I'd like to talk about the difference between two fundamentally different perspectives on infinity: the distinction between potential infinity and actual infinity. Everyone might agree that there are infinitely many numbers, but mathematicians and philosophers sometimes disagree about what exactly that means. According to the philosophy of potentialism, yes, there are infinitely many numbers. You can have more and more, as many as you like, but you will never have all of them as a completed totality. The potentialist says the numbers are infinite because you can always take more, but you cannot collect them all at once.
The actualist, by contrast, believes that the numbers are infinite because we can form that infinite collection as a completed totality and then proceed to do further work with it, further constructions, further reasoning. This is the dominant view today: almost all mathematicians are actualists. But that was not historically the case. For thousands of years, nearly all mathematicians, including the great ones, Gauss among them, were potentialists.
This potentialist outlook goes all the way back to Aristotle, who wrote: "For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different." This was built into the classical conception of mathematics, that some things were infinite, but only in a potentialist sense. You never completed the infinite collection; you never had infinitely many things all at once.
Consider the divisibility of a line segment. Given a line segment, one can divide it in various ways, chopping it into pieces, then adding more pieces to that division, gradually subdividing the segment into smaller and smaller parts. The segment is infinitely divisible in the potentialist sense: no matter how far the division has proceeded, one can always divide further. This is the potentialist understanding of infinite divisibility.
The actualist account, by contrast, would require that one has actually undertaken all of the divisions at once. This connects naturally to the conception of a line segment as constituted by the totality of points lying on it, a conception that is central to our contemporary understanding of the line segment, but one that is far from how mathematicians thought about it throughout most of the history of the subject. Here, then, is another instructive case.
Consider the infinitude of a line in geometry. A potentialist would say that a line is infinite because no matter how much of it you have, you can always extend it further. It is illuminating to read in Euclid's Elements, that ancient book of geometry, where one of his axioms states that every line can be extended further.
This is a strange claim from an actualist point of view. For the actualist, a line is already infinite; it does not make sense to extend it further, because it is already fully extended. To say that every line can be extended further is to admit a potentialist understanding of the line: you have just a line segment, and you can extend it as much as you like.
The line is thus potentially infinite: you can extend it further without limit, but you will never possess the whole thing at once. If you think about the classical geometric tools of straightedge and compass, the straightedge typically has a finite length, so you could never actually use it to produce a completed, actually infinite line. You can only extend it further, and that is precisely what Euclid had in mind: given two points on a line, you may extend it as far as you wish.
This is the potentialist conception in its clearest form. The line is infinite in the sense that you can always have more of it, but you will never have all of it.
Consider another case. In Archimedes' wonderful treatise, The Quadrature of the Parabola, he set himself the task of calculating the area of a parabolic segment. What he did was inscribe the largest possible triangle fitting within the segment, with its apex at the midpoint of the curved side, filling up most of the parabolic region. He then examined the two remaining spandrels and inscribed triangles within those as well.
If the principal triangle has area T, Archimedes argued that the two smaller triangles together have a height exactly one-fourth that of the original, and that by the formula for one-half base times height, those two triangles together contribute T/4. Inserting still smaller triangles into the remaining spandrels yields a further contribution of T/16, and so on. The area of the parabolic segment can therefore be expressed as the infinite sum T + T/4 + T/16 + ···, which equals 4T/3. The area of a parabolic segment is thus four-thirds times the area of the principal inscribed triangle.
Archimedes was a potentialist, and so he did not conceive of literally adding up infinitely many terms; yet he was still able to arrive at this result. Interestingly, our contemporary understanding of the value of an infinite sum is itself potentialist in character. We say that an infinite sum converges to a certain limit, meaning that for every epsilon, no matter how close an approximation one demands, taking a sufficiently large finite number of terms will achieve that degree of closeness.
This is a potentialist understanding of what it means to sum an infinite series. Indeed, one might argue that we lack any genuinely actualist conception of summing infinitely many terms, since we always define such sums in terms of the behavior of finite partial sums, which is, at its core, a potentialist framework.
