Results for 'Gödel’s completeness theorem'

999 found
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  1. The Church-Turing ‘Thesis’ as a Special Corollary of Gödel’s Completeness Theorem.Saul A. Kripke - 2013 - In B. J. Copeland, C. Posy & O. Shagrir (eds.), Computability: Turing, Gödel, Church, and Beyond. MIT Press.
    Traditionally, many writers, following Kleene (1952), thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing’s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis, what Turing (1936) calls “argument I,” has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing (...)
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  2.  9
    A New Proof of Ajtai’s Completeness Theorem for Nonstandard Finite Structures.Michal Garlík - 2015 - Archive for Mathematical Logic 54 (3-4):413-424.
    Ajtai’s completeness theorem roughly states that a countable structure A coded in a model of arithmetic can be end-extended and expanded to a model of a given theory G if and only if a contradiction cannot be derived by a proof from G plus the diagram of A, provided that the proof is definable in A and contains only formulas of a standard length. The existence of such model extensions is closely related to questions in complexity theory. In (...)
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  3.  35
    An Elementary Proof of Chang's Completeness Theorem for the Infinite-Valued Calculus of Lukasiewicz.Roberto Cignoli & Daniele Mundici - 1997 - Studia Logica 58 (1):79-97.
    The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.
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  4.  9
    Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.Steve Awodey & Jess Hughes - unknown
    Steve Awodey and Jesse Hughes. Modal Operators and the Formal Dual of Birkhoff's Completeness Theorem.
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  5.  22
    Completeness Theorem for Dummett's LC Quantified and Some of its Extensions.Giovanna Corsi - 1992 - Studia Logica 51 (2):317 - 335.
    Dummett's logic LC quantified, Q-LC, is shown to be characterized by the extended frame Q+, ,D, where Q+ is the set of non-negative rational numbers, is the numerical relation less or equal then and D is the domain function such that for all v, w Q+, Dv and if v w, then D v . D v D w . Moreover, simple completeness proofs of extensions of Q-LC are given.
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  6.  18
    How to Goedel a Frege-Russell: Goedel's Incompleteness Theorem.Geoffrey Hellman - 1981 - Noûs 15 (4):451-68.
  7.  25
    An Application of Kripke's Completeness Theorem for Intuitionism to Superconstructive Propositional Calculi.J. G. Anderson - 1969 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 15 (16-18):259-288.
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  8.  25
    Extensions of Gödel's Completeness Theorem and the Löwenheim-Skolem Theorem.Stephen L. Bloom - 1973 - Notre Dame Journal of Formal Logic 14 (3):408-410.
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  9.  3
    An Application of Kripke's Completeness Theorem for Intuitionism to Superconstructive Propositional Calculi.J. G. Anderson - 1969 - Mathematical Logic Quarterly 15 (16‐18):259-288.
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  10.  14
    Review: Ladislav Rieger, On Countable Generalised $|Sigma$-Algebras, with a New Proof of Godel's Completeness Theorem[REVIEW]Leon Henkin - 1955 - Journal of Symbolic Logic 20 (3):281-282.
  11.  9
    Rieger Ladislav. 0 Sčétnyh Obobščénnyh Σ-Algébrah I Novom Dokazatélstvé Téorémy Gédéla o Polnoté. Časopis Pro Pěstováni Matematiky a Fysiky , Vol. 1 No. 1 , Pp. 33–49.Rieger Ladislav. On Countable Generalised Σ-Algebras, with a New Proof of Gödel's Completeness Theorem. English Translation of the Preceding. Czechoslovak Mathematical Journal, Vol. 1 No. 1 , Pp. 29–40. [REVIEW]Leon Henkin - 1955 - Journal of Symbolic Logic 20 (3):281-282.
  12.  42
    Goedel's Theorem, the Theory of Everything, and the Future of Science and Mathematics.Douglas S. Robertson - 2000 - Complexity 5 (5):22-27.
  13.  30
    Alexander Abian. On the Solvability of Infinite Systems of Boolean Polynomial Equations. Colloquium Mathematicum, Vol. 21 , Pp. 27–30. - Alexander Abian. Generalized Completeness Theorem and Solvability of Systems of Boolean Polynomial Equations. Zeitschrift Für Mathematische Logik Und Grundlagen der Mathematik, Vol. 16 , Pp. 263–264. - Paul D. Bacsich. Injectivity in Model Theory. Colloquium Mathematicum, Vol. 25 , Pp. 165–176. - S. Bulman-Fleming. On Equationally Compact Semilattices. Algebra Universalis , Vol. 2 No. 2 , Pp. 146–151. - G. Grätzer and H. Lakser. Equationally Compact Semilattices. Colloquium Mathematicum, Vol. 20 , Pp. 27–30. - David K. Haley. On Compact Commutative Noetherian Rings. Mathematische Annalen, Vol. 189 , Pp. 272–274. - Ralph McKenzie. ℵ1-Incompactness of Z. Colloquium Mathematicum, Vol. 23 , Pp. 199–202. - Jan Mycielski. Some Compactifications of General Algebras. Colloquium Mathematicum, Vol. 13 No. 1 , Pp. 1–9. See Errata on Page 281 of Next Paper. - Jan. [REVIEW]Walter Taylor - 1975 - Journal of Symbolic Logic 40 (1):88-92.
  14.  14
    Post's Functional Completeness Theorem.Francis Jeffry Pelletier & Norman M. Martin - 1990 - Notre Dame Journal of Formal Logic 31 (3):462-475.
  15.  3
    A Post-Style Proof of Completeness Theorem for Symmetric Relatedness Logic S.Mateusz Klonowski - 2018 - Bulletin of the Section of Logic 47 (3):201.
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  16.  13
    What Could Self-Reflexiveness Be? Or Goedel’s Theorem Goes to Hollywood and Discovers That It’s All Done with Mirrors.Robert A. Schultz - 1980 - Semiotica 30 (1-2).
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  17. Goedel's Theorem and Models of the Brain: Possible Hemispheric Basis for Kant's Psychological Ideas.U. Fidelman - 1999 - Journal of Mind and Behavior 20 (1):43-56.
    Penrose proved that a computational or formalizable theory of the brainís cognitive functioning is impossible, but suggested that a physical non-computational and non-formalizable one may be viable. Arguments as to why Penroseís program is unrealizable are presented. The main argument is that a non-formalizable theory should be verbal. However, verbal paradoxes based on Cantorís diagonal processes show the impossibility of a consistent verbal theory of the brain comprising its arithmetical cognition. It is suggested that comprehensive theories of the human brain (...)
     
