  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.
