On Completed Infinity

What this is

This is a statement of a position I hold about the foundations of mathematics, written down mostly so that it exists somewhere outside my own head.

I am not a mathematician. I have no result to announce and nothing here is new — the position I describe has been held, in various forms, for a very long time. What follows is my own route to it, an honest account of the standard objection, and a statement of where I think the disagreement actually lies.

I want to be clear at the outset about what I am not claiming. I am not claiming Cantor made an error. I am not claiming standard set theory is inconsistent. Both of those claims are made frequently by amateurs and both are wrong. My disagreement is with a starting assumption, not with any inference drawn from it.

The intuition that started it

The usual proof that some infinities are larger than others is Cantor’s diagonal argument. Given any proposed list of the real numbers between 0 and 1, you construct a new real number by taking the first digit of the first number, the second digit of the second, and so on, and changing each one. The result differs from every number on the list, so no such list can be complete. The reals are therefore said to be uncountable, in contrast to the natural numbers, which are countable.

The obvious question is why the same construction cannot be run on the natural numbers. The answer given is that the diagonal construction produces an object with infinitely many nonzero digits, and every natural number has finitely many digits. The constructed object is therefore not a natural number, and no contradiction follows.

That answer is correct, and I accept it. But it raised a different question for me.

A finite number of digit places can only express finitely many distinct values. Ten digits per place, n places, at most 10ⁿ values. So if a set contains infinitely many distinct elements, it seemed to me that infinitely many digit places must be in use somewhere. Yet every natural number is said to have finitely many digits. Where, then, are the infinitely many digit places?

The standard reply, stated fairly

The reply is that I had conflated two different claims:

1. 1 Every natural number has finitely many digits. true
2. 2 There is a finite bound on how many digits a natural number may have. false

The first is true. The second is false. For any bound you propose, some natural number exceeds it. The set therefore uses unboundedly many digit places without any individual member using infinitely many. Each element is finite; the collection of lengths has no maximum.

Put another way: the set is not claimed to be infinite because some member is infinite. It is claimed to be infinite because the successor function never produces anything outside the set. There is no natural number n for which n + 1 fails to be a natural number.

This is a good answer to the question I asked, and my original argument does not survive it. If you accept the framework, the naturals are infinite and no individual number needs infinitely many digits.

Where I actually disagree

What the reply revealed is that my objection was never really about digits. It was about the word never in “the successor function never terminates.”

The same thing bothers me about the ordinary notation {1, 2, 3, ...}. The ellipsis stands in for a listing that has no end, and then the closing brace asserts that the listing is nonetheless one object. I take the point that this is only informal shorthand, and that the formal definition does not list anything: ℕ is defined as the smallest inductive set, a set containing ∅ and closed under the successor operation.

But that reformulation does not touch what I am objecting to. The closure property replaces the dots; it does not replace the existential claim. Something still has to assert that a set with that property exists, and in Zermelo–Fraenkel set theory that assertion is the Axiom of Infinity. So my objection is not to the notation, and it is not a misreading of the shorthand. It is to the step — in whatever notation — that closes the brace.

Two readings are available:

The first reading I find unproblematic. It says something about a process and its lack of a limit. The second reading asserts the existence of a completed object, and it is that assertion I do not accept.

My position is that these are the only two coherent options, and that standard mathematics occupies an uncomfortable middle position between them: the successor process is treated as completed, while every product of it remains finite. If the process is genuinely completed, I would expect the completion to be reflected somewhere in the resulting object. It is not. If it is not completed, then what we have is unboundedness, not infinity.

I recognise that mainstream mathematics does not see this as a middle position at all, and regards “infinitely many finite things” as perfectly coherent. I simply do not, and I have not encountered an argument that dissolves the difficulty rather than restating the framework in which it does not arise.

Put as plainly as I can, my commitment is this: every count is a whole number.

