By successive addition

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We now turn to a second philosophical argument in support of the premise that the universe began to exist, the argument from the impossibility of the formation of an actual infinite by successive addition. The argument may be simply formulated as follows:

2.21 A collection formed by successive addition cannot be an actual infinite.

2.22 The temporal series of events is a collection formed by successive addition.

2.23 Therefore, the temporal series of events cannot be an actual infinite.

This second argument is independent of the foregoing argument, for its conclusion is not incompatible with the existence of an actual infinite. It rather denies that a collection containing an actually infinite number of things can be formed by adding one member after another. If an actual infinite cannot be formed by successive addition, then the series of past events must be finite since that series is formed by successive addition of one event after another in time.

2.21. Formation of an actual infinite

Quite independent of the absurdities arising from the existence of an actually infinite number of things are the further difficulties arising as a result of the temporal formation of such a multitude through a process of successive addition. 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. For while it is true that 1 + 1 + 1 + . . . equals N0, the operation of addition signified by "+" is not applied successively but simultaneously or, better, timelessly. One does not add the addenda in temporal succession: 1 + 1 = 2, then 2 + 1 = 3, then 3 + 1 = 4, . . . , but rather all together. By contrast, we are concerned here with a temporal process of successive addition of one element after another.

The impossibility of the formation of an actual infinite by successive addition seems obvious in the case of beginning at some point and trying to reach infinity.15 For given any finite number n, n + 1 equals a finite number. Hence, N0 has no immediate predecessor; it is not the terminus of the natural number series but stands, as it were, outside it and is the

15. This despite the speculation concerning the possibility of supertasks, various thought experiments involving the completion of an infinite number of tasks in a finite time by performing each successive task during half the time taken to perform its immediate predecessor. The fatal flaw in all such scenarios is that the state at M + 1 is causally unconnected to the successive states in the M series of states. Since there is no last term in the M series, the state of reality at M + 1 appears mysteriously from nowhere. The absurdity of such supertasks underlines the metaphysical impossibility of trying to convert a potential into an actual infinite.

number of all the members in the series. Notice that the impossibility of forming an actual infinite by successive addition has nothing to do with the amount of time available. Sometimes it is wrongly alleged that the only reason an actual infinite cannot be formed by successive addition is because there is not enough time.16 But this is mistaken. While we can imagine an actually infinite series of events mapped onto a tenselessly existing infinite series of temporal intervals, such that each consecutive event is correlated with a unique consecutive interval, the question remains whether such a sequence of intervals can be instantiated, not tenselessly, but one interval after another. The very nature of the actual infinite precludes this. For regardless of the time available, a potential infinite cannot be turned into an actual infinite by any amount of successive addition since the result of every addition will always be finite. 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.

The question then arises whether, as a result of time's asymmetry, an actually infinite collection, although incapable of being formed by successive addition by beginning at a point and adding members, nevertheless could be formed by successive addition by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity. In this case, one is not engaged in the impossible task of trying to convert a potential into an actual infinite by successive addition. Rather at every point the series already is actually infinite, although allegedly successively formed.

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.17 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.

16. For example, Oppy's discussion of counting forward to infinity is predicated upon Dretske's assumption that if one never stops counting, then one does count to infinity (Oppy 2006a, p. 61; cf. Dretske 1965). Oppy fails so much as to mention, much less take account, of the difference between an actual and a potential infinite in this case. One who, having begun, never stops counting counts "to infinity" only in the sense that one counts potentially infinitely.

17. Richard Gale protests, "This argument depends on an anthropomorphic sense of 'going through' a set. The universe does not go through a set of events in the sense of planning which to go through first, in order to get through the second, and so on" (Gale 2007, pp. 92-3). Of course not; but on an A-Theory of time, the universe does endure through successive intervals of time. It arrives at its present event-state only by enduring through a series of prior event-states. Gale's framing the argument in terms of a "set of events" is maladroit since we are not talking about a set but about a series of events which elapse one after another.

It is unavailing to say that an infinite series of past events cannot be formed only in a finite time but that such formation is possible given infinite time, for that riposte only pushes the question back a notch: how can an actually infinite series of congruent temporal intervals successively elapse? Tenseless correlations are irrelevant here. Granted that the series of past events, if infinite, can be mapped one-to-one onto an equally infinite series of past temporal intervals, the question remains how such a temporal series can be lived through so as to arrive at the present.

