Existence of an actual infinite

Premise (2.11) asserts that an actual infinite cannot exist in the real world. It is frequently alleged that this sort of claim has been falsified by Cantor's work on the actual infinite and by subsequent developments in set theory, which provide a convincing demonstration of the existence of actual infinites. But this allegation is far too hasty. It not only begs the question against denials of the mathematical legitimacy of the actual infinite on the part of certain mathematicians (such as intuitionists), but, more seriously, it begs the question against anti-Platonist views of mathematical objects. These are distinct questions, all too

6. This criterion allows that there may be events of shorter duration prior to the first standard event. By stipulating as one's standard event a shorter interval, these can be made arbitrarily brief.

Mathematical objects

Mathematical objects

Meaningless question

Exist (realism)

Meaningless question

Do not exist (anti-realism)

Exist (realism)

Do not exist (anti-realism)

Physicalism Conceptualism

\ Deductivism / Structuralism

\ Fictionalism

Physicalism Conceptualism

Figure 3.1 Some metaphysical options concerning the existence of abstract objects.

Human

Divine

Figure 3.1 Some metaphysical options concerning the existence of abstract objects.

often conflated by recent critics of the argument (Sobel 2004, pp. 181-9, 198;9; Oppy 2006a, pp. 291-3; cf. Craig, 2008). Most non-Platonists would not go to the intuitionistic extreme of denying mathematical legitimacy to the actual infinite - hence, Hilbert's defiant declaration, "No one shall be able to drive us from the paradise that Cantor has created for us" (Hilbert 1964, p. 141) - rather they would simply insist that acceptance of the mathematical legitimacy of certain notions does not imply an ontological commitment to the reality of various objects. Thus, in Hilbert's view, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought. . . . The role that remains for the infinite to play is solely that of an idea" (Hilbert 1964, p. 151). Cantor's system and axiomatized set theory may be taken to be simply a universe of discourse, a mathematical system based on certain adopted axioms and conventions, which carries no ontological commitments. In view of the plethora of alternatives to Platonism (Figure 3.1), critics of the argument cannot justifiably simply assume that the language of mathematics commits us ontologically to mind-independent entities, especially to such obscure objects as sets.

On antirealist views of mathematical objects such as Balaguer's fictionalism (Balaguer 1998, pt. II; 2001, pp. 87-114; Stanford Encyclopedia of Philosophy 2004b), Yablo's figuralism (Yablo 2000, pp. 275-312; 2001, pp. 72-102; 2005, pp. 88-115), Chihara's constructibilism (Chihara 1990, 2004; 2005, pp. 483-514), or Hellman's Modal structuralism (Hellman 1989; 2001, pp. 129-57; 2005, pp. 536-62), mathematical discourse is not in any way abridged, but there are, notwithstanding, no mathematical objects at all, let alone an infinite number of them. The abundance of nominalist (not to speak of conceptualist)

alternatives to Platonism renders the issue of the ontological status of mathematical entities at least a moot question. The Realist, then, if he is to maintain that mathematical objects furnish a decisive counterexample to the denial of the existence of the actual infinite, must provide some overriding argument for the reality of mathematical objects, as well as rebutting defeaters of all the alternatives consistent with classical mathematics - a task whose prospects for success are dim, indeed. It is therefore open to the mutakallim to hold that while the actual infinite is a fruitful and consistent concept within the postulated universe of discourse, it cannot be transposed into the real world.

The best way to support (2.11) is by way of thought experiments that illustrate the various absurdities that would result if an actual infinite were to be instantiated in the real world.7 Benardete, who is especially creative and effective at concocting such thought experiments, puts it well: "Viewed in abstracto, there is no logical contradiction involved in any of these enormities; but we have only to confront them in concreto for their outrageous absurdity to strike us full in the face" (Benardete 1964, p. 238).8

Let us look at just one example: David Hilbert's famous brainchild "Hilbert's Hotel."9 As a warm-up, let us first imagine a hotel with a finite number of rooms. Suppose,