Let us consider another classical mathematical result that exhibits the distinction between potential and actual infinity: the infinitude of primes. The prime numbers, 2, 3, 5, 7, 11, 13, and so on, are those numbers whose only factors are one and themselves. This is a common definition, but it does not quite give the right answer in the case of the number one, which also has only itself and one as factors among the positive integers, yet is not generally counted as prime. Mathematicians have decided to exclude one from the primes because including it would ruin a great many theorems.
For example, consider the unique factorization of integers into primes: every positive integer factors uniquely into a product of primes. If one were counted as prime, this uniqueness would fail, since we could write 12 as 2 × 2 × 3, but also as 2 × 2 × 3 × 1, or 2 × 2 × 3 × 1 × 1 × 1, appending as many ones as we like. A cleaner definition, therefore, is that a positive integer is prime if it has exactly two distinct factors, namely one and itself, or equivalently, that it is greater than one and divisible only by one and itself.
How do we know there are infinitely many primes? Suppose we have some finite list of prime numbers P1, P2, P3, …, PN. Form the number obtained by multiplying all of them together and adding one: P1 × P2 × P3 × … × PN + 1. This number is not divisible by any prime on the list, because dividing it by any Pi leaves a remainder of one. Yet every integer greater than one has at least one prime factor: either it is prime itself, or it factors into primes. Therefore, this new number has a prime factor Q that does not appear anywhere on our original list, and so the list can be extended.
This argument shows that any finite list of primes can be extended to a larger finite list, and that larger list can be extended again, and so on without end. Notice that this is an inherently potentialist way of expressing the infinitude of primes: we do not assert that the primes form a completed infinite collection, but rather that no finite list of primes can be the whole story. This is close to how Euclid argues in The Elements.
One sometimes encounters a different version of the argument, framed as a proof by contradiction. The mathematician says: suppose, towards a contradiction, that there are only finitely many primes. List them all, multiply them together, add one, and observe that the resulting number has a prime factor not on the list, a contradiction. Therefore, there must be infinitely many primes. This version carries an actualist flavor: it presupposes that the primes could, in principle, form a completed finite totality, and derives a contradiction from that assumption.
There is a notable exception to the pervasive potentialist outlook that dominated mathematics for thousands of years, and that exception is Galileo. Galileo argued against this potentialist orthodoxy and offered several arguments in favor of actual infinity. His first claim was that it is simply not true that we cannot conceive of completed infinities. We can, in fact, bring the completed infinite case into actuality.
To illustrate this, Galileo proposed imagining a sequence of regular polygons with an increasing number of sides: a square, then a hexagon, then an octagon, then a forty-sided polygon, and so on. A potentialist can describe this process naturally. One can always add more sides, producing a hundred-sided polygon, a thousand-sided polygon, and so forth. But Galileo then asked whether it is conceivable to actualize the limiting case of a polygon with infinitely many sides. His answer was yes: that is precisely what a circle is. A circle, he argued, can be understood as a polygon with infinitely many sides, and this is an actualist conception, since circles are definite objects in classical geometry, drawable with a compass and fully realized, not merely potential.
Galileo also offered a second, quite striking argument. Suppose one is a potentialist about the infinitude of the primes, holding that there can always be more and more primes, but that one never possesses all of them at once. On this view, one might have five primes in one's possession, or a hundred, or a thousand. Each of these is a distinct possibility available right now, not as something realized, but as a genuine abstract possibility.
Here is the crucial point: those distinct possibilities are themselves different things that one already has, right now, in some sense. There is the possibility of having five primes, the possibility of having a hundred primes, the possibility of having a thousand primes, and so on, and these are all distinct. Therefore, the potentialist is already committed, right now, to an actual infinity of different things, namely the infinity of distinct possibilities themselves. This argument, originating with Galileo, suggests that every potentialist is implicitly committed to an actual infinity, not of realized objects, but of abstract possibilities.
