The Ouroboros Goodstein Principle

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 $n=b^{e}a+r$, 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 $\mathsf{KP}$.