An actual infinite cannot be formed by successive addition
Craig’s second philosophical argument grants for the sake of argument that there can exist an actual infinite in reality. Nevertheless, Craig argues that the past, due to the nature of how it has been formed, cannot be an actual infinite. The basic argument can be formulated as follows:
- A collection formed by successive addition cannot be an actual infinite.
- The temporal series of events is a collection formed by successive addition.
- Therefore, the temporal series of events cannot be an actual infinite.
To begin, we need to understand what is meant by “successive addition.” Craig explains,
By “successive addition,” one means the accrual of one new element at a (later) time. The temporality of the process of accrual is critical here…[W]e are concerned here with a temporal process of successive addition of one element after another (Blackwell Companion, pp. 117).
The basic idea is that we add an element to a collection at one point in time. We then add a second element to the collection at a later point in time. We then add a third at a yet later point in time. And so forth. We are not adding multiple elements simultaneously; rather, we are adding each of the elements sequentially.
With this understanding, we proceed to an evaluation of the argument.
Defense of first premise
That an actually infinite collection cannot be formed by successive addition seems to follow from the nature of successive addition and the actual infinite. It seems clear that the formation of an actual infinite by successive addition by beginning at some point is impossible. For given any number N, N + 1 is a finite number. So, no matter how many successive additions are made, one cannot get from a beginning point to another point infinitely far away. As Craig writes,
One sometimes, therefore, speaks of the impossibility of counting to infinity, for no matter how many numbers one counts, one can always count one more number before arriving at infinity. One sometimes speaks instead of the impossibility of traversing the infinite. The difficulty is the same: no matter how many steps one takes, the addition of one more step will not bring one to a point infinitely distant from one’s starting point (Blackwell Companion, pp. 118).
But what about forming an actually infinite collection by successive addition by never beginning but ending at a point? In this case, the series of elements is, at every point, already actually infinite. Thus, the difficulty of moving from a starting point to another point infinitely far away is avoided. Craig responds to this possibility as follows:
Although the problems will be different, the formation of an actually infinite collection by never beginning and ending at some point seems scarcely less difficult than the formation of such a collection by beginning at some point and never ending. If one cannot count to infinity, how can one count down from infinity? If one cannot traverse the infinite by moving in one direction, how can one traverse it by moving in the opposite direction? In order for us to have “arrived” at today, temporal existence has, so to speak, traversed an infinite number of prior events. But before the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infinitum. One gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd (ibid., pp. 118).
So, it seems impossible to form an actually infinite collection by successive addition. The first premise of the argument, therefore, is confirmed.
Another defense of the first premise that Craig gives appeals to the Tristram Shandy Paradox (henceforth, TSP). The paradox was originally discussed by the twentieth-century philosopher Bertrand Russell. Tristram Shandy is the title character from the book The Life and Opinions of Tristram Shandy, Gentleman by Laurence Stern. In the novel, Tristram Shandy writes his autobiography so slowly that it takes him an entire year to record the events of a single day. The paradox arises when we ask the question, Can Tristram Shandy ever finish writing his autobiography? For the sake of the paradox, we are assuming that Tristram Shandy is immortal and thus has infinite time to write his autobiography.
Russell answered the question in the affirmative. His reasoning was that, given infinite time, we can put the days and years that go by in infinite time into a correspondence so that to each day there would correspond one year, and both are infinite. Thus, there is no day that Tristram Shandy would fail to write in his autobiography.
But, as Craig observes,
Such an assertion is misleading, however. The fact that every part of the autobiography will be eventually written does not imply that the whole autobiography will be eventually written, which was, after all, Tristram Shandy’s concern. For every part of the autobiography there is some time at which it will be completed, but there is not some time at which every part of the autobiography will be completed. Given an A-Theory of time, though he write forever, Tristram Shandy would only get farther and farther behind, so that instead of finishing his autobiography, he would progressively approach a state in which he would be infinitely far behind (Blackwell Companion, pp. 120-121).
