9 found
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  1.  11
    A Marriage of Brouwer’s Intuitionism and Hilbert’s Finitism I: Arithmetic.Takako Nemoto & Sato Kentaro - 2022 - Journal of Symbolic Logic 87 (2):437-497.
    We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity principle (...)
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  2.  7
    Finite sets and infinite sets in weak intuitionistic arithmetic.Takako Nemoto - 2020 - Archive for Mathematical Logic 59 (5-6):607-657.
    In this paper, we consider, for a set \ of natural numbers, the following notions of finitenessFIN1:There are a natural number l and a bijection f between \\);FIN5:It is not the case that \\), and infinitenessINF1:There are not a natural number l and a bijection f between \\);INF5:\\). In this paper, we systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, (...)
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  3.  12
    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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  4.  9
    Determinacy of Wadge Classes and Subsystems of Second Order Arithmetic.Takako Nemoto - 2009 - Mathematical Logic Quarterly 55 (2):154-176.
    In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi-rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep-Det* ↔ WKL0, 2. Bisep-Det* ↔ ATR0 + (...)
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  5.  14
    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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  6.  11
    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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  7.  21
    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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  8.  11
    On the independence of premiss axiom and rule.Hajime Ishihara & Takako Nemoto - 2020 - Archive for Mathematical Logic 59 (7-8):793-815.
    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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  9.  14
    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.
    We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF. We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs. We also show that the finitary NID is equivalent to its binary fragment and that the elementary NID is equivalent to a variant of NID based on the notion (...)
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