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.


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