Goodstein’s argument [1] is essentially that the hereditary representation $m_{[b]}$ in Peano Arithmetic [2], of any given natural number $m$ in the natural number base $b$ can be mirrored in Cantor’s Ordinal Arithmetic [3], and used to yield a `finite’ [4], decreasing, sequence of transfinite ordinals, each of which is not smaller [5] than the `finite’ 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 (Goodstein’s Theorem); second that this number-theoretic proposition is unprovable in any formal system of Peano Arithmetic, but expresses a truth under the standard interpretation of the Arithmetic that appeals necessarily to transfinite reasoning (Kirby-Paris Theorem [6]); and third that Goodstein’s Theorem is, in a sense, a proposition that under such interpretation expresses a more natural independence phenomenon than Gödel’s Theorem on formally unprovable, but interpretatively true, sentences of any formal system of Peano Arithmetic.

However we note first that Gödel’s reasoning can be carried out in a weak Arithmetic such as Robinson’s system Q [7]), which does not admit mathematical induction.

The truth of the unprovable Gödel sentence could thus be reasonably argued as being even more intuitive than the truth—under the standard interpretation—of any number-theoretic assertion of Peano Arithmetic that necessarily appeals to mathematical induction.

Moreover both truths are classically accepted as constructive and intuitionistically unobjectionable.

We note further that Goodstein’s Theorem involves a non-constructive—hence non-verifiable—concept of mathematical truth [8] that, prima facie, is of a higher order of intuition—in a manner of speaking—than that required to see that Gödel’s formally unprovable sentence is a true number-theoretic assertion of Peano Arithmetic under its standard interpretation.[9]

If, therefore, the proof of Goodstein’s Theorem is to be considered as having established both an unprovable proposition of Peano Arithmetic that is true under its standard interpretation, and a more natural independence phenomenon than Gödel’s, then such truth too must reasonably be a consequence of some constructive—and intuitionistically unobjectionable—interpretation of Goodstein’s reasoning.

NOTES

Return to 1: Goodstein, R. L. 1944. On the Restricted Ordinal Theorem. J. Symb. Logic 9, 33-41, 1944.

Return to 2: By Peano Arithmetic we mean the arithmetic of the natural numbers that is definable in a formal number theory such as Mendelson’s theory S (See Mendelson, Elliott. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton, p102).

Return to 3: By Cantor’s Ordinal Arithmetic we mean the arithmetic of the ordinal numbers that are definable in a formal set theory such as ZFC or NBG as detailed in, for instance, Elliott Mendelson (1964 ed.) Introduction to Mathematical Logic, p.160.

Return to 4: We note that for a sequence of ordinals to be termed as finite it must be a well-defined set in Ordinal Arithmetic.

Return to 5: In the sense in which this relation is defined in Ordinal Arithmetic.

Return to 6: Kirby, L. and Paris, J. 1982. Accessible independence results for Peano arithmetic. Bulletin of the London Math. Soc. 4, 1982, pp.285-293.

Return to 7: Robinson, R. An essentially undecidable axiom system. In Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950. (Graves et al, editors). 1952. American Mathematical Society, Providence, RI, pp. 729-730.

Return to 8: Necessarily so, according to a reasonable interpretation of the Kirby-Paris Theorem.

Return to 9: Because, as Gödel pertinently notes in his seminal 1931 paper, the truth of his unprovable sentence under its standard interpretation is meta-mathematically verifiable constructively, in an intuitionistically unobjectionable manner, in Peano Arithmetic (See Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In Martin Davis (ed.). 1965. The Undecidable. Raven Press, New York., p26).