Craig then modifies the TSP so that instead of Tristram Shandy writing endlessly into the future, we suppose that he has been writing from eternity past at the rate of one day per year. Craig explains,
Should not Tristram Shandy now be infinitely far behind? For if he has lived for an infinite number of years, Tristram Shandy has recorded an equally infinite number of past days. Given the thoroughness of his autobiography, these days are all consecutive days. At any point in the past or present, therefore, Tristram Shandy has recorded a beginningless, infinite series of consecutive days. But now the question arises: Which days are these? Where in the temporal series of events are the days recorded by Tristram Shandy at any given point? The answer can only be that they are days infinitely distant from the present. For there is no day on which Tristram Shandy is writing which is finitely distant from the last recorded day…
But there is no way to traverse the temporal interval from an infinitely distant event to the present, or, more technically, for an event which was once present to recede to an infinite temporal distance. Since the task of writing one’s autobiography at the rate of 1 year per day seems obviously coherent, what follows from the Tristram Shandy story is that an infinite series of past events is absurd (ibid., pp. 121).
Thus, the TSP provides further support that the traversal of an infinite past (via successive addition) is impossible.
Critique of first premise
Graham Oppy, drawing from the philosopher Fred Dretske, considers the possibility of traversing the infinite (or counting to infinity). The idea is that one can indeed traverse the infinite (or count to infinity) if one counts and never stops counting. Oppy writes,
“One counts to infinity just in case, for each finite number N, one counts past N. But unless one stops counting, one will eventually reach any given finite N (Philosophical Perspectives on Infinity, pg. 61).
But as Craig points out, Oppy does not take into account here the important distinction between an actual infinite and a potential infinite. Craig writes, “One who, having begun, never stops counting counts ‘to infinity’ only in the sense that one counts potentially infinitely” (Blackwell Companion, pp. 118, footnote). Thus, Oppy’s contention provides no support to showing the possibility of forming an actually infinite collection by successive addition.
Elsewhere, Oppy engages with this reply and offers additional commentary on, and counterarguments against, the first premise of Craig’s argument as follows:
On behalf of the…[first] premise of this sub-argument – that is, the claim that a collection formed by successive addition cannot be an actual infinite – Craig notes that it is tantamount to the claim that it is impossible to count to infinity. He offers the following illustration of what he takes to be the central difficulty: ‘Suppose we imagine a man running through empty space on a path of stone slabs, a path constructed such that when the man’s foot strikes the last slab, another appears immediately in front of him. It is clear that, even if the man runs for eternity, he will never run across all of the slabs. For every time his foot strikes the last slab, a new one appears in front of him, ad infinitum. The traditional cognomen for this is the impossibility of traversing the infinite.’ (104)
In Craig’s example, the question is not whether the man can run across all of the slabs, but rather whether he can run across infinitely many slabs. For, if he achieves the latter task and yet not the former, he will still have completed an actual infinite by successive addition. If we suppose that the rate at which the slabs appear is constant, then, in any finite amount of time, only finitely many slabs appear: there is no time at which infinitely many slabs have been crossed. However, if the man runs for an infinite amount of time – that is, if, for each n, there is an nth slab that the man crosses – it is nonetheless true that infinitely many slabs are crossed: there is an actually infinite collection that is formed by successive addition. (Of course, Craig will resist this way of characterising matters: given his view that the future is not real, he will insist that it is at best true that infinitely many slabs will be crossed: the collection that is formed here by successive addition is at best “potentially infinite”.)