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  18. A Surreptitious Change in the Designation of a Term: The Foundation of Goedel's Theorem of the Non-Demonstrability of Non-Contradictoriness-A New Metalinguistic Exposition and Philosophical Considerations.F. RivettiBarbo - 1996 - Rivista di Filosofia Neo-Scolastica 88 (1):95-128.
     
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  19. Baldwin, JT and Holland, K., Constructing Ω-Stable Struc-Tures: Model Completeness (1–3) 159–172 Berarducci, A. And Servi, T., An Effective Version of Wilkie's Theorem of the Complement and Some Effective o-Minimality Results (1–3) 43–74. [REVIEW]R. Downey, A. Li, G. Wu, M. Dzˇamonja & S. Shelah - 2004 - Annals of Pure and Applied Logic 125:173.
  20.  15
    A New Proof of Sahlqvist's Theorem on Modal Definability and Completeness.G. Sambin & V. Vaccaro - 1989 - Journal of Symbolic Logic 54 (3):992-999.
  21.  21
    A Simpler Proof of Sahlqvist's Theorem on Completeness of Modal Logics.Giovanni Sambin - 1980 - Bulletin of the Section of Logic 9 (2):50-54.
  22. Hallden's Theorem on Post Completeness.Krister Segerberg - 1973 - In Sören Halldén (ed.), Modality, Morality and Other Problems of Sense and Nonsense. Lund, Gleerup. pp. 206--9.
     
