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A WALK WITH GOODSTEIN

Part of: Computability and recursion theory Proof theory and constructive mathematics

Published online by Cambridge University Press:  17 January 2024

DAVID FERNÁNDEZ-DUQUE Open the ORCID record for DAVID FERNÁNDEZ-DUQUE [Opens in a new window]  and
ANDREAS WEIERMANN Open the ORCID record for ANDREAS WEIERMANN [Opens in a new window]
DAVID FERNÁNDEZ-DUQUE
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BARCELONA BARCELONA, SPAIN E-mail: fernandez-duque@ub.edu
ANDREAS WEIERMANN
Affiliation:
DEPARTMENT OF MATHEMATICS: ANALYSIS LOGIC AND DISCRETE MATHEMATICS GHENT UNIVERSITY GHENT, BELGIUM E-mail: Andreas.Weiermann@UGent.be
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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 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 $\mathsf {I}\Sigma _n$ of Peano arithmetic.

Keywords

Goodstein’s theoremproofs of independencefast-growing functions

MSC classification

Primary: 03F40: Gödel numberings and issues of incompleteness
Secondary: 03D20: Recursive functions and relations, subrecursive hierarchies 03D60: Computability and recursion theory on ordinals, admissible sets, etc.
Type
Article
Information
Bulletin of Symbolic Logic , Volume 30 , Issue 1 , March 2024 , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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