Ultrafinitism is the philosophical view that only comparatively small or accessible numbers actually exist. According to this position, the extremely large numbers that mathematicians conventionally take themselves to be describing, such as 10100 or a googolplex, are not actually meaningful. They do not exist as numbers, but merely as descriptions of numbers.
An ultrafinitist may grant that we can write down such expressions, but would insist that these expressions fail to denote any existing number. They are finite strings of symbols that are ultimately meaningless and fail to refer. This may sound like a strange philosophical position, but there are genuine mathematicians who hold it and find it a helpful framework for expressing their vision of what mathematics is about, though it must be said that most mathematicians are not ultrafinitists.
One might naturally ask whether an ultrafinitist, by denying the existence of a number like a googolplex, is thereby committed to the existence of a largest number. Some versions of ultrafinitism do indeed commit to a largest number, while other versions do not. The criticism that positing a largest number is absurd, because one could always imagine a number system with one more element, is actually too quick, precisely because those other versions of ultrafinitism are not committed to any such largest number.
This story was related by Harvey Friedman, the logician, who described a conversation he had with Alexander Esenin-Volpin, the famous ultrafinitist. Friedman put to Esenin-Volpin the following challenge: consider the sequence of numbers 21, 22, 23, 24, and so on, up to 2100. One can easily imagine writing out this list. The question is the obvious draw-the-line objection: at what point in this sequence do the numbers cease to have platonic reality?
Friedman recounts that he raised precisely this objection with Esenin-Volpin during one of his lectures, and Esenin-Volpin asked him to be more specific. Friedman then proceeded to start with 21 and asked whether that number was real, to which Esenin-Volpin virtually immediately said yes. He then asked about 22, and again the answer was yes, but with a perceptible delay. At 23, yes again, but with still more delay. This continued for a few more steps until it became obvious how Esenin-Volpin was handling the objection: he was prepared to always say yes, but he was going to take 2100 times as long to answer yes to 2100 as he would to answer yes to 21, and therefore there was no way to get very far with that line of objection.
The picture of ultrafinitism that emerges from this kind of attitude is that numbers become progressively blurrier as they grow larger, and their existence is called more and more into question. Though perhaps, if one waited long enough, they would eventually come into focus. This raises the question: what, exactly, is the theory of ultrafinitism?
It turns out to be somewhat difficult to pin down an exact theory expressing the idea of ultrafinitism. Part of the motivation behind ultrafinitism is the sense that there is something innocent about adding or even multiplying numbers that is not so innocent about exponentiation, which takes us into the realm of much larger numbers that we could never hope to reach by counting. One can write down formal theories of arithmetic that capture this feature, theories that prove addition and multiplication are totally defined, while exponentiation is possibly not total. In other words, exponentiation does not always give a meaningful result, even though in those theories there is no sharp cutoff for where it fails.
One of the standard axiomatizations of arithmetic is called Peano arithmetic, formulated in the language of addition, multiplication, and the ordering relation. It lays down the basic axioms about addition and multiplication, including the recursive formulas that define multiplication in terms of addition. Peano arithmetic also includes what is called the induction scheme: if a statement expressible in the language of arithmetic is true of zero, and whenever it is true of a number n it is also true of n plus one, then the statement is true of all numbers. Using this induction scheme, one can prove essentially all of elementary number theory within the Peano system.
In particular, in Peano arithmetic one can prove that exponentiation is total, that there exists a function satisfying the recursive definition of exponentiation. For example, one defines two to the zero as one, and two to the n plus one as two to the n multiplied by two. Peano arithmetic proves that all such recursive definitions are successful, and therefore exponentiation exists as a total function. This means Peano arithmetic is not an ultrafinitist theory.
