Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity. Peter writes:
(ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . . If for some reason oak trees could not exist, then an acorn cannot be said to have the potential to become an oak tree. Similarly, if the proponent of syllogism S1 thinks that actual infinity is ruled out by some conceptual, logical, or metaphysical necessity, then he is committed to hold that there cannot be a potential infinity either. Thus, in order for something to have the potential to be such-and-such, it is required that it is at least possible for actual such-and-such to exist.
This is a very fruitful misunderstanding! For it allows us to clarify the different senses of 'potential' and 'actual' as applied to the analysis of change and to the topic of infinity. First of all, Peter is completely correct in what he says in the first two sentences of the above quotation. The essence of what he is saying may be distilled in the following principle
If actual Fs are impossible, then potential Fs are also impossible.
But this irreproachable principle is misapplied if 'F' is instantiated by 'infinity.' If an actual infinity is impossible, it does not follow that a potential infinity is impossible. For when we say that a series, say, is potentially infinite we precisely mean to exclude its being actually infinite. A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree. On the contrary, it is a series which, no matter how much time elapses, is never completed. An actually infinite series, by contrast, is complete at every instant.
Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ). If these numbers form a set, call it N, then N will of course be actually infinite. A set is a single, definite object, a one-over-many, distinct from each of its members and from all of them. N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers.
It is worth noting, as I have noted before, that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.' This is because the phrase 'potentially infinite set' is nonsense. It is nonsense because a set is a definite object whose definiteness derives from its having exactly the members it has. In the case of the natural numbers, if they form a set, then that set will have a transfinite cardinality. Cantor refers to that cardinality as aleph-zero or aleph-nought.
But surely it is not obvious that the natural numbers form a set. Suppose they don't. Then the natural number series, though infinite, will be merely potentially infinite. What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series. No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting. The numbers are not 'out there' waiting to be counted; they are created by the counting. In that sense, their infinity is merely potential. But if the naturals are an actual infinity, then they are not created but labeled.
Or consider a line segment. One can divide it repeatedly and in principle 'infinitely.' But if one does so is one creating divisions or recognizing divisions that exist already? If the former, then the infinity of divisions is merely potential; if the latter, it is actual.
Peter seems worried by the fact that no human or nonhuman adding machine can enumerate all of the natural numbers. But this is no problem at all. If there is an actual infinity of natural numbers, then it is obvious that a complete enumeration is impossible: the first transfinite ordinal omega has aleph-nought predecessors. If there is only a potential infinity of naturals, then as many enumerations have taken place, that is the last number created.
Peter seems not to be taking seriously the notion of potential infinity by simply assuming that the naturals must form an infinite set. He doesn't take it seriously because he confuses the use of 'potential' in the context of an analysis of change, where change is the reduction of potency to act, with the use of 'potential' in discussions of infinity.
But now I'm having second thoughts. I want to say that from the fact that a line segment is infinitely divisible, it does not follow that it is actually divided into continuum-many points. But what about the number of possible dividings? If that is a finite number, one that reflects the ability of some divider, then how can the segment be infinitely divisible? But if the number of possible dividings is a transfinite number, then it seems we have re-introduced an actual infinity, namely, an actual infinity of possible dividings. In other words, infinite divisibility seems to require an actual infinity of possible dividings. Or does it?