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  23. Can Gödel's Incompleteness Theorem Be a Ground for Dialetheism?Seungrak Choi - 2017 - Korean Journal of Logic 20 (2):241-271.
    Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of (...)
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  24.  11
    Completeness Theorem for Propositional Probabilistic Models Whose Measures Have Only Finite Ranges.Radosav Dordević, Miodrag Rašković & Zoran Ognjanović - 2004 - Archive for Mathematical Logic 43 (4):557-563.
    A propositional logic is defined which in addition to propositional language contains a list of probabilistic operators of the form P ≥s (with the intended meaning ‘‘the probability is at least s’’). The axioms and rules syntactically determine that ranges of probabilities in the corresponding models are always finite. The completeness theorem is proved. It is shown that completeness cannot be generalized to arbitrary theories.
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  25. The Emperor's Real Mind -- Review of Roger Penrose's The Emperor's New Mind: Concerning Computers Minds and the Laws of Physics.Aaron Sloman - 1992 - Artificial Intelligence 56 (2-3):355-396.
    "The Emperor's New Mind" by Roger Penrose has received a great deal of both praise and criticism. This review discusses philosophical aspects of the book that form an attack on the "strong" AI thesis. Eight different versions of this thesis are distinguished, and sources of ambiguity diagnosed, including different requirements for relationships between program and behaviour. Excessively strong versions attacked by Penrose (and Searle) are not worth defending or attacking, whereas weaker versions remain problematic. Penrose (like Searle) regards the notion (...)
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  26.  28
    An Intuitionistic Completeness Theorem for Classical Predicate Logic.Victor N. Krivtsov - 2010 - Studia Logica 96 (1):109-115.
    This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel's completeness theorem for classical predicate logic.
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  27. How Godel's Theorem Supports the Possibility of Machine Intelligence.Taner Edis - 1998 - 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 (...)
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  28.  18
    A Strong Completeness Theorem for the Gentzen Systems Associated with Finite Algebras.Àngel J. Gil, Jordi Rebagliato & Ventura Verdú - 1999 - Journal of Applied Non-Classical Logics 9 (1):9-36.
    ABSTRACT In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. (...)
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  29. The Noneffectivity of Arslanov’s Completeness Criterion and Related Theorems.Sebastiaan A. Terwijn - forthcoming - Archive for Mathematical Logic:1-11.
    We discuss the effectivity of Arslanov’s completeness criterion. In particular, we show that a parameterized version, similar to the recursion theorem with parameters, fails. We also discuss the effectivity of another extension of the recursion theorem, namely Visser’s ADN theorem, as well as that of a joint generalization of the ADN theorem and Arslanov’s completeness criterion.
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  30.  43
    Godel's Theorem and Mechanism.David Coder - 1969 - 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 (...)
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  31.  16
    Analytic Completeness Theorem for Singular Biprobability Models.Radosav S. Đordević - 1993 - Mathematical Logic Quarterly 39 (1):228-230.
    The aim of the paper is to prove tha analytic completeness theorem for a logic LAs with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.
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  32.  12
    Analytic Completeness Theorem for Absolutely Continuous Biprobability Models.Radosav S. Đorđević - 1992 - Mathematical Logic Quarterly 38 (1):241-246.
    Hoover [2] proved a completeness theorem for the logic L[MATHEMATICAL SCRIPT CAPITAL A]. The aim of this paper is to prove a similar completeness theorem with respect to product measurable biprobability models for a logic Lmath image with two integral operators. We prove: If T is a ∑1 definable theory on [MATHEMATICAL SCRIPT CAPITAL A] and consistent with the axioms of Lmath image, then there is an analytic absolutely continuous biprobability model in which every sentence in (...)
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  33.  18
    Satisfying Predicates: Kleene's Proof of the Hilbert–Bernays Theorem.Gary Ebbs - 2015 - History and Philosophy of Logic 36 (4):346-366.
    The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of (...)
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  34.  57
    Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL∀.Franco Montagna & Hiroakira Ono - 2002 - Studia Logica 71 (2):227-245.
    The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames (...)
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  35. Completeness and Representation Theorem for Epistemic States in First-Order Predicate Calculus.Serge Lapierre & François Lepage - 1999 - Logica Trianguli 3:85-109.
    The aim of this paper is to present a strongly complete first order functional predicate calculus generalized to models containing not only ordinary classical total functions but also arbitrary partial functions. The completeness proof follows Henkin’s approach, but instead of using maximally consistent sets, we define saturated deductively closed consistent sets . This provides not only a completeness theorem but a representation theorem: any SDCCS defines a canonical model which determine a unique partial value for every (...)
     