But what if we suppose that the time lapse between slabs decreases according to a geometric ratio, and that the man is replaced by a bouncing ball whose height of bounce decreases according to the same geometric ratio? If the ball hits the first slab at one minute to twelve, the second slab at ½ minute to twelve, the third slab at ¼ minute to twelve, and so on, then the ball can come to rest on a slab at twelve, having made infinitely many bounces on different slabs in the interval between one minute to twelve and twelve. In this example, we have a process – the bouncing of the ball – that plainly does form an actual infinite by successive addition. Consequently, we don’t need to challenge Craig’s view about the reality of the future in order to reject the second premise of the argument under discussion: there are perfectly ordinary processes that involve formation of an actual infinite by successive addition in not obviously impossible worlds (in which space and time are composed of points, and there are no quantum or thermodynamical effects to rule out the precise application of classical kinematics to the motion of a bouncing ball). Since Craig has – for the purposes of this argument – renounced the claim that there cannot be actual infinities, it is quite unclear what reason we are supposed to have for rejecting this counter-example to the alleged impossibility of forming actual infinities by successive addition (Arguing About Gods, pg. 143-144).
Here, Oppy recognizes the important distinction between an actual infinite and a potential infinite and seems to more or less concede the point that Craig’s response succeeds on the supposition of a presentist view of time. But he then raises a further difficulty. If successive addition can be carried out in such a way that each addition is carried out in half the time it took to carry out the previous addition, then there will be an actually infinite number of successive additions carried out over a finite time interval. (The completion of an infinite number of tasks in a finite amount of time is called a supertask). An actually infinite collection will then have been formed by successive addition, thus implying that the first premise of Craig’s argument is false. An actual infinite can be formed by successive addition after all.
How might we reply to Oppy’s argument? There are a couple of responses.
First, it might be argued that time is not structured in such a way that in a continuous stretch of time there are an actually infinite number of subintervals of time. In order for Oppy’s suggested supertask with the bouncing ball to be possible, a given interval of time must be composed of an actually infinite number of subintervals of time. So, if an interval of time does not have this property, then the supertask cannot be performed. The question is, is there a good reason to think that time cannot have this kind of structure? I think that there is indeed. An argument involving what is called the Grim Reaper Paradox can be pressed against the possibility of time having this kind of structure. In general, the argument would rule out the possibility of supertasks. It can be simply formulated as follows:
- If supertasks are possible, then the Grim Reaper Paradox is possible.
- The Grim Reaper Paradox is not possible.
- Therefore, supertasks are not possible.
We will discuss the Grim Reaper Paradox later on.
Second, it is not clear that Oppy’s supertask is even relevant to an infinite series of past events. Indeed, it is not clear that supertasks in general shed any light on the possibility of there being an infinite series of past events. For with respect to an allegedly infinite past, we are talking about the difficulty of traversing an infinite distance, not traversing a finite distance that is composed of an infinite number of ever-smaller sub-distances. As Craig writes,
The question is not whether it is possible to traverse infinitely many (progressively shorter) distances but whether it is possible to traverse an infinite distance. Thus, the problem of traversing an infinite distance comprising an infinite number of equal, actual intervals to arrive at our present location cannot be dismissed [along these lines] (Blackwell Companion, pp. 119).
If need be, we can simply modify the premises of the argument by changing successive addition to non-accelerating successive addition, which would render Oppy's objection irrelevant and would allow the argument would go through as before. For, as Craig says, the problem is with traversing an actually infinite number of equal intervals.
For both of these reasons, therefore, Craig’s first premise seems to be secure, despite Oppy’s objections.
St. Thomas Aquinas considered the argument that if there were an infinite series of past events, then the present event could not have occurred. St. Thomas formulates the argument as follows:
Further, if the world always was, the consequence is that infinite days preceded this present day. But it is impossible to pass through an infinite medium. Therefore we should never have arrived at this present day; which is manifestly false (Summa Theologica, Pt. I, Q. 46, Art. 2).
His rejoinder to the argument is as follows:
Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection is founded on the idea that, given two extremes, there is an infinite number of mean terms (ibid.).
What is St. Thomas saying here? Essentially, he is saying that from the present moment to any given moment in the past (let us call each of these intervals segments of time), a merely finite distance separates the two moments. So, since traversing finite distances is unproblematic and each of the past segments of time is finite, it follows that traversing an infinite past (which is composed of these segments of time) is unproblematic. The philosophers J.L. Mackie and J. Howard Sobel argue similarly (The Miracle of Theism, pg. 93; Logic and Theism, pg. 182).
