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Summary Godelian arguments use Godel's incompleteness theorems to argue against the possibility of human-level computer intelligence.  Godel proved that any number system strong enough to do arithmetic would contain true propositions that were impossible to prove within the system. Let G be such a proposition, and let the relevant system correspond to a computer.  It seems to follow that no computer can prove G (and so know G is true), but humans can know that G is true (by, as it were, moving outside of the number system and seeing that G has to be true to preserve soundness).  So, it appears that humans are more powerful than computers restricted to just implementations of number systems.  This is the essence of Godelian arguments. Many replies to these arguments have been put forward.  An obvious reply is that computers can be programmed to be more than mere number systems and so can step outside number systems just like humans can.  
Key works Probably the central paper using Godelian arguments against AI is Lucas 1961. Another good paper is Benacerraf 1967.  For what is often regarded as the classic reply to Lucas, see Putnam 1960.
Introductions Penrose 1994 and Penrose 1989.
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  1. Samuel Alexander (2013). This Sentence Does Not Contain the Symbol X. The Reasoner 7 (9):108.
    A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".
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  2. Samuel A. Alexander (2014). A Machine That Knows Its Own Code. Studia Logica 102 (3):567-576.
    We construct a machine that knows its own code, at the price of not knowing its own factivity.
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  3. Paul Benacerraf (1967). God, the Devil, and Godel. The Monist 51 (January):9-32.
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  4. Ronald Marcel Biron (1992). Roger Penrose's "the Emperor's New Mind": Implications for Critical Thinking. Dissertation, Boston University
    The study examines Roger Penrose's arguments against strong AI and for non-algorithmic thinking as presented in his book, The Emperor's New Mind. The primary question addressed is whether or not to accept the computational hypothesis--that human thought is algorithmic. The study concludes that Penrose offers neither convincing logical proof nor sufficient scientific evidence to reject the hypothesis, but does present a comprehensive position that is very suggestive. It is then argued that this position combined with the criticisms voiced against pure (...)
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  5. Marcel Bodea (2001). An Analysis On A Machine Type Gödel. Studia Philosophica 1.
    The formal domain of abstract mathematical systems and the propositions provable in them can be illustrated using a ‘machine language’: The Gödel’s machine. This article was written primarily as an investigation and introduction to incompleteness theorems in a way wich is a simplifying factor. We tried to show how a certain computing machine can have very general features in subject to Gödel’s argument. We belive that the proofs we give are unusually simple. We then can turn to some incompletness arguments (...)
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  6. Damjan Bojadziev (1997). Mind Versus Godel. In Matjaz Gams & M. Wu Paprzycki (eds.), Mind Versus Computer. IOS Press 202-210.
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  7. G. Bouwman (2013). Samaria in lucas—handelingen. Bijdragen 34 (1):40-59.
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  8. G. Lee Bowie (1982). Lucas' Number is Finally Up. Journal of Philosophical Logic 11 (3):279-85.
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  9. David L. Boyer (1983). R. Lucas, Kurt Godel, and Fred Astaire. Philosophical Quarterly 33 (April):147-59.
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  10. Selmer Bringsjord, A Refutation of Penrose's Godelian Case Against Artificial Intelligence.
    Having, as it is generally agreed, failed to destroy the computational conception of mind with the G\"{o}delian attack he articulated in his {\em The Emperor's New Mind}, Penrose has returned, armed with a more elaborate and more fastidious G\"{o}delian case, expressed in and 3 of his {\em Shadows of the Mind}. The core argument in these chapters is enthymematic, and when formalized, a remarkable number of technical glitches come to light. Over and above these defects, the argument, at best, is (...)
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  11. Selmer Bringsjord & H. Xiao (2000). A Refutation of Penrose's New Godelian Case Against the Computational Conception of Mind. Journal of Experimental and Theoretical Artificial Intelligence 12.
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  12. Donald E. Broadbent (ed.) (1993). The Simulation of Human Intelligence. Blackwell.
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  13. Alan Bundy & Roger Penrose (1990). On the Nature of Mathematical Judgement Reply to Penrose. Edinburgh University.
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  14. A. Buti (2002). John Lucas, Godel and Mechanical Intelligence. Rivista di Filosofia Neo-Scolastica 94 (4):637-674.
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  15. A. Butti (2004). Penrose and the Renewal of Goedelian Argument. Rivista di Filosofia Neo-Scolastica 96 (4):669-697.
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  16. Whiteley C. H. (1962). Minds, Machines and Godel : A Reply to Mr Lucas. Philosophy 37 (139):61-.
