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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  Smullyan 1992 Raatikainen 2013 Smorynski 1977
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  1. David Auerbach (1994). Saying It With Numerals. Notre Dame Journal of Formal Logic 35 (1):130-146.
    This article discusses the nature of numerals and the plausibility of their special semantic and epistemological status as proper names of numbers. Evidence is presented that minimizes the difference between numerals and other devices of direct reference. The availability of intensional contexts within formalised metamathematics is exploited to shed light on the relation between formal numerals and numerals.
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  2. David Auerbach (1992). How to Say Things with Formalisms. In Michael Detlefsen (ed.), Proof, logic, and formalization. Routledge 77--93.
  3. David D. Auerbach (1985). Intensionality and the Gödel Theorems. Philosophical Studies 48 (3):337--51.
  4. J. Bagaria (2003). A Short Guide To Second Incompleteness Theorem. Teorema: International Journal of Philosophy 22 (3).
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  5. Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
    An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...)
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  6. Cezary Cieslinski (2002). Heterologicality and Incompleteness. 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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  7. Irving M. Copi (1950). Gödel and the Synthetic a Priori: A Rejoinder. Journal of Philosophy 47 (22):633-636.
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  8. Irving M. Copi (1949). Modern Logic and the Synthetic a Priori. Journal of Philosophy 46 (8):243-245.
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  9. Gabriella Crocco & Eva-Maria Engelen (eds.) (forthcoming). Kurt Gödel: Philosopher-Scientist. 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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  10. Michael Detlefsen (2001). What Does Gödel's Second Theorem Say. 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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  11. 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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  12. Michael Detlefsen (1995). The Mechanization of Reason. 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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  13. Michael Detlefsen (1993). Hilbert's Formalism. 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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  14. Michael Detlefsen (ed.) (1992). Proof, Logic, and Formalization. Routledge.
    Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the (...)
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  15. Michael Detlefsen (1990). On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem. 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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  16. Michael Detlefsen (1979). On Interpreting Gödel's Second Theorem. 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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  17. Eva-Maria Engelen (forthcoming). 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. 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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  18. Juliet Floyd (2006). Bays, Steiner, and Wittgenstein's “Notorious” Paragraph About the Gödel Theorem. Journal of Philosophy 103 (2):101-109.
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  19. Juliet Floyd (2002). Prosa versus Demonstração: Wittgenstein sobre Gödel, Tarski e a Verdade. 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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  20. Juliet Floyd (2002). Wittgenstein sobre Gödel, Tarski e a Verdade. Revista Portuguesa de Filosofia 58 (3):605-632.
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  21. Juliet Floyd (2001). Prose Versus Proof: Wittgenstein on Gödel, Tarski and Truth. 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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  22. Juliet Floyd (2001). Prose Versus Proof: Wittgenstein on Gödel, Tarski and Truth†: Articles. 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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  23. Juliet Floyd & Hilary Putnam (2006). Bays, Steiner, and Wittgenstein’s “Notorious” Paragraph About the Gödel Theorem. Journal of Philosophy 103 (2):101-110.
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  24. Juliet Floyd & Hilary Putnam (2000). A Note on Wittgenstein's "Notorious Paragraph" About the Gödel Theorem. Journal of Philosophy 97 (11):624-632.
  25. Kurt Godel, The modern development of the foundations of mathematics in the light of philosophy.
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  26. Albert Johnstone (2003). Self-Reference and Gödel's Theorem: A Husserlian Analysis. [REVIEW] Husserl Studies 19 (2):131-151.
    A Husserlian phenomenological approach to logic treats concepts in terms of their experiential meaning rather than in terms of reference, sets of individuals, and sentences. The present article applies such an approach in turn to the reasoning operative in various paradoxes: the simple Liar, the complex Liar paradoxes, the Grelling-type paradoxes, and Gödel’s Theorem. It finds that in each case a meaningless statement, one generated by circular definition, is treated as if were meaningful, and consequently as either true or false, (...)
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  27. Jeffrey Ketland (2005). Deflationism and the Gödel Phenomena: Reply to Tennant. Mind 114 (453):75-88.
    Any (1-)consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a nontrivial fashion. The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence. We are concerned to show that (...)
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  28. Hitoshi Kitada (2011). Gēderu Fukanzensei Hakken E No Michi. Gendai Sūgakusha.
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  29. Saul Kripke (2014). The Road to Gödel. In Jonathan Berg (ed.), Naming, Necessity and More: Explorations in the Philosophical Work of Saul Kripke. Palgrave Macmillan
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  30. Nicolás F. Lori & Alex H. Blin (2010). 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] 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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  31. Carlos Montemayor & Rasmus Grønfeldt Winther (2015). Review of Space, Time, and Number in the Brain. [REVIEW] Mathematical Intelligencer 37 (2):93-98.
    Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...)
