Anyway, supertasks do not just provide us with intuition-pumps against the standard view. I ended my 2003 by addressing the particularly plausible, and therefore (for me) troubling, supertask of a classical ball bouncing À0 times (see Earman and Norton 1996: 235), and this final section will therefore reject it as an argument that endless sequences are actually rather than potentially infinite. (I will then end by glancing more briefly at three other possible arguments.)
Consider an ideal rigid ball B, bouncing upon a surface inelastically (i.e. after each bounce it attains a reduced height above the surface) in such a way that the time-intervals between its bounces are 1, ½, ¼, and so forth, so that B bounces À0 times in a finite time (cf. Bostock 1973: 43). Although there are a lot of bounces, they are essentially similar to each other, and so B is extremely simple. And since the existence of B’s À0 bounces (and hence the presence of À0 points in space-time) seems to follow almost immediately, this supertask seems to illustrate that the natural numbers form an actual infinitude (in what I would regard as the relevantly physical sense).
Nonetheless, if physical continua do not contain À0 points (e.g. if they are all of the form C-II, of my 2005), then B is simply unrealistic, and hence shows us nothing about reality. So note that B is unrealistic—real balls always stop bouncing after a finite number of bounces. If we regard B as a hypothetically physical ball (rather than as a purely mathematical entity), we might improve upon it by making it flexible rather than rigid, so that after a few bounces the ball would merely wobble upon the surface, À0 times. And we might further improve our model by making the ball’s surface fuzzy, and by adding thermal vibrations into the background, so that once the wobbles become relatively insignificant they effectively cease to exist. Adding a finitistic limitation to the possible number of B’s bounces would just make B more realistic (and would therefore be far from ad hoc). And as it is, B is only a simple model of a real ball when it is used in a finitistic way (e.g. for modelling a collision between two balls).
More needs to be said, of course. We get B’s À0 bounces via the repetition of one perfectly acceptable (if rather idealised) bounce, which leaves B essentially as it was before that bounce, thereby making any number of repetitions seem perfectly acceptable—but is À0 a sufficiently proper number?
This thought-experiment, far from showing that it is, simply assumes that it is. Admittedly B’s À0 bounces appear to show that it is, but appearances can be deceptive, especially when it comes to infinitudes, which are naturally envisaged as what they are not—very big finitudes. When we first considered B bouncing inelastically upon a surface we surely did not consider B’s À0 bounces in their entirety, which was why B’s À0 bounces seemed so straightforward. If we do direct our attention to that totality, we find them far from straightforward (see below); whereas we surely thought only of a few bounces and then (since each was essentially the same as the first) a short period of time in which the rest of B’s bounces were assumed to occur. We therefore simply assumed what this scenario was supposed to illustrate.
Since we know that similarly endless repetitions can generate pluralities that are paradoxical (e.g. Burali-Forti’s paradox) hence, because it is often assumed that the simplest infinite sequences would not be that strange (being as simple as 1, 2, 3, …), it may be worth emphasising our inability to visualise most finite numbers of bounces (after which À0 bounces would remain to be seen). Prima facie it seems that we can imagine that B is able to bounce endlessly without too much difficulty, but in fact for us to imagine that we are clearly seeing that B’s À0 bounces are possible we have to ignore the fact that we are ignoring the À0 bounces in question, which we must do because we get dizzy just trying to imagine a typical natural number—a fact that is often obscured by the simplicity of ‘1, 2, 3, …’; so consider some relatively big numbers. E.g. the famous googol, beyond which most calculators don’t go, is g = 10100. And to illustrate the aforementioned dizziness, let a doodol (so-called as it is less meaningful) be defined as follows.
The function j (of natural numbers n) given by j(n) = nn generates a rapidly increasing sequence 1, 4, 27, ....
Writing j(j(n)) as j2(n), and similarly for higher powers of j, a function f may be defined by f(n) = jj(n)(n).
Finally, by defining a function y recursively, by the rule y(n + 1) = fy(n)(y(n)), starting with y(1) = ff(g)(g), we get the doodol, d = yy(g)(g). Although d has such a simple definition, it is unimaginably huge. And yet it is also unimaginably tiny, e.g. compared to the relatively big b = yy(d)(d), and almost all the natural numbers are vastly bigger than b of course (every single natural number being in the first no more than 0% of the totality of all and only those same natural numbers in their natural ordering).
Now, we can probably imagine B bouncing 10 times, and it is likely that we could even (if rather tediously) visualise 100 bounces (if we don’t worry about metrical inaccuracies). But clearly we cannot visualise B bouncing g times, we can only form a distinct picture of the first few bounces and then skip to g bounces, justifying that skip by the similarity of the bounces. Such a skip is conceptually fine for g bounces, and even for (the far more unrealistic) d bounces, and similarly for b bounces, and so forth—but what are our reasons for thinking it fine for À0 bounces?