7. Ludwig Wittgenstein nicely enunciated this strategy when, in response to Hilbert's solemn declaration, he quipped, "I wouldn't dream of trying to drive anyone from this paradise. I would do something quite different: I would try to show you that it is not a paradise - so that you'll leave of your own accord. I would say, 'You're welcome to this; just look about you' . . ." (Wittgenstein 1976, p. 103). But here the strategy is employed on behalf of metaphysical, not mathematical, finitism. Oppy objects that such puzzles show, at most, that certain kinds of actual infinities cannot exist, but that this conclusion cannot be generalized (Oppy 2006b, p. 140). The difficulty with this attempt to blunt the force of the absurdities is twofold: (i) nothing in the various situations seems to be metaphysically impossible apart from the assumption of an actual infinite and (ii) the absurdities are not tied to the particular kinds of objects involved.

8. He has in mind especially what he calls paradoxes of the serrated continuum, such as the following:

Here is a book lying on the table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper - 1/2 inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? 1/4 inch thick. And the third page of the book, how thick is this third sheet of paper? 1/8 inch thick, &c. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one-half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now - slowly - lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze. (Benardete 1964, pp. 236-7).

To our mind this conclusion itself is evidently metaphysically absurd. Although Oppy, following Hazen (1993), offers expansions of the story so that someone opening the book will have some sort of visual experience, rather than as it were, a blank (Oppy 2006a, pp. 83-5), that does not negate the conclusion that there is nothing there to see since there is no last page. Benardete imagines what would happen if we tried to touch the last page of the book. We cannot do it. Either there will be an impenetrable barrier at M + 1, which seems like science fiction, or else our fingers will penetrate through an infinity of pages without first penetrating a page, which recalls Zeno's paradoxes in spades, since the pages are actual entities. What makes paradoxes such as these especially powerful, as Benardete points out, is that no process or supertask is involved here; each page is an actual entity having a finite thickness (none is a degenerate interval) which could be unbound from the others and all the pages scattered to the four winds, so that an actual infinity of pages would exist throughout space. If such a book cannot exist, therefore, neither can an actual infinite.

9. The story of Hilbert's Hotel is related in Gamow (1946, p. 17).

furthermore, that all the rooms are occupied. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full," and that is the end of the story. But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are occupied. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4, and so on out to infinity. As a result of these room changes, room #1 now becomes vacant, and the new guest gratefully checks in. But remember, before he arrived, all the rooms were occupied! Equally curious, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys - how can there not be one more person in the hotel than before?

But the situation becomes even stranger. For suppose an infinity of new guests show up at the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. Because any natural number multiplied by two always equals an even number, all the guests wind up in even-numbered rooms. As a result, all the odd-numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were occupied! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be a single person more in the hotel than before.

But Hilbert's Hotel is even stranger than the German mathematician made it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one fewer person in the hotel? Not according to infinite set theory! Suppose the guests in rooms #1, 3, 5, . . . check out. In this case an infinite number of people has left the hotel, but by Hume's Principle, there are no fewer people in the hotel. In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any fewer people in the hotel. Now suppose the proprietor does not like having a half-empty hotel (it looks bad for business). No matter! By shifting guests in even-numbered rooms into rooms with numbers half their respective room numbers, he transforms his half-vacant hotel into one that is completely full. In fact, if the manager wanted double occupancy in each room, he would have no need of additional guests at all. Just carry out the dividing procedure when there is one guest in every room of the hotel, then do it again, and finally have one of the guests in each odd-numbered room walk next door to the higher even-numbered room, and one winds up with two people in every room!

One might think that by means of these maneuvers the proprietor could always keep this strange hotel fully occupied. But one would be wrong. For suppose that the persons in rooms #4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that as many guests checked out this time as when the guests in rooms #1, 3, 5, . . . checked out! Can anyone believe that such a hotel could exist in reality?

Hilbert's Hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible.

