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  1. Douglas S. Bridges, Hajime Ishihara & Maarten McKubre‐Jordens (2013). Uniformly Convex Banach Spaces Are Reflexive—Constructively. Mathematical Logic Quarterly 59 (4-5):352-356.
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  2. Hajime Ishihara (2013). Relating Bishopʼs Function Spaces to Neighbourhood Spaces. 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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  3. Josef Berger, Hajime Ishihara, Erik Palmgren & Peter Schuster (2012). A Predicative Completion of a Uniform Space. Annals of Pure and Applied Logic 163 (8):975-980.
  4. Hajime Ishihara (2012). Two Subcategories of Apartness Spaces. Annals of Pure and Applied Logic 163 (2):132-139.
  5. Hajime Ishihara (2012). The Uniform Boundedness Theorem and a Boundedness Principle. Annals of Pure and Applied Logic 163 (8):1057-1061.
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  6. Hajime Ishihara & Peter Schuster (2011). On the Contrapositive of Countable Choice. 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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  7. Vasco Brattka, Hajime Ishihara, Matthias Schröder & Ning Zhong (2008). Editorial: Math. Log. Quart. 5/2008. Mathematical Logic Quarterly 54 (5):453-453.
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  8. Robin Havea, Hajime Ishihara & Luminita Vîta (2008). Separation Properties in Neighbourhood and Quasi-Apartness Spaces. 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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  9. Hajime Ishihara & Peter Schuster (2008). A Continuity Principle, a Version of Baire's Theorem and a Boundedness Principle. 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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  10. Peter Aczel, Laura Crosilla, Hajime Ishihara, Erik Palmgren & Peter Schuster (2006). Binary Refinement Implies Discrete Exponentiation. 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 re.nement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary re.nement implies that the class of detachable subsets of a set form a set. Binary re.nement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was su.cient to prove that the (...)
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  11. Hajime Ishihara (2006). Weak König's Lemma Implies Brouwer's Fan Theorem: A Direct Proof. 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. Hajime Ishihara (2006). Reverse Mathematics in Bishop's Constructive Mathematics. Philosophia Scientiae:43-59.
  13. Hajime Ishihara, Ray Mines, Peter Schuster & Luminiţa Vîţă (2006). Quasi-Apartness and Neighbourhood Spaces. 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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  14. Hajime Ishihara & Erik Palmgren (2006). Quotient Topologies in Constructive Set Theory and Type Theory. 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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  15. Josef Berger & Hajime Ishihara (2005). Brouwer's Fan Theorem and Unique Existence in Constructive Analysis. 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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  16. Douglas Bridges, Hajime Ishihara, Peter Schuster & Luminiţa Vîţa (2005). Strong Continuity Implies Uniform Sequential Continuity. Archive for Mathematical Logic 44 (7):887-895.
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  17. Laura Crosilla, Hajime Ishihara & Peter Schuster (2005). On Constructing Completions. 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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  18. Hajime Ishihara & Peter Schuster (2004). Compactness Under Constructive Scrutiny. 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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  19. Hajime Ishihara & Satoru Yoshida (2002). A Constructive Look at the Completeness of the Space (ℝ). 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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  20. Hajime Ishihara & Satoru Yoshida (2002). A Constructive Look at the Completeness of the Space $\Mathcal{D} (\Mathbb{R})$. 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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  21. Douglas Bridges & Hajime Ishihara (1998). A Definitive Constructive Open Mapping Theorem? 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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  22. Hajime Ishihara, Bakhadyr Khoussainov & Anil Nerode (1998). Decidable Kripke Models of Intuitionistic Theories. 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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  23. Douglas S. Bridges & Hajime Ishihara (1994). Complements of Intersections in Constructive Mathematics. 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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  24. Douglas Bridges & Hajime Ishihara (1994). Absolute Continuity and the Uniqueness of the Constructive Functional Calculus. 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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  25. Hajime Ishihara (1992). Continuity Properties in Constructive Mathematics. 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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  26. Hajime Ishihara (1991). Continuity and Nondiscontinuity in Constructive Mathematics. 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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  27. Hajime Ishihara (1991). Constructive Compact Operators on a Hilbert Space. 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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  28. Hajime Ishihara (1990). An Omniscience Principle, the König Lemma and the Hahn‐Banach Theorem. Mathematical Logic Quarterly 36 (3):237-240.
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