I suspect that most of us will simply think of À0 as being sufficiently like such unimaginably big numbers as d or b whenever that would not be obviously contradictory (since we usually try to think within set-theoretical frameworks; cf. Lavine 1994); but of course, that hardly eliminates the possibility that À0 is not (in such hypothetically physical scenarios) sufficiently like d (which is the very possibility that needs to be eliminated, if B is to give us an argument for the actual infinitude of the natural numbers). After all, not only is À0 famously counter-intuitive (i.e. known to be unlike such numbers as d and b in ways that surprise us), one relevant thing that À0 and d do have in common is the utter obscurity of that many unrealistic bounces of B (whence this scenario can hardly give us a clear illustration of the actual infinitude of the natural numbers).
Nonetheless, the classical physics of B is very simple, so that it seems (at least prima facie) to be a coherent possibility. But although B is admittedly a very simple thing, conceptually, note that so too is, e.g., time-travel. Having heard of space-time, I can easily picture time-travel as being like space-travel. But nonetheless, time-travel may also, upon further thought, involve me in complications and paradoxes to such an extent that I come to think of it as either horrendously complicated or just plain unrealistic. It may not; but on the other hand time-travel might, for all we really know, be physically impossible, and whether or not it is impossible is pretty independent of the simplicity of my original picture of it as being like space-travel (whence any conclusions about time that might be drawn from such a picture should be treated with suspicion).
Similarly, we ought to consider B’s À0 bounces in their entirety, and only then see just how simple they are (and even then we ought to take care when concluding anything about the abstract structure of actual continua). Since the time-reverse of B coming to rest would involve it beginning not to bounce but to have already bounced À0 times (in the manner of the counter-intuitive w*-sequences mentioned above), it seems that such actually infinite totalities are indeed more complicated than they first appear. Conversely, the hypothesis of their impossibility is less incoherent than it is likely to appear at first. In particular, note that the metaphysical necessity of some finite limit to such physical repetitions (which would be the case were continua of the form C-II, of my 2005) is not the same as the necessity of some particular limit; and of course, it so happens that the real world does indeed—for some reason—contain such finite limits.
Another (and perhaps more relevant) example, of a plausible concept that is so simple that we naturally take it to be possible (and therefore to be giving us an actual mathematical structure), is given by naïve set theory. It was very surprising when Russell’s simple paradox showed it to be impossible. And of course, standard set theory (the foundation of mainstream mathematics) cannot be shown to be consistent—personally, I believe that it is (or rather, that such theories can be) consistent, and I wonder only if it is true (in the sense of it being the right foundation for our applicable mathematics); but nonetheless, for all that we will ever know (and recall that our intuitions are famously unreliable in such areas) even standard set theory may be a logical impossibility. So, we should not reject too easily the epistemic possibility that classical physics is metaphysically impossible.
A similar confusion, of epistemic possibility with metaphysical possibility, forms the core of our second argument for the actual infinitude of the natural numbers:
It is fatuous to suppose that we know a priori that the stars in the heavens cannot possibly go on and on forever but that at some point in space they must come to an end. The great argument in favour of aleph-null as a cardinal […] is that it is quite intelligible and possible that there is no last star in the heavens […] that it is possible for an infinite number of stars, or apples, to exist […] aleph-null is a number in the original primitive sense, zero is not. (Benardete 1964: 31)
Now, while it is admittedly possible that the set-theoretic À0 is a primitive cardinal number, a similarly epistemic possibility is that it is not (e.g. because the integers form a potential infinitude in the sense of C-II). And while the sense of ‘possible’ in which it is possible that infinitely many stars coexist in this universe is epistemic, for this to be an argument for the actual infinitude of the natural numbers it would have to be metaphysical, i.e. À0 would have to be a primitive number (e.g. a physical possibility for the number of some coexisting objects).
Benardete was surely not equivocating, but nonetheless the above quote is not so much an argument as a potentially misleading assertion. Similarly (and thirdly), it has been argued that, since God could know all the natural numbers individually, and since surely such a God is at least possible, hence the natural numbers must form an actual infinitude, for such knowledge of them to be possible (cf. Russell 1936: 143-4). The first italicised ‘possible’ is again epistemic, while the second is presumably metaphysical, but even given such a God the natural numbers need not form an actual infinitude—if they did not, then God would simply know that they did not and (if rather more mysteriously) how they did not.
In fact, it is as well for the standard view that our knowledge of the nature of such omniscience is far too weak for this argument to have any force, because a very similar argument could be made for the actual infinitude of the proper class of cardinal numbers, whereas (although such things remain rather mysterious) it is standard to associate actual infinities with sets (and potential infinities with proper classes, e.g. see Hart 1976) in view of the famous set-theoretical paradoxes.
