A new proof of Goodstein's Theorem

Abstract

Goodstein sequences are numerical sequences in which a natural number m, expressed as the complete normal form to a given base a, is modified by increasing the value of the base a by one unit and subtracting one unit from the resulting expression. As initially defined, the first term of the Goodstein sequence is the complete normal form of m to base 2. Goodstein's Theorem states that, for all natural numbers, the Goodstein sequence eventually terminates at zero. Goodstein's Theorem was originally proved using the well-ordered properties of transfinite ordinals. The theorem was also shown to be unprovable-in-PA (Peano Arithmetic) using transfinite induction and Godel's Second Incompleteness Theorem. This article describes a proof of Goodstein's Theorem in first-order arithmetic that contradicts the theorem's unprovability-in-PA. The proof uses mathematical induction and is applied (via the super-exponential function) to a generalized version of the Goodstein sequences. Such a proof demonstrates the inconsistency of classical set theory, more precisely the combination of the Zermelo-Fraenkel axioms and the axiom of choice (ZFC).