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  36.  47
    On the Proof of Solovay's Theorem.Dick Jongh, Marc Jumelet & Franco Montagna - 1991 - Studia Logica 50 (1):51 - 69.
    Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof be replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic (the so-called Rosser logic of Gauspari-Solovay, 1979) so that any arithmetical system for which this logic is sound is strong enough to carry out (...)
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  37.  13
    On the Proof of Solovay's Theorem.Dick de Jongh, Marc Jumelet & Franco Montagna - 1991 - Studia Logica 50 (1):51-69.
    Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular $\text{I}\Delta _{0}+\text{EXP}$ . The (...)
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  38.  17
    Bell’s Theorem, Realism, and Locality.Peter Lewis - 2019 - In Alberto Cordero (ed.), Philosophers Look at Quantum Mechanics. Springer Verlag.
    According to a recent paper by Tim Maudlin, Bell’s theorem has nothing to tell us about realism or the descriptive completeness of quantum mechanics. What it shows is that quantum mechanics is non-local, no more and no less. What I intend to do in this paper is to challenge Maudlin’s assertion about the import of Bell’s proof. There is much that I agree with in the paper; in particular, it does us the valuable service of demonstrating that Einstein’s (...)
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  39.  6
    Krivine's Intuitionistic Proof of Classical Completeness.Stefano Berardi & Silvio Valentini - 2004 - Annals of Pure and Applied Logic 129 (1-3):93-106.
    In 1996, Krivine applied Friedman's A-translation in order to get an intuitionistic version of Gödel completeness result for first-order classical logic and countable languages and models. Such a result is known to be intuitionistically underivable 559), but Krivine was able to derive intuitionistically a weak form of it, namely, he proved that every consistent classical theory has a model. In this paper, we want to analyze the ideas Krivine's remarkable result relies on, ideas which where somehow hidden by the (...)
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  40.  15
    On Theorems of Gödel and Kreisel: Completeness and Markov's Principle.D. C. McCarty - 1994 - Notre Dame Journal of Formal Logic 35 (1):99-107.
    In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how to deduce (...)
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  41.  21
    Bull's Theorem by the Method of Diagrams.Giovanna Corsi - 1999 - Studia Logica 62 (2):163-176.
    We show how to use diagrams in order to obtain straightforward completeness theorems for extensions of K4.3 and a very simple and constructive proof of Bull's theorem: every normal extension of S4.3 has the finite model property.
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  42. Resplendent Models and $${\Sigma_1^1}$$ -Definability with an Oracle.Andrey Bovykin - 2008 - Archive for Mathematical Logic 47 (6):607-623.
    In this article we find some sufficient and some necessary ${\Sigma^1_1}$ -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal (...)
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  43. Recursive Functions and Metamathematics Problems of Completeness and Decidability, Gödel's Theorems.Roman Murawski - 1999
     
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  44.  10
    A Completeness Theorem for Open Maps.A. Joyal & I. Moerdijk - 1994 - Annals of Pure and Applied Logic 70 (1):51-86.
    This paper provides a partial solution to the completeness problem for Joyal's axiomatization of open and etale maps, under the additional assumption that a collection axiom holds.
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  45.  4
    Goedel's Property Abstraction and Possibilism.Randoph Rubens Goldman - 2014 - Australasian Journal of Logic 11 (2).
    Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of positive property. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes positive property as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" Pure attribution seems likely to be related to the Leibnizian concept of perfection.By (...)
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  46. Goedel's Property Abstraction and Possibilism.Randoph Rubens Goldman - 2017 - Australasian Journal of Logic 14 (3).
    Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of _positive property_. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes _positive property_ as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" _Pure attribution_ seems likely to be related to the Leibnizian concept of perfection. (...)
     
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  47. Bell's Theorem: What It Takes.Jeremy Butterfield - 1992 - British Journal for the Philosophy of Science 43 (1):41-83.
    I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour (...)
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  48.  61
    Nonstandard Models and Kripke's Proof of the Gödel Theorem.Hilary Putnam - 2000 - Notre Dame Journal of Formal Logic 41 (1):53-58.
    This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke. Today we know purely algebraic techniques that can be used to give direct proofs of the existence of nonstandard models in a style with which ordinary mathematicians feel perfectly comfortable--techniques that do not even require knowledge of the Completeness Theorem or even require that logic itself be axiomatized. Kripke used these techniques to establish incompleteness by (...)
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  49.  33
    Brouwer's Fan Theorem and Unique Existence in Constructive Analysis.Josef Berger & Hajime Ishihara - 2005 - Mathematical Logic Quarterly 51 (4):360-364.
    Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. (...)
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  50.  33
    Weak König's Lemma Implies Brouwer's Fan Theorem: A Direct Proof.Hajime Ishihara - 2006 - Notre Dame Journal of Formal Logic 47 (2):249-252.
    Classically, weak König's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.
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