Goodstein’s argument in set theory

The set-theoretical form of the argument due to Goodstein is essentially that:

(a) if we take the value of $x$ in the Goodstein Functional Sequence $G_{o}(m_{o})_{[x]}$ over the finite ordinals to be the first limit ordinal $\omega$,

(b) and consider the—necessarily decreasing in this case—ordinal sequence (corresponding to the conditionally decreasing natural number sequence $G(m)_{[z]}$):

$G_{o}(m_{o})_{[\omega]} \equiv \{g_{1}(m_{o})_{[2_{o}\ \hookrightarrow\ \omega]},\ g_{2}(m_{o})_{[3_{o}\ \hookrightarrow\ \omega]},\ g_{3}(m_{o})_{[4_{o}\ \hookrightarrow\ \omega]}, \ldots \}$

(c) then—since the ordinal numbers are well-ordered, and contain a subset of $\omega$ that can be put in a 1-1 correspondence with the set of natural numbers—we may conclude that $\omega > n$ for any given natural number $n$ in some putative non-standard model of first order PA;

(d) hence we need not bother to establish a proof that some natural number $z > n$, too, always exists for all non-zero terms of any Goodstein sequence over the natural numbers in the model,

(e) and, since $G(m)$ and $G_{o}(m_{o})_{[\omega]}$ can always be put in a 1-1 correspondence meta-mathematically, we may meta-mathematically conclude that every Goodstein sequence over the natural numbers terminates finitely over the structure $\mathbb{N}$ of the natural numbers.

What if some Goodstein’s natural number sequence does not converge?

However we note that if there is no natural number $z$ such that $z > n$ for all non-zero terms of some Goodstein sequence, then:

(i) For any given natural number $n$, we can always find a natural number $z > n$ such that the first $n$ terms of the sequence $G(m)_{[z]}$ are a strictly descending sequence of natural numbers in $\mathbb{N}$;

(ii) For any given natural number $n$, we cannot always find a natural number $z > n$ such that the first $n$ terms of the sequence $G_{o}(m_{o})_{[\omega]}$ are a strictly descending sequence of ordinal numbers in $\mathbb{M}$.

The ordinal-based proof of Goodstein’s Theorem is thus the postulation that $G(m)_{[z]}$ and $G_{o}(m_{o})_{[\omega]}$ can always be put in a 1-1 correspondence, and so the above is a contradiction from which we may conclude that there is always some natural number $z$ such that $z > n$ for all non-zero terms of the Goodstein sequence $G(m)$!

The trap of Skolem’s (apparent) paradox

Such a postulation, however, ignores the cautionary remarks by Thoraf Skolem (about unrestrictedly corresponding putative mathematical entities meta-mathematically across domains of different axiom systems) in a 1922 address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, where Skolem first improved upon both the argument and statement of Löwenheim’s 1915 theorem [1] —subsequently labelled as the (downwards) Löwenheim-Skolem Theorem [2].

(Downwards) Löwenheim-Skolem Theorem: If a first-order proposition is satisfied in any domain at all, then it is already satisfied in a denumerably infinite domain.

Skolem then drew attention to a [3] :

“… peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities, of higher number classes, and so forth. How can it be, then, that the entire domain $B$ can already be enumerated by means of the finite positive integers? The explanation is not difficult to find. In the axiomatization, set’ does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set $M$ of the domain $B$ is non-denumerable in the sense of the axiomatization; for this means merely that within $B$ there occurs no one-to-one mapping $\Phi$ of $M$ onto $Z_{o}$ (Zermelo’s number sequence). Nevertheless there exists the possibility of numbering all objects in $B$, and therefore also the elements of $M$, by means of the positive integers; of course such an enumeration too is a collection of certain pairs, but this collection is not a set’ (that is, it does not occur in the domain $B$).”

Notes

Return to 1: Leopold Löwenheim. 1915. On possibilities in the calculus of relatives. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts, p.235, Theorem 2.

Return to 2: Thoralf Skolem. 1922. Some remarks on axiomatized set theory. Text of an address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, 4-7 August 1922. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts, p.293.