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https://plato.stanford.edu/entries/set-theory/ states outright that set theory "can be defined as the mathematical theory of the actual—as opposed to potential—infinite," and the article on constructive set theory says, "Predicativity is also often seen as related to the time-honoured distinction between actual and potential infinity. Predicative (and thus, in particular, constructive) theories are often seen as avoiding reference to actual infinity, and only committing to potential infinity" (sec. 1.3.3). So it would appear that the notion of the distinction is relevant to the philosophy of set theory, but because it adverts to a philosophical context, it is such that the question is not posed within, and thus not solvable within, set theories as such.

That being said, I was thinking about Kant's reply to the ontological argument scheme, specifically the place where he says something like, "A hundred possible thalers contains no more than a hundred actual thalers." At the risk of anachronism, I will say that this looks like a statement that the set of n possible dollars is equal in cardinality to the set of n actual dollars. (In fact, the general rule of modality in Kant requires that the cardinality of the set of n necessary dollars (so to say) would be equal to the cardinality of the other such sets.)*

So now, allowing that this is true, it seems that with respect to cardinality itself, then, actual and potential infinity collapse into each other. That is, the cardinality C of a set's potential infinity, if C is actually potentially infinite, is just actually infinite. This might be thought of along the lines of things like the basic modal entailment "that X is actual implies that X is possible" or "that X is possible means that X is actually possible," alongside (IIRC) the Barcan formula (if X is actually possible, then X is possibly actual). In effect, we can preface a variable with one possibility operator first, and then preface that operator by an infinite stream of actuality operators, but this stream will reduce (by the rules of iterated modal operators) to the potential case (and vice versa, if we start from an actual X and then "infer" an infinitely iterated potentiality "behind" it). And indeed we can go on to arbitrarily interpose different infinite streams of these operators, and the cardinality of the set of the actual operations will be potentially and therefore actually infinite. "QED"

*[There seems to be a way to link this idea with the notion of existential quantification re: e.g. Quine's "to be is to be the value of a bound variable," i.e. there is a relationship between the concept "existent" and "having nonzero cardinality." But I'm not sure how to parse this seeming link.]

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  • I don't understand what question "is not posed within, and thus not solvable within" set theory (first paragraph). Thanks for a clarification. – Mr. White Jun 10 '20 at 15:37
  • What does "The cardinality of a set's potential infinity" mean? – Noah Schweber Jun 10 '20 at 15:55
  • Thanks for your question! As I understand it, the descriptors of “actual” versus “potential” infinities refer to conceptually different descriptions of sets, rather than measures over those sets. A set that is defined according some indefinitely extensible applications of finite steps would be “potentially infinite”, whereas one that explicitly involves the axiom of infinity or a non-constructive proof by contradiction would be an “actually infinite” set. Finitists acknowledge potentially infinite sets without allowing that any sets are “actually infinite”. – Sofie Selnes Jun 10 '20 at 17:13
  • It is unclear what Kant's remark about finite cardinalities has to do with either actual or potential infinite. That aside, I think you are mixing extensional possibilities, which are indeed "actual" in the sense intended in potential/actual distinction, with Aristotle's potentiality, which is intensional in modern terms. Semantics of possible worlds explicitly embraces actual infinite from the start, but has little to do with potentiality. And in predicative and constructive theories that attempt to capture potential infinite the notion of (actual) infinite cardinality is not definable. – Conifold Jun 11 '20 at 6:00
  • Let's start with "it is possible to increase a set by X." This converts to "there is a possible set that has been increased by X." So "it is possible to increase a set infinitely" becomes "there is a possible set that has been increased infinitely." Now, these possible sets would be abstract objects, so we are not saying that they imply that everything whatsoever that is potentially infinite is also actually so; just that, in terms of questions about the nature of set theory, this is so <e.g. intuitionists are being "superstitious" about infinity?> – Kristian Berry Jun 11 '20 at 15:56
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I draw my answer partially from The Philosophy of Set Theory by Mary Tiles.

PREFACE TO THE QUESTION

The question 'what is the nature of infinity' is an ontological one, because it is the attempt to wrap the mind around the processes of number systems which extend subitization, which is a basic psychological faculty, to number systems which is a theoretical activity. When counting, certain present questions themselves to the naive thinker, such as 'what is the biggest number'. Intuitionally, most physically real things have maxima and minima. There is a tallest family member, or there is a longest finger. It is natural to presume there is a biggest number. Of course, it quickly becomes apparent that that any number given can be incremented by 1. It seems you are asking about what is the essence of infinity? At metaphysical play then is the same ontological muddle that lies behind Tiles question on page 1. "Did Cantor discover... transfinite sets... or did he (with a little help from his friends) create it?" and her statement on page 2. "Cantor's continuum hypothesis cannot be proved from the standardly accepted axioms of set theory."

So, by asking the question, let's note that you set aside finitism as a position. It's only worth noting because some might seek to undermine the meaningfulness of your question by questioning the basis for your question. But let's review cardinality.