Partisans of the actual infinite might concede the absurdity of a Hilbert's Hotel but maintain that this case is somehow peculiar and, therefore, its metaphysical impossibility warrants no inference that an actual infinite is metaphysically impossible. This sort of response might seem appropriate with respect to certain absurdities involving actual infinites; for example, those imagining the completion of a so-called supertask, the sequential execution of an actually infinite number of definite and discrete operations in a finite time. But when it comes to situations involving the simultaneous existence of an actually infinite number of familiar macroscopic objects, then this sort of response seems less plausible.10 If a (denumerably) actually infinite number of things could exist, they could be numbered and manipulated just like the guests in Hilbert's Hotel. Since nothing hangs on the illustration's involving a hotel, the metaphysical absurdity is plausibly attributed to the existence of an actual infinite. Thus, thought experiments of this sort show, in general, that it is impossible for an actually infinite number of things to exist in reality.

At this point, the actual infinitist has little choice but, in Oppy's words, simply to "embrace the conclusion of one's opponent's reductio ad absurdum argument" (Oppy 2006a, p. 48). Oppy explains, "these allegedly absurd situations are just what one ought to expect if there were . . . physical infinities" (Oppy 2006a, p. 48).

Oppy's response, however, falls short: it does nothing to prove that the envisioned situations are not absurd but only serves to reiterate, in effect, that if an actual infinite could exist in reality, then there could be a Hilbert's Hotel, which is not in dispute. The problem cases would, after all, not be problematic if the alleged consequences would not ensue! Rather the question is whether these consequences really are absurd.

Sobel similarly observes that such thought experiments bring into conflict two "seemingly innocuous" principles, namely,

(i) There are not more things in a multitude M than there are in a multitude M' if there is a one-to-one correspondence of their members.

(ii) There are more things in M than there are in M' if M' is a proper submultitude of M.

We cannot have both of these principles along with

(iii) An infinite multitude exists.

10. Oppy, for example, makes the point that having a hotel with an infinite number of occupied rooms does not commit one to the possibility of accommodating more guests by shifting guests about - maybe the hotel's construction hinders the guests' movements or the guests die off before their turn to move comes round. But as a Gedankenexperiment Hilbert's Hotel can be configured as we please without regard to mere physical possibilities.

For Sobel, the choice to be taken is clear: "The choice we have taken from Cantor is to hold on to (i) while restricting the proper submultiplicity condition to finite multiplicities. In this way we can 'have' comparable infinite multitudes" (Sobel 2004, pp. 186-7; cf. Mackie 1982, p. 93).

But the choice taken from Cantor of which Sobel speaks is a choice on the part of the mathematical community to reject intuitionism and finitism in favor of axiomatic infinite set theory. Finitism would too radically truncate mathematics to be acceptable to most mathematicians. But, as already indicated, that choice does not validate metaphysical conclusions. The metaphysician wants to know why, in order to resolve the inconsistency among (i)-(iii), it is (ii) that should be jettisoned (or restricted). Why not instead reject or restrict to finite multiplicities (i), which is a mere set-theoretical convention? More to the point, why not reject (iii) instead of the apparently innocuous (i) or (ii)? It certainly lacks the innocuousness of those principles, and giving it up would enable us to affirm both (i) and (ii). Remember: we can "have" comparable infinite multiplicities in mathematics without admitting them into our ontology.

Sobel thus needs some argument for the falsity of (ii). Again, it is insufficient merely to point out that if (i) and (iii) are true, then (ii) is false, for that is merely to reiterate that if an actual infinite were to exist, then the relevant situations would result, which is not in dispute.

Take Hilbert's Hotel. Sobel says that the difficulties with such a hotel are practical and physical; "they bring out the physical impossibility of this particular infinity of concurrent real things, not its logical impossibility" (Sobel 2004, p. 187). But the claim is not that such a hotel is logically impossible but metaphysically impossible. As an illustrative embodiment of transfinite arithmetic based on the axiomatic set theory, Hilbert's Hotel will, of necessity, be as logically consistent as that system; otherwise it would be useless as an illustration. But it also vividly illustrates the absurd situations to which the real existence of an infinite multitude can lead. The absurdity is not merely practical and physical; it is ontologically absurd that a hotel exist which is completely full and yet can accommodate untold infinities of new guests just by moving people around.

