Search results for 'Incompleteness' (try it on Scholar)

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  1. Francesco Berto (2009). There's Something About Gödel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.score: 24.0
    The Gödelian symphony -- Foundations and paradoxes -- This sentence is false -- The liar and Gödel -- Language and metalanguage -- The axiomatic method or how to get the non-obvious out of the obvious -- Peano's axioms -- And the unsatisfied logicists, Frege and Russell -- Bits of set theory -- The abstraction principle -- Bytes of set theory -- Properties, relations, functions, that is, sets again -- Calculating, computing, enumerating, that is, the notion of algorithm -- Taking numbers (...)
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  2. H. Gaifman (2000). What Godel's Incompleteness Result Does and Does Not Show. Journal of Philosophy 97 (8):462-471.score: 24.0
    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 (...)
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  3. James W. Garson (2010). Expressive Power and Incompleteness of Propositional Logics. Journal of Philosophical Logic 39 (2):159-171.score: 24.0
    Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules (however formulated) underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the (...)
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  4. Robert F. Hadley (2008). Consistency, Turing Computability and Gödel's First Incompleteness Theorem. Minds and Machines 18 (1):1-15.score: 24.0
    It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with (...)
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  5. Christopher Gauker (2001). T-Schema Deflationism Versus Gödel’s First Incompleteness Theorem. Analysis 61 (270):129–136.score: 24.0
    I define T-schema deflationism as the thesis that a theory of truth for our language can simply take the form of certain instances of Tarski's schema (T). I show that any effective enumeration of these instances will yield as a dividend an effective enumeration of all truths of our language. But that contradicts Gödel's First Incompleteness Theorem. So the instances of (T) constituting the T-Schema deflationist's theory of truth are not effectively enumerable, which casts doubt on the idea that (...)
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  6. Roman Murawski (1997). Gödel's Incompleteness Theorems and Computer Science. Foundations of Science 2 (1):123-135.score: 24.0
    In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.
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  7. Marcelo Tsuji, Newton C. A. Costdaa & Francisco A. Doria (1998). The Incompleteness of Theories of Games. Journal of Philosophical Logic 27 (6):553-568.score: 24.0
    We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.
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  8. 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.score: 24.0
    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 (...)
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  9. Panu Raatikainen (1998). On Interpreting Chaitin's Incompleteness Theorem. Journal of Philosophical Logic 27 (6):569-586.score: 24.0
    The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure (...)
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  10. Laureano Luna & Alex Blum (2008). Arithmetic and Logic Incompleteness: The Link. The Reasoner 2 (3):6.score: 24.0
    We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.
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  11. Cristian S. Calude (2002). Incompleteness, Complexity, Randomness and Beyond. Minds and Machines 12 (4):503-517.score: 24.0
    Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.
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  12. Paul H. Dembinski (2011). The Incompleteness of the Economy and Business: A Forceful Reminder. [REVIEW] Journal of Business Ethics 100 (S1):29-40.score: 24.0
    Many different but related arguments developed in the Caritas in Veritate converge on one central, yet not clearly stated, conclusion or thesis: economic and business activities are ‘incomplete’. This article will explore the above-mentioned ‘incompleteness’ thesis or argument from three different perspectives: the role, the practice and the purpose of economic and business activities in contemporary societies. In doing so, the paper will heavily draw on questions and, still not fully learned, lessons derived from the present financial and economic (...)
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  13. Nicholas Harrigan & Robert W. Spekkens (2010). Einstein, Incompleteness, and the Epistemic View of Quantum States. Foundations of Physics 40 (2):125-157.score: 24.0
    Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something (...)
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  14. Albert Visser (2012). The Second Incompleteness Theorem and Bounded Interpretations. Studia Logica 100 (1-2):399-418.score: 24.0
    In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for (...)
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  15. Juliana Bueno-Soler (2013). Multimodal Incompleteness Under Weak Negations. Logica Universalis 7 (1):21-31.score: 24.0
    This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.
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  16. Newton C. A. Da Costa (2012). Gödel's Incompleteness Theorems and Physics. Principia 15 (3):453-459.score: 24.0
    This paper is a summary of a lecture in which I presented some remarks on Gödel’s incompleteness theorems and their meaning for the foundations of physics. The entire lecture will appear elsewhere. doi: http://dx.doi.org/ 10.5007 / 1808-1711.2011v15n3p453.
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  17. Marcelo Tsuji, Newton C. A. Da Costa & Francisco A. Doria (1998). The Incompleteness of Theories of Games. Journal of Philosophical Logic 27 (6):553 - 568.score: 24.0
    We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.
