Formally, Goodstein’s ordinal-based argument is that since there are no infinitely descending sequences of ordinals, the sequence of ordinal numbers:
can be shown to terminate finitely for any given finite ordinal in any putative model of a formal Ordinal Arithmetic.
Hence the following proposition—where denotes the term of the Goodstein ordinal sequence —would hold in every model of the Arithmetic:
Goodstein’s Theorem over the natural numbers is then the conclusion that:
holds for any given natural number 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 .
This interpretation explicitly assumes the existence of Cantor’s first limit-ordinal in the domain of the interpretation so that the PA-formula , for instance, transforms as the ZF formula:
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 such that for all natural numbers .
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
Return to 1: Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton, p192.