The argument of Goodstein’s Theorem ^{[1]} is that:

(i) The natural number considerations involved in the construction of Goodstein’s sequence can all be formalised over the finite ordinals (sets) in any putative model of a first order Ordinal Arithmetic;

(ii) Any such Ordinal Arithmetic does allow us to *postulate* the existence of an ordinal—Cantor’s first limit ordinal —such that:

(a) if ^{[2]} —say —is the sequence of finite ordinals in that corresponds to Goodstein’s natural number sequence in ,

(b) and —say —the corresponding Goodstein Functional Sequence over ,

(c) then for any given natural number :

If in ,

then ;

(iii) The sequence of ordinals cannot descend infinitely in ;

(iv) Hence terminates finitely in .

If Ordinal Arithmetic is consistent (i.e., it has a model), then such a must `exist’ and the above argument is valid in . However, Goodstein’s Theorem is the conclusion that Goodstein’s sequence must *therefore* terminate finitely in !

Prima facie such a conclusion from the ordinal-based reasoning challenges belief insofar as—at heart—the argument *essentially* appears to be that, since Goodstein’s natural number sequence obviously `terminates finitely’ if, and only if, it is bounded above in ^{[3]} with respect to the arithmetical relation `‘, we may conclude the existence of such a bound since Goodstein’s ordinal sequence :

(a) *is* bounded above by in ;

(b) `terminates finitely’ with respect to the ordinal relation `‘;

(c) can be put in a 1-1 correspondence with ;

and since the natural numbers can be put into a 1-1 correspondence with the finite ordinals!

Such disbelief is justified since the above invalidly presumes that the structure of the natural numbers is isomorphic to the sub-structure of the finite ordinals in the structure of the ordinals below , and so the property of `terminating finitely’ with respect to the ordinal relation ` in any putative model of a first order Ordinal Arithmetic must interpret as the property of `terminatingly finitely’ with respect to the arithmetical relation ` in any model of PA.

**NOTES**

Return to 1: See, for instance, Andrès Eduardo Caicedo. 2007. *Goodstein’s Theorem*. Revista Colombiana de Matemàticas, Volume 41(2007)2, pp. 381-391.

Return to 2: Notation: We denote by the ordinal corresponding to the natural number ; by `‘ and `‘ the function/relation letters in ordinal arithmetic corresponding to the function/relation letters `‘ and `‘ in Peano Arithmetic, etc.

Return to 3: Although we do not address the question here, it can be shown without appealing to any transfinite considerations that cannot oscillate for any natural number .

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