Goodstein’s Theorem

Formally, Goodstein’s ordinal-based argument is that since there are no infinitely descending sequences of ordinals, the sequence of ordinal numbers:

$G_{o}(m_{o})_{[\omega]} \equiv \{g_{1}(m_{o})_{[2_{o}\ \hookrightarrow\ \omega]},\ g_{2}(m_{o})_{[3_{o}\ \hookrightarrow\ \omega]},\ g_{3}(m_{o})_{[4_{o}\ \hookrightarrow\ \omega]}, \ldots \}$

can be shown to terminate finitely for any given finite ordinal $m_{o}$ in any putative model $\mathbb{M}$ of a formal Ordinal Arithmetic.

Hence the following proposition—where $g_{y}(x)$ denotes the $y^{th}$ term of the Goodstein ordinal sequence $G_{o}(x)$—would hold in every model of the Arithmetic:

$(\forall x)((x \in \omega) \rightarrow (\exists y)( (y \in \omega) \wedge g_{y}(x) = 0_{o}))$

Goodstein’s Theorem over the natural numbers is then the conclusion that:

$(\exists y)(g_{y}(m)) = 0$

holds for any given natural number $m$ in the standard interpretation of the first order Peano Arithmetic PA.

However this argument implicitly assumes that every putative model of a formal Ordinal Arithmetic is a model of PA.

The case against unrestrictedly concluding arithmetical properties from set-theoretical reasoning

Now there is a suitable interpretation of the primitive symbols of PA in ZF which transforms the axioms of PA into theorems of ZF whilst preserving its rules of inference [1].

This interpretation explicitly assumes the existence of Cantor’s first limit-ordinal $\omega$ in the domain of the interpretation so that the PA-formula $[(\forall x)(f(x))]$, for instance, transforms as the ZF formula:

$[(\forall x)((x \in \omega) \rightarrow (f(x)))]$.

Every putative model of  the arithmetic of the ordinals in ZF is then a model of the transformed axioms of PA.

The question is: Can we assume that some model of  the arithmetic of the ordinals in ZF is also a model of PA?

Now we shall show that this is impossible since, in any model of first-order PA, every element in the domain of the interpretation—except 0—is necessarily a successor.

Hence no model of PA can have an initial non-successor limit-ordinal $\omega$ such that $\omega > n$ for all natural numbers $n \geq 1$.

Thus Goodstein’s argument arguably conflates a postulated interpretation of PA which transforms some formulas of PA into provable formulas of ZF with an interpretation of PA under which these formulas are held to be true over $\mathbb{N}$

Notes

Return to 1: Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton, p192.