**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:

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 ^{[1]}.

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

**Notes**

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

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