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  1. Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
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  • An independence result for (II11-CA)+BI.Wilfried Buchholz - 1987 - Annals of Pure and Applied Logic 33 (C):131-155.
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  • Elementary descent recursion and proof theory.Harvey Friedman & Michael Sheard - 1995 - Annals of Pure and Applied Logic 71 (1):1-45.
    We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.
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  • A Uniform Approach to Fundamental Sequences and Hierarchies.Wilfried Buchholz, Adam Cichon & Andreas Weiermann - 1994 - Mathematical Logic Quarterly 40 (2):273-286.
    In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.
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  • 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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  • (1 other version)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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  • Transfinite induction within Peano arithmetic.Richard Sommer - 1995 - Annals of Pure and Applied Logic 76 (3):231-289.
    The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than 0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic techniques. A variety of additional results that come as an application of such techniques are mentioned.
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  • Proof-theoretic analysis of termination proofs.Wilfried Buchholz - 1995 - Annals of Pure and Applied Logic 75 (1-2):57-65.
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