In response to this, it must be said that St. Thomas and these other philosophers seem to be committing the fallacy of composition here. The argument, to reiterate, is that, since we are able to traverse (pass through) the interval from any particular past moment to the present moment, it follows that we can therefore traverse the entire infinite past up to the present moment. But this is as fallacious as saying that because we can lift each brick of the Great Wall of China, we can therefore lift the entire Great Wall of China, which clearly does not follow. So, just because we can get from any particular moment in the past to the present does not mean that the entire infinite past can be traversed. The objection, therefore, fails.
Jimmy Akin criticizes the first premise of the argument by way of presenting a dilemma: Either the argument assumes a beginning to the temporal series of events (in which case, since starting at one definite point and ending at another constitutes a finite distance, the argument would then be presupposing the finitude of the past and so would be question-begging) or else the first premise of the argument is simply false. The reason the first premise would be false if there is no assumption of a beginning point is that if we started out with an actually infinite collection and then added to that collection by successive addition, we would technically form a new collection that is actually infinite, and we would have thus formed such a collection by successive addition. “Depending on how you interpret it,” writes Akin, “the argument [thus] either commits a fallacy or uses a false premise.” (“Traversing the Infinite?”).
With respect to the first horn of the dilemma, Craig insists that the argument does not presuppose a starting point, not even an infinitely distant starting point. Craig writes,
But, in fact, no proponent of the kalam argument of whom we are aware has assumed that there was an infinitely distant starting point in the past. The fact that there is no beginning at all, not even an infinitely distant one, seems only to make the problem worse, not better. To say that the infinite past could have been formed by successive addition is like saying that someone has just succeeded in writing down all the negative numbers, ending at -1 (Blackwell Companion, pp. 119-120).
As Craig elsewhere says, attempting such a feat would be like trying to jump out of a bottomless pit (Time and Eternity, pg. 229).
With respect to the second horn of the dilemma, Craig anticipates this and responds as follows:
The only way a collection to which members are being successively added could be actually infinite would be for it to have an infinite tenselessly existing “core” to which additions are being made. But then, it would not be a collection formed by successive addition, for there would always exist a surd infinite, itself not formed successively but simply given, to which a finite number of successive additions have been made. Clearly, the temporal series of events cannot be so characterized, for it is by nature successively formed throughout. Thus, prior to any arbitrarily designated point in the temporal series, one has a collection of past events up to that point which is successively formed and completed and cannot, therefore, be actually infinite (Blackwell Companion, pp. 124-125).
And given a presentist ontology of time, there can be no infinite tenselessly existing core. Akin rejects presentism and so thinks that there can be such a core. The debate in this case, therefore, comes down to the fundamental nature of time. If presentism is true, then it seems that Craig has the upper hand. If presentism is false, then it seems that Akin has the upper hand. Since presentism seems to be the most plausible view of time, Craig’s argument appears to win out. Akin’s objection, therefore, does not succeed.
Defense of second premise
In defense of the second premise of the argument, the truth of the premise follows from the nature of time. On presentism, the series of temporal events is formed by the successive addition of one event after another as time sequentially keeps on ticking. For a full defense of the premise, we would need to provide a substantive defense of presentism, which is beyond the scope of this blog post. However, it can be pointed out that presentism is the commonsense view of time and therefore has the presumption of innocence (so to speak) in its favor.
Critique of second premise
Opponents of the second premise are generally going to be those who reject presentism and hold to some form of the B-Theory of time. For if the B-Theory is correct, then the past was not formed by successive addition, but was rather formed all at once. This follows because all events (past, present, and future) exist all at once on this view of time. Again, critiquing the B-Theory of time is beyond the scope of this blog post.
