How Definitive the Standard Interpretation of Goodstein's Argument?

Goodstein's argument is, essentially, that the hereditary representation, <>, of any given natural number in the natural number base , 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, (). The standard interpretation of this argument is, firstly, that () must, therefore, converge; and, secondly, that this conclusion - Goodstein's Theorem - is unprovable in Peano Arithmetic, but true under its standard interpretation. 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 highlight why the standard interpretation of Goodstein's argument ought not to be accepted as definitive