How definitive is the standard interpretation of Goodstein's argument?

Abstract

Goodstein's argument is essentially that the hereditary representation m_{[b]} of any given natural number m in the natural number base b can be mirrored in Cantor Arithmetic, and used to well-define a finite decreasing sequence of transfinite ordinals, each of which is not smaller than the ordinal corresponding to the corresponding member of Goodstein's sequence of natural numbers G(m). The standard interpretation of this argument is first that G(m) must therefore converge; and second that this conclusion---Goodstein's Theorem---is unprovable in Peano Arithmetic but true under the standard interpretation of the Arithmetic. We argue however that even assuming Goodstein's Theorem is indeed unprovable in PA, its truth must nevertheless be an intuitionistically unobjectionable consequence of some constructive interpretation of Goodstein's reasoning. We consider such an interpretation, and construct a Goodstein functional sequence to highlight why the standard interpretation of Goodstein's argument ought not to be accepted as a definitive property of the natural numbers.