Even if presentism should turn out to be false and the B-Theory of time is actually true, however, the second premise can be modified so as to still get the desired conclusion, viz., the universe began to exist. The philosopher Andrew Loke has argued that even on the B-Theory, although time in and of itself would not be formed by successive addition (since all moments of time exist tenselessly), nevertheless, a conscious being’s experience of the passage of time would be formed by successive addition (The Teleological and Kalam Cosmological Arguments Revisited, pg. 211-214). To elaborate, we can suppose that there exists a conscious being that has a series of experiences, one after another as he experiences the illusory passage of time. Even if his experience of the passage of time is illusory, the experiences of the events of time themselves still exist in his consciousness, and these experiences accumulate via successive addition. He can in principle count them as they occur. Now, although we are supposing that moments of time do not objectively pass but rather exist all at once tenselessly, the moments of time nevertheless are experienced successively by the conscious being. Consequently, if past time is infinite and the conscious being has always existed, then he has at present (his present) had an actually infinite number of experiences that he has been able to count. If it is the case, therefore, that an actual infinite cannot be formed by successive addition, then since the conscious being’s experiences have accumulated via successive addition, it follows that he cannot have had an actually infinite number of experiences. From this fact, it follows that past time cannot be infinite, i.e., the universe began to exist. Thus, the present argument for the beginning of the universe can be modified so as to succeed even on the supposition of the B-Theory of time.
Of course, if the B-Theory is false and presentism is correct, then the second premise goes through without any modifications needed. Or does it? One objection that has been raised by Graham Oppy against the claim that time is formed by successive addition (even given presentism) is that, given the possibility of time being continuous in nature, time possibly has the structure of the real numbers rather than the natural numbers. In such a case, time would be formed by continuous addition or accretion rather than successive addition. Further, the set of past events would be uncountably infinite as opposed to countably infinite; consequently, the set of past events would not be a series, since a series is essentially discrete in nature. Time is not, on such a view, made up of discrete moments forming a series “like beads on a string,” as Oppy puts it (“Time, Successive Addition, and Kalam Cosmological Arguments”, pp. 185). If this is right, then the second premise of Crag’s argument is false. What’s more, the traversal of a continuous interval of time entails traversing infinitely many points of time that compose said interval. So, traversal of the infinite seems to be possible via continuous addition even if it is not via successive addition.
The main idea seems to be something like this: If time is continuous, then a given interval of time is composed of an actually infinite number of subintervals of time. Time would then accumulate via continuous addition rather than via successive addition. So, the second premise of Craig’s argument would be false. Perhaps Craig could simply modify the argument by replacing “successive addition” with “continuous addition” as follows:
- A collection formed by continuous addition cannot be an actual infinite.
- The temporal series of events is a collection formed by continuous addition.
- Therefore, the temporal series of events cannot be an actual infinite.
In this case, however, Oppy would argue that while the second premise is now true on a continuous view of time, the first premise becomes false. This can be seen as follows: The succession of each second (the choice of second as a unit of time here is arbitrary) of time involves the traversal of an actually infinite number of sub-seconds of time. Since seconds of time clearly are capable of passing, the traversal of an actual infinite by continuous addition is possible. After all, if we are traversing an infinite number of subintervals of time in order to traverse the interval of which those subintervals are a part, then we are ipso facto traversing an actual infinite. Since passing from one second to the next is simply a part of such continuous addition, there does not seem to be any barrier to the traversal of infinitely many seconds (or any other unit) of time by continuous addition. So, the set of past events is formed by continuous addition rather than successive addition, and it is possible to traverse an infinite collection by continuous addition. So, if time is continuous in this way, Craig’s argument fails.
