About this topic
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. added 2020-03-18
    Reseña de ‘Soy un Bucle Extraño’ ( I am a Strange Loop) de Douglas Hofstadter (2007) (reseña revisado 2019).Michael Richard Starks - 2020 - In Comprender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política, Economía, Historia y Literatura - Artículos y reseñas 2006-2019. Las Vegas, NV USA: Reality Press. pp. 265-282.
    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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  2. added 2020-03-13
    Revisão de ‘Eu sou um Loop Estranho’ (I am a Strange Loop) por Douglas Hofstadter (2007) (revisão revisada 2019).Michael Richard Starks - 2020 - In Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política, Economia, História e Literatura - Artigos e Avaliações 2006-2019. Las Vegas, NV USA: Reality Press. pp. 251-268.
    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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  3. added 2020-01-25
    Incompleteness and Computability. An Open Introduction to Gödel's Theorems.Richard Zach - 2019
    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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  4. added 2019-12-21
    Wolpert, Chaitin e Wittgenstein em impossibilidade, incompletude, o paradoxo do mentiroso, o teísmo, os limites da computação, um princípio de incerteza mecânica não quântica e o universo como computador — o teorema final na teoria da máquina de Turing (revisado 2019).Michael Richard Starks - 2019 - In Delírios Utópicos Suicidas no Século XXI Filosofia, Natureza Humana e o Colapso da Civilization- Artigos e Comentários 2006-2019 5ª edição. Las Vegas, NV USA: Reality Press. pp. 183-187.
    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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  5. added 2019-12-21
    O que significa paraconsistente, indecível, aleatório, computável e incompleto?- Uma revisão da ‘Godel’s Way: exploits into an undecidable world’ (Maneira de Godel: façanhas em um mundo indecidível) por Gregory Chaitin, Francisco A Doria, Newton C.A. da costa 160P (2012) (revisão revisada 2019).Michael Richard Starks - 2019 - In Delírios Utópicos Suicidas no Século XXI Filosofia, Natureza Humana e o Colapso da Civilization- Artigos e Comentários 2006-2019 5ª edição. Las Vegas, NV USA: Reality Press. pp. 168-182.
    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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  6. added 2019-12-19
    Revisão de ‘Eu sou um Loop Estranho’ (I am a Strange Loop) por Douglas Hofstadter (2007) (revisão revisada 2019).Michael Richard Starks - 2019 - In Delírios Utópicos Suicidas no Século XXI Filosofia, Natureza Humana e o Colapso da Civilization- Artigos e Comentários 2006-2019 5ª edição. Las Vegas, NV USA: Reality Press. pp. 112-128.
    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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  7. added 2019-12-05
    ¿Qué significa paraconsistente, indescifrable, aleatorio, computable e incompleto? Una revisión de’ la Manera de Godel: explota en un mundo indecible’ (Godel’s Way: Exploits into an Undecidable World) por Gregory Chaitin, Francisco A Doria, Newton C.A. da Costa 160p (2012) (revisión revisada 2019).Michael Richard Starks - 2019 - In Delirios Utópicos Suicidas en el Siglo 21 La filosofía, la naturaleza humana y el colapso de la civilización Artículos y reseñas 2006-2019 4a Edición. Las Vegas, NV USA: Reality Press. pp. 263-277.
    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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  8. added 2019-12-05
    Wolpert, Chaitin y Wittgenstein sobre la imposibilidad, la incompletitud, la paradoja mentirosa, el teísmo, los límites de la computación, un principio de incertidumbre mecánica no cuántica y el universo como computadora, el teorema definitivo en la teoría de la máquina de Turing (revisado en 2019).Michael Richard Starks - 2019 - In Delirios Utópicos Suicidas en el Siglo 21 La filosofía, la naturaleza humana y el colapso de la civilización Artículos y reseñas 2006-2019 4a Edición. Las Vegas, NV USA: Reality Press. pp. 278-282.
    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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  9. added 2019-12-04
    Reseña de ‘I am a Strange Loop’ (Soy un Lazo Extraño) de Douglas Hofstadter (2007) (revisión revisada 2019).Michael Richard Starks - 2019 - In Delirios Utópicos Suicidas en el Siglo 21 La filosofía, la naturaleza humana y el colapso de la civilización Artículos y reseñas 2006-2019 4a Edición. Las Vegas, NV USA: Reality Press. pp. 205-221.
