но не нашли ни одной цитаты, и поэтому я представляю это очень краткое резюме. Вольперт доказал некоторые потрясающие невозможности или теоремы неполноты (1992 до 2008-см arxiv dot org) на пределы выводов (вычисления), которые настолько общие они не зависят от устройства делать вычисления, и даже независимо от законов физики, поэтому они применяются через компьютеры, физика, и (...)

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26. Nel 'Godel's Way' tre eminenti scienziati discutono questioni come l'indecidibilità, l'incompletezza, la casualità, la computabilità e la paracoerenza. Affronto questi problemi dal punto di vista di Wittgensteinian che ci sono due questioni fondamentali che hanno soluzioni completamente diverse. Ci sono le questioni scientifiche o empiriche, che sono fatti sul mondo che devono essere studiati in modo osservante e filosofico su come il linguaggio può essere usato in modo intelligibilmente (che include alcune domande in matematica e logica), che devono essere decise (...)

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27. 「ゴーデルの道」では、3人の著名な科学者が、デシッド不能、不完全性、ランダム性、計算可能性、パラコンシステンションなどの問題について議論しています。私は、ウィトゲンシュタイニアンの視点から、全く異なる 解決策を持つ2つの基本的な問題があることをこれらの問題に取り組んでいます。科学的または経験的な問題は、言語がどのように理解的に使用できるか(数学と論理に特定の質問を含む)、特定の文脈で実際にどのように 単語を使用するかを調べて決定する必要がある、観察的および哲学的な問題を調査する必要がある世界に関する事実です。私たちがプレイしている言語ゲームについて明確になると、これらのトピックは他の人と同じように 普通の科学的、数学的な質問であると見なされます。ウィトゲンシュタインの洞察はめったに等しくなく、決して上回ることはなく、彼がブルーブックスとブラウンブックスを口述した80年前と同じくらい適切です。失敗 にもかかわらず、本当に完成した本ではなく一連のノートは、半世紀以上にわたって物理学、数学、哲学の出血エッジで働いてきたこれらの3人の有名な学者の作品のユニークな源です。ダ・コスタとドリアは、普遍的な計 算に書いて以来、ウォルパート(以下または私の記事を参照)によって引用されています(ウォルパートとヤナフスキーの「理由の外側の限界」の私のレビューを参照)、,そして彼の多くの成果の中で、ダ・コスタはパラ コンシタンションのパイオニアです。 現代の2つのシス・エムスの見解から人間の行動のための包括的な最新の枠組みを望む人は、私の著書「ルートヴィヒ・ヴィトゲンシュタインとジョン・サールの第2回(2019)における哲学、心理学、ミンと言語の論 理的構造」を参照することができます。私の著作の多くにご興味がある人は、運命の惑星における「話す猿--哲学、心理学、科学、宗教、政治―記事とレビュー2006-2019 第3回(2019)」と21世紀4日(2019年)の自殺ユートピア妄想st Century 4th ed (2019)などを見ることができます。 .

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28. 'Godel's Way'에서 세 명의 저명한 과학자들은 부정성, 불완전성, 임의성, 계산성 및 파라불일치와 같은 문제에 대해 논의합니다. 나는 완전히 다른 해결책을 가지고 두 가지 기본 문제가 있다는 비트 겐슈타인의 관점에서 이러한 문제에 접근. 과학적 또는 경험적 문제가 있다, 관찰 하 고 철학적 문제 언어를 어떻게 이해할 수 있는 (수학 및 논리에 특정 질문을 포함) 에 대 한 조사 해야 하는 세계에 대 한 사실,우리가 실제로 특정 컨텍스트에서 단어를 사용 하는 방법을 보고 하 여 결정 될 필요가. 우리가 어떤 언어 게임을 하고 (...)

