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.

Bhupinder Singh Anand

Advertisements