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  32. E. Nelson (2002). Mathematics and the Mind. In Kunio Yasue, Marj Jibu & Tarcisio Della Senta (eds.), No Matter, Never Mind. John Benjamins 731-737.
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  33. Karl-Georg Niebergall & Matthias Schirn (2002). Hilbert's Programme and Gödel's Theorems. Dialectica 56 (4):347–370.
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  34. T. Parent, Paradox with Just Self-Reference.
    If a semantically open language allows self-reference, one can show there is a predicate which is both satisfied and unsatisfied by a self-referring term. The argument requires diagonalization on substitution instances of a definition-scheme for the predicate "x is Lagadonian." (The term 'Lagadonian' is adapted from David Lewis). Briefly, a self-referring term is counted as “Lagadonian” if the initial variable in the schema is replaced with the term itself. But the same term is not counted as Lagadonian if this variable (...)
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  35. Jaroslav Peregrin, Gödel, Truth & Proof.
    In this paper I would like to indicate that this interpretation of Gödel goes far beyond what he really proved. I would like to show that to get from his result to a conclusion of the above kind requires a train of thought which is fuelled by much more than Gödel's result itself, and that a great deal of the excessive fuel should be utilized with an extra care.
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  36. Duccio Pianigiani (2008). Una Guida Ai Risultati di Incompletezza di Kurt Gödel. Ets.
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  37. Panu Raatikainen (2013). Gödel's Incompleteness Theorems. The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (Ed.).
    Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...)
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  38. Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
    Gödel began his 1951 Gibbs Lecture by stating: “Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics.” (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel’s first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, (...)
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  39. Peter Roeper (2003). Giving an Account of Provability Within a Theory. Philosophia Mathematica 11 (3):332-340.
    This paper offers a justification of the ‘Hilbert-Bernays Derivability Conditions’ by considering what is required of a theory which gives an account of provability in itself.
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  40. Paul Sagal (1989). Reflexive Consistency Proofs and Gödel's Second Theorem. Philosophia Mathematica (1):58-60.
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  41. Nathan U. Salmon (2005). Metaphysics, Mathematics, and Meaning. Oxford University Press.
    Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.
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  42. G. Sereny (2011). How Do We Know That the Godel Sentence of a Consistent Theory Is True? Philosophia Mathematica 19 (1):47-73.
    Some earlier remarks Michael Dummett made on Gödel’s theorem have recently inspired attempts to formulate an alternative to the standard demonstration of the truth of the Gödel sentence. The idea underlying the non-standard approach is to treat the Gödel sentence as an ordinary arithmetical one. But the Gödel sentence is of a very specific nature. Consequently, the non-standard arguments are conceptually mistaken. In this paper, both the faulty arguments themselves and the general reasons underlying their failure are analysed. The analysis (...)
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  43. S. Shapiro (2006). Review of T. Franzen, Godel's Theorem: An Incomplete Guide to its Use and Abuse. [REVIEW] Philosophia Mathematica 14 (2):262-264.
    This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...)
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  44. Peter Smith, Incompleteness – the Very Idea.
    Why these notes? After all, I’ve written An Introduction to Gödel’s Theorems. Surely that’s more than enough to be going on with? Ah, but there’s the snag. It is more than enough. In the writing, as is the way with these things, the book grew far beyond the scope of the lecture notes from which it started. And while I hope the result is still pretty accessible to someone prepared to put in the time and effort, there’s a lot more (...)
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  45. Peter Smith, Godel Without (Too Many) Tears.
    odel’s Theorems (CUP, heavily corrected fourth printing 2009: henceforth IGT ). Surely that’s more than enough to be going on with? Ah, but there’s the snag. It is more than enough. In the writing, as is the way with these things, the book grew far beyond the scope of the lecture notes from which it started. And while I hope the result is still pretty accessible to someone prepared to put in the time and effort, there is – to be (...)
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  46. Peter Smith (2007). An Introduction to Gödel's Theorems. 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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  47. Craig Smorynski (2010). Review of P. Smith, An Introduction to Gödel's Theorems. [REVIEW] Philosophia Mathematica 18 (1):122-127.
    (No abstract is available for this citation).
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  48. Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
    Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The (...)
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  49. Michael Starks, Wolpert, Chaitin and Wittgenstein on Impossibility, Incompleteness, the Limits of Computation, Theism and the Universe as Computer-the Ultimate Turing Theorem.
    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.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, and even (...)
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  50. Alfred Tarski (1968). Undecidable Theories. Amsterdam, North-Holland Pub. Co..
    This book is well known for its proof that many mathematical systems - including lattice theory and closure algebras - are undecidable. It consists of three treatises from one of the greatest logicians of all time: "A General Method in Proofs of Undecidability," "Undecidability and Essential Undecidability in Mathematics," and "Undecidability of the Elementary Theory of Groups.".
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