To obtain an ultrafinitist theory, one must weaken the induction axiom. If we want a theory that treats addition and multiplication as unproblematic while regarding exponentiation as suspect, we must deny certain instances of induction, since the full induction principle suffices to prove that exponentiation is total. It turns out that if we restrict induction to statements of a particularly simple syntactic form, we obtain the theory known as IΔ₀. This theory asserts the induction scheme, the I, only for Δ₀ assertions, which are assertions involving only bounded quantifiers.
By limiting the induction principle to assertions that quantify only over numbers smaller than a given bound, one obtains a theory in which addition and multiplication are provably total, but exponentiation is not. In this sense, IΔ₀ is a formal ultrafinitist theory of arithmetic: addition and multiplication work out exactly as expected, while exponentiation does not.
Let me draw a distinction between two kinds of potentialism: potentialism as an epistemological phenomenon versus potentialism as an ontological phenomenon. Epistemology is the study of knowledge and its nature, so to be a potentialist in the epistemological sense means that it is our knowledge that has a potential character. We can never know the entirety of the infinitude of primes, even though, in the ontological sense, in terms of the actual existence of abstract objects, there may be no difficulty with the infinitude of primes existing as a completed object. On this view, the limitation is in us: we can only ever possess finite knowledge at any given time.
The contemporary understanding of the dispute between potential infinity and actual infinity has taken on an increasingly modal character. Philosophers working in this area now emphasize that the essence of potentialism is not really about infinity, or about whether a collection is completed, but rather about whether the mathematical universe one inhabits is itself a finished totality. On the potentialist outlook, it is a mistake to regard the mathematical realm as a completed, finished whole. Instead, we should think in terms of universe fragments: pieces of the mathematical universe that are successively larger. Those fragments may already contain completed infinities within them, but if they do not constitute the whole mathematical universe, the perspective remains potentialist, even while admitting completed infinities inside each fragment.
The potentialist perspective, then, is that the mathematical realm we inhabit is unfinished, and that we must continually extend it to larger and larger realms, larger and larger universe fragments. The picture is one of possible worlds growing without bound, with no single fragment ever constituting a completed whole, regardless of what mathematical objects exist within any one of them. This kind of structure, in which possible worlds are ordered by extension, is precisely the setting for a modal logic analysis. Modal logic is the abstract study of possibility and necessity, and such a picture admits a natural modal semantics.
If we are situated at some world U, we say that U regards a statement φ as possible if there exists a larger world V in which φ is true; that is, something is possible at a world if it becomes true in some larger world. Conversely, U regards φ as necessary if every larger world makes φ true. This is the basic idea of the modal semantics appropriate to this setting, and one recognizes immediately that when universe fragments are ordered by growth in this way, the resulting framework embodies a fundamentally potentialist perspective on the nature of mathematical truth.
Consider a specific example of what might be called Aristotelian potentialism. Imagine the natural numbers zero, one, two, and so on, arranged in increasing order, where the possible worlds are simply the initial segments of this sequence. The worlds are, in a sense, growing: one may have more and more numbers, as many as one likes, but no single world ever contains all of them. Every world contains only the numbers up to some finite point.
We can now bring the potentialist modal vocabulary to bear on this picture. Consider the statement that every number N has a successor, N plus one. This statement is not true in any of these finite fragments, because in a given world the largest number has no successor within that world. One must pass to a larger world to find it. The correct potentialist rendering of the statement is therefore: for every X, it is possible that there exists a Y such that Y equals X plus one. In other words, every number possibly has a successor.
We can in fact say something stronger. Not only is it true at every world that each X possibly has a successor, but we can say it is necessarily true that every individual possibly has a successor. From any world, no matter which other world one passes to, every individual in that world possibly has a successor. This modal vocabulary thus allows one to express, with considerable precision, the nature of the potentialism under discussion, articulating the potentialist principles in terms of possibility and necessity, and distinguishing between different varieties of potentialism one might wish to consider.
