39 found
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  1. Uniformly Convex Banach Spaces Are Reflexive—Constructively.Douglas S. Bridges, Hajime Ishihara & Maarten McKubre‐Jordens - 2013 - Mathematical Logic Quarterly 59 (4-5):352-356.
    We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.
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  2.  9
    Reverse Mathematics in Bishop’s Constructive Mathematics.Hajime Ishihara - 2006 - Philosophia Scientae:43-59.
  3. A Constructive Look at the Completeness of the Space $\Mathcal{D} (\Mathbb{R})$.Hajime Ishihara & Satoru Yoshida - 2002 - Journal of Symbolic Logic 67 (4):1511-1519.
    We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space $\mathcal{D} (\mathbb{R})$ of test functions is equivalent to the principle BD-N which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.
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  4.  21
    Strong Continuity Implies Uniform Sequential Continuity.Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa - 2005 - Archive for Mathematical Logic 44 (7):887-895.
    Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.
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  5. Binary Refinement Implies Discrete Exponentiation.Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2006 - Studia Logica 84 (3):361-368.
    Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to prove that the (...)
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  6.  35
    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. Furthermore, (...)
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  7.  10
    Reverse Mathematics in Bishop’s Constructive Mathematics.Hajime Ishihara - 2006 - Philosophia Scientiae:43-59.
  8.  19
    Two Subcategories of Apartness Spaces.Hajime Ishihara - 2012 - Annals of Pure and Applied Logic 163 (2):132-139.
  9.  7
    On the independence of premiss axiom and rule.Hajime Ishihara & Takako Nemoto - forthcoming - Archive for Mathematical Logic:1-23.
    In this paper, we deal with a relationship among the law of excluded middle, the double negation elimination and the independence of premiss rule ) for intuitionistic predicate logic. After giving a general machinery, we give, as corollaries, several examples of extensions of \ and \ which are closed under \ but do not derive the independence of premiss axiom.
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  10.  28
    On Constructing Completions.Laura Crosilla, Hajime Ishihara & Peter Schuster - 2005 - Journal of Symbolic Logic 70 (3):969-978.
    The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel and (...)
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  11.  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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  12.  32
    Continuity Properties in Constructive Mathematics.Hajime Ishihara - 1992 - Journal of Symbolic Logic 57 (2):557-565.
    The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.
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  13.  22
    An Omniscience Principle, the König Lemma and the Hahn-Banach Theorem.Hajime Ishihara - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (3):237-240.
  14.  9
    Some Principles Weaker Than Markov’s Principle.Makoto Fujiwara, Hajime Ishihara & Takako Nemoto - 2015 - Archive for Mathematical Logic 54 (7-8):861-870.
    We systematically study several principles and give a principle which is weaker than disjunctive Markov’s principle. We also show that the principle is underivable and strictly weaker than MP∨ in certain extensions of the system EL of elementary analysis.
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  15.  21
    An Omniscience Principle, the König Lemma and the Hahn‐Banach Theorem.Hajime Ishihara - 1990 - Mathematical Logic Quarterly 36 (3):237-240.
  16.  7
    Compactness Under Constructive Scrutiny.Hajime Ishihara & Peter Schuster - 2004 - Mathematical Logic Quarterly 50 (6):540-550.
    How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, (...)
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  17.  14
    Quasi-Apartness and Neighbourhood Spaces.Hajime Ishihara, Ray Mines, Peter Schuster & Luminiţa Vîţă - 2006 - Annals of Pure and Applied Logic 141 (1):296-306.
    We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.
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  18.  36
    A Constructive Look at the Completeness of the Space (ℝ).Hajime Ishihara & Satoru Yoshida - 2002 - Journal of Symbolic Logic 67 (4):1511-1519.
    We show, within the framework of Bishop's constructive mathematics, that (sequential) completeness of the locally convex space $\mathcal{D} (\mathbb{R})$ of test functions is equivalent to the principle BD-N which holds in classical mathemtatics, Brouwer's intuitionism and Markov's constructive recursive mathematics, but does not hold in Bishop's constructivism.
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  19.  20
    A Note on the Godel-Gentzen Translation.Hajime Ishihara - 2000 - Mathematical Logic Quarterly 46 (1):135-138.
    We give a variant of the Gödel-Gentzen-negative translation, and a syntactic characterization which entails conservativity result for formulas.
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  20.  18
    Continuity and Nondiscontinuity in Constructive Mathematics.Hajime Ishihara - 1991 - Journal of Symbolic Logic 56 (4):1349-1354.
    The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that "every mapping is strongly extensional", "every sequentially nondiscontinuous mapping is sequentially continuous", and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tsĕitin theorem.
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  21.  10
    Non-Deterministic Inductive Definitions and Fullness.Takako Nemoto & Hajime Ishihara - 2016 - In Peter Schuster & Dieter Probst (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science. De Gruyter. pp. 163-170.
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  22.  11
    A Note on the Independence of Premiss Rule.Hajime Ishihara & Takako Nemoto - 2016 - Mathematical Logic Quarterly 62 (1-2):72-76.
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  23.  11
    Quotient Topologies in Constructive Set Theory and Type Theory.Hajime Ishihara & Erik Palmgren - 2006 - Annals of Pure and Applied Logic 141 (1):257-265.
