A walk with Goodstein
David Fernández-Duque, Andreas Weiermann
TL;DR
This work analyzes Goodstein's theorem through the lens of optimal notation systems. It introduces norm-minimization and base-change maximality as criteria for canonical normal forms and shows that base-change maximality yields strong termination results across multiple notation systems, including a weak multiplicative variant and an extension to elementary functions. By linking termination times to Hardy functions and exploring phase transitions with respect to fragments of arithmetic, the authors establish robust independence results and generalize Goodstein-type phenomena beyond the classical exponential base. The findings suggest that base-change maximality is a powerful, broadly applicable tool for proving termination and independence results in ordinal- and fast-growing-function regimes, with potential extensions to Ackermann-level systems and beyond.
Abstract
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 grow very quickly, 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 explore notions of optimality for such notation systems and apply them to the classical Goodstein process, to a weaker variant based on multiplication rather than exponentiation, and to a stronger variant based on the Ackermann function. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein's result.