70 found
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  1.  28
    Epsilon substitution method for ID1.Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 121 (2-3):163-208.
    Hilbert proposed the epsilon substitution method as a basis for consistency proofs. Hilbert's Ansatz for finding a solving substitution for any given finite set of transfinite axioms is, starting with the null substitution S0, to correct false values step by step and thereby generate the process S0,S1,… . The problem is to show that the approximating process terminates. After Gentzen's innovation, Ackermann 162) succeeded to prove termination of the process for first order arithmetic. Inspired by G. Mints as an Ariadne's (...)
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  2.  30
    Proof theory for theories of ordinals—I: recursively Mahlo ordinals.Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 122 (1-3):1-85.
    This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od of the system O of ordinal diagrams in Arai 353). This paper is the first published one in which a proof-theoretic analysis à la Gentzen–Takeuti of recursively large ordinals is expounded.
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  3.  23
    A slow growing analogue to buchholz' proof.Toshiyasu Arai - 1991 - Annals of Pure and Applied Logic 54 (2):101-120.
    In this, journal, W. Buchholz gave an elegant proof of a characterization theorem for provably total recursive functions in the theory IDv for the v-times iterated inductive definitions . He characterizes the classes of functions by Hardy functions. In this note we will show that a slow growing analogue to the theorem can be obtained by a slight modification of Buchholz' proof.
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  4.  32
    Epsilon substitution method for theories of jump hierarchies.Toshiyasu Arai - 2002 - Archive for Mathematical Logic 41 (2):123-153.
    We formulate epsilon substitution method for theories (H)α0 of absolute jump hierarchies, and give two termination proofs of the H-process: The first proof is an adaption of Mints M, Mints-Tupailo-Buchholz MTB, i.e., based on a cut-elimination of a specially devised infinitary calculus. The second one is an adaption of Ackermann Ack. Each termination proof is based on transfinite induction up to an ordinal θ(α0+ ω)0, which is best possible.
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  5.  45
    Some results on cut-elimination, provable well-orderings, induction and reflection.Toshiyasu Arai - 1998 - Annals of Pure and Applied Logic 95 (1-3):93-184.
    We gather the following miscellaneous results in proof theory from the attic.1. 1. A provably well-founded elementary ordering admits an elementary order preserving map.2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21.3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic.4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE.5. 5. Intuitionistic fixed (...)
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  6.  52
    Derivability conditions on Rosser's provability predicates.Toshiyasu Arai - 1990 - Notre Dame Journal of Formal Logic 31 (4):487-497.
  7.  26
    Proof theory for theories of ordinals II: Π3-reflection.Toshiyasu Arai - 2004 - Annals of Pure and Applied Logic 129 (1-3):39-92.
    This paper deals with a proof theory for a theory T3 of Π3-reflecting ordinals using the system O of ordinal diagrams in Arai 1375). This is a sequel to the previous one 1) in which a theory for recursively Mahlo ordinals is analyzed proof-theoretically.
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  8.  31
    Derivatives of normal functions and $$\omega $$ ω -models.Toshiyasu Arai - 2018 - Archive for Mathematical Logic 57 (5-6):649-664.
    In this note the well-ordering principle for the derivative \ of normal functions \ on ordinals is shown to be equivalent to the existence of arbitrarily large countable coded \-models of the well-ordering principle for the function \.
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  9.  56
    Wellfoundedness proofs by means of non-monotonic inductive definitions II: first order operators.Toshiyasu Arai - 2010 - Annals of Pure and Applied Logic 162 (2):107-143.
  10.  29
    Ordinal diagrams for recursively Mahlo universes.Toshiyasu Arai - 2000 - Archive for Mathematical Logic 39 (5):353-391.
    In this paper we introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9].
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  11.  45
    Wellfoundedness proofs by means of non-monotonic inductive definitions I: Π₂⁰-operators.Toshiyasu Arai - 2004 - Journal of Symbolic Logic 69 (3):830-850.
    In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.
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  12.  35
    Proof theory of weak compactness.Toshiyasu Arai - 2013 - Journal of Mathematical Logic 13 (1):1350003.
    We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.
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  13.  60
    Ordinal diagrams for Π3-reflection.Toshiyasu Arai - 2000 - Journal of Symbolic Logic 65 (3):1375 - 1394.