Oppy is prepared, if need be, simply to bite the bullet: "There can, after all, be a hotel in which infinitely many new guests are accommodated, even though all the rooms are full, via the simple expedient of moving the guests in room N to room 2N (for all N)" (Oppy 2006a, p. 53). So asserting does nothing to alleviate one's doubts that such a hotel is absurd. And would Oppy say something similar about what would happen when an infinite number of guests depart?11 In transfinite arithmetic, inverse operations of subtraction and division with infinite quantities are prohibited because they lead to contradictions; as Sobel says, "Of course, as operations and properties are extended from finite to transfinite cardinals, some arithmetic principles are left confined to the finite" (Sobel 2007). But in reality, one cannot stop people from checking out of a hotel if they so desire! In this case, one does wind up with logically impossible situations, such

11. Oppy suggests using J. Conway's recently developed constructions called surreal numbers to define operations of subtraction and division of transfinite numbers (Oppy 2006b, p. 140), but he explicitly denies that such non-canonical theories can be applied "to real-world problems, if one wishes to treat one's models with full ontological seriousness" (Oppy 2006a, p. 272). Oppy does not show, nor does he think, that the results of operations on sur-reals would be any less counterintuitive when translated into the concrete realm.

as subtracting identical quantities from identical quantities and finding nonidentical differences.12

In response to the absurdities springing from performing inverse operations with infinite quantities, David Yandell has insisted that subtraction of infinite quantities does not yield contradictions. He writes,

Subtracting the even positive integers from the set of positive integers leaves an infinite set, the odd positive integers. Subtracting all of the positive integers greater than 40 from the set of positive integers leaves a finite (forty-membered) set. Subtracting all of the positive integers from the set of positive integers leaves one with the null set. But none of these subtractions could possibly lead to any other conclusion than each leads to. This alleged contradictory feature of the infinite seems not to generate any actual contradictions. (Yandell 2003, p. 132)

It is, of course, true that every time one subtracts all the even numbers from all the natural numbers, one gets all the odd numbers, which are infinite in quantity. But that is not where the contradiction is alleged to lie. Rather the contradiction lies in the fact that one can subtract equal quantities from equal quantities and arrive at different answers. For example, if we subtract all the even numbers from all the natural numbers, we get an infinity of numbers, and if we subtract all the numbers greater than three from all the natural numbers, we get only four numbers. Yet in both cases we subtracted the identical number of numbers from the identical number of numbers and yet did not arrive at an identical result. In fact, one can subtract equal quantities from equal quantities and get any quantity between zero and infinity as the remainder. For this reason, subtraction and division of infinite quantities are simply prohibited in transfinite arithmetic - a mere stipulation which has no force in the nonmathematical realm.

Sometimes it is said that we can find concrete counterexamples to the claim that an actually infinite number of things cannot exist, so that Premise (2.11) must be false. For example, Walter Sinnott-Armstrong asserts that the continuity of space and time entails the existence of an actually infinite number of points and instants (Craig & SinnottArmstrong 2003, p. 43). This familiar objection gratuitously assumes that space and time are composed of real points and instants, which has never been proven. Mathematically, the objection can be met by distinguishing a potential infinite from an actual infinite. While one can continue indefinitely to divide conceptually any distance, the series of subintervals thereby generated is merely potentially infinite, in that infinity serves as a limit that one endlessly approaches but never reaches. This is the thoroughgoing Aristotelian position on the infinite: only the potential infinite exists. This position does not imply that minimal time atoms, or chronons, exist. Rather time, like space, is infinitely divisible in the sense that division can proceed indefinitely, but time is never actually infinitely divided, neither does one arrive at an instantaneous point. If one thinks of a geometrical line as logically

12. It will not do, in order to avoid the contradiction, to assert that there is nothing in transfinite arithmetic that forbids using set difference to form sets. Indeed, the thought experiment assumes that we can do such a thing. Removing all the guests in the odd-numbered rooms always leaves an infinite number of guests remaining, and removing all the guests in rooms numbered greater than four always leaves three guests remaining. That does not change the fact that in such cases identical quantities minus identical quantities yields nonidentical quantities, a contradiction.