    In Philosophy for April 1961 Mr J. R. Lucas argues that Gödel's theorem proves that Mechanism is false. I wish to dispute this view, not because I maintain that Mechanism is true, but because I do not believe that this issue is to be settled by what looks rather like a kind of logical conjuring-trick. In my discussion I take for granted Lucas's account of Gödel's procedure, which I am not competent to criticise.
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  17. Peter Carruthers & Peter K. Smith (eds.) (1996). Theories of Theories of Mind. Cambridge University Press.
    Theories of Theories of Mind brings together contributions by a distinguished international team of philosophers, psychologists, and primatologists, who between them address such questions as: what is it to understand the thoughts, feelings, and intentions of other people? How does such an understanding develop in the normal child? Why, unusually, does it fail to develop? And is any such mentalistic understanding shared by members of other species? The volume's four parts together offer a state of the art survey of the (...)
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  18. David Chalmers (1995). Minds, Machines, And Mathematics A Review of Shadows of the Mind by Roger Penrose. [REVIEW] Psyche 2.
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  19. David J. Chalmers (1996). Minds, Machines, and Mathematics. Psyche 2:11-20.
    In his stimulating book SHADOWS OF THE MIND, Roger Penrose presents arguments, based on Gödel's theorem, for the conclusion that human thought is uncomputable. There are actually two separate arguments in Penrose's book. The second has been widely ignored, but seems to me to be much more interesting and novel than the first. I will address both forms of the argument in some detail. Toward the end, I will also comment on Penrose's proposals for a "new science of consciousness".
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  20. C. T. K. Chari (1963). Further Comments on Minds, Machines and Godel. Philosophy 38 (April):175-8.
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  21. C. Chihara (1972). On Alleged Refutations of Mechanism Using Godel's Incompleteness Results. Journal of Philosophy 69 (September):507-26.
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  22. David Coder (2003). Gödel's Theorem and Mechanism. Etica E Politica 5 (1):1.
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  23. David Coder (1969). Godel's Theorem and Mechanism. Philosophy 44 (September):234-7.
    In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is not true (...)
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  24. Jon Cogburn & Jason Megill (2010). Are Turing Machines Platonists? Inferentialism and the Computational Theory of Mind. Minds and Machines 20 (3):423-439.
    We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.
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  25. A. G. Cohn & J. R. Thomas (eds.) (1986). Artificial Intelligence and Its Applications. John Wiley and Sons.
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  26. Jack Copeland (1998). Turing's o-Machines, Searle, Penrose, and the Brain. Analysis 58 (2):128-138.
    In his PhD thesis (1938) Turing introduced what he described as 'a new kind of machine'. He called these 'O-machines'. The present paper employs Turing's concept against a number of currently fashionable positions in the philosophy of mind.
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  27. Julien Offray de La Mettrie & Ann Thomson (1996). Machine Man and Other Writings. Monograph Collection (Matt - Pseudo).
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  28. Jacek Dębiec (1996). Od Platona do Penrose'a. Zagadnienia Filozoficzne W Nauce 19.
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  29. Daniel C. Dennett (1989). Murmurs in the Cathedral: Review of R. Penrose, The Emperor's New Mind. [REVIEW] Times Literary Supplement (September) 29.
    The idea that a computer could be conscious--or equivalently, that human consciousness is the effect of some complex computation mechanically performed by our brains--strikes some scientists and philosophers as a beautiful idea. They find it initially surprising and unsettling, as all beautiful ideas are, but the inevitable culmination of the scientific advances that have gradually demystified and unified the material world. The ideologues of Artificial Intelligence (AI) have been its most articulate supporters. To others, this idea is deeply repellent: philistine, (...)
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  30. Daniel C. Dennett (1978). Brainstorms. MIT Press.
    This collection of 17 essays by the author offers a comprehensive theory of mind, encompassing traditional issues of consciousness and free will.
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  31. Daniel C. Dennett (1978). The Abilities of Men and Machines. In Brainstorms. MIT Press
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  32. Michael Detlefsen (1998). Mind in the Shadows. Studies in History and Philosophy of Science Part B 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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  33. José Vara Donado (1983). Una sugerencia: lección originaria en Lucas 11, 11 - 12. Salmanticensis 30 (2):225-229.
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  34. Virgil DrĂghici (2001). The Incompleteness And The Argument Lucas-Penrose. Studia Philosophica 1.
    This paper intends to make explicit, in a logical-mathematical frame, the conceptual mechanism of the Lucas/Penrose argument. It also offers some remarks about consequences stated on the basis of this argument.