    In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...)
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  14.  26
    Consistency proof via pointwise induction.Toshiyasu Arai - 1998 - Archive for Mathematical Logic 37 (3):149-165.
    We show that the consistency of the first order arithmetic $PA$ follows from the pointwise induction up to the Howard ordinal. Our proof differs from U. Schmerl [Sc]: We do not need Girard's Hierarchy Comparison Theorem. A modification on the ordinal assignment to proofs by Gentzen and Takeuti [T] is made so that one step reduction on proofs exactly corresponds to the stepping down $\alpha\mapsto\alpha [1]$ in ordinals. Also a generalization to theories $ID_q$ of finitely iterated inductive definitions is proved.
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  15.  14
    Proof theory for theories of ordinals II:< i> Π_< sub> 3-reflection.Toshiyasu Arai - 2004 - Annals of Pure and Applied Logic 129 (1):39-92.
  16.  33
    Epsilon Substitution Method for [image] -FIX.Toshiyasu Arai - 2006 - Journal of Symbolic Logic 71 (4):1155 - 1188.
    In this paper we formulate epsilon substitution method for a theory $\Pi _{2}^{0}$-FIX for non-monotonic $\Pi _{2}^{0}$ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].
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  17.  29
    A bounded arithmetic AID for Frege systems.Toshiyasu Arai - 2000 - Annals of Pure and Applied Logic 103 (1-3):155-199.
    In this paper we introduce a system AID of bounded arithmetic. The main feature of AID is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems F in Buss 3–29). We show that AID proves the soundness of F , and conversely any Σ 0 b -theorem in AID yields boolean sentences of which F has polysize proofs. Further we define Σ 1 b -faithful interpretations between AID+Σ 0 b -CA and (...)
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  18.  19
    A Simplified Ordinal Analysis of First-Order Reflection.Toshiyasu Arai - 2020 - Journal of Symbolic Logic 85 (3):1163-1185.
    In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system$OT$is introduced based on$\psi $-functions. Provable$\Sigma _{1}$-sentences on$L_{\omega _{1}^{CK}}$are bounded through cut-elimination on operator controlled derivations.
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  19. On the Slowly Well Orderedness of ɛo.Toshiyasu Arai - 2002 - Mathematical Logic Quarterly 48 (1):125-130.
    For α < ε0, Nα denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number-theoretic function f let B denote the length n of the longest descending chain of ordinals <ε0 such that for all i < n, Nαi ≤ f . Simpson [2] called ε0 as slowly well ordered when B is totally defined for f = K · . Let |n| denote the binary length of the (...)
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  20.  33
    Variations on a theme by Weiermann.Toshiyasu Arai - 1998 - Journal of Symbolic Logic 63 (3):897-925.
    Weiermann [18] introduces a new method to generate fast growing functions in order to get an elegant and perspicuous proof of a bounding theorem for provably total recursive functions in a formal theory, e.g., in PA. His fast growing function θαn is described as follows. For each ordinal α and natural number n let T α n denote a finitely branching, primitive recursive tree of ordinals, i.e., an ordinal as a label is attached to each node in the tree so (...)
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  21.  22
    Quick cut-elimination for strictly positive cuts.Toshiyasu Arai - 2011 - Annals of Pure and Applied Logic 162 (10):807-815.
    In this paper we show that the intuitionistic theory for finitely many iterations of strictly positive operators is a conservative extension of Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic.
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  22.  31
    Intuitionistic fixed point theories over set theories.Toshiyasu Arai - 2015 - Archive for Mathematical Logic 54 (5-6):531-553.
    In this paper we show that the intuitionistic fixed point theory FiXi over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.
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  23.  27
    Predicatively computable functions on sets.Toshiyasu Arai - 2015 - Archive for Mathematical Logic 54 (3-4):471-485.
    Inspired from a joint work by A. Beckmann, S. Buss and S. Friedman, we propose a class of set-theoretic functions, predicatively computable set functions. Each function in this class is polynomial time computable when we restrict to finite binary strings.
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  24.  18
    Lifting proof theory to the countable ordinals: Zermelo-Fraenkel set theory.Toshiyasu Arai - 2014 - Journal of Symbolic Logic 79 (2):325-354.