prior to any points which one may care to specify on it rather than as a construction built up out of points (itself a paradoxical notion13), then one's ability to specify certain points, like the halfway point along a certain distance, does not imply that such points actually exist independently of our specification of them. As Grunbaum emphasizes, it is not infinite divisibility as such which gives rise to Zeno's paradoxes; the paradoxes presuppose the postulation of an actual infinity of points ab initio. ". . . [A]ny attribution of (infinite) 'divisibility' to a Cantorian line must be based on the fact that ab initio that line and the intervals are already 'divided' into an actual dense infinity of point-elements of which the line (interval) is the aggregate. Accordingly, the Cantorian line can be said to be already actually infinitely divided" (Grunbaum 1973, p. 169). By contrast, if we think of the line as logically prior to any points designated on it, then it is not an ordered aggregate of points nor actually infinitely divided. Time as duration is then logically prior to the (potentially infinite) divisions we make of it. Specified instants are not temporal intervals but merely the boundary points of intervals, which are always nonzero in duration. If one simply assumes that any distance is already composed out of an actually infinite number of points, then one is begging the question. The objector is assuming what he is supposed to prove, namely that there is a clear counterexample to the claim that an actually infinite number of things cannot exist.

Some critics have charged that the Aristotelian position that only potential, but no actual, infinites exist in reality is incoherent because a potential infinite presupposes an actual infinite. For example, Rudy Rucker claims that there must be a "definite class of possibilities," which is actually infinite in order for the mathematical intuitionist to regard the natural number series as potentially infinite through repetition of certain mathematical operations (Rucker 1980, p. 66). Similarly, Richard Sorabji asserts that Aristotle's view of the potentially infinite divisibility of a line entails that there is an actually infinite number of positions at which the line could be divided (Sorabji 1983, pp. 210-3, 322-4).

13. See Craig (1985). Consider, for example, the many variations on the Grim Reaper Paradox (Benardete 1964, pp. 259-61; Hawthorne 2000; Oppy 2006a, pp. 63-6, 81-3). There are denumerably infinitely many Grim Reapers (whom we may identify as gods, so as to forestall any kinematic objections). You are alive at 12:00 p.m. Grim Reaper 1 will strike you dead at 1:00 p.m. if you are still alive at that time. Grim Reaper 2 will strike you dead at 12:30 p.m. if you are still alive then. Grim Reaper 3 will strike you dead at 12:15 p.m., and so on. Such a situation seems clearly conceivable but leads to an impossibility: you cannot survive past 12:00 p.m. and yet you cannot be killed at any time past 12:00 p.m. Oppy's solution to a similar paradox concerning infinitely many deafening peals, viz. that there is no particular peal responsible for your deafness but that the collective effect of infinitely many peals is to bring about deafness (Oppy 2006a, p. 83), not only involves a most bizarre form of retro-causation (Benardete 1964, p. 259) but is also in any case inapplicable to the Grim Reaper version since once you are dead no further Grim Reaper will swing his scythe, so that collective action is out of the question. The most plausible way to avert such paradoxes is by denying that time and space are constructions out of an actually infinite number of points. (My thanks to Alexander Pruss for drawing my attention to this version of the paradox.)

Moreover, on an A-Theory of time, according to which temporal becoming is an objective feature of reality, treating time as composed of instants (degenerate temporal intervals of zero duration) seems to land one in Zeno's clutches since temporal becoming would require the actualization of consecutive instants, which is incoherent. For a good discussion, see Grunbaum (1950-1, pp. 143-86). Grunbaum succeeds in defending the continuity of time only at the expense of sacrificing temporal becoming, which his interlocutors James and Whitehead would not do. See further Craig (2000c).