In reply to this, we note (as we did in our reply to one of Oppy’s aforementioned objections to the first premise of Craig’s argument) that the structure of time that Oppy’s objection relies on is ruled out by the Grim Reaper Paradox (see below). Intervals of time are not composed of an actually infinite number of subintervals of time; rather, an interval of time is infinitely divisible in the sense that one can keep dividing the interval in half potentially infinitely. To further support this point, Edward Feser, drawing from Zeno’s paradoxes (originally developed by the ancient Greek philosopher Zeno), notes that the notion of a continuum of time or space that is composed of an actually infinite number of points is fraught with metaphysical difficulties. Feser writes,
[T]he parts of which a continuous object is purportedly composed would be either extended or unextended, and either supposition leads to absurdity. Suppose first that the parts are unextended. These unextended parts are either at a distance from each other or they are not. If they are at a distance from one another, then they would not form a continuum, but would rather be a series of discrete things (like the dots in a dotted line, only without even the minute extension such dots have). Suppose then that they are not at a distance from one another, but are instead in contact. Then, since they have no extension at all and thus lack any extreme or middle parts, they will exactly coincide with one another (like a single dot, only once again without even the minute extension of such a dot). All these parts together, no matter how many of them there are, will be as unextended as an individual part. In that case, too, then, they will not form a true continuum.
So, if a continuous object is made up of parts, they will have to be extended parts. Now these purported extended parts would either be finite in number or infinite. They cannot be finite, however, because anything extended, no matter how small, can always be divided at least in principle into yet smaller extended parts, and those parts into yet smaller extended parts ad infinitum. So if a continuous object is made up of extended parts, they will have to be infinite in number. But the more extended parts a thing has, however minute those parts, the larger it is. Hence if a continuous object is made up of an infinite number of extended parts, it will be of infinite size. This will be so of every continuous object, however small it might seem…But this is absurd. Hence a continuous object can no more be made up of extended parts than it can be made up of unextended parts (Aristotle’s Revenge, pg. 204-205).
Must we, therefore, deny the reality of continua in nature? No, we needn’t do so. Instead, we can make the important distinction between actuality and potentiality. The parts of a continuum are in the continuum potentially but not actually. Feser writes,
Applying the theory of actuality and potentiality, [the proponent of continua] argues that what the paradoxes really show is that the parts of a continuum are in it only potentially rather than actually. That is not to say that they are not there at all. A potentiality is not nothing, but rather a kind of reality. That is why a wooden block (for example) is divisible despite being continuous or uninterrupted in a way a stack of blocks is not. But until it is actually divided, the parts are not actual…Affirming that reality includes both potentialities as well as actualities allows us to acknowledge the reality of the parts of a continuum while at the same time avoiding paradox (ibid., pg. 205).
This solution to the paradoxes of continua implies that intervals of (continuous) time are not composed of an actually infinite number of points of time. Rather, such intervals are merely infinitely divisible in principle. And the divisibility that is in mind here is a potential infinite rather than an actual infinite. As David S. Oderberg says, “[T]he KCA supporter need not deny that there are natural continua, including temporal ones, if that entails only that there are potential infinities…There may well be natural continua…but all this means is that infinite divisibility is to be found in nature, perhaps both spatially and temporally” (“The Kalam Cosmological Argument Neither Bloodied nor Bowed: A Response to Graham Oppy”, pp. 193).
So, time turns out not to have the structure that Oppy’s objection relies on. Thus, Oppy’s objection does not succeed.
The Grim Reaper Paradox
As a third philosophical argument for the finitude of the past, we will briefly consider the Grim Reaper Paradox. The Grim Reaper Paradox (henceforth, GRP) was originally developed by the philosopher José Benardete. It has been used to defend the second premise of the KCA by philosophers such as Alexander R. Pruss and Robert C. Koons. The key result that is argued to follow from the GRP (especially by Pruss) is the thesis of causal finitism, the view that there cannot be an actually infinite causal series such that infinitely many causes sequentially produce a given effect. The paradox is described in the following paragraph.
