Fast Goodstein Walks

Fernández–Duque, David; Weiermann, Andreas

doi:10.48550/arxiv.2111.15328

Cited by 1 publication

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“…For notation systems based on the Ackermann function [1], normal forms are already quite cumbersome, so normal form-free Goodstein principles would lead to substantially more accessible independent statements. We have already applied base-change maximality to fast-growing hierarchies [7], where once again normal forms can be quite complex. We believe that, going forward, the analysis of norm-minimality and, especially, base-change maximality will become an essential ingredient in the study of new and ever-more-powerful Goodstein principles.…”

Section: Proof We Use Transfinite Induction Belowmentioning

confidence: 99%

A Walk With Goodstein

FERNÁNDEZ-DUQUE,

WEIERMANN

2024

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Goodstein's principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein's principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein's result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments IΣn of Peano arithmetic.

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Section: Proof We Use Transfinite Induction Belowmentioning

confidence: 99%