The Ouroboros Goodstein Principle

TL;DR

This paper introduces a new ordinal assignment for a variant of Goodstein's process, enabling a clearer understanding of its termination and independence properties within foundational systems.

Contribution

It presents a novel ordinal assignment that proves termination and independence simultaneously, clarifying the process's proof-theoretic strength.

Findings

The new ordinal assignment guarantees process termination.

It establishes independence from certain set-theoretic systems.

The analysis delineates restrictions affecting proof-theoretic strength.

Abstract

In arXiv:2508.14768, a variant of Goodstein's original process was recently introduced which, given a set $B\subseteq \mathbb{N}$ of bases, writes each $n\in\mathbb{N}$ in $B$-normal form, namely $n=b^ea+r$, where $b\in B$ the greatest base below $n$. The numbers $e$ and $r$ are then recursively written in $B$-normal form, and finally each base of $B$ is replaced by a corresponding base of some other set $C\subseteq \mathbb{N}$. The resulting process was shown to terminate and to be independent of $\mathsf{KP}$, but the proofs relied on two different ordinal assignments: one monotone but not tight enough to establish independence, and another suitable for independence but not monotone and thus ineffective for proving termination. We introduce a new ordinal assignment that simultaneously yields termination and independence, thereby revealing the `true' ordinals associated with the numbers in the process. This assignment allows us to investigate which restrictions to impose on the process in order for the proof-theoretic strength of its termination to lie between the systems $\mathsf{RCA}_0$, $\mathsf{ACA}_0$, $\mathsf{ATR}_0$ and $\mathsf{KP}$.