A simple finitary proof of Goodstein's Theorem

TL;DR

The paper presents a straightforward finitary proof demonstrating that all Goodstein sequences terminate, based on an initial assumption that is shown to be false, thus confirming the sequences' finiteness.

Contribution

It offers a simple, finitary proof of Goodstein's Theorem by examining the properties of specific sequences and their termination behavior.

Findings

The initial assumption about the existence of certain natural numbers is false.

All Goodstein sequences over natural numbers terminate finitely.

The proof clarifies the logical structure behind Goodstein's Theorem.

Abstract

We assumed that, for every natural number k, there is a natural number u such that the (k-1)th term of G(u) is k^k, and that G(u) terminates finitely. It immediately follows that every Goodstein Sequence G(m) over the natural numbers must terminate finitely. The assumption is easily seen to be false since there is no m such that the third term of g(m) is 4^4.