Returning to a discussion we were having back in August of 2010, I want to see if I can get Peter Lupu to agree with me on one point: It is not obvious or compellingly arguable (arguable in a 'knock-down' way) that there are infinite sets. Given my aporetic concerns, which Peter thoroughly understands, I will be satisfied if I can convince him that the italicized sentence is true, and therefore that the thesis that the infinite in mathematics is potential only is respectable and defensible and has never been shown definitively to be false. Let us start with a datanic claim that no one can reasonably deny:
1. There are infinitely many natural numbers.
If anyone were to deny (1) I would show him the door. For anyone who denied (1) would show by his denial that he did not grasp the sense of 'natural number.' The question, however, is whether from (1) we can validly infer
2. There is a set of natural numbers.
If there is such a set, then of course it is an infinite set, an actually infinite set. (Talk of potentially infinite sets is nonsense as I have argued in previous posts.) So, if the inference from (1) to (2) is valid, we have a knock-down proof of actual infinity. For if there are infinite sets then there are actual infinities, completed infinities.
Now I claim that it is obvious that (2) does not follow from (1). For it might be that the naturals do not form a set. A set is a one-over-many, a definite single object distinct from each of its members and from all of them. It should be obvious, then, that from the fact that there ARE many Fs it does not straightaway follow that there IS a single thing comprising these many Fs. This is especially clear in the case of infinitely many Fs.
But from Logic 101 we know that an invalid argument can have a true conclusion. So, despite the fact that (2) does not follow from (1), it might still be the case that (2) is true. I might be challenged to say what (1) could mean if it does not entail (2). Well, I can say that however many numbers we have counted, we can count more. If we have counted up to n, we can add 1 and arrive at n + 1. The procedure is obviously indefinitely iterable. That means: there is no definite n such one can perform the procedure only n times. One can perform it indefinitely many times. Accordingly, 'infinitely many' behaves differently than 'finitely many.' If something can be done only finitely many times, then there is some finite n such that n is the number of times the thing can be done. But 'infinitely many' does not require us to say that that there is some definite transfinite cardinal which is the number of times a thing that can be done infinitely many times can be done. For 'infinitely many' can be construed to mean: indefinitely many.
On this approach, the naturals do not form a single complete object, the set N, but are such that their infinity is an endless task. The German language allows a cute way of putting this: Die Zahlen sind nicht gegeben, sondern aufgegeben. In Aristotelian terms, the infinity of the naturals is potential not actual. But if you find these words confusing, as Peter does, they can be avoided. A wise man never gets hung up on words.
Now if I understood him aright, one of Peter's objections is that the approach I am sketching implies that there is a last number, one than which there is no greater. But it has no such implication. For the very sense of 'natural number' rules out there being a last number, and this sense is understood by all parties to the dispute. There cannot be a last number precisely because of the very meaning of 'number.' Every natural number is such that it has an immediate successor. But from this it does not follow that there is a set of natural numbers. For 'has an immediate successor' needn't be taken to mean that each number has now a successor; it can be taken to mean that each number at which we have arrived by computation is such that an immediate successor can be computed by adding 1.
But Peter has a stronger objection, one that I admit has force. His objection in nuce is that potential infinity presupposes actual infinity. Peter points out that my explanation of what it means to say that the naturals are potentially infinite makes use of words like 'can.' Thus above I said, "however many numbers we have counted, we can count more." This 'can' refers either to the abilities of men or machines or else it refers to abstract possibilities of counting not tied to the powers of men or machines.
Consider the second idea, the more challenging of the two. Suppose the universe ceases to exist at a time t right after some huge but finite n has been computed. Now n cannot be the last number for the simple reason that there cannot be a last number. This 'cannot' is grounded in the very sense of 'natural number.' So it must be possible that 1 be added to n to generate its successor. And it must be possible that 1 be added to n + 1 to generate its successor, and so on. So Peter could say to me, "Look, you have gotten rid of an actual infinity of numbers but at the expense of introducing an actual infinity of unrealized possibilities of adding 1: the possibility P1 of adding 1 to n; the possibility P2 of adding 1 to n + 1, etc."
The objection is not compelling. For I can maintain that the unrealized possibilities P1, P2, . . . Pn, . . . all 'telescope,' i.e., collapse into one generic possibility of adding 1. P1 is the possibility of adding 1 to n and P2 is the possibility of adding 1 to the last number computed just before the universe ceases to exist.
What I'm proposing is that 'Every natural number has an immediate successor' is true solely in virtue of the sense or meaning of 'natural number.' Its being true does not require that there be, stored up in Plato's Heaven, a completed actual infinity of naturals, a set of same. Since I have decidedly Platonic sympathies, I would welcome a refutation of this proposal.