Classifying Phase Transition Thresholds for Goodstein Sequences and Hydra Games

Meskens, Frederik; Weiermann, Andreas

doi:10.1007/978-3-319-10103-3_16

Cited by 1 publication

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“…These parametrized Goodstein principles are detailed in Section 9. Meskens and Weiermann [17] have also shown independence results for based on Goodstein principles, albeit our approach is quite different: we consider Goodstein processes where only the initial base is modified, while they consider sequences with slowly changing bases. Our presentation is mostly self-contained, so that in particular Goodstein’s original theorem and its independence from Peano arithmetic are obtained via our methods.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%