Suppose you are alive at 12:00 A.M. And suppose there are infinitely many grim reapers (GRs). Suppose that at 12:30 A.M., GR 1 will strike you dead if you are still alive. Suppose also at 12:15 A.M., GR 2 will strike you dead if you are still alive. At 12:07.5 A.M., GR 3 will strike you dead if you are still alive. At 12:03.75 A.M., GR 4 will strike you dead if you are still alive. At 12:01.875 A.M., GR 5 will strike you dead if you are still alive. And so on ad infinitum. If a GR sees that you are dead, he will do nothing. Now, we ask a question: Are you still alive at 12:30 A.M.? On the one hand, you must be dead because some GR must have killed you. This follows because GR 1 would have killed you if you were still alive at 12:30 A.M., GR 2 would have killed you if you were still alive at 12:15 A.M, and so on. You thus could not be alive at 12:30 A.M. On the other hand, you can’t be dead (assuming nothing other than a GR killed you) because no GR could have killed you! This follows because GR 1 couldn’t have killed you because GR 2 would have beaten him to the punch (or slice?). But GR 2 couldn’t have killed you because GR 3 would have beaten him to the punch. And so on. So, we are left with the conclusion that you can’t possibly be alive at 12:30 A.M. because some GR must have killed you, and yet you must still be alive at 12:30 A.M. because no GR could have killed you. This is a logical contradiction.
A plausible resolution of the paradox is to propose that it is metaphysically impossible for there to be an infinite chain of causes going into the past that impinge upon a single effect in the present. It appears that this is what is ultimately going on in the GRP. The upshot of this resolution is that this would give us strong philosophical grounds for thinking that an infinite past is metaphysically impossible since given an infinite past, we would have an infinite chain of causes going into the past (past events) that impinge upon a single effect (a present event). In other words, the causal structure of the grim reaper scenario is isomorphic to the causal structure of an infinite past. So, if we reject the possibility of the grim reaper scenario, we should similarly reject the possibility of an infinite past.
Building on this reasoning, we can formulate an argument for the proposition that the universe began to exist as follows:
- If the universe did not begin to exist, then there is an infinite past.
- If there is an infinite past, then there is an infinite chain of causes going into the past that impinge upon a present effect (and the GRP is possible).
- There cannot be an infinite chain of causes going into the past that impinge upon a present effect (the GRP is impossible).
- Therefore, there is not an infinite past (2, 3).
- Therefore, the universe began to exist (1, 4).
One preliminary issue to deal with is that we have previously applied the GRP to proving that time is not structured in such a way that over a given interval of time, there are an actually infinite number of subintervals of time. And this is because the GRP as presently formulated can only get off the ground if time is structured in this way. So, it seems that once we reject time having this structure, the GRP can no longer be used to show that past time cannot be infinite. For so long as past time is not structured in this way, the GRP is already ruled out without the need to hold to a finite past.
This problem can be remedied by modifying the GRP as follows: Suppose that the past is infinite and that you have always existed throughout the entire infinite past up to the present day. Now, suppose there are infinitely many GRs such that today a GR will strike you dead if you are still alive, another GR would have struck you dead if you were still alive yesterday, another GR would have struck you dead if you were still alive a day before that, and so on. The contradiction that arises is that you cannot possibly be alive today because some GR must have killed you, and yet you must still be alive today because no GR could have killed you. This revision is suggested by Pruss (Infinity, Causation, and Paradox, pg. 55-56). On this revised scenario, the GRP is clearly isomorphic to an infinite past. Hence, if the past is infinite, the GRP is possible. But the GRP is impossible. Therefore, the past is finite, i.e., the universe began to exist.
Objections
Some philosophers, such as Graham Oppy and John Hawthorne have suggested an alternative resolution of the GRP. Oppy actually discusses a different but very similar paradox developed by Benardete (Philosophical Perspectives on Infinity, pg. 81-83), but his suggestion can equally apply to the GRP. Appropriating Oppy’s thoughts and suitably modifying them to apply to the GRP, Oppy suggests that while it is true that no individual grim reaper killed you, nevertheless perhaps the entire collection of grim reapers killed you. This reply, though ingenious, seems quite implausible. If no single grim reaper killed you with his scythe, how could all the grim reapers have collectively killed you? How could an infinite number of reapers individually not doing anything collectively kill you? Consider the following example. You start a calculator on the value “0.” You then add zero to the current sum. The resultant sum is, of course, still 0. Suppose you add zero again. The sum is still 0. Suppose you add zero infinitely more times. Clearly, the sum will still be 0. The bottom line: a whole lot of nothings (even infinitely many nothings) don’t add up to something.