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29. 나는 컴퓨터로 계산과 우주의 한계에 대한 많은 최근의 토론을 읽었습니다, polymath 물리학자 및 결정 이론가 데이비드 울퍼트의 놀라운 작품에 대한 몇 가지 의견을 찾을 수 있기를 바라고 있지만 하나의 인용을 발견하지 않은 그래서 나는이 매우 간단한 요약을 제시. Wolpert는 계산을 수행하는 장치와 는 별개이며 물리학법칙과는 무관하므로 컴퓨터, 물리학 및 인간의 행동에 적용되므로 추론(계산)에 대한 제한에 대해 놀라운 불가능또는 불완전성 정리(1992년에서 2008년 참조 arxiv dot org)를 입증했습니다. 그들은 캔터의 대각선화, 거짓말쟁이 역설 및 세계관을 사용하여 튜링 머신 이론의 궁극적 인 정리가 될 (...)

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30. पादरी Hofstadter द्वारा कट्टरपंथी प्रकृतिवाद के चर्च से नवीनतम उपदेश. अपने बहुत अधिक प्रसिद्ध (या अपने अथक दार्शनिक त्रुटियों के लिए कुख्यात) काम Godel, Escher, बाख की तरह, यह एक सतही प्रशंसनीयता है, लेकिन अगर एक समझता है कि यह बड़े पैमाने पर वैज्ञानिकता है जो दार्शनिक लोगों के साथ वास्तविक वैज्ञानिक मुद्दों घोला जा सकता है (यानी, केवल असली मुद्दों क्या भाषा का खेल हम खेलना चाहिए रहे हैं) तो लगभग सभी अपनी रुचि गायब हो जाता है. मैं विकासवादी (...)

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31. ¿Cómo utilizar el Teorema de Herbrand para decidir la validez de razonamientos en lenguaje de primer orden, en conformidad con el Teorema de Indecidibilidad de Church?Franklin Galindo & María Alejandra Morgado - 2019 - Apuntes Filosóficos: Revista Semestral de la Escuela de Filosofía 18 (55):67-86.
This article’s objetive is to present four application examples of Herbrand’s theorem to decide the validity of reasoning on first order language, in accordance whit Church’s Undecidability’s theorem. Also, to tell which is the principal problem around it. The logical resolution calculus will be worked on this article, which is a method used in artificial intelligence.

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32. I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how language works. (...)

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33. I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...)

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34. On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans. Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents (...)

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35. Eu li muitas discussões recentes sobre os limites da computação e do universo como computador, na esperança de encontrar alguns comentários sobre o trabalho surpreendente do físico polimatemático e teórico da decisão David Wolpert, mas não encontrei uma única citação e assim que eu apresento este muito breve Resumo. Wolpert provou alguma impossibilidade impressionante ou teoremas da incompletude (1992 a 2008-Veja arxiv dot org) nos limites à inferência (computação) que são tão gerais que são independentes do dispositivo que faz a (...)

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36. Último sermón de la iglesia del naturalismo fundamentalista por el pastor Hofstadter. Al igual que su mucho más famoso (o infame por sus incesantemente errores filosóficos) trabajo Godel, Escher, Bach, tiene una plausibilidad superficial, pero si se entiende que se trata de un científico rampante que mezcla problemas científicos reales con los filosóficos (es decir, el sólo los problemas reales son los juegos de idiomas que debemos jugar) entonces casi todo su interés desaparece. Proporciono un marco para el análisis basado (...)

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37. Em "Godel's Way", três cientistas eminentes discutem questões como a undecidability, incompletude, aleatoriedade, computabilidade e paraconsistência. Eu abordar estas questões do ponto de vista Wittgensteinian que existem duas questões básicas que têm soluções completamente diferentes. Há as questões científicas ou empíricas, que são fatos sobre o mundo que precisam ser investigados observacionalmente e questões filosóficas sobre como a linguagem pode ser usada inteligìvelmente (que incluem certas questões em matemática e lógica), que precisam ser decidido por olhar uma como nós realmente (...)

