Infinity and Proofs for the Existence of God
In the Spring 1991 issue of this journal, Ronald Tacelli argued for the thesis “that an eternal or everlasting world entails an actual quantitative infinite.” Tacelli's argumentation shows that he is principally concerned with a world which is everlasting because its past is without beginning--he has not much to say about the future. So, as I understand it, his thesis can be put in the form of the following conditional: if the world is without beginning, then the series of past events is an actual quantitative infinite.
This is a significant thesis, for, coupled with a claim about the impossibility of there existing or having existed an actual infinity of anything (books, days, dogs, etc.) in reality, it makes for a powerful argument for the claim that our universe has had a beginning in time. The argument looks like this: If the world is without beginning, then there has been an actual infinity of past events. But an actual infinity of past events is impossible. Therefore, the world is not without beginning. (modus tollens) One need only add another premise that whatever has a beginning must be caused by something other than itself (a version of the principle of sufficient reason), in order to complete an argument for the existence of God, i.e., of a first cause of the universe.
The impossibility of an actual infinity of anything in reality can be made intuitively obvious in a number of ways. While it is true that mathematicians are quite comfortable talking about actually infinite sets of numbers, some philosophers are uncomfortable talking about actual infinite sets of real, extramental objects. For example, a mathematician is apt to say that the set of positive integers is actually infinite, as well the set of negative integers. That is, each set contains an actual infinity of members. But note that the union of these two sets also contains an actual infinity of members as well; there are infinitely many positive and negative integers. This leads to the mathematical curiosity that in the case of infinite sets, a part can equal the whole: The set of positive integers contains just as many members as the set of positive and negative integers. But while one might be dazzled by this mathematical curiosity, one is soon confounded if one thinks that there can be an actual infinity of non-numerical, extramental objects as well.
Consider a library with infinitely many books. One morning the librarian checks his holdings and is pleased to see all is in order; no books have been lost. The library still contains infinitely many books. That night, however, a thief makes her way into the library and steals all the even numbered books. The next morning our cautious librarian checks his holdings again, and is relieved to find that he has just as many books as he had the day before. Even though the thief took infinitely many books from the library, infinitely many books remain. This strikes many as counter-intuitive. No matter what sense can be made of the actual infinite in mathematics, when it comes to libraries it makes no sense at all. If a thief takes infinitely many books out of the library, then, indeed, the library ought to have fewer books than before her clandestine operation.
Similar counter-intuitive results follow from a universe which contains an actual infinity of past events. It would seem, for example, that if the past is without beginning, then the universe can never be any older than it already is. For even if another ten million years should pass, the universe will have existed an infinite number of days before that passage of that ten million years as well as after it. Just as one cannot diminish the number of books in a library that has infinitely many of them, one cannot add one day to a universe the past of which is infinite.
What is happening in these examples; why do the proposed extramental, actual infinities strike us as impossibilities? In each case, one looks at the books in the library or the past days of the universe as elements of a set. The set itself is understood to be a certain kind of whole or totality. We then discover something odd about these totalities; they do not meet our ordinary expectations about wholes and parts. We expect wholes to be greater than their parts. But it turns out that a whole which has infinitely many members is equal to certain of its parts. We expect wholes to get bigger when we add something to them, but infinite totalities remain the same size, infinite, no matter how much we add to them. We understand mathematicians when they speak this way: We can readily see that the even numbers can be put in one-to-one correspondence with the set of even and odd numbers. If, on this basis, the mathematician wants to say there are as many even numbers as there are even and odd numbers, so be it. But we don't want librarians to speak this way because it violates our ordinary experience with parts and wholes in the physical world. We know that when we take even one book from the library, there are fewer books remaining within the library. We know that with each passing day, we are really one day older.
I would agree, then, that a real, actual infinity of past events is impossible. I do not think, however, that the conclusion that the universe must have had a beginning in time follows; for I do not think that a beginningless past necessarily entails that there has been an actual infinity of past events. That is to say, I think Tacelli's thesis “if the world is without beginning, then the series of past events is an actual quantitative infinite” is wrong. I want to argue that a beginningless past entails only that there has been a potential infinity of past events. I order to make my case, I wish first to discuss the meaning of the key phrases in this debate, actual infinity and potential infinity.