There is an alternative reply to the present objection that modifies the GRP scenario. This modification has been suggested by Pruss. Instead of imagining infinitely many grim reapers, we are instead to imagine infinitely many jolly givers (JGs). And instead of trying to kill you like the grim reapers, the JGs try to put an orange in your Christmas stocking. As Pruss explains:
You hang up a stocking at midnight. There is an infinite sequence of Jolly Givers, each with a different name, and each of which has exactly one orange. There are no other oranges in the world, nor anything that would make an orange. When a JG’s alarm goes off, it checks if there is anything in the stocking. If there is, it does nothing. If there is nothing in the stocking, it puts its orange in the stocking. The alarm times are the same as in the previous story (“The Paradox of the Jolly Givers”).
In this variant, Oppy’s suggestion that it is the entire collection of JGs that puts an orange in the stocking leads to a violation of the principle of ex nihilo nihil fit (out of nothing, nothing comes). For, no JG gave up his orange and yet there is an orange in the stocking. Hence, the orange came into being uncaused from nothing, which is metaphysically absurd. The alternative explanation suggested by Oppy, therefore, fails. It should also be noted that this scenario can easily be modified so as to get around the problem of time not being structured in such a way that over a given interval of time, there are an actually infinite number of subintervals of time. The modification would parallel the foregoing modification of the original GRP to get around this problem.
Jimmy Akin objects to the GRP by arguing that it tacitly presupposes a first reaper in the series of reapers, which is already impossible because, since there is a last reaper and an interval with a first member and a last member cannot be infinite, there cannot be a first reaper, given that we are supposing that there are an infinite number of reapers. As Akin writes (using “Fred” in place of “you”):
The resolution of this paradox is fairly straightforward. It has envisioned a situation where Fred begins alive and then will be killed by the first grim reaper he encounters.
The problem is that—if the series of grim reapers is infinite—then it must have no beginning.
To suppose that an infinite series of whole numbers has both a first and last member involves what I’ve called the First-and-Last Fallacy.
· Infinite series can have no beginning ( . . . -3, -2, -1, 0)
· They can have no end (0, 1, 2, 3 . . .)
· Or they can lack both a beginning and an end ( . . . -3, -2, -1, 0, 1, 2, 3 . . . )
But if a series has both a beginning and an end, then it’s finite.
The series of reapers set to kill Fred has an end—Reaper 0—but if that’s the case, it cannot have a beginning.
This means that there is no first grim reaper that Fred encounters, just as there is no “first negative number.”
The idea of a first negative number involves a logical contradiction, and therefore the…Grim Reaper paradox is proposing a situation that cannot exist (“Grim Reapers, Paradoxes, and Infinite History”).
In response, it must be said that the GRP does not presuppose a first reaper. It’s precisely the fact that there is no first reaper that generates the paradox. One advantage of the GRP over the previously canvassed arguments for the beginning of the universe is that it does not involve purely quantitative operations but instead involves explicitly causal processes. In the GRP, we are no longer talking about the difficulties of there existing an actual infinite or of traversing an actual infinite. Instead, we are talking about the difficulty of an infinite number of causes going back into the past that each contribute to a single effect in the present. What the GRP shows is that such a causal series leads to logical contradiction. It is precisely the fact that there is no first reaper that makes the situation causally impossible. There must be a first reaper, but there is no first reaper. Contradiction. The way to resolve the paradox is to say that there must be a first reaper. Consequently, the causal series must be finite, not infinite. This is the thesis of causal finitism. Akin’s objection, therefore, is unpersuasive.
Overall, then, I conclude that—in contrast to his first argument—Craig's second philosophical argument for the beginning of the universe is sound. Additionally, the Grim Reaper Paradox furnishes us with another sound argument for the beginning of the universe.
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