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  35. Dorothy Edgington (1972). LUCAS, J. R.-"The Concept of Probability". [REVIEW] Philosophy 47:375.
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  36. Taner Edis (1998). How Godel's Theorem Supports the Possibility of Machine Intelligence. Minds and Machines 8 (2):251-262.
    Gödel's Theorem is often used in arguments against machine intelligence, suggesting humans are not bound by the rules of any formal system. However, Gödelian arguments can be used to support AI, provided we extend our notion of computation to include devices incorporating random number generators. A complete description scheme can be given for integer functions, by which nonalgorithmic functions are shown to be partly random. Not being restricted to algorithms can be accounted for by the availability of an arbitrary random (...)
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  37. Eva-Maria Engelen (2013). Hat Kurt Gödel Thomas von Aquins Kommentar zu Aristoteles' De anima rezipiert? Philosophia Scientiæ 17 (17-1):167-188.
    La recherche d’une réponse à la question qui constitue le titre a conduit à des éclaircissements concernant la réception critique d’œuvres philosophiques majeures par Kurt Gödel. Cela illustre la manière dont il utilise des argumentations philosophiques d’auteurs classiques et les change en des aspects nouveaux pour sa propre argumentation philosophique. Dans le cas qui nous concerne, Gödel emploie un argument classique d’Aristote pour l’immatérialité de l’âme afin d’ajouter certains éléments à son propre raisonnement concernant l’inexhaustibilité des mathématiques, le problème corps-esprit, (...)
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  38. George L. Farre (1999). Penrose, Roger. The Large, the Small and the Human Mind. Review of Metaphysics 53 (1):191-193.
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  39. S. Feferman (1996). Penrose's Godelian Argument. Psyche 2:21-32.
    In his book Shadows of the Mind: A search for the missing science of con- sciousness [SM below], Roger Penrose has turned in another bravura perfor- mance, the kind we have come to expect ever since The Emperor’s New Mind [ENM ] appeared. In the service of advancing his deep convictions and daring conjectures about the nature of human thought and consciousness, Penrose has once more drawn a wide swath through such topics as logic, computa- tion, artificial intelligence, quantum physics (...)
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  40. Solomon Feferman (1995). Penrose's Gödelian Argument A Review of Shadows of the Mind by Roger Penrose. [REVIEW] Psyche 2.
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  41. Shingo Fujita (2003). Use and Misuse of Godel's Theorem. Annals of the Japan Association for Philosophy of Science 12 (1):1-14.
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  42. H. Gaifman (2000). What Godel's Incompleteness Result Does and Does Not Show. Journal of Philosophy 97 (8):462-471.
    In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument (...)
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  43. Matjaz Gams (ed.) (1997). Mind Versus Computer: Were Dreyfus and Winograd Right? Amsterdam: IOS Press.
  44. Jay L. Garfield (1990). Foundations of Cognitive Science: The Essential Readings. New York: Paragon House.
  45. A. George & Daniel J. Velleman (2000). Leveling the Playing Field Between Mind and Machine: A Reply to McCall. Journal of Philosophy 97 (8):456-452.
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  46. F. H. George (1962). Minds, Machines and Godel: Another Reply to Mr. Lucas. Philosophy 37 (January):62-63.
    I Would like to draw attention to the basic defect in the argument used by Mr J. R. Lucas . Mr Lucas there states that Gödel's theorem shows that any consistent formal system strong enough to produce arithmetic fails to prove, within its own structure, theorems that we, as humans , can nevertheless see to be true. From this he argues that ‘minds’ can do more than machines, since machines are essentially formal systems of this same type, and subject to (...)
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  47. F. H. George (1962). Minds, Machines and Gödel: Another Reply to Mr. Lucas. Philosophy 37 (139):62 - 63.
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  48. Brie Gertler (2004). Simulation Theory on Conceptual Grounds. ProtoSociology 20:261-284.
    I will present a conceptual argument for a simulationist answer to (2). Given that our conception of mental states is employed in attributing mental states to others, a simulationist answer to (2) supports a simulationist answer to (1). I will not address question (3). Answers to (1) and (2) do not yield an answer to (3), since (1) and (2) concern only our actual practices and concepts. For instance, an error theory about (1) and (2) would say that our practices (...)
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  49. Barbara Giolito (2003). Guest Editor’s Preface: Introduction to Lucas's Argument Against Mechanism by Means of Gödel's Incompleteness Theorem. Etica E Politica 5 (1):1-4.
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  50. Barbara Giolito (2003). An Interview with John Randolph Lucas. Etica E Politica 5 (1):1.
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