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