If this line of argument were successful, it would, indeed, be a tour de force since it would show mathematical thought from Aristotle to Gauss to be not merely mistaken or incomplete but incoherent in this respect. But the objection is not successful. For the claim that a physical distance is, say, potentially infinitely divisible does not entail that the distance is potentially divisible here and here and here and. . . . Potential infinite divisibility (the property of being susceptible of division without end) does not entail actual infinite divisibility (the property of being composed of an infinite number of points where divisions can be made). The argument that it does is guilty of a modal operator shift, inferring from the true claim

(1) Possibly, there is some point at which x is divided to the disputed claim

(2) There is some point at which x is possibly divided.

But it is coherent to deny the validity of such an inference. Hence, one can maintain that a physical distance is potentially infinitely divisible without holding that there is an infinite number of positions where it could be divided.

Rucker also argues that there are probably, in fact, physical infinities (Rucker 1980, p. 69). If the mutakallim says, for example, that time is potentially infinite, then Rucker will reply that the modern, scientific worldview sees the past, present, and future as merely different regions coexisting in space-time. If he says that any physical infinity exists only as a temporal (potentially infinite) process, Rucker will rejoin that it is artificial to make physical existence a by-product of human activity. If there are, for example, an infinite number of bits of matter, this is a well-defined state of affairs which obtains right now regardless of our apprehension of it. Rucker concludes that it seems quite likely that there is some form of physical infinity.

Rucker's conclusion, however, clearly does not follow from his arguments. Time and space may well be finite. But could they be potentially infinite? Concerning time, even if Rucker were correct that a tenseless four-dimensionalism is correct, that would provide no reason at all to think the space-time manifold to be temporally infinite: there could well be finitely separated initial and final singularities. In any case, Rucker is simply incorrect in saying that "the modern, scientific worldview" precludes a theory of time, according to which temporal becoming is a real and objective feature of reality. Following McTaggart, contemporary philosophers of space and time distinguish between the so-called A-Theory of time, according to which events are temporally ordered by tensed determinations of past, present, and future, and temporal becoming is an objective feature of physical reality, and the so-called B-Theory of time, according to which events are ordered by the tenseless relations of earlier than, simultaneous with, and later than, and temporal becoming is purely subjective. Although some thinkers have carelessly asserted that relativity theory has vindicated the B-Theory over against its rival, such claims are untenable. One could harmonize the A-Theory and relativity theory in at least three different ways: (1) distinguish metaphysical time from physical or clock time and maintain that while the former is A-Theoretic in nature, the latter is a bare abstraction therefrom, useful for scientific purposes and quite possibly B-Theoretic in character, the element of becoming having been abstracted out; (2) relativize becoming to reference frames, just as is done with simultaneity; and (3) select a privileged reference frame to define the time in which objective becoming occurs, most plausibly the cosmic time, which serves as the time parameter for hypersurfaces of homogeneity in space-time in the General Theory of Relativity. And concerning space, to say that space is potentially infinite is not to say, with certain constructivists, that it depends on human activity (nor again, that there are actual places to which it can extend), but simply that space expands limitlessly as the distances between galaxies increase with time. As for the number of bits of matter, there is no incoherence in saying that there is a finite number of bits or that matter is capable of only a finite number of physical subdivisions, although mathematically one could proceed to carve up matter potentially ad infinitum. The sober fact is that there is just no evidence that actual infinities are anywhere instantiated in the physical world. It is therefore futile to seek to rebut (2.11) by appealing to clear counterexamples drawn from physical science.

2.12. An infinite regress of events as an actual infinite

The second premise states that an infinite temporal regress of events is an actual infinite. The point seems obvious enough, for if there has been a sequence composed of an infinite number of events stretching back into the past, then the set of all events in the series would be an actually infinite set.

But manifest as this may be to us, it was not always considered so. The point somehow eluded Aristotle himself, as well as his scholastic progeny, who regarded the series of past events as a potential infinite. Aristotle contended that since things in time come to exist sequentially, an actual infinite never exists at any one moment; only the present thing actually exists (Physics 3.6.206a25-206b1). Similarly, Aquinas, after confessing the impossibility of the existence of an actual infinite, nevertheless proceeded to assert that the existence of an infinite regress of past events is possible (Summa Theologiae 1.a.7.4.). This is because the series of past events does not exist in actuality. Past events do not now exist, and hence do not constitute an infinite number of actually existing things. The series is only potentially infinite, not actually infinite, in that it is constantly increasing by the addition of new events.