    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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  10. added 2019-11-29
    Reseña de ‘Soy un Bucle Extraño’ ( I am a Strange Loop) de Douglas Hofstadter.Michael Richard Starks - 2019 - In Observaciones Sobre Imposibilidad, Incompleta, Paracoherencia,Indecisión,Aleatoriedad, Computabilidad, Paradoja E Incertidumbre En Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, Dacosta, Godel, Searle, Rodych, Berto,Floyd, Moyal-Sharrock Y Yanofsky. Las Vegas, NV USA: Reality Press. pp. 21-43.
    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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  11. added 2019-09-04
    Remarques sur l'impossibilité l'incomplétude, la paracohérence l'indécision, le hasard, la calculabilité, le paradoxe et l'incertitude - dans Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria da Costa, Godel, Searle, Rodych, Berto Floyd, Moyal-Sharrock et Yanofsky.Michael Richard Starks - 2019 - Las Vegas, NV USA: Reality Press.
    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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  12. added 2019-08-28
    Pernyataan tentang kemustahilan, ketidaklengkapan, Paraconsistency,Undecidability, Randomness, Komputabilitas, paradoks, dan ketidakpastian dalam Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock dan Yanofsky.Michael Richard Starks - 2019 - Las Vegas, NV USA: Reality Press.
    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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  13. added 2019-08-27
    اظهارات در مورد عدم امکان ، بی کامل بودن ، پاراستشتها، Undecidability ، اتفاقی ، Computability ، پارادوکس ، و عدم قطعیت در Chaitin ، ویتگنشتاین ، Hofstadter ، Wolpert ، doria ، دا کوستا ، گودل ، سرل ، رودیچ ، برتو ، فلوید ، مویال-شرراک و یانفسکی.Michael Richard Starks - 2019 - Las Vegas, NV USA: Reality Press.
    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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  14. added 2019-08-25
    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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  15. added 2019-08-16
    Remarks on Wittgenstein, Gödel, Chaitin, Incompleteness, Impossiblity and the Psychological Basis of Science and Mathematics.Michael Richard Starks - 2019 - In Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal. Las Vegas, NV USA: Reality Press. pp. 24-38.
    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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  16. added 2019-06-20
    Proof That Wittgenstein is Correct About Gödel.P. Olcott - manuscript
    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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  17. added 2019-06-03
    Saving Proof From Paradox: Gödel’s Paradox and the Inconsistency of Informal Mathematics.Fenner Stanley Tanswell - 2016 - In Peter Verdee & Holger Andreas (eds.), Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Springer. pp. 159-173.
    In this paper I shall consider two related avenues of argument that have been used to make the case for the inconsistency of mathematics: firstly, Gödel’s paradox which leads to a contradiction within mathematics and, secondly, the incompatibility of completeness and consistency established by Gödel’s incompleteness theorems. By bringing in considerations from the philosophy of mathematical practice on informal proofs, I suggest that we should add to the two axes of completeness and consistency a third axis of formality and informality. (...)
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  18. added 2019-04-26
    Gödel's Theorems and Platonism (Comment on Penrose).Michael Detlefsen - 2011 - In John Polkinghorne (ed.), Meaning in Mathematics. Oxford University Press. pp. 46-47..
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  19. added 2019-04-20
    Philosophy of Logic – Reexamining the Formalized Notion of Truth.Pete Olcott - manuscript
    Tarski "proved" that there cannot possibly be any correct formalization of the notion of truth entirely on the basis of an insufficiently expressive formal system that was incapable of recognizing and rejecting semantically incorrect expressions of language. -/- The only thing required to eliminate incompleteness, undecidability and inconsistency from formal systems is transforming the formal proofs of symbolic logic to use the sound deductive inference model.
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  20. added 2019-03-03
    Provability, Mechanism, and the Diagonal Problem.Graham Leach-Krouse - 2016 - In Leon Horsten & Philip Welch (eds.), Gödel's Disjunction: the Scope and Limits of Mathematical Knowledge. Oxford, UK: pp. 211-240.
  21. added 2019-02-24
    Wolpert, Chaitin and Wittgenstein on Impossibility, Incompleteness, the Liar Paradox, Theism, the Limits of Computation, a Non-Quantum Mechanical Uncertainty Principle and the Universe as Computer—the Ultimate Theorem in Turing Machine Theory (Revised 2019).Michael Starks - 2019 - In Suicidal Utopian Delusions in the 21st Century -- Philosophy, Human Nature and the Collapse of Civilization -- Articles and Reviews 2006-2019 4th Edition Michael Starks. Las Vegas, NV USA: Reality Press. pp. 294-299.
