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The Foundations of Intuitionistic Mathematics

Amsterdam: North-Holland Pub. Co. (1965)

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  1. Constructivity in Geometry.Richard Vesley - 1999 - History and Philosophy of Logic 20 (3-4):291-294.
    We review and contrast three ways to make up a formal Euclidean geometry which one might call constructive, in a computational sense. The starting point is the first-order geometry created by Tarski.
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  • Relative and Modified Relative Realizability.Lars Birkedal & Jaap van Oosten - 2002 - Annals of Pure and Applied Logic 118 (1-2):115-132.
    The classical forms of both modified realizability and relative realizability are naturally described in terms of the Sierpinski topos. The paper puts these two observations together and explains abstractly the existence of the geometric morphisms and logical functors connecting the various toposes at issue. This is done by advancing the theory of triposes over internal partial combinatory algebras and by employing a novel notion of elementary map.
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  • Spreads or Choice Sequences?H. C. M. De Swart - 1992 - History and Philosophy of Logic 13 (2):203-213.
    Intuitionistically. a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights?in particular, the so-called continuity principles?which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer?s reconstruction of mathematics. The predicate ?α (...)
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  • Godel's Functional Interpretation.J. Avigad & S. Feferman - 1998 - In Samuel R. Buss (ed.), Handbook of Proof Theory. Elsevier.
  • Analyzing Realizability by Troelstra's Methods.Joan Rand Moschovakis - 2002 - Annals of Pure and Applied Logic 114 (1-3):203-225.
    Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. Troelstra discovered principles ECT 0 and GC 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Building on Troelstra's results and using his methods, we introduce the notions of Church domain and domain of continuity in order to demonstrate the optimality of “almost negativity” in ECT 0 and GC 1 ; strengthen “double negation shift” DNS 0 to DNS (...)
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  • Proof Theory and Constructive Mathematics.Anne S. Troelstra - 1977 - In Jon Barwise & H. Jerome Keisler (eds.), Handbook of Mathematical Logic. North-Holland Pub. Co.. pp. 973--1052.
  • A Transfer Theorem in Constructive P-Adic Algebra.Deirdre Haskell - 1992 - Annals of Pure and Applied Logic 58 (1):29-55.
    The main result of this paper is a transfer theorem which describes the relationship between constructive validity and classical validity for a class of first-order sentences over the p-adics. The proof of one direction of the theorem uses a principle of intuitionism; the proof of the other direction is classically valid. Constructive verifications of known properties of the p-adics are indicated. In particular, the existence of cylindric algebraic decompositions for the p-adics is used.
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  • La Descente Infinie, L’Induction Transfinie Et le Tiers Exclu.Yvon Gauthier - 2009 - Dialogue 48 (1):1.
    ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite (...)
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  • Realizability and Intuitionistic Logic.J. Diller & A. S. Troelstra - 1984 - Synthese 60 (2):253 - 282.
  • The Theory of Empirical Sequences.Carl J. Posy - 1977 - Journal of Philosophical Logic 6 (1):47 - 81.
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  • Brouwer's Conception of Truth.Casper Storm Hansen - 2016 - Philosophia Mathematica 24 (3):379-400.
    In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.
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  • A New Realizability Notion for Intuitionistic Analysis.B. Scarpellini - 1977 - Mathematical Logic Quarterly 23 (7‐12):137-167.
  • A Topological Model for Intuitionistic Analysis with Kripke's Scheme.M. D. Krol - 1978 - Mathematical Logic Quarterly 24 (25‐30):427-436.
  • Intuitionistische Kennzeichnung der endlichen Spezies.Fritz Homagk - 1971 - Studia Logica 28 (1):41-63.
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  • Essay Review.M. Detlefsen - 1988 - History and Philosophy of Logic 9 (1):93-105.
    S. SHAPIRO (ed.), Intensional Mathematics (Studies in Logic and the Foundations of Mathematics, vol. 11 3). Amsterdam: North-Holland, 1985. v + 230 pp. $38.50/100Df.
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  • A Very Strong Intuitionistic Theory.Sergio Bernini - 1976 - Studia Logica 35 (4):377 - 385.
  • Lifschitz Realizability for Intuitionistic Zermelo–Fraenkel Set Theory.Ray-Ming Chen & Michael Rathjen - 2012 - Archive for Mathematical Logic 51 (7-8):789-818.
    A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The machinery (...)
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  • Metric Spaces in Synthetic Topology.Andrej Bauer & Davorin Lešnik - 2012 - Annals of Pure and Applied Logic 163 (2):87-100.
  • Reference and Perspective in Intuitionistic Logics.John Nolt - 2006 - Journal of Logic, Language and Information 16 (1):91-115.
    What an intuitionist may refer to with respect to a given epistemic state depends not only on that epistemic state itself but on whether it is viewed concurrently from within, in the hindsight of some later state, or ideally from a standpoint “beyond” all epistemic states (though the latter perspective is no longer strictly intuitionistic). Each of these three perspectives has a different—and, in the last two cases, a novel—logic and semantics. This paper explains these logics and their semantics and (...)
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  • Brouwer and Souslin on Transfinite Cardinals.John P. Burgess - 1980 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 26 (14-18):209-214.
  • A Formally Constructive Model for Barrecursion of Higher Types.Bruno Scarpellini - 1972 - Mathematical Logic Quarterly 18 (21‐24):321-383.
  • A Topological Model for Intuitionistic Analysis with Kripke's Scheme.M. D. Krol - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (25-30):427-436.
  • A New Realizability Notion for Intuitionistic Analysis.B. Scarpellini - 1977 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 23 (7-12):137-167.
  • A Formally Constructive Model for Barrecursion of Higher Types.Bruno Scarpellini - 1972 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 18 (21-24):321-383.
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