(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

A. A mathematical physicist’s conception of thinking about infinity in consistent ways

John Baez is a mathematical physicist, currently working at the math department at U. C. Riverside in California, and also at the Centre for Quantum Technologies in Singapore.

Baez is not only academically active in the areas of network theory and information theory, but also socially active in promoting and supporting the Azimuth Project, which is a platform for scientists, engineers and mathematicians to collaboratively do something about the global ecological crisis.

In a recent post—Large Countable Ordinals (Part 1)—on the Azimuth Blog, Baez confesses to a passionate urge to write a series of blogs—that might even eventually yield a book—about the infinite, reflecting both his fascination with, and frustration at, the challenges involved in formally denoting and talking meaningfully about different sizes of infinity:

“I love the infinite. … It may not exist in the physical world, but we can set up rules to think about it in consistent ways, and then it’s a helpful concept. … Cantor’s realization that there are different sizes of infinity is … part of the everyday bread and butter of mathematics.”

B. Why thinking about infinity in a consistent way must be constrained by an objective, evidence-based, perspective

I would cautiously submit however that (as I briefly argue in this blogpost), before committing to any such venture, whether we can think about the “different sizes of infinity” in “consistent ways“, and to what extent such a concept is “helpful“, are issues that may need to be addressed from an objective, evidence-based, computational perspective in addition to the conventional self-evident, intuition-based, classical perspective towards formal axiomatic theories.

C. Why we cannot conflate the behaviour of Goodstein’s sequence in Arithmetic with its behaviour in Set Theory

Let me suggest why by briefly reviewing—albeit unusually—the usual argument of Goodstein’s Theorem (see here) that every Goodstein sequence over the natural numbers must terminate finitely.

1. The Goodstein sequence over the natural numbers

First, let $g(1, m, [2]), g(2, m, [3]), g(3, m, [4]), \ldots$, be the terms of the Goodstein sequence $G(m)$ for $m$ over the domain $N$ of the natural numbers, where $[i+1]$ is the base in which the hereditary representation of the $i$‘th term of the sequence is expressed.

Some properties of Goodstein’s sequence over the natural numbers

We note that, for any natural number $m$, R. L. Goodstein uses the properties of the hereditary representation of $m$ to construct a sequence $G(m) \equiv \{g(1, m, [2]),\ g(2, m, [3]), \ldots\}$ of natural numbers by an unusual, but valid, algorithm.

Hereditary representation: The representation of a number as a sum of powers of a base $b$, followed by expression of each of the exponents as a sum of powers of $b$, etc., until the process stops. For example, we may express the hereditary representations of $266$ in base $2$ and base $3$ as follows:

$226_{[2]} \equiv 2^{8_{[2]}}+2^{3_{[2]}}+2 \equiv 2^{2^{(2^{2^{0}}+2^{0})}}+2^{2^{2^{0}}+2^{2^{0}}}+2^{2^{0}}$

$226_{[3]} \equiv 2.3^{4_{[3]}}+2.3^{3_{[3]}}+3^{2_{[3]}}+1 \equiv 2.3^{(3^{3^{0}}+3^{0})}+2.3^{3^{3^{0}}}+3^{2.3^{0}}+3^{0}$

We shall ignore the peculiar manner of constructing the individual members of the Goodstein sequence, since these are not germane to understanding the essence of Goodstein’s argument. We need simply accept for now that $G(m)$ is well-defined over the structure $N$ of the natural numbers, and has, for instance, the following properties:

$g(1, 226, [2]) \equiv 2^{2^{2+1}}+2^{2+1}+2$

$g(2, 226, [3]) \equiv (3^{3^{3+1}}+3^{3+1}+3)-1$

$g(2, 226, [3]) \equiv 3^{3^{3+1}}+3^{3+1}+2$

$g(3, 226, [4]) \equiv (4^{4^{4+1}}+4^{4+1}+2)-1$

$g(3, 226, [4]) \equiv 4^{4^{4+1}}+4^{4+1}+1$

If we replace the base $[i+1]$ in each term $g(i, m, [i+1])$ of the sequence $G(m)$ by $[n]$, we arrive at a corresponding sequence of, say, Goodstein’s functions for $m$ over the domain $N$ of the natural numbers.

Where, for instance:

$g(1, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+n$

$g(2, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+2$

$g(3, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+1$

It is fairly straightforward (see here) to show that, for all $i \geq 1$:

Either $g(i, m, [n]) > g(i+1, m, [n])$, or $g(i, m, [n]) = 0$.

Clearly $G(m)$ terminates in $N$ if, and only if, there is a natural number $k > 0$ such that, for any $i > 0$, we have either that $g(i, m, [k]) > g(i+1, m, [k])$ or that $g(i, m, [k]) = 0$.