To give further examples, consider the statement that any two numbers have a sum: for every X and Y, there exists a Z such that Z equals X plus Y. In the initial-segment potentialism described here, this is again not true at any single world, since the required sum Z may not yet be present and one must pass to a larger world to find it. The correct formulation is therefore: necessarily, for any two numbers X and Y, it is possibly the case that there exists a Z which is their sum. One could write the same for products.
This raises the possibility of finer distinctions. A particular version of potentialism might be committed to the principle that necessarily any two numbers possibly have a sum, and possibly have a product, while remaining agnostic, or even negative, about whether they possibly have an exponential. The choice of which such modal principles to endorse is precisely what distinguishes one form of potentialism from another.
Consider closure under addition and multiplication. How might we express the infinitude of primes in this modal vocabulary? To say that there are infinitely many primes, one natural formulation is: for every number, there is a prime above it. In the modal setting, this becomes: necessarily, for every number, possibly there is a prime larger than that number. Moreover, in the Aristotelian potentialism case, primality itself requires no modal operators to express. If a number is present in a universe fragment, then all smaller numbers are present as well, and the question of whether that number is prime reduces to whether it admits a non-trivial factorization into smaller numbers, a purely local, non-modal question.
Now let us consider forms of potentialism that are not linear. One motivation comes from an issue raised in an earlier lecture. The number googolplex, that is, 10 to the 10 to the 100, is very easy to describe, yet most of the numbers smaller than it are beyond our comprehension. Those smaller numbers are, in effect, random strings of decimal digits of googolplex length; it would take longer than the age of the universe merely to pronounce them. In a potentialist framework where numbers come into existence, there is therefore a sense in which we might have access to a googolplex earlier than we have access to many of those smaller numbers.
This is not as strange as it first appears. We know a great deal about the googolplex: it is even, its prime factorization is 2 to the 10 to the 100 times 5 to the 10 to the 100, it is not a multiple of seven, and so on. By contrast, Aristotelian potentialism insists that whenever a number is present, all smaller numbers are already present, so the googolplex could never arrive before its predecessors. This observation opens the door to what might be called a nonlinear form of potentialism.
In nonlinear potentialism, a universe fragment might contain some numbers, say, the numbers up to ten, together with certain larger numbers, without containing everything in between. Different fragments can coexist and overlap in various ways, though one might still require that any two such fragments can be amalgamated into a single larger fragment. This picture of arbitrary finite set potentialism in arithmetic has modal features that differ in important respects from those of Aristotelian potentialism, and it is to those differences that we now turn.
Consider how we would express the claim that x is even. One natural formulation says: there exists a y such that y + y = x. In Aristotelian potentialism, no modal operators are needed here, because if we already have x and x is even, we will already have the witnessing y — it is a smaller number and therefore present. If x is not even, then no such y exists at all.
In arbitrary set potentialism, however, we may have an x that is even without yet having the y that doubles to it. We may not yet have x/2 in our current fragment. In that case, the unmodalized formula would fail to express evenness correctly, since the number is even even though the witness is absent. We should therefore insert a diamond operator, making the existential claim potentially rather than actually satisfied. This illustrates that the correct translation of mathematical concepts into modal terms is sensitive to the specific modal properties of the potentialist conception in use.
One can then begin writing down the basic principles that all potentialist conceptions are likely to share. Suppose we have a potentialist understanding of arithmetic: a collection of universe fragments, each included in larger ones, growing in some order, not necessarily linear. A first natural principle is that if φ holds necessarily, then φ holds. This corresponds to the accessibility relation being reflexive: every world counts as a sub-world of itself, so if φ holds in all accessible worlds, it holds in the current world, which is one of them.
A second principle is that if φ holds necessarily, then it is necessary that φ holds necessarily, that is, □φ → □□φ. This corresponds to transitivity of the accessibility relation. If φ holds in every world accessible from the current world U, and we move to one of those worlds, does □φ still hold there? Yes: any world accessible from that intermediate world is a world reachable from U in two steps, and since φ holds throughout, □φ is preserved. Transitivity simply says that if U can access a larger world, and that world can access a still larger one, then U can access the still larger world directly.