These are misleading phrases. We ordinarily take the term ‘actual’ to refer to something which is really existing or which is a determinate part of reality, and the term ‘potential’ to refer to what can exist or can be a part of reality, but is not yet a part of that reality. Given our ordinary use of these terms, some have argued that it is wrongheaded to consider the past actual, since it no longer exists. They would rule out the past's being an actual infinity on this basis alone; for how can there be an actual infinity of non-existing past events? But others have responded that it is equally wrongheaded to consider the past as some sort of potential, for surely the past is no longer a mere possibility which can be, but is not yet.
I think that it makes sense to speak each past event as actual even though past events no longer exist in the present, for I think actual can mean something more than existing in the present. A past event is actual in the sense that it has really occurred; it is a definite and determinate part of reality which, once it has occurred, can never be otherwise. But even though I think it makes sense to speak of the past as actual in this way, I still do not think that the past events of a beginningless universe constitute an actual infinity.
The term ‘actual’ in the phrase ‘actual infinity’ means something more than “having really occurred” or “being a determinate part of reality.” Fr. Tacelli's paper gives us a valuable insight into what the term ‘actual’ means in this context. He states that by ‘actual quantitative infinite’ he means “a limitless many which nevertheless comprise a completed set.” So the actual. quantitative infinite is infinite because it is limitless; it is quantitative because it speaks of the many; and it is actual because it is a completed set.
Tacelli's use of the terms ‘actual’ and ‘complete’ in this context mirrors that of William Lane Craig, whom Tacelli cites on the first page of his article. According to Craig, the characteristic feature of an actually infinite set is that it constitutes “a determinate whole actually possessing an infinite number of members.” Tacelli's completed set is Craig's determinate whole. What both uses of the term ‘actual’ have in common is that they speak of the infinite as some sort of totality which is all together, whole or complete. Thus, when one speaks of the set of natural numbers as an actual infinity, one must envision all of the natural numbers collected together as a whole at once and immediately, with no number lying outside of this collection so that it is inconceivable that any other natural number should be added to the set. The set of natural numbers is like a well-made chair. No part is missing; it is an integral whole. Yet it is unlike the chair inasmuch as it has infinitely many parts.
What then is a potential infinite? An example will be useful here. Suppose that matter, a Thanksgiving turkey for example, is infinitely divisible. Suppose also that an immortal master carver exists, who is so skilled in his craft that he is capable of cutting any slice of turkey, no matter how thin, in half again. On Thanksgiving morning, he is slicing turkey and placing these slices in a pile on a platter. He cuts a nice thick slice, and he cuts it in half. He places one half of the slice on the platter, and cuts the remaining half in half again. He places this half of the half on the platter, and slices the remaining half in half again. He continues in this fashion for eternity.
Note that the original slice is infinite inasmuch as an indefinite number of slices can be taken from it. The pile of slices on the platter is infinite in the sense that the number of slices it contains is increasing indefinitely, even though each new slice is getting thinner and thinner. But neither the original slice nor the pile of slices is actually infinite, though for different reasons.
The original slice is not actually infinite because it is a whole with determinate limits; it has a finite magnitude. The original slice is only potentially infinite, inasmuch as indefinitely many slices can be taken from it without end. The pile is at any given moment actually finite as well, since at any moment it will contain only finitely many slices of turkey. However, it is potentially infinite in the sense that it can continue to grow indefinitely without end. But from this point of view it is not a completed whole. Rather, it is a whole always in the process of being completed.
A potential infinite, then, is an indeterminate collection, which, because it is ever in the process of increasing or decreasing, fails to be a completed whole or totality. As soon as this process ceases, it finds itself to be actually finite. It is infinite only with respect to process, and this is to say, with respect to its state of incompleteness. An actual infinite, as Tacelli and Craig define it, is at once unlimited and yet is a completed whole or totality.
Now I want to take a closer look at Tacelli's argument. He points out that if the world began one year ago, then the series or set of events terminating in the present must be finite. But if the universe is infinitely old, “the set of past events--a set which terminates in what is happening now--must be in that case infinite.” So far, I agree with Tacelli. The real point to be established is that this infinite series is an actual infinite in the sense defined above. Tacelli attempts to establish this point by stating:
And it must be an actually infinite set--in this sense: that its infinity has already been achieved. For just as, if the universe were finite, the series of past events must have happened and been completed--must have been “gone through”--in order for the present event to have been reached, so, too, this must have happened if the universe is eternal or everlasting. But then the past must comprise a completed infinite set of events: a set which terminates in this event, and to which other events--somehow!--are being added.