  25.  18
    Proof-theoretic strengths of weak theories for positive inductive definitions.Toshiyasu Arai - 2018 - Journal of Symbolic Logic 83 (3):1091-1111.
  26.  22
    Conservations of first-order reflections.Toshiyasu Arai - 2014 - Journal of Symbolic Logic 79 (3):814-825.
  27. On the Slowly Well Orderedness of ɛo.Toshiyasu Arai - 2002 - Mathematical Logic Quarterly 48 (1):125-130.
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  28.  17
    A Sneak Preview of Proof Theory of Ordinals.Toshiyasu Arai - 2012 - Annals of the Japan Association for Philosophy of Science 20:29-47.
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  29.  13
    Cut-elimination for ω1.Toshiyasu Arai - 2018 - Annals of Pure and Applied Logic 169 (12):1246-1269.
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  30.  17
    Cut-eliminability in Second Order Logic Calculus.Toshiyasu Arai - 2018 - Annals of the Japan Association for Philosophy of Science 27:45-60.
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  31.  34
    Exact bounds on epsilon processes.Toshiyasu Arai - 2011 - Archive for Mathematical Logic 50 (3-4):445-458.
    In this paper we show that the lengths of the approximating processes in epsilon substitution method are calculable by ordinal recursions in an optimal way.
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  32.  3
    Finitary Reductions for Local Predicativity, I: Recursively Regular Ordinals.Toshiyasu Arai, Wilfried Buchholz & Sergei Tupailo - 2002 - Bulletin of Symbolic Logic 8 (3):437.
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  33.  4
    Goodstein Sequences Based on a Parametrized Ackermann–Péter Function.Toshiyasu Arai, Stanley S. Wainer & Andreas Weiermann - 2021 - Bulletin of Symbolic Logic 27 (2):168-186.
    Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA,$\Sigma ^1_1$-DC$_0$, ATR$_0$, up to ID$_1$. The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal (...)
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  34.  30
    Jeremy Avigad. Update procedures and the 1-consistency of arithmetic. Mathematical Logic Quarterly, vol. 48 , pp. 3–13.Toshiyasu Arai - 2003 - Bulletin of Symbolic Logic 9 (1):45-47.
  35.  17
    Nested PLS.Toshiyasu Arai - 2011 - Archive for Mathematical Logic 50 (3-4):395-409.
    In this note we will introduce a class of search problems, called nested Polynomial Local Search (nPLS) problems, and show that definable NP search problems, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{b}_{1}}$$\end{document}-definable functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T^{2}_{2}}$$\end{document} are characterized in terms of the nested PLS.
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  36.  19
    Non‐elementary speed‐ups in logic calculi.Toshiyasu Arai - 2008 - Mathematical Logic Quarterly 54 (6):629-640.
    In this paper we show some non-elementary speed-ups in logic calculi: Both a predicative second-order logic and a logic for fixed points of positive formulas are shown to have non-elementary speed-ups over first-order logic. Also it is shown that eliminating second-order cut formulas in second-order logic has to increase sizes of proofs super-exponentially, and the same in eliminating second-order epsilon axioms. These are proved by relying on results due to P. Pudlák.
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  37.  29
    On the consistency proofs.Toshiyasu Arai - 2007 - Journal of the Japan Association for Philosophy of Science 34 (2):91-99.
  38.  17
    Proof-theoretic strengths of the well-ordering principles.Toshiyasu Arai - 2020 - Archive for Mathematical Logic 59 (3-4):257-275.
    In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions \ on ordinals is shown to be equal to the least fixed point of \. Moreover corrections to the previous paper are made.
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  39.  13
    Wellfoundedness proof with the maximal distinguished set.Toshiyasu Arai - 2023 - Archive for Mathematical Logic 62 (3):333-357.
    In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal \(\sup _{N is an upper bound for the proof-theoretic ordinal of a set theory \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)\). In this paper we show that a second order arithmetic \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) proves the wellfoundedness up to \(\psi _{\varOmega _{1}}(\varepsilon _{\varOmega _{{\mathbb {S}}+N+1}})\) for each _N_. It is easy to interpret \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) in \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)\).