    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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  22. added 2019-02-24
    Review of 'The Outer Limits of Reason' by Noson Yanofsky 403p (2013) (Review Revised 2019).Michael Starks - 2019 - In Suicidal Utopian Delusions in the 21st Century -- Philosophy, Human Nature and the Collapse of Civilization -- Articles and Reviews 2006-2019 4th Edition Michael Starks. Las Vegas, NV USA: Reality Press. pp. 299-316.
    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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  23. added 2018-07-08
    Gödel’s Disjunction: The Scope and Limits of Mathematical Knowledge. [REVIEW]Panu Raatikainen - 2018 - History and Philosophy of Logic 39 (4):401-403.
  24. added 2018-05-11
    What is Mathematics: Gödel's Theorem and Around (Edition 2015).Karlis Podnieks - manuscript
    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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  25. added 2018-03-12
    Exploring Randomness.Panu Raatikainen - 2001 - Notices of the AMS 48 (9):992-6.
    Review of "Exploring Randomness" (200) and "The Unknowable" (1999) by Gregory Chaitin.
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  26. added 2018-02-22
    The Arithmetization of Syntax and the New Paradoxes of Self-Reference.T. Parent - manuscript
    In this paper, I recreate a paradox from my earlier work (“Paradox with just self-reference”) albeit entirely within the language of arithmetic. The paradox might suggest that Robinson arithmetic and its extensions are unsound; however, I claim instead that the metalanguage may be to blame, owing to the lack of restrictions on the arithmetization of syntax. If so, then the moral would be to restrict arithmetization in the metalanguage somehow, rather than distrust the arithmetical theory in the object language.
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  27. added 2018-02-17
    Proof, Logic and Formalization.Michael Detlefsen (ed.) - 1992 - Routledge.
    The mathematical proof is the most important form of justification in mathematics. It is not, however, the only kind of justification for mathematical propositions. The existence of other forms, some of very significant strength, places a question mark over the prominence given to proof within mathematics. This collection of essays, by leading figures working within the philosophy of mathematics, is a response to the challenge of understanding the nature and role of the proof.
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  28. added 2017-11-09
    Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or (...)
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  29. added 2016-12-08
    Kurt Gödel: Philosopher-Scientist.Gabriella Crocco & Eva-Maria Engelen (eds.) - 2016 - Marseille: Presses universitaires de Provence.
    This volume represents the beginning of a new stage of research in interpreting Kurt Gödel’s philosophy in relation to his scientific work. It is more than a collection of essays on Gödel. It is in fact the product of a long enduring international collaboration on Kurt Gödel’s Philosophical Notebooks (Max Phil). New and significant material has been made accessible to a group of experts, on which they rely for their articles. In addition to this, Gödel’s Nachlass is presented anew by (...)
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  30. added 2016-12-08
    An Introduction to Gödel's Theorems.Peter Smith - 2007 - Cambridge University Press.
    In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the (...)
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  31. added 2016-10-05
    Proving Unprovability.Bruno Whittle - 2017 - Review of Symbolic Logic 10 (1):92–115.
    This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e. S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g. 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness (...)
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  32. added 2016-08-14
    Gödel and the Synthetic a Priori: A Rejoinder.Irving M. Copi - 1950 - Journal of Philosophy 47 (22):633-636.
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  33. added 2016-08-14
    Modern Logic and the Synthetic a Priori.Irving M. Copi - 1949 - Journal of Philosophy 46 (8):243-245.
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  34. added 2016-08-08
    A Note on Wittgenstein’s “Notorious Paragraph” About the Gödel Theorem.Juliet Floyd & Hilary Putnam - 2000 - Journal of Philosophy 97 (11):624-632.
  35. added 2016-08-07
    Bays, Steiner, and Wittgenstein’s “Notorious” Paragraph About the Gödel Theorem.Juliet Floyd & Hilary Putnam - 2006 - Journal of Philosophy 103 (2):101-110.
  36. added 2016-08-07
    Prosa versus Demonstração: Wittgenstein sobre Gödel, Tarski e a Verdade.Juliet Floyd - 2002 - Revista Portuguesa de Filosofia 58 (3):605 - 632.
    O presente artigo procede, em primeiro lugar, a um exame das evidências disponíveis referentes à atitude de Wittgenstein em relação ao, bem como conhecimento do, primeiro teorema da incompletude de Gödel, incluindo as suas discussões com Turing, Watson e outros em 1937-1939, e o testemunho posterior de Goodstein e Kreisel Em segundo lugar, o artigo discute a importância filosófica e histórica da atitude de Wittgenstein em relação ao teorema de Gödel e outros teoremas da lógica matemática, contrastando esta atitude com (...)
