Generalizing Goodstein's theorem and Cichon's independence proof

TL;DR

This paper extends classical theorems by applying advanced ordinal analysis techniques to stronger logical systems, demonstrating their broader applicability.

Contribution

It generalizes Goodstein's theorem and Cichon's independence proof to the system _0, using recent results and a method adaptable to stronger notation systems.

Findings

Generalization of Goodstein's theorem to _0

Extension of Cichon's independence proof to _0

Method applicable to stronger ordinal notation systems

Abstract

We generalize Goodstein's theorem (Goodstein 1944) and Cichon's independence proof (Cichon 1983) to $\Pi^1_1-\mathrm{CA}_0$ using results from (Wilken 2026). The method is generalizable to stronger notation systems that provide unique terms for ordinals and enjoy Bachmann property.