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  18. C. Chihara (1972). On Alleged Refutations of Mechanism Using Godel's Incompleteness Results. Journal of Philosophy 69 (September):507-26.score: 21.0
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  19. Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai (2012). On Proofs of the Incompleteness Theorems Based on Berry's Paradox by Vopěnka, Chaitin, and Boolos. Mathematical Logic Quarterly 58 (4‐5):307-316.score: 21.0
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  20. L. Sacchetti (2002). Incompleteness and Fixed Points. Mathematical Logic Quarterly 48 (1):15-28.score: 21.0
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  21. Makoto Kikuchi (1994). A Note on Boolos' Proof of the Incompleteness Theorem. Mathematical Logic Quarterly 40 (4):528-532.score: 21.0
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  22. Dolph Ulrich (1992). On the Incompleteness of a Descending Chain of Extensions of Implicational S5. Mathematical Logic Quarterly 38 (1):321-323.score: 21.0
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  23. H. P. Barendregt (1976). The Incompleteness Theorems. Rijksuniversiteit Utrecht, Mathematisch Instituut.score: 21.0
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  24. Gregory J. Chaitin (1970). Computational Complexity and Godel's Incompleteness Theorem. [Rio De Janeiro,Centro Técnico Científico, Pontifícia Universidade Católica Do Rio De Janeiro.score: 21.0
     
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  25. C. Cieslinski (2002). Heterologicality and Incompleteness. Mathematical Logic Quarterly 48 (1):105-110.score: 21.0
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  26. Katsumasa Ishii (2003). A Note on the First Incompleteness Theorem. Mathematical Logic Quarterly 49 (2):214-216.score: 21.0
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  27. Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.score: 18.0
    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 (...)
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  28. Glen Hoffmann (2007). The Semantic Theory of Truth: Field's Incompleteness Objection. Philosophia 35 (2):161-170.score: 18.0
    According to Field’s influential incompleteness objection, Tarski’s semantic theory of truth is unsatisfactory since the definition that forms its basis is incomplete in two distinct senses: (1) it is physicalistically inadequate, and for this reason, (2) it is conceptually deficient. In this paper, I defend the semantic theory of truth against the incompleteness objection by conceding (1) but rejecting (2). After arguing that Davidson and McDowell’s reply to the incompleteness objection fails to pass muster, I argue that, (...)
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  29. Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.score: 18.0
    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 (...)
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  30. Yi-Zhuang Chen (2004). Edgar Morin's Paradigm of Complexity and Gödel's Incompleteness Theorem. World Futures 60 (5 & 6):421 – 431.score: 18.0
    This article shows that in two respects, Gödel's incompleteness theorem strongly supports the arguments of Edgar Morin's complexity paradigm. First, from the viewpoint of the content of Gödel's theorem, the latter justifies the basic view of complexity paradigm according to which knowledge is a dynamic, unfinished process, and develops by way of self-criticism and self-transcendence. Second, from the viewpoint of the proof procedure of Gödel's theorem, the latter confirms the complexity paradigm's circular line of inference through which is formed (...)
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  31. Ray Buchanan & Gary Ostertag (2005). Has the Problem of Incompleteness Rested on a Mistake? Mind 114 (456):889-913.score: 18.0
    A common objection to Russell's theory of descriptions concerns incomplete definite descriptions: uses of (for example) ‘the book is overdue’ in contexts where there is clearly more than one book. Many contemporary Russellians hold that such utterances will invariably convey a contextually determined complete proposition, for example, that the book in your briefcase is overdue. But according to the objection this gets things wrong: typically, when a speaker utters such a sentence, no facts about the context or the speaker's communicative (...)
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  32. Richard Woodward (2012). Fictionalism and Incompleteness. Noûs 46 (4):781-790.score: 18.0
    The modal fictionalist faces a problem due to the fact that her chosen story seems to be incomplete—certain things are neither fictionally true nor fictionally false. The significance of this problem is not localized to modal fictionalism, however, since many fictionalists will face it too. By examining how the fictionalist should analyze the notion of truth according to her story, and, in particular, the role that conditionals play for the fictionalist, I develop a novel and elegant solution to the (...) problem. (shrink)
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  33. John Byron Manchak, Observational Indistinguishability and Geodesic Incompleteness.score: 18.0
    It has been suggested by Clark Glymour that the spatio-temporal structure of the universe might be underdetermined by all observational data that could ever, theoretically, be gathered. It is possible for two spacetimes to be observationally indistinguishable (OI) yet topologically distinct. David Malament extended the argument for the underdetermination of spacetime structure by showing that under quite general conditions (such as the absence of any closed timelike curves) a spacetime will always have an OI counterpart (at least in weak sense). (...)
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  34. G. Longo (2011). Reflections on Concrete Incompleteness. Philosophia Mathematica 19 (3):255-280.score: 18.0
    How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and (...)
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  35. Stewart Shapiro (2002). Incompleteness and Inconsistency. Mind 111 (444):817-832.score: 18.0
    Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, (...)
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  36. A. O. Barut, M. Božić & Z. Marić (1988). Joint Probabilities of Noncommuting Operators and Incompleteness of Quantum Mechanics. Foundations of Physics 18 (10):999-1012.score: 18.0
    We use joint probabilities to analyze the EPR argument in the Bohm's example of spins.(1) The properties of distribution functions for two, three, or more noncommuting spin components are explicitly studied and their limitations are pointed out. Within the statistical ensemble interpretation of quantum theory (where only statements about repeated events can be made), the incompleteness of quantum theory does not follow, as the consistent use of joint probabilities shows. This does not exclude a completion of quantum mechanics, going (...)
