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  1. Dynamic Ordinal Analysis.Arnold Beckmann - 2003 - Archive for Mathematical Logic 42 (4):303-334.
    Dynamic ordinal analysis is ordinal analysis for weak arithmetics like fragments of bounded arithmetic. In this paper we will define dynamic ordinals – they will be sets of number theoretic functions measuring the amount of sΠ b 1(X) order induction available in a theory. We will compare order induction to successor induction over weak theories. We will compute dynamic ordinals of the bounded arithmetic theories sΣ b n (X)−L m IND for m=n and m=n+1, n≥0. Different dynamic ordinals lead to (...)
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  • Intuitionistic Fixed Point Theories for Strictly Positive Operators.Christian Rüede & Thomas Strahm - 2002 - Mathematical Logic Quarterly 48 (2):195-202.
    In this paper it is shown that the intuitionistic .xed point theory equation image for α times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and equation image sentences over the theory equation image for α times iterated arithmetic comprehension without set parameters.This generalizes results previously due to Buchholz [5] and Arai [2].
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  • 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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  • 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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  • 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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  • Some Interesting Connections Between the Slow Growing Hierarchy and the Ackermann Function.Andreas Weiermann - 2001 - Journal of Symbolic Logic 66 (2):609-628.
    It is shown that the so called slow growing hierarchy depends non trivially on the choice of its underlying structure of ordinals. To this end we investigate the growth rate behaviour of the slow growing hierarchy along natural subsets of notations for $\Gamma_0$. Let T be the set-theoretic ordinal notation system for $\Gamma_0$ and $T^{tree}$ the tree ordinal representation for $\Gamma$. It is shown in this paper that $_{\alpha \in T}$ matches up with the class of functions which are elementary (...)
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  • 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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  • Proof Systems for BAT Consequence Relations.Pawel Pawlowski - 2018 - Logic Journal of the IGPL 26 (1):96-108.
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  • The Omega-Rule Interpretation of Transfinite Provability Logic.David Fernández-Duque & Joost J. Joosten - 2018 - Annals of Pure and Applied Logic 169 (4):333-371.
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