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38. 关于在柴廷、维特根斯坦、霍夫施塔特、沃尔珀特、多里亚、达科斯塔、戈德尔、西尔、罗迪赫、贝托、弗洛伊德、贝托、弗洛伊德、莫亚尔-沙罗克和亚诺夫斯基.Michael Richard Starks - 2019 - Las Vegas, NV USA: Reality Press.
人们普遍认为，不可能性、不完整性、不一致性、不可度、随机性、可预见性、悖论、不确定性和理性极限是完全不同的科学物理或数学问题，在常见。我认为，它们主要是标准的哲学问题（即语言游戏），这些问题大多在80 多年前由维特根斯坦解决。 -/- "在这种情况下，我们'想说'当然不是哲学，而是它的原材料。因此，例如，数学家倾向于对数学事实的客观性和现实性说的，不是数学哲学，而是哲学处理的东西。维特根斯坦 PI 234 -/- "哲学家们经常看到科学的方法，他们不可抗拒地试图以科学的方式提问和回答问题。这种倾向是形而上学的真正源泉，将哲学家带入完全的黑暗之中。 维特根斯坦 -/- 我简要地总结了现代两位最杰出的学生路德维希·维特根斯坦和约翰·西尔关于故意的逻辑结构（思想、语言、行为）的一些主要发现，作为我的起点Wittgenstein 的基本发现——所有真正的"哲学"问题都是相同的——关于在特定上下文中如何使用语言的困惑，因此所有解决方案都是一样的——研究如何在相关上下文中使用语言，使其真实性条件（满意度或 COS 条件）是明确的。基本问题是，人们可以说什么，但一个人不能意味着（状态明确COS）任何任意的话语和意义只有在非常具体的上下文中才可能。 -/- 在两种思想体系的现代视角（被推广为"思维快，思维慢"）的框架内，我从维特根斯坦人的角度剖析了一些主要评论员关于这些问题的一些著作，并采用了一个新的表意向性和新的双系统命名法。 我表明，这是一个强大的启发式描述这些假定的科学，物理或数学问题的真实性质，这是真正最好的处理作为标准哲学问题，如何使用语言（语言游戏在维特根斯坦的术语）。 -/- 我的论点是，这里突出特征的意向表（理性、思想、思想、语言、个性等）或多或少地准确地描述了，或者至少作为启发式，我们思考和行为的方式，所以它包含不只是哲学和心理学，但其他一切（历史，文学，数学，政治等） 。特别要注意，我（以及西尔、维特根斯坦和其他人）认为，故意和理性包括有意识的审议语言系统2和无意识的自动预语言系统1行为或反射。 .

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39. He leído muchas discusiones recientes sobre los límites de la computación y el universo como computadora, con la esperanza de encontrar algunos comentarios sobre el increíble trabajo del físico polimatemático y teórico de la decisión David Wolpert pero no han encontrado una sola citación y así que presento esta muy breve Resumen. Wolpert demostró algunos teoremas sorprendentes de imposibilidad o incompletos (1992 a 2008-ver arxiv dot org) en los límites de la inferencia (computación) que son tan generales que son independientes (...)

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40. معمولا تصور می شود که عدم امکان ، بی کامل بودن ، پارامونشتها ، Undecidability ، اتفاقی ، قابلیت های مختلف ، پارادوکس ، عدم قطعیت و محدودیت های دلیل ، مسائل فیزیکی و ریاضی علمی و یا با داشتن کمی یا هیچ چیز در مشترک. من پیشنهاد می کنم که آنها تا حد زیادی مشکلات فلسفی استاندارد (به عنوان مثال ، بازی های زبان) که عمدتا توسط ویتگنشتاین بیش از 80 سال پیش حل و فصل شد. -/- "آنچه ما (...)

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41. Hal ini sering berpikir bahwa kemustahilan, ketidaklengkapan, Paraconsistency, Undecidability, Randomness, komputasi, Paradox, ketidakpastian dan batas alasan yang berbeda ilmiah fisik atau matematika masalah memiliki sedikit atau tidak ada dalam Umum. Saya menyarankan bahwa mereka sebagian besar masalah filosofis standar (yaitu, Permainan bahasa) yang sebagian besar diselesaikan oleh Wittgenstein lebih dari 80years yang lalu. -/- "Apa yang kita ' tergoda untuk mengatakan ' dalam kasus seperti ini, tentu saja, bukan filsafat, tetapi bahan baku. Jadi, misalnya, apa yang seorang matematikawan cenderung mengatakan (...)

