The Ouroboros Goodstein Principle
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908902821462016 |
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| author | Fernández-Duque, David Morreel, Milan Weiermann, Andreas |
| author_facet | Fernández-Duque, David Morreel, Milan Weiermann, Andreas |
| contents | 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 $\mathsf{KP}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_19981 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Ouroboros Goodstein Principle Fernández-Duque, David Morreel, Milan Weiermann, Andreas Logic 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 $\mathsf{KP}$. |
| title | The Ouroboros Goodstein Principle |
| topic | Logic |
| url | https://arxiv.org/abs/2603.19981 |