These Aristotelian thinkers are clearly presupposing an A-Theory of time and an ontology of presentism, according to which the only temporal items which exist are those that presently exist. On a B-Theory of tenseless time, since there is ontological parity among all events, there can be no question that an infinite temporal regress of events is composed of an actually infinite number of events.14 Since all events are equally real, the fact that they exist (tenselessly) at different times loses any significance. The question, then, is whether events' temporal distribution over the past on a presentist ontology precludes our saying that the number of events in a beginningless series of events is actually infinite.

Now we may take it as a datum that the presentist can accurately count things that have existed but no longer exist. He knows, for example, how many US presidents there have been up through the present incumbent, what day of the month it is, how many shots Oswald squeezed off, and so forth. He knows how old his children are and can reckon how many billion years have elapsed since the Big Bang, if there was such an event. The

14. Some philosophers of time, such as C. D. Broad and Michael Tooley, have defended a sort of hybrid A/B-Theory, according to which the past and present are on an ontological par, the past being a growing space-time block. On such a view, a beginningless series of past events is also, uncontroversially, actually infinite.

nonexistence of such things or events is no hindrance to their being enumerated. Indeed, any obstacle here is merely epistemic, for aside from considerations of vagueness there must be a certain number of such things. So in a beginningless series of past events of equal duration, the number of past events must be infinite, for it is larger than any natural number. But then the number of past events must be N0, for ^ is not a number but an ideal limit. Aquinas' own example of a blacksmith working from eternity who uses one hammer after another as each one breaks furnishes a good example of an actual infinite, for the collection of all the hammers employed by the smith is an actual infinite. The fact that the broken hammers still exist is incidental to the story; even if they had all been destroyed after being broken, the number of hammers broken by the smith is the same. Similarly, if we consider all the events in an infinite temporal regress of events, they constitute an actual infinite.

The question arises whether on the A-Theory the series of future events, if time will go on forever, is not also actually infinite. Intuitively, it seems clear that the situation is not symmetrical, but this is notoriously difficult to express. It might rightly be pointed out that on presentism there are no future events and so no series of future events. Therefore, the number of future events is simply zero, not N0. (By this statement, one means not that there are future events, and that their number is 0, but that there just are no future events.) But on presentism, the past is as unreal as the future and, therefore, the number of past events could, with equal justification, be said to be zero. It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infinite number of events, just as in a beginningless past there have been an actually infinite number of events. But in a sense that assertion is false; for there never will be an actually infinite number of events since it is impossible to count to infinity. The only sense in which there will be an infinite number of events is that the series of events will go toward infinity as a limit. But that is the concept of a potential infinite, not an actual infinite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infinite, that is to say, finite but indefinitely increasing toward infinity as a limit. The situation, significantly, is not symmetrical: as we have seen, the series of events earlier than any arbitrarily selected future event cannot properly be regarded as potentially infinite. So when we say that the number of past events is infinite, we mean that prior to today, N0 events have elapsed. But when we say that the number of future events is infinite, we do not mean that N0 events will elapse, for that is false. Ironically, then, it turns out that the series of future events cannot be actually infinite regardless of the infinity of the past or the metaphysical possibility of an actual infinite, for it is the objectivity of temporal becoming that makes the future potentially infinite only.

Because the series of past events is an actual infinite, all the absurdities attending the existence of an actual infinite apply to it. For example, if the series of past events is actually infinite, then the number of events that have occurred up to the present is no greater than the number that have occurred at any point in the past. Or again, if we number the events beginning in the present, then there have occurred as many odd-numbered events as events. If we mentally take away all the odd-numbered events, there are still an infinite number of events left over; but if we take away all the events greater than three, there are only four events left, even though in both cases we took away the same number of events.

2.13. Conclusion

Since an actual infinite cannot exist and an infinite temporal regress of events is an actual infinite, we may conclude that an infinite temporal regress of events cannot exist. Therefore, since the temporal regress of events is finite, the universe began to exist.

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