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 $\mathbb{M}$ 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 $\omega$—such that:

(a) if $\{g_{k}(m_{o})\}$ [2] —say $G_{o}(m_{o})$—is the sequence of finite ordinals in $\mathbb{M}$ that corresponds to Goodstein’s natural number sequence $G(m)$ in $\mathbb{N}$,

(b) and $\{g_{k}(x)_{[(k_{o}+_{o}1_{o})\ \hookrightarrow\ x]}: k > 0\}$—say $G_{o}(m_{o})_{[x]}$—the corresponding Goodstein Functional Sequence over $\mathbb{M}$,

(c) then for any given natural number $k > 0$:

If $g_{k}(m_{o})_{[(k_{o}+_{o}1_{o})\ \hookrightarrow\ \omega]} >_{o} 0_{o}$ in $G_{o}(m_{o})_{[\omega]}$,

then $g_{k}(m_{o})_{[(k_{o}+_{o}1_{o})\ \hookrightarrow\ \omega]} >_{o} g_{k+1}(m_{o})_{[(k_{o}+_{o}2_{o})\ \hookrightarrow\ \omega]}$;

(iii) The sequence $\{g_{1}(m_{o})_{[2_{o}\ \hookrightarrow\ \omega]},\ g_{2}(m_{o})_{[3_{o}\ \hookrightarrow\ \omega]}, \ldots\}$ of ordinals cannot descend infinitely in $\mathbb{M}$;

(iv) Hence $G_{o}(m_{o})$ terminates finitely in $\mathbb{M}$.

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

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 $G(m)$ obviously terminates finitely’ if, and only if, it is bounded above in $\mathbb{N}$ [3] with respect to the arithmetical relation $>$‘, we may conclude the existence of such a bound since Goodstein’s ordinal sequence $G_{o}(m_{o})$:

(a) is bounded above by $\omega$ in $\mathbb{M}$;

(b) terminates finitely’ with respect to the ordinal relation $>_{o}$‘;

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

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 $\mathbb{N}$ of the natural numbers is isomorphic to the sub-structure of the finite ordinals in the structure of the ordinals below $\epsilon_{0}$, and so the property of terminating finitely’ with respect to the ordinal relation $>_{o}$ 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 $m_{o}$ the ordinal corresponding to the natural number $m$; by $+_{o}$‘ and $>_{o}$‘ 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 $G(m)$ cannot oscillate for any natural number $m$.

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