Fourthly consider the more common thought that, were standard set theory inconsistent, we would probably have noticed by now, whence it is probably consistent. Consequently, the thought continues, since standard set theory is the foundation of mainstream mathematics, which is assumed throughout modern science, we ought to assume that endless sequences are actually infinite.
But, to begin with, note that the foundational inconsistency of general relativity, with quantum mechanics, remains with modern science. We might (conceivably) have had general relativity without quantum mechanics, and we might then have been talking of its consistency in similar terms. As it is, we say that physics is a work in progress. So, even were an inconsistency in standard set theory discovered, surprising as that would be (no less so than when Russell discovered one in naïve set theory), we would just build a new set theory around that problem; we would hardly feel the need to question the actuality of the transfinites (or at least, no more than we did following Russell’s paradox). (Cf. how many will view my unpublished as an argument against quantum-mechanical propensities.) Conversely, does the last century of set theory indicate that endless sequences are actually infinite?
After all, logical consistency is hardly sufficient for metaphysical (or semantic) truth anyway. Pure mathematics contains many consistent structures that do not correspond very directly with physical reality; and although the (implicit) assumption of set theory within the sciences gives us a pragmatic reason for (effectively) working within set theory, there are few reasons why endless sequences would really be actually infinite (as indicated by the failings of the previous three arguments, for example).
Furthermore, the relevant distinction (between actual and potential infinities) is indeed metaphysical rather than logical. Whereas set theory is essentially a mathematical model of mathematics; so consider such well-known examples of mathematical modelling as Ptolemaic astrophysics, with its ability to model observed reality as accurately as desired, by consistently adding epicycles (in accordance with the then-current intuitions, in favour of circular celestial motions, and a stationary Earth)—cf. our ability to add set-theoretical axioms as desired—or Newtonian astrophysics, which is also presumably consistent (and coherent with widespread physical intuitions) and yet is presumably false.
From such examples it seems that the fact that a certain sort of structure (that is presumably consistent) is being assumed within science is not necessarily an indicator of truth (or of future utility); and although there is considerable empirical evidence that modern scientific theories are better than previous ones, whence we might expect them to be closer to the truth (and more useful), the peculiarly infinitary presumptions of standard set theory have not been so severely tested.
Furthermore a similar thought (to this fourth reason) might give us an argument in favour of potential infinity, via the consistency of constructive mathematics. Although one difference is that such mathematics is not widely assumed within modern science, surely the question ought to be whether or not we really need (in our sciences) the specifically set-theoretical aspects of our mathematics. And since the century of mathematics has been dominated by set theory, it is surely too early to say. (And note that although constructive mathematics is widely associated with an Intuitionism that appears relatively incoherent, the Logicism that was widely associated with set theory in its early days also seems relatively incoherent.)
In short, although the last century of mathematics might suggest (inconclusively) the consistency of set theory, it says little about its truth (and thence its future utility).
Finally, perhaps I ought to apologise for my tedious doodol, d = yy(g)(g). Not only did it not prove anything, you may not have found it very illuminating. Still, I find that thinking of such natural numbers gives me a feel for what we are talking about, when we talk about the natural numbers, so rather than apologise I shall end by continuing in that same vein! I shall construct the initial part of a rapidly increasing sequence, Sn. Such sequences are a good way to approach the vastness of the totality of the natural numbers, because compared to such sequences the operations that give us greater numbers (e.g. adding 1 to, multiplying by 2, squaring, and most notably, taking 2 to the power of) become relatively insignificant (cf. Lavine 1994). In particular, b will seem hardly greater than d once we have got S3.
Since we already have b = yy(d)(d), we have the start of a sequence (g, d, b, …) generated by the function, say k, such that k(1) = g, and k(n + 1) = yy(k(n))(k(n)). And if we change the name of k to ‘[1]’ we can define a whole sequence of functions by [p + 1](n) = [p][p](n)(n), for positive integers p. E.g., [2](n) = kk(n)(n). Then (much as I began above with j, the function that takes n to nn) we may begin again with the function that takes n to [n][n](n)(n), say j2. And repeating the above steps, through f2, y2, k2 = [1]2, and [p]2, we get to another new beginning, with j3. And so forth; from which we might take the [p]q for natural numbers p and q (renaming j as j1, etc.), in order to consider their diagonal function. Again it is useful to rename, this time replacing [p]q by [p|q], and to make a fresh start with the function that takes n to [n|n][n|n](n)(n), say K (whence we might move towards defining [p|q|r]; and thence, through [2||2], [3|||3|||3] and so forth, towards defining S4.) So we have S3 = KK(d)(d), the successor to S2 = d and S1 = g in some rapidly increasing sequence Sn.