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.

Bhupinder Singh Anand

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