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  1. Accessible Recursive Functions.Stanley S. Wainer - 1999 - Bulletin of Symbolic Logic 5 (3):367-388.
    The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can a (...)
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  • Goodstein Sequences for Prominent Ordinals Up to the Ordinal of Π11 -CAo.Andreas Weiermann & Gunnar Wilken - 2013 - Annals of Pure and Applied Logic 164 (12):1493-1506.
    We introduce strong Goodstein principles which are true but unprovable in strong impredicative theories like IDn.
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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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  • 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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  • Induction and Inductive Definitions in Fragments of Second Order Arithmetic.Klaus Aehlig - 2005 - Journal of Symbolic Logic 70 (4):1087 - 1107.
    A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae (...)
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  • Proof-Theoretic Analysis of Termination Proofs.Wilfried Buchholz - 1995 - Annals of Pure and Applied Logic 75 (1-2):57-65.
  • Inductive Definitions Over a Predicative Arithmetic.Stanley S. Wainer & Richard S. Williams - 2005 - Annals of Pure and Applied Logic 136 (1-2):175-188.
    Girard’s maxim, that Peano Arithmetic is a theory of one inductive definition, is re-examined in the light of a weak theory EA formalising basic principles of Nelson’s predicative Arithmetic.
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  • Investigations on Slow Versus Fast Growing: How to Majorize Slow Growing Functions Nontrivially by Fast Growing Ones. [REVIEW]Andreas Weiermann - 1995 - Archive for Mathematical Logic 34 (5):313-330.
    Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π 1 l −TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast growing) extended Grzegorczyk (...)
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