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42. En ' Godel’s Way ', tres eminentes científicos discuten temas como la indecisión, la incompleta, la aleatoriedad, la computabilidad y la paracoherencia. Me acerco a estas cuestiones desde el punto de vista de Wittgensteinian de que hay dos cuestiones básicas que tienen soluciones completamente diferentes. Existen las cuestiones científicas o empíricas, que son hechos sobre el mundo que necesitan ser investigados Observacionalmente y cuestiones filosóficas en cuanto a cómo el lenguaje se puede utilizar inteligiblemente (que incluyen ciertas preguntas en matemáticas (...)

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43. It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than as (...)

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44. Último sermão da Igreja do naturalismo fundamentalista pelo pastor Hofstadter. Como o seu muito mais famoso (ou infame por seus erros filosóficos implacáveis) Godel, Escher, Bach, ele tem uma plausibilidade superficial, mas se se compreende que este é um scientismo desenfreado que mistura questões científicas reais com os filosóficos (ou seja, o somente as edições reais são que jogos da língua nós devemos jogar) então quase todo seu interesse desaparece. Eu forneci um quadro para análise baseada na psicologia evolutiva e (...)

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45. Incompleteness and Computability: An Open Introduction to Gödel's Theorems.Richard Zach - 2019 - Open Logic Project.
Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

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46. Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge. [REVIEW]Panu Raatikainen - 2018 - History and Philosophy of Logic 39 (4):401-403.
Austrian-born Kurt Gödel is widely considered the greatest logician of modern times. It is above all his celebrated incompleteness theorems—rigorous mathematical results about the necessary limits...

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47. Decidable and undecidable fragments in First order logic.Ricardo José Da Silva & Franklin Galindo - 2017 - Apuntes Filosóficos 26 (50):90-113.
The present paper has three objectives: Presenting an actualization of a proof of the decidability of monadic predicates logic in the contemporary model theory context; Show examples of decidable and undecidable fragments inside First order logic, offering an original proof of the following theorem: Any formula of First of order logic is decidable if its prenex normal form is in the following form: ∀x1,…,∀xn∃y1,…,∃ymφ; Presenting a theorem that characterizes the validity of First order logic by the tautologicity of Propositional logic, (...)

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48. El Programa original de David Hilbert y el Problema de la Decibilidad.Franklin Galindo & Ricardo Da Silva - 2017 - Episteme NS: Revista Del Instituto de Filosofía de la Universidad Central de Venezuela 37 (1):1-23.
En este artículo realizamos una reconstrucción del Programa original de Hilbert antes del surgimiento de los teoremas limitativos de la tercera década del siglo pasado. Para tal reconstrucción empezaremos por mostrar lo que Torretti llama los primeros titubeos formales de Hilbert, es decir, la defensa por el método axiomático como enfoque fundamentante. Seguidamente, mostraremos como estos titubeos formales se establecen como un verdadero programa de investigación lógico-matemático y como dentro de dicho programa la inquietud por la decidibilidad de los problemas (...)

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49. Church's Undecidability Theorem (1936): Formulation and presentation of the main ideas of its demonstration.Franklin Galindo & Ricardo José Da Silva - 2017 - Apuntes Filosóficos 26 (50):8-31.
Church's Undecidability Theorem is one of the meta-theoretical results of the mid-third decade of the last century, which along with other limiting theorems such as those of Gödel and Tarski have generated endless reflections and analyzes, both within the framework of the formal sciences, that is, mathematics, logic and theoretical computation, as well as outside them, especially the philosophy of mathematics, philosophy of logic and philosophy of mind. We propose, as a general purpose of this article, to formulate Church's Undecidability (...)