Later, Tacelli makes the same point a little differently:
All past events must have occurred before this present event. For this present event is present not merely to remote days we may at our leisure imagine. It is ex hypothesi present to a beginningless past; and all past events must actually have happened for this present one to be.
In each of these two passages, Tacelli seems to be making this argument: In a beginningless universe, past events--simply in virtue of their being past events--have actually occurred. Each must have been actually completed in order to get to the present. Therefore, all must have been actually completed. And since there are infinitely many actually completed past events, the set of all past events must be infinite and whole or complete, that is, must constitute an actual infinity.
My objection to this argument is as follows: It is true that each past event in a beginningless universe must have been completed, for this is what it means to say that it is past. But here complete means something like having already occurred, being finished, used up, over. In this way, one might say that a baseball game is completed when it is over. Moreover, if every event in the past is complete or over, then the series of past events must be over too; just as the baseball game is over if every inning is over. But does this mean that the series of past events is an actual infinite? It does not; for to say that an infinite set is actually infinite says something other than that each of its members is complete.
When applied to the notion of infinity, as we have seen, actual means complete in the sense of constituting some sort of whole or totality. Now I agree that a baseball game is over when all of its innings are over, but I do not consider the game to be a whole or totality simply because all of its innings are over. This is a necessary, but not a sufficient condition for speaking about a whole game of baseball. To be rightly considered as some sort of whole or totality, a series of events like a baseball game must have not only an end, but a beginning and middle as well. It must have a first and a fifth inning. Since a beginningless universe lacks a beginning and a middle, the series of events which constitute this universe cannot constitute a whole. And therefore, the series of past events is not, in this case, an actual infinity because by actual infinity, we meant precisely something which was both an infinity and a totality.
To put my argument somewhat differently, even in a beginningless universe, the past cannot be an actual infinity, that is, an infinite totality, simply because the notion of an infinite totality is an incoherent notion, analogous to the notion of a square circle. A whole or totality is that which is complete in the sense of having a beginning, a middle, and an end. A beginningless universe does not, by definition, have a beginning. Therefore, it cannot constitute a totality. I do not dispute the fact that the past events of a beginningless universe are complete in the sense that they have actually occurred, each and every one of them. But I do deny that taken together, they can in any meaningful sense be considered to be a kind of whole or totality, which is what the term ‘actual’ in the phrase ‘actual infinity’ means.
My position is, I think, born out by the way in which we ordinarily use the terms ‘whole’ and ‘infinite’. If one reflects carefully on those sorts of things within one's experience which one considers to be wholes of a certain sort, note that each of these wholes has certain limits and is a whole precisely in virtue of those limits. A story is a whole story because of its beginning, middle, and end. A wall is a whole wall because of its top, sides, and bottom. A wall which had no top simply would not be a whole wall. That which is infinite is that which lacks a limit, an so fails to be a whole in some respect.
Tacelli's argument fails because it uses the term ‘actual’ in two different ways. He says that past events are said to be actual in the sense of being completed or over; but from this, he moves to the claim that the infinite series of events is actual in the sense of constituting some sort of whole. This second claim does not follow from the first, and must be false. For to say that something is both infinite because beginningless and a complete whole is to say that it is without beginning and has a beginning.
In a universe which lacks a beginning the past is infinite, but only potentially so. To say that a beginningless past is potentially infinite is only to say that prior to any event,there is some other event, or that before each event there is another event. I say “prior to any event” and “before each event” because I want to be making a claim about past events considered individually. I want to avoid talking about “the set” of past events, “the series” of past events, or “the past” considered as the collection of all past events, because phrases such as ‘the set’, ‘the series’, and ‘the past’ falsely suggest that we can talk about past events as some sort of totality--which is just what cannot be done in a beginningless universe.
Unlike an actual infinity of past events, a potential infinity of events could exist. There is nothing immediately counter-intuitive about a universe in which each event is preceded by some other event. First, between any two events in such a universe, there will be a determinate, finite relationship. No two events would be infinitely separated from one another. So a determinate causal relationship could exist between any two events in this universe. Second, one could not speak of the universe as getting any older in such a scenario, if by getting older one understands a day being added to the total sum of one's days. This definition of ‘older’ simply could not applied to an universe without beginning, for it presupposes that past days form some sort of total whole to which something can be added--a supposition which is incompatible with supposing that the universe to be without beginning. One could adjust one's definition, however. To say that the universe at T2 is older than the universe at T1 is simply to say that the “set,” loosely conceived, of events at T1 is a proper subset of the set of events at T2. This is an odd use of the term ‘older’, but it is a coherent one.