There is also the axiom known as K, after Kripke, which states that if an implication holds necessarily and its antecedent holds necessarily, then its consequent holds necessarily. If a certain conditional is true in every world accessible from the current one, and the hypothesis of that conditional is likewise true in every such world, then the conclusion must hold in all those worlds as well. Together, these axioms, reflexivity, transitivity, and K, constitute the modal system known as S4. The upshot is that with only these very primitive requirements on the modal structure, the full theory S4 is valid for any potentialist conception.
Consider some of the more specific principles that arise in particular potentialist conceptions. In Aristotelian potentialism, the possible worlds, the universe fragments, are linearly ordered, so given any two, one extends the other. This linearity yields validities beyond S4. Consider, for example, the statement "possibly necessary φ implies necessarily possibly φ." Suppose we are at a universe fragment where possibly necessary φ holds. That means we can enlarge it to a world where necessary φ holds, a world in which φ is true in all further extensions. Returning to the original world and asking whether necessarily possibly φ holds there, the answer is yes: from the original world, any extension can itself be extended to include this world where necessary φ was holding, and so φ will be true there. Thus possibly necessary φ implies necessarily possibly φ.
This principle is also valid in the other potentialist conception mentioned earlier, namely arbitrary set potentialism, which is not linear, and the argument is perhaps even clearer in that setting. If possibly necessary φ holds at a world, we can pass to a world where necessary φ holds, meaning every further extension satisfies φ. Now take any world extending the original one; by the amalgamation property, there is a larger world encompassing both it and the world where necessary φ holds. Since that larger world extends the one satisfying necessary φ, it must satisfy φ, and therefore the intermediate world satisfies possibly φ. Since this reasoning applies to any extension of the original world, the original world satisfies necessarily possibly φ. This principle is not provable in S4, yet it is valid in both of these potentialist conceptions. It is the central axiom of the system known as S4.2.
There is one further principle that distinguishes these two conceptions from each other. It is the axiom known as .3. These names are admittedly terrible, but the first writers on modal logic assigned them and the terminology has stuck. The axiom states: if φ is possible and ψ is possible, then either it is possible that φ is true while ψ remains possible, or it is possible that ψ is true while φ remains possible. In other words, if two things are each possible, there is a larger world in which one of them is true and the other is still possible.
This axiom expresses a kind of linearity. In Aristotelian potentialism, where the worlds are linearly ordered, if φ and ψ are each true at some later point, then one of them must come first. Whichever comes first realizes the relevant clause: if φ is true first, then ψ is still possible at that point; if ψ comes first, then φ is still possible. Thus S4.3 is valid in any linear potentialist system.
However, S4.3 fails in arbitrary set potentialism, precisely because that framework is not linear. One can construct incompatible possibilities: for instance, φ might assert that the number 17 exists but 16 does not, while ψ asserts that 16 exists but has no successor. Both are individually possible, yet each makes the other impossible, so there is no world in which one holds while the other remains possible. This is a direct violation of S4.3, even though S4.3 is valid in the Aristotelian conception.
The broader point, though somewhat technical, is that this modal perspective gives us the power to distinguish between genuinely different kinds of potentialism. It is not simply a matter of potentialism versus actualism; there is an extraordinarily rich variety of potentialist conceptions, and many of the distinctions among them can be articulated precisely through modal principles. This applies not only to potentialism about numbers or sets, but to any mathematical theory whatsoever. In joint work with a graduate student, Vojtěch Vopěnka, published under the title Modal Model Theory, we examine modal graph theory, modal group theory, modal field theory, and more. For any collection of mathematical structures, one can form the collection of models of the relevant theory and treat it as a potentialist framework for that subject.
Conceptions of potentialism that involve a linear hierarchy of possible worlds generally validate S4.3, for the reasons already discussed. But there are other conceptions in which a possible world has two distinct extensions growing in different directions, yet some still larger world encompasses both, with further branching always admitting a common upper bound. This is what we call convergent potentialism: given any world in the potentialist system, if it accesses two worlds, there is a further, larger world that encompasses both of them. One can prove that the modal principle .2 is always valid in such a convergent system, giving validities of S4.2.