In the case of a finite set, cardinality is a bijection between members of the set and a specific member of the naturals, which is closely related to the notion of a sequence which is a correspondence between the order of the elements of the set and the naturals. The question arises, however, is it possible to have a cardinality of a set that is not finite. To this Cantor suggests yes, and offers the notion of using the naturals which are infinite as a measure of other infinite sets by establishing a bijection between their respective members. Thus, the cardinality of an infinite set is a question of whether or not a bijection can exist between the naturals and the set in question. With this bijection as a basis, a new ordering can occur using aleph-nought, and cardinality can be extended beyond a bijection with a specific natural number per se. This is accomplished with the diagonal proof and leads to the the cardinal numbers.

Now, the question of possible and actual infinity should also be reviewed, because unlike cardinal numbers, it is a genuine question in regards to the philosophy of math. From page 25. "Aristotle points out, there is an incompatibility between the notion of a potential infinity and that of a totality... Indeed, there is here the source of another notion of infinity - the absolutely infinite, that than which nothing can be greater." This notion arises from Ancient Greek inquiry into the nature of time, divisibility of matter, and the nature of the universe. Again she cites Aristotle and says on page 26 "The concept of potentially infinity is essentially linked to the idea of a process of construction, of generation, or simply coming to be." She points to the tension between 'universe' which is everything or is the ultimate completeness and infinity which is the sense of introducing new things. On page 27, she goes on to say "If Aristotle is right... if the only viable sense of 'infinite' is that of the potentially infinite, then the universe must either be finite or not a completed whole, not a unity." And further along says on page 28 "comprehensible notions of infinity are intimately bound up with views on the nature of time... between metaphysics of 'being' and that of 'becoming'.

ANSWER TO THE QUESTION

So you say:

*[There seems to be a way to link this idea with the notion of existential quantification re: e.g. Quine's "to be is to be the value of a bound variable," i.e. there is a relationship between the concept "existent" and "having nonzero cardinality." But I'm not sure how to parse this seeming link.]

So the relationship you are seeing in Quine's statement is a metaphysical presumption. That which exists, is that which can be counted, and you are asking after the notion of the existence of things that have transfinite cardinality, that is, cardinality where size is not denoted by bijection with the naturals directly, but by bijections to a bijection with the naturals or reals and so on. Certainly one can accept as first principles Quine's quantification as a condition denoting 'reality' and the transfinites as an extension of a naive cardinality. But instead of using the language that "potential infinity collapses into actual infinity", I'd suggest you say that the existence of 'actual infinity' is a symbolism for the entire process of 'potential infinity' when working in an atemporal formalism.

As far as your statements:

we can preface a variable with one possibility operator first, and then preface that operator by an infinite stream of actuality operators, but this stream will reduce (by the rules of iterated modal operators) to the potential case (and vice versa, if we start from an actual X and then "infer" an infinitely iterated potentiality "behind" it)

Here, as far as I know, you are moving into idiosyncratic or novel formalisms by invoking modal logic notation in regards to statements of infinity and sets. It is standard to assume possibility in quantifier logic with statements like ∃e∀S, where S is an infinite set, but not use modal operators. You certainly could do so, but it would be considered original material as far as I know. If I'm wrong, I'd love to be pointed in the direction of the theory.

So, consider the extended reals. Built into the notion of the interval (-∞,∞) is the idea that if there exists n in the interval, it is actually necessary for any, and indeed every n that there are larger or smaller numbers, not to be named explicitly but constructed by possibility. Hence, '∞' is not a cardinality strictly speaking, but a notation which would translate into the concept of 'possibility'.

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I think you are confusing, or at least striking a false equivalence between the two terms: potential and possible.

To be fair, in any other context potential and possible would be equivalent terms but here it has a strict meaning:

A sequence of events, numbers, or perhaps even changes that tends to infinity is called a potential infinity. For example, you can imagine an ever-extending ruler.

A better example, yet, would be a recursive function: Fibonacci Sequence. If you put the function in a computer it will have to complete n+1 steps to get to the nth term (depending on how you define it). Similarly, then, it will have to finish an infinite amount of calculations to get the "final" (which there isn't any) term. In other words, the computer will keep calculating the function term by term and slowly tend towards the infinity. [This is highly informal, if you need a rigorous analysis of this do ask for it but it suffices for now].

On the other hand, then, an actual infinity just is a completed infinity. It is something that doesn't quite tend towards, but just is, in every moment, an infinitude. Naturally, then, what else can best describe this infinity other than infinite sets? Basically, "complete" infinities (seems like a contradiction in term but it's really not).

{0, 1, 2, 3, ...}

For example, the set of positive integers, mentioned above, can be considered an actual infinity because it just contains every positive integer and naturally positive integers are not finite.

Basically the difference is on the "change"/"tending towards" notion. Actual infinities just are, whereas potential infinities tend towards or yearn to reach infinitude.

Aside,

How useful is this dichotomy, or how actual are sets they are up for debate. I think the stanford encyclopedia is a little misleading here, but that's not relevant.

Finally,

You made a point about equal cardinalities between sets implying potential infinities are actual infinities, and therefore, you argued, it entails modal collapse.

Modal collapse is a different issue. That's not really relevant here because, like i said, potential here doesn't (at all) mean possible. This is an unfortunate result of our linguistic expression. That said, a set of potentially infinite element is not one-to-one and onto the set of natural numbers (by definition).

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