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GOODSTEIN SEQUENCES BASED ON A PARAMETRIZED ACKERMANN–PÉTER FUNCTION

Part of: Computability and recursion theory Proof theory and constructive mathematics

Published online by Cambridge University Press:  02 July 2021

TOSHIYASU ARAI ,
STANLEY S. WAINER  and
ANDREAS WEIERMANN
TOSHIYASU ARAI
Affiliation:
UNIVERSITY OF TOKYOTOKYO, JAPANE-mail: tosarai@ms.u-tokyo.ac.jp
STANLEY S. WAINER
Affiliation:
UNIVERSITY OF LEEDSLEEDS, UKE-mail: s.s.wainer@leeds.ac.uk
ANDREAS WEIERMANN
Affiliation:
UNIVERSITY OF GHENTGHENT, BELGIUME-mail: andreas.weiermann@ugent.be
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Abstract

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 form representations of positive integers.

Keywords

proof-theoretic ordinalsfast-growing functionsHardy hierarchyindependence results

MSC classification

Primary: 03F15: Recursive ordinals and ordinal notations
Secondary: 03F25: Relative consistency and interpretations 03F30: First-order arithmetic and fragments 03F35: Second- and higher-order arithmetic and fragments 03D55: Hierarchies
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 2 , June 2021 , pp. 168 - 186
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Copyright
© The Association for Symbolic Logic 2021

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