Ackermann and Goodstein go functorial

TL;DR

This paper explores variants of Goodstein's theorem linked to arithmetical comprehension and transfinite recursion, revealing their equivalence over weak base theories and connecting ordinal hierarchies with the Ackermann function.

Contribution

It introduces new variants of Goodstein's theorem that entail complex infinite objects and relates the Veblen hierarchy to an extension of the Ackermann function.

Findings

Variants of Goodstein's theorem are equivalent to arithmetical comprehension and transfinite recursion.

The Veblen hierarchy is closely related to an extension of the Ackermann function.

These variants necessarily involve the existence of complex infinite objects.

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.