I do not offer that as an argument. It is the position itself, in one line, and it can no more be proved than the Axiom of Infinity can. What it does is make the disagreement easy to see. If there were a single completed object containing every natural number, its count could not be a whole number: whatever whole number k is proposed, the numbers 1 through k + 1 are all members, so k is too small. Standard mathematics answers by admitting a count that is not a whole number. I answer by declining the object.

The honest status of the disagreement

The thing that took me longest to find out, and that I think is worth stating plainly, is this:

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The existence of a completed infinite set is not a theorem of standard mathematics. It is an axiom.

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In Zermelo–Fraenkel set theory it is called the Axiom of Infinity, and it asserts, in effect, that a set containing all the natural numbers exists. It cannot be derived from the other axioms. If you remove it, the remaining axioms are satisfied by the hereditarily finite sets — a perfectly coherent universe in which every set is finite. Rejecting completed infinity does not produce a contradiction. It produces a different, smaller mathematics.

So the disagreement is not over a derivation. It is over a starting point, which was adopted because of what could be built on top of it.

I think that is a legitimate reason to adopt an axiom. I do not think it is the same thing as being shown that the axiom describes something. And since I do not think a completed infinite totality is coherent, I do not think it does — which means I am disagreeing with the people who hold otherwise, not merely working in a different framework from them. I would rather say that plainly than pretend the difference is one of taste.

What accepting this costs

I want to be honest that the cost is real and I am not in a position to pay it.

Standard analysis — limits, continuity, integration, measure theory — is constructed on the real numbers, which are built from completed infinite sets. Most of modern probability and much of theoretical physics inherits this. It is sometimes suggested that constructive or predicative mathematics rescues most of this, and a surprising amount can indeed be recovered in weak systems. Those systems are usually written as though the natural numbers were a completed totality, but for much of what they prove this turns out not to matter. Solomon Feferman argued that the mathematics science actually uses needs nothing stronger than ordinary arithmetic, and work by Harvey Friedman and Stephen Simpson showed that much of what large parts of analysis prove about whole numbers can be proved again without any completed infinity at all.

The cost is still real, but narrower than it first appears. What is lost is Cantor’s hierarchy of infinities, some generality, and a small number of facts about ordinary whole numbers whose only known proofs pass through the infinite. Goodstein’s theorem, which says that a certain process on whole numbers always comes to an end, is the best-known. Strict finitism, which doubts even that counting can always go on, gives up considerably more.

I am not proposing that mathematics should be rebuilt. I am recording that I do not believe one of its starting assumptions, while continuing to use the mathematics that rests on it, in the same way one might use a map known to be a projection.

The tradition this belongs to

I arrived at this on my own and found out only afterwards that it has a name. Anyone reaching the same place should know the following names earlier than I did:

• Aristotle drew the distinction between potential and actual infinity in the Physics, and accepted only the former.
• Gauss in an 1831 letter to Schumacher, objected to the use of an infinite magnitude as a completed thing, calling it a manner of speaking rather than a legitimate object.
• Poincaré was a sustained critic of Cantorian set theory.
• Brouwer founded intuitionism, which admits potential but not actual infinity.
• Weyl in Das Kontinuum, attempted a predicative reconstruction of analysis out of similar misgivings.
• Yessenin-Volpin and, more recently, Zeilberger and Nelson have defended stricter forms — ultrafinitism — in which even very large finite numbers are treated with suspicion.

The relevant survey articles are the Stanford Encyclopedia of Philosophy entries on finitism, intuitionism, and the philosophy of mathematics. They are free, and they are better than anything I could write.

Summary

• Cantor’s diagonal argument is a valid derivation within ZFC. I deny one of its premises, so I do not accept its conclusion. This is not a claim that the derivation contains an error.
• My original objection, that an infinite set requires infinitely many digit places, does not survive contact with the distinction between unbounded and infinite. I record it here because it is what led me to the real question.
• The real question is whether a completed infinite totality is coherent. I do not think it is. A process with no stopping point cannot, in my view, be treated as having stopped.
• This is a disagreement about an axiom, not a claim of error.
• The position has a long history and I make no claim to originality.

Anyone who wants to tell me where this goes wrong is welcome to. I would rather know.