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  37. Harvey Friedman, Fromal Statements of Godel's Second Incompleteness Theorem.score: 18.0
    Informal statements of Gödel's Second Incompleteness Theorem, referred to here as Informal Second Incompleteness, are simple and dramatic. However, current versions of Formal Second Incompleteness are complicated and awkward. We present new versions of Formal Second Incompleteness that are simple, and informally imply Informal Second Incompleteness. These results rest on the isolation of simple formal properties shared by consistency statements. Here we do not address any issues concerning proofs of Second Incompleteness.
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  38. Haim Gaifman, Gödel's Incompleteness Results.score: 18.0
    This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We suggest this as a (...)
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  39. Paolo Mancosu (1999). Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems. History and Philosophy of Logic 20 (1):33-45.score: 18.0
    What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.
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  40. Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.score: 18.0
    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 (...)
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  41. Sungho Choi (2008). The Incompleteness of Dispositional Predicates. Synthese 163 (2):157 - 174.score: 18.0
    Elizabeth Prior claims that dispositional predicates are incomplete in the sense that they have more than one argument place. To back up this claim, she offers a number of arguments that involve such ordinary dispositional predicates as ‘fragile’, ‘soluble’, and so on. In this paper, I will first demonstrate that one of Prior’s arguments that ‘is fragile’ is an incomplete predicate is mistaken. This, however, does not immediately mean that Prior is wrong that ‘fragile’ is an incomplete predicate. On the (...)
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  42. Solomon Feferman, The Nature and Significance of Gödel's Incompleteness Theorems.score: 18.0
    What Gödel accomplished in the decade of the 1930s before joining the Institute changed the face of mathematical logic and continues to influence its development. As you gather from my title, I’ll be talking about the most famous of his results in that period, but first I want to indulge in some personal reminiscences. In many ways this is a sentimental journey for me. I was a member of the Institute in 1959-60, a couple of years after receiving my PhD (...)
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  43. E. Sober & M. Steel (2013). Screening-Off and Causal Incompleteness: A No-Go Theorem. British Journal for the Philosophy of Science 64 (3):513-550.score: 18.0
    We begin by considering two principles, each having the form causal completeness ergo screening-off. The first concerns a common cause of two or more effects; the second describes an intermediate link in a causal chain. They are logically independent of each other, each is independent of Reichenbach's principle of the common cause, and each is a consequence of the causal Markov condition. Simple examples show that causal incompleteness means that screening-off may fail to obtain. We derive a stronger result: (...)
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  44. John Bell (2008). Incompleteness in a General Setting (Vol 13, Pg 21, 2007). Bulletin of Symbolic Logic 14 (1):21 - 30.score: 18.0
    Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without (...)
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  45. Solomon Feferman, The Impact of the Incompleteness Theorems on Mathematics.score: 18.0
    In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about (...)
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  46. Martin Montminy (2011). Indeterminacy, Incompleteness, Indecision, and Other Semantic Phenomena. Canadian Journal of Philosophy 41 (1):73-98.score: 18.0
    This paper explores the relationships between Davidson's indeterminacy of interpretation thesis and two semantic properties of sentences that have come to be recognized recently, namely semantic incompleteness and semantic indecision.1 More specifically, I will examine what the indeterminacy thesis entails for sentences of the form 'By sentence S (or word w), agent A means that m' and 'Agent A believes that p.' My primary goal is to shed light on the indeterminacy thesis and its consequences. I will distinguish two (...)
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  47. Solomon Feferman, Incompleteness: The Proof and Paradox of Kurt Gödel.score: 18.0
    Like Heisenberg’s uncertainty principle, Gödel’s incompleteness theorem has captured the public imagination, supposedly demonstrating that there are absolute limits to what can be known. More specifically, it is thought to tell us that there are mathematical truths which can never be proved. These are among the many misconceptions and misuses of Gödel’s theorem and its consequences. Incompleteness has been held to show, for example, that there cannot be a Theory of Everything, the so-called holy grail of modern physics. (...)
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  48. Adrian Heathcote (1990). Unbounded Operators and the Incompleteness of Quantum Mechanics. Philosophy of Science 57 (3):523-534.score: 18.0
    A proof is presented that a form of incompleteness in Quantum Mechanics follows directly from the use of unbounded operators. It is then shown that the problems that arise for such operators are not connected to the non- commutativity of many pairs of operators in Quantum Mechanics and hence are an additional source of incompleteness to that which allegedly flows from the..
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  49. Peter Smith, The First Incompleteness Theorem.score: 18.0
    • How to construct a ‘canonical’ Gödel sentence • If PA is sound, it is negation imcomplete • Generalizing that result to sound p.r. axiomatized theories whose language extends LA • ω-incompleteness, ω-inconsistency • If PA is ω-consistent, it is negation imcomplete • Generalizing that result to ω-consistent p.r. axiomatized theories which extend Q..
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  50. M. J. Cresswell (1995). Incompleteness and the Barcan Formula. Journal of Philosophical Logic 24 (4):379 - 403.score: 18.0
    A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...)
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