Third, one could not reach the present moment in such a universe, if one had to traverse past events an event at a time. But one does not have to worry about reaching the present, for, fortunately, the present is just where one happens to find oneself anyway. Fourth, one cannot speak of “the past” in such a universe in the way one ordinarily speaks of the past. When I speak of my past life, for example, I generally have in mind some whole made determinate by its beginning, my conception, and its end, the present. The past is history; and history, as story, has narrative unity, which unity is dependent upon making determinate a beginning, a middle, and an end. A beginningless universe cannot have history or a past in this sense. At best, one can arbitrarily select some event and consider it as if it were some first event. So just as the term ‘older’ will means something different when predicated of a beginningless universe, so does the phrase ‘the past’. ‘The past’ means only that which is already over.
Finally, is important to note that the potential infinity of a beginningless universe is not completely analogous to the potential infinity of the ever increasing pile of turkey-slices produced by the immortal master carver. At any given instant, the pile is finite. One can arrest the process of carving and contemplate the pile at any given instant as a whole, that is, as possessing some first and last slice. But even if one were to witness the last event of a beginningless universe, one could never step back and see it as a whole or gather all of its events into a totality because it lacks a beginning. A beginningless universe is, therefore, infinite at every moment of its existence; that is, every event has been preceded by another event ad infinitum.
This notable difference has lead some, including Fr. Tacelli, to speak about the past of a beginningless universe as an actual infinity, since there actually have been infinitely many past events in a universe without beginning (quite unlike the pile of turkey slices). But to say that there actually have been infinitely many past events is to say something like there have really been or truly been infinitely many events. With this much I have agreed. But this does not entail that these past events, which have really been, constitute a totality. By claiming that the past events of a beginningless universe constitute only a potential infinity, I have not intended to cast doubt upon the ontological status of past events as past, i.e., as really having occurred. I intend only to deny that they form a totality.
This denial is significant, however, for it means that the proof that the universe must have a beginning (and, consequently, must have been created by God) is flawed. Past events in a beginningless universe fail to constitute just that sort of infinity, an infinite totality, required in order to make the proof work. Although a beginningless universe is odd and quite impossible to imagine, it is not for that reason impossible. Actual infinities are impossible--conceptually impossible--because the notion of an infinite totality is incoherent. Potential infinities are conceptually consistent. Though they can not be comprehended as wholes, they are intelligible: One need only state that prior to each event, there is another event. Here one's attention is not upon the whole of past events; one simply makes a claim about each past event. So if the universe is without beginning, no contradiction follows, and no easy argument for the finiteness of time itself is open to us.
This is not to say, however, that a beginningless universe must be a godless one. In fact, a universe without beginning affords an interesting argument for the existence of God. Let me conclude by sketching out briefly the sort of argument for which it allows:
If the universe is without beginning, then prior to each event, there is another event. If “the past” is such that prior to each event, there is another event, then “the past” could not have transpired an event at a time. No infinite, not even a potential one, can be established by successive addition--as is evident in the case of the ever increasing pile of turkey slices, which never adds up to an infinity. But “the past” is, by definition, actual in the sense that it has been; and there must be a cause of its having been. Nothing within the infinite “series” of past events can be the cause of its having been, since past events are finite and transpire successively. Therefore, the cause of a potential infinity of past events must lie outside those events and, from outside of time, cause there to be a potential infinity of events in time. But an atemporal cause of events in time is what we mean by the term ‘God.’ So if “the past” is without beginning, it must have been caused by God; and if it so happens that it does have a beginning in time, it must also have been created by God. But “the past” has either had a beginning or it has not. Therefore God exists.
St. Anselm College
Manchester, New Hampshire
Tacelli, R. K., “Does the Eternity of the World Entail an Actual Infinite,” Lyceum 3.1 (1991), p. 15.
Craig, William L., The Kalam Cosmological Argument (New York: Barnes and Noble Books, 1979), p. 69.