Finally, there is a much more radical conception of potentialism, one more closely connected with ultra-finitism and with pluralist views in the foundations of mathematics, the multiverse view in the philosophy of set theory, for instance. This is the radical branching perspective on potentialism, in which convergence fails entirely. A world may have various extensions that simply never come together again; the branching continues indefinitely, with those branches remaining permanently separated. The world grows in such a way that what happens along one branch affects what remains possible, and no common extension is guaranteed.
One can exhibit concrete cases of this phenomenon. If one looks at models of Zermelo–Fraenkel set theory ordered under end extension, the resulting potentialist system is of exactly this radical branching kind, and the principle .2 is not valid in that modal framework. In such cases the validities collapse to exactly S4, and for the models of ZFC under end extension one obtains precisely S4 and nothing beyond.
The potentialist translation works as follows. Given a statement φ in the actualist language, we produce a corresponding statement φ◇ in the potentialist language by replacing every existential quantifier with "possibly there exists" and every universal quantifier with "necessarily for all." In other words, we insert a diamond before every existential statement and a box before every universal statement. That is the potentialist translation.
We were already performing this translation earlier. When I said that every number has a successor, what I actually wrote was: necessarily, for every x, possibly there exists a y such that y equals x plus one. That is precisely the potentialist translation of the actualist statement.
Now consider a convergent potentialist system, one in which any two worlds you can reach can always be assembled together into a single larger world. In such a system, the possible worlds are converging toward a limit model M: from the actualist perspective, one can simply take the union of all those worlds to obtain it. This is in contrast to radical branching potentialism, where the worlds can diverge in entirely different directions and never come together, so no limit model exists.
In the convergent case, one can prove a fundamental biconditional: the limit model M satisfies a statement φ if and only if the worlds in the system satisfy the potentialist translation φ◇. For example, the actualist assertion that every number has a successor is equivalent to the potentialist assertion that, from any fragment, necessarily every number possibly has a successor. These two statements track each other exactly.
This biconditional establishes a translation between actualist and potentialist assertions: one can pass freely back and forth between them. I have argued that this makes convergent potentialism implicitly actualist: if every truth the actualist wishes to assert can already be expressed within the potentialist framework using only potentialist ontology and potentialist concepts, then there is nothing genuinely at stake in the dispute between the two positions. The disagreement is, in this sense, deflationary.
Crucially, this biconditional fails entirely in the radical branching case. When the potentialist conception allows the worlds to branch in ways that never reconverge, no potentialist translation exists that converts that form of potentialism into actualism. This suggests that a philosopher who is genuinely committed to potentialism and wishes to preserve it as a substantively distinct position should look more carefully at radical branching potentialism as expressing a fuller and more authentic potentialist view, precisely because the convergent version collapses back into actualism via the potentialist translation.
Let me close by mentioning more explicitly the sea change that occurred in mathematics at the end of the nineteenth century. Before that time, almost everyone was a potentialist. With the rise of set theory, particularly through the work of Dedekind and Cantor, and the deeper understanding of infinity that grew out of that work, mathematicians came to recognize that it was not actually problematic to conceive of completed infinities. On the contrary, this actualist understanding of infinity proved to be a great source of mathematical insight, powerful, useful, and ultimately transformative for the entire subject.
David Hilbert famously declared, "Let no one cast us from the paradise that Cantor has created for us." Part of what he was expressing is precisely this perspective: that mathematics could embrace actual infinity, completed totalities, and further constructions proceeding on top of those infinities. By now, this is entirely routine. Almost everyone working in mathematics is an actualist, building infinities on top of infinities on top of infinities. Mathematicians today find great inspiration from this actualist perspective, and it has become the default foundation from which the subject continues to grow.