The fractal Goodstein principle

TL;DR

This paper introduces a generalized fractal Goodstein process that extends the original by using a hierarchy of bases, demonstrating its termination independently of strong set-theoretic systems.

Contribution

It defines a new hierarchical process generalizing the Goodstein process and proves its termination without relying on strong set-theoretic assumptions.

Findings

The process always terminates.

Termination is independent of Kripke-Platek set theory.

Termination does not depend on Bachmann-Howard strength theories.

Abstract

The original Goodstein process is based on writing numbers in hereditary $b$-exponential normal form: that is, each number $n$ is written in some base $b\geq 2$ as $n=b^ea+r$, with $e$ and $r$ iteratively being written in hereditary $b$-exponential normal form. We define a new process which generalises the original by writing expressions in terms of a hierarchy of bases $B$, instead of a single base $b$. In particular, the `digit' $a$ may itself be written with respect to a smaller base $b'$. We show that this new process always terminates, but termination is independent of Kripke-Platek set theory, or other theories of Bachmann-Howard strength.