However, since we cannot, equally clearly, immediately conclude from the axioms of the first-order Peano Arithmetic PA that such a $k$ must exist merely from the definition of the $G(m)$ sequence in $N$, we cannot immediately conclude from the above argument that $G(m)$ must terminate finitely in $N$.

2. The Goodstein sequence over the finite ordinal numbers

Second, let $g_{o}(1, m, [2_{o}]), g_{o}(2, m, [3_{o}]), g_{o}(3, m, [4_{o}]), \ldots$, be the terms of the Goodstein sequence $G_{o}(m)$ over the domain $\omega$ of the finite ordinal numbers $0_{o}, 1_{o}, 2_{o}, \ldots$, where $\omega$ is Cantor’s least transfinite ordinal.

If we replace the base $[(i+1)_{o}]$ in each term $g_{o}(i, m, [(i+1)_{o}])$ of the sequence $G_{o}(m)$ by $[c]$, where $c$ ranges over all ordinals upto $\varepsilon_{0}$, it is again fairly straightforward to show that:

Either $g_{o}(i, m, [c]) >_{o} g_{o}(i+1, m, [c])$, or $g_{o}(i, m, [c]) = 0_{o}$.

Clearly, in this case too, $G_{o}(m)$ terminates in $\omega$ if, and only if, there is an ordinal $k_{o}>_{o} 0_{o}$ such that, for all finite $i > 0$, we have either that $g_{o}(i, m, [k_{o}]) >_{o} g_{o}(i+1, m, [k_{o}])$, or that $g_{o}(i, m, [k_{o}]) =_{o} 0_{o}$.

3. Goodstein’s argument over the transfinite ordinal numbers

If we, however, let $c =_{o} \omega$ then—since the ZF axioms do not admit an infinite descending set of ordinals—it now immediately follows that we cannot have:

$g_{o}(i, m, [\omega]) >_{o} g_{o}(i+1, m, [\omega])$ for all $i > 0$.

Hence $G_{o}(m)$ must terminate finitely in $\omega$, since we must have that $g(i, m, [\omega]) =_{o} 0_{o}$ for some finite $i > 0$.

4. The intuitive justification for Goodstein’s Theorem

The intuitive justification—which must implicitly underlie any formal argument—for Goodstein’s Theorem then is that, since the finite ordinals can be meta-mathematically seen to be in a $1-1$ correspondence with the natural numbers, we can conclude from (2) above that every Goodstein sequence over the natural numbers must also terminate finitely.

5. The fallacy in Goodstein’s argument

The fallacy in this conclusion is exposed if we note that, by (2), $G_{o}(m)$ must terminate finitely in $\omega$ even if $G(m)$ did not terminate in $N$!

6. Why we need to heed Skolem’s cautionary remarks

Clearly, if we heed Skolem’s cautionary remarks (reproduced here) about unrestrictedly corresponding conclusions concerning elements of different formal systems, then we can validly only conclude that the relationship of ‘terminating finitely’ with respect to the ordinal inequality ‘$>_{o}$‘ over an infinite set $S_{0}$ of finite ordinals in any putative interpretation of a first order Ordinal Arithmetic cannot be obviously corresponded to the relationship of ‘terminating finitely’ with respect to the natural number inequality ‘$>$‘ over an infinite set $S$ of natural numbers in any interpretation of PA.

7. The significance of Skolem’s qualification

The significance of Skolem’s qualification is highlighted if we note that we cannot force PA to admit a constant denoting a ‘completed infinity’, such as Cantor’s least ordinal $\omega$, into either PA or into any interpretation of PA without inviting inconsistency.

(The proof is detailed in Theorem 4.1 on p.7 of this preprint. See also this blogpage).

8. PA is finitarily consistent

Moreover, the following paper, due to appear in the December 2016 issue of Cognitive Systems Research, gives a finitary proof of consistency for the first-order Peano Arithmetic PA:

9. Why ZF cannot have an evidence-based interpretation

It also follows from the above-cited CSR paper that ZF axiomatically postulates the existence of an infinite set which cannot be evidenced as true even under any putative interpretation of ZF.

10. The appropriate conclusion of Goodstein’s argument

So, if a ‘completed infinity’ cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency, it would follow in Russell’s colourful phraseology that the appropriate conclusion to be drawn from Goodstein’s argument is that:

(i) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(ii) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of notional interest.

Which raises the issue not only of whether we can think about the different sizes of infinity in a consistent way, but also to what extent we may need to justify that such a concept is helpful to an emerging student of mathematics.

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