Giant and illusionary giant Goodstein principles

TL;DR

This paper examines various natural Goodstein principles linked to Ackermann functions, revealing some are proof-theoretically strong and unprovable in certain systems, while others appear strong but are actually weaker.

Contribution

It distinguishes between genuinely strong and illusionary Goodstein principles related to Ackermann functions, providing proof-theoretic strength analysis.

Findings

Giant Goodstein principles are unprovable in PA and ID_1.

Illusionary giant principles are weaker than they appear.

The analysis clarifies the proof-theoretic strength of these principles.

Abstract

We analyze several natural Goodstein principles which themselves are defined with respect to the Ackermann function and the extended Ackermann function. These Ackermann functions are well established canonical fast growing functions labeled by ordinals not exceeding $\varepsilon_0$. Among the Goodsteinprinciples under consideration, the giant ones, will be proof-theoretically strong (being unprovable in $\mathrm{PA}$ in the Ackermannian case and being unprovable in $\mathrm{ID}_1$ in the extended Ackermannian case) whereas others, the illusionary giant ones, will turn out to be comparatively much much weaker although they look strong at first sight.