Vol. 313, No. 2, 2021

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Ackermann and Goodstein go functorial

Juan P. Aguilera, Anton Freund, Michael Rathjen and Andreas Weiermann

Vol. 313 (2021), No. 2, 251–291
DOI: 10.2140/pjm.2021.313.251
Abstract

We present variants of Goodstein’s theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits.

Keywords
Ackermann function, Goodstein's theorem, number representation, reverse mathematics, Veblen hierarchy, well-ordering principles
Mathematical Subject Classification
Primary: 03B30, 03F15, 03F40, 11A67
Milestones
Received: 15 January 2021
Revised: 22 June 2021
Accepted: 2 July 2021
Published: 12 October 2021
Authors
Juan P. Aguilera
Vakgroep Wiskunde
Ghent University
Ghent
Belgium
Institut für Diskrete Mathematik und Geometrie
Vienna University of Technology
Vienna
Austria
Anton Freund
Department of Mathematics
Technical University of Darmstadt
Darmstadt
Germany
Michael Rathjen
Department of Pure Mathematics
University of Leeds
Leeds
United Kingdom
Andreas Weiermann
Vakgroep Wiskunde
Ghent University
Ghent
Belgium