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  37. added 2016-08-07
    Wittgenstein sobre Gödel, Tarski e a Verdade.Juliet Floyd - 2002 - Revista Portuguesa de Filosofia 58 (3):605-632.
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  38. added 2016-08-07
    Prose Versus Proof: Wittgenstein on Gödel, Tarski and Truth.Juliet Floyd - 2001 - Philosophia Mathematica 9 (3):280-307.
    A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g., Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of Tarski's semantical (...)
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  39. added 2016-08-07
    Prose Versus Proof: Wittgenstein on Gödel, Tarski and Truth†: Articles.Juliet Floyd - 2001 - Philosophia Mathematica 9 (3):280-307.
    1) A survey of current evidence available concerning Wittgenstein's attitude toward, and knowledge of, Gödel's first incompleteness theorem, including his discussions with Turing, Watson and others in 1937–1939, and later testimony of Goodstein and Kreisel; 2) Discussion of the philosophical and historical importance of Wittgenstein's attitude toward Gödel's and other theorems in mathematical logic, contrasting this attitude with that of, e.g. , Penrose; 3) Replies to an instructive criticism of my 1995 paper by Mark Steiner which assesses the importance of (...)
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  40. added 2016-03-15
    Heterologicality and Incompleteness.Cezary Cieśliński - 2002 - Mathematical Logic Quarterly 48 (1):105-110.
    We present a semantic proof of Gödel's second incompleteness theorem, employing Grelling's antinomy of heterological expressions. For a theory T containing ZF, we define the sentence HETT which says intuitively that the predicate “heterological” is itself heterological. We show that this sentence doesn't follow from T and is equivalent to the consistency of T. Finally we show how to construct a similar incompleteness proof for Peano Arithmetic.
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  41. added 2015-06-14
    What Does Gödel's Second Theorem Say?Michael Detlefsen - 2001 - Philosophia Mathematica 9 (1):37-71.
    We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We (...)
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  42. added 2015-06-14
    Mind in the Shadows.Michael Detlefsen - 1998 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 29 (1):123-136.
    This is a review of Penrose's trilogy, The Emperor's New Mind, Shadows of the Mind and The Large the Small and the Human Mind.
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  43. added 2015-06-14
    The Mechanization of Reason.Michael Detlefsen - 1995 - Philosophia Mathematica 3 (1).
    Introduction to a special issue of Philosophia Mathematica on the mechanization of reasoning. Authors include: M. Detlefsen, D. Mundici, S. Shanker, S. Shapiro, W. Sieg and C. Wright.
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  44. added 2015-06-14
    Hilbert's Formalism.Michael Detlefsen - 1993 - Revue Internationale de Philosophie 47 (186):285-304.
    Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.
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  45. added 2015-06-14
    On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  46. added 2015-06-14
    On Interpreting Gödel's Second Theorem.Michael Detlefsen - 1979 - Journal of Philosophical Logic 8 (1):297 - 313.
    In this paper I have considered various attempts to attribute significance to Gödel's second incompleteness theorem (G2 for short). Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are false. Two others (an argument suggested by Beth, Cohen and ??? and Resnik's Interpretation), I argue, are groundless.
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  47. added 2015-04-16
    A Short Guide To Second Incompleteness Theorem.J. Bagaria - 2003 - Teorema: International Journal of Philosophy 22 (3).
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  48. added 2015-03-02
    The Road to Gödel.Saul Kripke - 2014 - In Jonathan Berg (ed.), Naming, Necessity and More: Explorations in the Philosophical Work of Saul Kripke. Palgrave-Macmillan.
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  49. added 2015-01-07
    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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  50. added 2014-10-15
    Application of Quantum Darwinism to Cosmic Inflation: An Example of the Limits Imposed in Aristotelian Logic by Information-Based Approach to Gödel’s Incompleteness. [REVIEW]Nicolás F. Lori & Alex H. Blin - 2010 - Foundations of Science 15 (2):199-211.
    Gödel’s incompleteness applies to any system with recursively enumerable axioms and rules of inference. Chaitin’s approach to Gödel’s incompleteness relates the incompleteness to the amount of information contained in the axioms. Zurek’s quantum Darwinism attempts the physical description of the universe using information as one of its major components. The capacity of quantum Darwinism to describe quantum measurement in great detail without requiring ad-hoc non-unitary evolution makes it a good candidate for describing the transition from quantum to classical. A baby-universe (...)
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