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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. 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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  5. 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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  6. Kurt Godel, The modern development of the foundations of mathematics in the light of philosophy.
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  7. Albert Johnstone (2003). Self-Reference and Gödel's Theorem: A Husserlian Analysis. [REVIEW] Husserl Studies 19 (2):131-151.
  8. 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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  9. Hitoshi Kitada (2011). Gēderu Fukanzensei Hakken E No Michi. Gendai Sūgakusha.
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  10. 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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  11. Carlos Montemayor & Rasmus Grønfeldt Winther (forthcoming). Review of Space, Time, and Number in the Brain. [REVIEW] Mathematical Intelligencer.
    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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  12. 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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  13. Karl-Georg Niebergall & Matthias Schirn (2002). Hilbert's Programme and Gödel's Theorems. Dialectica 56 (4):347–370.
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  14. 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 something akin to diagonalization on substitution instances of a definition-scheme (*): ‘x is Lagadonian iff, in the g(t)th substitution instance of (*), x = t’. (Given a substitution instance of (*), let t be the term replacing 'x' and let g(t) be the Godel code for t.) Assuming an appropriate enumeration of the instances, (...)
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  15. T. Parent, Self-Reference is Sufficient for Paradox.
    This is a much less technical argument for the same conclusion from my “Paradox with just Self-Reference,” viz., that if self-reference is unconstrained, paradox will result. I first show that in classical logic, expressions must be seen as linguistic types rather than tokens. (Otherwise, ‘this very term = this very term’ is a false instance of the Law of Identity.) But then, one can derive a contradiction from the premise ‘This sentence is not derived’, or from the premise ‘ ‘this (...)
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  16. 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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  17. Duccio Pianigiani (2008). Una Guida Ai Risultati di Incompletezza di Kurt Gödel. Ets.
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  18. Panu Raatikainen, 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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  19. 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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  20. 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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  21. Paul Sagal (1989). Reflexive Consistency Proofs and Gödel's Second Theorem. Philosophia Mathematica (1):58-60.
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  22. 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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  23. 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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  24. 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.
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  25. Peter Smith, Incompleteness – the Very Idea.
    Why these notes? After all, I’ve written An Introduction to Gödel’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 (...)
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  26. 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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  27. Peter Smith (2013). 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 (and most misunderstood) 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 (...)
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  28. 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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  29. 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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  30. Alfred Tarski (1968/2010). 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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  31. Neil Tennant (2008). Carnap, Gödel, and the Analyticity of Arithmetic. Philosophia Mathematica 16 (1):100-112.
    Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture. The author is grateful to Michael Friedman for valuable comments. Part of the research (...)
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  32. N. Tennant (2010). Deflationism and the Godel Phenomena: Reply to Cieslinski. Mind 119 (474):437-450.
    I clarify how the requirement of conservative extension features in the thinking of various deflationists, and how this relates to another litmus claim, that the truth-predicate stands for a real, substantial property. I discuss how the deflationist can accommodate the result, to which Cieslinski draws attention, that non-conservativeness attends even the generalization that all logical theorems in the language of arithmetic are true. Finally I provide a four-fold categorization of various forms of deflationism, by reference to the two claims of (...)
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  33. Neil Tennant (2001). On Turing Machines Knowing Their Own Gödel-Sentences. Philosophia Mathematica 9 (1):72-79.
    Storrs McCall appeals to a particular true but improvable sentence of formal arithmetic to argue, by appeal to its irrefutability, that human minds transcend Turing machines. Metamathematical oversights in McCall's discussion of the Godel phenomena, however, render invalid his philosophical argument for this transcendentalist conclusion.
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  34. Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
    In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...)
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  35. Roy Wagner (2009). S(Zp, Zp): Post-Structural Readings of Gödel's Proof. Polimetrica.
    Acknowledgement At one time I was labelled a mathematical prodigy. But due to insufficient luck, talent or motivation I wasn't as successful as my teachers ...
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