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Predicatively unprovable termination of the Ackermannian Goodstein process


Authors: Toshiyasu Arai, David Fernández-Duque, Stanley Wainer and Andreas Weiermann
Journal: Proc. Amer. Math. Soc. 148 (2020), 3567-3582
MSC (2010): Primary 03F40, 03D20, 03D60
DOI: https://doi.org/10.1090/proc/14813
Published electronically: April 27, 2020
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Abstract: The classical Goodstein process gives rise to long but finite sequences of natural numbers whose termination is not provable in Peano arithmetic. In this manuscript we consider a variant based on the Ackermann function. We show that Ackermannian Goodstein sequences eventually terminate, but this fact is not provable using predicative means.


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Additional Information

Toshiyasu Arai
Affiliation: University of Tokyo, 7-Chome-3-1 Hongo, Bunkyo City, Tokyo 113-8654, Japan
Email: tosarai@ms.u-tokyo.ac.jp

David Fernández-Duque
Affiliation: Ghent University, Krijgslaan 281, S8, 9000 Gent, Belgium
Email: david.fernandezduque@ugent.be

Stanley Wainer
Affiliation: University of Leeds, Woodhouse Lane, Leeds LS2 9JT, United Kingdom
Email: s.s.wainer@leeds.ac.uk

Andreas Weiermann
Affiliation: Ghent University, Krijgslaan 281, S8, 9000 Gent, Belgium
Email: andreas.weiermann@ugent.be

DOI: https://doi.org/10.1090/proc/14813
Received by editor(s): May 31, 2019
Received by editor(s) in revised form: August 2, 2019
Published electronically: April 27, 2020
Additional Notes: The second and fourth authors were supported in part by the Hausdorff Institute for Mathematics (Bonn, Germany) and the Fonds Wetenschappelijk Onderzoek (Flanders, Belgium).
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society