    The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the predicative methods available in constructive type theory and constructive set theory.
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  24.  23
    A Predicative Completion of a Uniform Space.Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster - 2012 - Annals of Pure and Applied Logic 163 (8):975-980.
  25.  8
    Equivalents of the Finitary Non-Deterministic Inductive Definitions.Ayana Hirata, Hajime Ishihara, Tatsuji Kawai & Takako Nemoto - 2019 - Annals of Pure and Applied Logic 170 (10):1256-1272.
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  26.  16
    Decidable Kripke Models of Intuitionistic Theories.Hajime Ishihara, Bakhadyr Khoussainov & Anil Nerode - 1998 - Annals of Pure and Applied Logic 93 (1-3):115-123.
    In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T.
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  27.  15
    The Binary Expansion and the Intermediate Value Theorem in Constructive Reverse Mathematics.Josef Berger, Hajime Ishihara, Takayuki Kihara & Takako Nemoto - 2019 - Archive for Mathematical Logic 58 (1-2):203-217.
    We introduce the notion of a convex tree. We show that the binary expansion for real numbers in the unit interval ) is equivalent to weak König lemma ) for trees having at most two nodes at each level, and we prove that the intermediate value theorem is equivalent to \ for convex trees, in the framework of constructive reverse mathematics.
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  28.  12
    Constructive Compact Operators on a Hilbert Space.Hajime Ishihara - 1991 - Annals of Pure and Applied Logic 52 (1-2):31-37.
    In this paper, we deal with compact operators on a Hilbert space, within the framework of Bishop's constructive mathematics. We characterize the compactness of a bounded linear mapping of a Hilbert space into C n , and prove the theorems: Let A and B be compact operators on a Hilbert space H , let C be an operator on H and let α ϵ C . Then α A is compact, A + B is compact, A ∗ is compact, CA (...)
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  29.  12
    Consistency of the Intensional Level of the Minimalist Foundation with Church’s Thesis and Axiom of Choice.Hajime Ishihara, Maria Emilia Maietti, Samuele Maschio & Thomas Streicher - 2018 - Archive for Mathematical Logic 57 (7-8):873-888.
    Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, (...)
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  30.  9
    Complements of Intersections in Constructive Mathematics.Douglas S. Bridges & Hajime Ishihara - 1994 - Mathematical Logic Quarterly 40 (1):35-43.
    We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and (...)
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  31.  12
    Embedding Classical in Minimal Implicational Logic.Hajime Ishihara & Helmut Schwichtenberg - 2016 - Mathematical Logic Quarterly 62 (1-2):94-101.
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  32.  15
    A Definitive Constructive Open Mapping Theorem?Douglas Bridges & Hajime Ishihara - 1998 - Mathematical Logic Quarterly 44 (4):545-552.
    It is proved, within Bishop's constructive mathematics , that, in the context of a Hilbert space, the Open Mapping Theorem is equivalent to a principle that holds in intuitionistic mathematics and recursive constructive mathematics but is unlikely to be provable within BISH.
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  33.  14
    On the Contrapositive of Countable Choice.Hajime Ishihara & Peter Schuster - 2011 - Archive for Mathematical Logic 50 (1-2):137-143.
    We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for ${\Sigma_{2}^{0}}$ -formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.
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  34.  14
    Absolute Continuity and the Uniqueness of the Constructive Functional Calculus.Douglas Bridges & Hajime Ishihara - 1994 - Mathematical Logic Quarterly 40 (4):519-527.
    The constructive functional calculus for a sequence of commuting selfadjoint operators on a separable Hilbert space is shown to be independent of the orthonormal basis used in its construction. The proof requires a constructive criterion for the absolute continuity of two positive measures in terms of test functions.
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  35.  20
    A Continuity Principle, a Version of Baire's Theorem and a Boundedness Principle.Hajime Ishihara & Peter Schuster - 2008 - Journal of Symbolic Logic 73 (4):1354-1360.
    We deal with a restricted form WC-N' of the weak continuity principle, a version BT' of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N'. BT' + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.
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  36.  10
    Editorial: Math. Log. Quart. 5/2008.Vasco Brattka, Hajime Ishihara, Matthias Schröder & Ning Zhong - 2008 - Mathematical Logic Quarterly 54 (5):453-453.
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  37.  11
    Relating Bishopʼs Function Spaces to Neighbourhood Spaces.Hajime Ishihara - 2013 - Annals of Pure and Applied Logic 164 (4):482-490.
    We extend Bishopʼs concept of function spaces to the concept of pre-function spaces. We show that there is an adjunction between the category of neighbourhood spaces and the category of Φ-closed pre-function spaces. We also show that there is an adjunction between the category of uniform spaces and the category of Ψ-closed pre-function spaces.
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  38.  8
    Separation Properties in Neighbourhood and Quasi-Apartness Spaces.Robin Havea, Hajime Ishihara & Luminiţa Vîţă - 2008 - Mathematical Logic Quarterly 54 (1):58-64.
    We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.
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  39.  7
    The Uniform Boundedness Theorem and a Boundedness Principle.Hajime Ishihara - 2012 - Annals of Pure and Applied Logic 163 (8):1057-1061.