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  40. Three papers on proof theory.W. Buchholz, S. Tupailo & Toshiyasu Arai - 2002 - Bulletin of Symbolic Logic 8 (3):437-438.
  41. REVIEWS-The logic of provability.G. Japaridze, D. De Jongh & Toshiyasu Arai - 2000 - Bulletin of Symbolic Logic 6 (4):472-472.
  42.  24
    2001 european summer meeting of the association for symbolic logic logic colloquium'01.Itay Neeman, Alexander Leitsch, Toshiyasu Arai, Steve Awodey, James Cummings, Rod Downey & Harvey Friedman - 2002 - Bulletin of Symbolic Logic 8 (1):111-180.
  43. REVIEWS-Realizability.A. Troelstra & Toshiyasu Arai - 2000 - Bulletin of Symbolic Logic 6 (4):470-471.
     
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  44.  23
    Consistency Proof via Pointwise Induction.Andreas Weiermann & Toshiyasu Arai - 2002 - Bulletin of Symbolic Logic 8 (4):536.
  45.  15
    Ideas in the epsilon substitution method for -FIX.Toshiyasu Arai - 2005 - Annals of Pure and Applied Logic 136 (1-2):3-21.
    Hilbert proposed the epsilon substitution method as a basis for consistency proofs. Hilbert’s Ansatz for finding a solving substitution for any given finite set of transfinite axioms is, starting with the null substitution S0, to correct false values step by step and thereby generate the process S0,S1,…. The problem is to show that the approximating process terminates. After Gentzen’s innovation, Ackermann [W. Ackermann, Zur Widerspruchsfreiheit der Zahlentheorie, Math. Ann. 117 162–194] succeeded in proving the termination of the process for the (...)
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  46.  17
    Epsilon substitution method for< i> ID< sub> 1(< i> Π_< sub> 1< sup> 0∨< i> Σ< sub> 1< sup> 0). [REVIEW]Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 121 (2):163-208.
  47.  15
    Review: Samuel R. Buss, Handbook of Proof Theory: The Lengths of Proofs. [REVIEW]Toshiyasu Arai - 2000 - Bulletin of Symbolic Logic 6 (4):473-475.
  48.  79
    Gerhard Jäger and Robert F. Stärk. A proof-theoretic framework for logic programming. Handbook of proof theory, edited by Samuel R. Buss, Studies in logic and the foundations of mathematics, vol. 137, Elsevier, Amsterdam etc. 1998, pp. 639–682. [REVIEW]Toshiyasu Arai - 2000 - Bulletin of Symbolic Logic 6 (4):475-476.
  49.  60
    Wilfried Buchholz. Notation systems for infinitary derivations_. Archive for mathematical logic, vol. 30 no. 5–6 (1991), pp. 277–296. - Wilfried Buchholz. _Explaining Gentzen's consistency proof within infinitary proof theory_. Computational logic and proof theory, 5th Kurt Gödel colloquium, KGC '97, Vienna, Austria, August 25–29, 1997, Proceedings, edited by Georg Gottlob, Alexander Leitsch, and Daniele Mundici, Lecture notes in computer science, vol. 1289, Springer, Berlin, Heidelberg, New York, etc., 1997, pp. 4–17. - Sergei Tupailo. _Finitary reductions for local predicativity, I: recursively regular ordinals. Logic Colloquium '98, Proceedings of the annual European summer meeting of the Association for Symbolic Logic, held in Prague, Czech Republic, August 9–15, 1998, edited by Samuel R. Buss, Petr Háajek, and Pavel Pudlák, Lecture notes in logic, no. 13, Association for Symbolic Logic, Urbana, and A K Peters, Natick, Mass., etc., 2000, pp. 465–499. [REVIEW]Toshiyasu Arai - 2002 - Bulletin of Symbolic Logic 8 (3):437-439.
  50.  56
    Matt Fairtlough and Stanley S. Wainer. Hierarchies of provably recursive functions. Handbook of proof theory, edited by Samuel R. Buss, Studies in logic and the foundations of mathematics, vol. 137, Elsevier, Amsterdam etc. 1998, pp. 149–207. [REVIEW]Toshiyasu Arai - 2000 - Bulletin of Symbolic Logic 6 (4):466-467.
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