1. To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words. And to keep it simple, let's confine ourselves to the natural numbers (0 plus the positive integers). The issue is whether or not the naturals form a set. I hope it is clear that if the naturals form a set, that set will not have a finite cardinality! Were someone to claim that there are 463 natural numbers, he would not be mistaken so much as completely clueless as to the very sense of 'natural number.' But from the fact that there is no finite number which is the number of natural numbers, it does not follow that there is a set of natural numbers.
2. So the dispute is between the Platonists -- to give them a name -- who claim that the naturals form a set and the Aristotelians -- to give them a name -- who claim that the naturals do not form a set. Both hold of course that the naturals are in some sense infinite since both deny that the number of naturals is finite. But whereas the Platonists claim that the infinity of naturals is completed, the Aristotelians claim that it is incomplete. To put it another way, the Platonists -- good Cantorians that they are -- claim that the naturals, though infinite, are a definite totality whereas the Aristoteleans claim that the naturals are infinite in the sense of indefinite. The Platonists are claiming that there are definite infinities, finite infinities -- which has an oxymoronic ring to it. The Aristotelians stick closer to ordinary language. To illustrate, consider the odds and evens. For the Platonists, they are infinite disjoint subsets of the naturals. Their being disjoint from each other and non-identical to their superset shows that for the Platonists there are definite infinities.
3. Suppose 0 has a property P. Suppose further that if some arbitrary natural number n has P, then n + 1 has P. From these two premises one concludes by mathematical induction that all n have P. For example, we know that 0 has a successor, and we know that if arbitrary n has a successor, then n +1 has a successor. From these premises we conclude by mathematical induction that all n have a successor.
4. Brightly claims in effect that to champion the Aristotelean position is to deny mathematical induction. But I don't see it. Note that 'all' can be taken either distributively or collectively. It is entirely natural to read 'all n have a successor' as 'each n has a successor' or 'any n has a successor.' These distributivist readings do not commit us to the existence of a set of naturals. Thus we needn't take 'all n have a successor' to mean that the set of naturals is such that each member of it has a successor.
5. Brightly writes, "My understanding of 'there is' and 'for all' requires a pre-existing domain of objects, which is why, perhaps, I have to think of the natural numbers as forming a set." Suppose that the human race will never come to an end. Then we can say, truly, 'For every generation, there will be a successor generation.' But it doesn't follow that there is a set of all these generations, most of which have not yet come into existence. Now if, in this example, the universal quantification does not require an actually infinite set as its domain, why is there a need for an actually infinite set as the domain for the universal quantification, 'Each n has a successor'?
6. When we say that each human generation has a successor, we do not mean that each generation now has a successor; so why must we mean by 'every n has a successor' that each n now has a successor? We could mean that each n is such that a successor for it can be constructed or computed. And wouldn't that be enough to justify mathematical induction?
Addendum 8/15/2010 11:45 AM. I see that I forgot to activate Comments before posting last night. They are on now.
It occurred to me this morning that I might be able to turn the tables on Brightly by arguing that actual infinity poses a problem for mathematical induction. If the naturals are actually infinite, then each of them enjoys a splendid Platonic preexistence vis-a-vis our computational activities. They are all 'out there' in Plato's heaven/Cantor's paradise. Now consider some stretch of the natural number series so far out that it will never be reached by any computational process before the Big Crunch or the Gnab Gib, or whatever brings the whole shootin' match crashing down. How do we know that the naturals don't get crazy way out there? How can we be sure that the inductive conclusion For all n, P(n) holds? Ex hypothesi, no constructive procedure can reach out that far. So if the numbers exist out there, but we cannot reach them by computation, how do we know they behave themselves, i.e. behave as they behave closer to home? This won't be a problem for the constructivist, but it appears to be a problem for the Platonist.