The arguments against the formation of an actual infinite by successive addition bear a clear resemblance to Zeno's celebrated paradoxes of motion, in particular the Stadium and Dichotomy paradoxes, the Stadium in the case of beginning at some point and never ending and the Dichotomy in the case of never beginning and ending at some point. In the Dichotomy Paradox, Zeno argued that before Achilles could cross the stadium, he would have to cross halfway; but before he could cross halfway, he would have to cross a quarter of the way; but before he could cross a quarter of the way, he would have to cross an eighth of the way, and so on to infinity. It is evident that Achilles could not arrive at any point. In the case of the infinite past, we cannot speak meaningfully of halfway through the past or a quarter of the way, and so on since there is no beginning point, as there is in Achilles' case. But the metrical distances traversed are not essential to the conundrum insofar as the series of past events is concerned since the essential point holds that before traversing any interval there will always be a prior interval to be traversed first.

Now although Zeno's paradoxes have proved very stubborn, scarely anybody has really believed that motion is impossible. Is the argument against the impossibility of traversing an infinite past, as some critics allege, no more plausible than Zeno's paradoxes? This cannot be said because the allgation fails to reckon with two crucial disanalogies of the case of an infinite past to Zeno's paradoxes: whereas in Zeno's thought experimentsm the intervals traversed are potential and unequal, in the case of an infinite past the intervals are actual and equal. The claim that Achilles must pass through an infinite number of halfway points in order to cross the stadium already assumes that the whole interval is a composition of an infinite number of points, whereas Zeno's opponents, like Aristotle, take the line as a whole to be conceptually prior to any divisions which we might make in it. Moreover, Zeno's intervals, being unequal, sum to a merely finite distance, whereas the intervals in an infinite past sum to an infinite distance. 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 on the basis of the argument's resemblance in certain respects to Zeno's puzzles.

It is surprising that a number of critics, such as Mackie and Sobel, have objected that the argument illicitly presupposes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But if the past is infinite, they say, then there would be no starting point whatever, not even an infinitely distant one. Nevertheless, from any given point in the past, there is only a finite distance to the present, which is easily "traversed" (Mackie 1982, p. 93; Sobel 2004, p. 182). 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. And how is the claim that from any given moment in the past there is only a finite distance to the present even relevant to the issue? For the question is how the whole series can be formed, not a finite portion of it. Do Mackie and Sobel think that because every finite segment of the series can be formed by successive addition the whole infinite series can be so formed? That is as logically fallacious as saying that because every part of an elephant is light in weight, the whole elephant is light in weight, or in other words, to commit the fallacy of composition. The claim that from any given moment in the past there is only a finite distance to the present is simply irrelevant.

Wholly apart from these Zenonian arguments, the notion that the series of past events could be actually infinite is notoriously difficult. Consider, for example, al-Ghazali's thought experiment involving two beginningless series of coordinated events. He envisions our solar system's existing from eternity past, the orbital periods of the planets being so coordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many. If they have been orbiting from eternity, which planet has completed the most orbits? The correct mathematical answer is that they have completed precisely the same number of orbits. But this seems absurd, for the longer they revolve, the greater becomes the disparity between them, so that they progressively approach a limit at which Jupiter has fallen infinitely far behind Saturn. Yet, being now actually infinite, their respective completed orbits are somehow magically identical. Indeed, they will have "attained" infinity from eternity past: the number of completed orbits is always the same. Moreover, Ghazali asks, will the number of completed orbits be even or odd? Either answer seems absurd. We might be tempted to deny that the number of completed orbits is either even or odd. But post-Cantorian transfinite arithmetic gives a quite different answer: the number of orbits completed is both even and odd! For a cardinal number n is even if there is a unique cardinal number m such that n = 2m, and n is odd if there is a unique cardinal number m such that n = 2m + 1. In the envisioned scenario, the number of completed orbits is (in both cases!) N0, and N0 = 2 N0 = 2 N0 + 1. So Jupiter and Saturn have each completed both an even and an odd number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time. This seems absurd.18

Or consider the case of Tristram Shandy, who, in the novel by Sterne, writes his autobiography so slowly that it takes him a whole year to record the events of a single day. Tristram Shandy laments that at this rate he can never come to an end.

According to Russell, if Tristram Shandy were immortal and did not weary of his task, "no part of his biography would have remained unwritten," since by Hume's Principle to each day there would correspond 1 year, and both are infinite (Russell, 1937, p. 358). 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

18. Oppy's discussion of al-Ghazali's problem just fails to connect with the problem as we understand it (Oppy 2006a, pp. 49-51), probably because Oppy takes its point to be that there is a logical contradiction with respect to the number of orbits completed (Oppy 2006a, p. 8), so that he spends most of his space arguing that given Cantorian assumptions there is no unequivocal sense in which the number of orbits both is and is not same. Temporal becoming is left wholly out of account.

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.

But now turn the story about: suppose Tristram Shandy has been writing from eternity past at the rate of 1 day per year. 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.

This may be seen through an incisive analysis of the Tristram Shandy Paradox given by Robin Small (1986, pp. 214-5). He points out that if Tristram Shandy has been writing for 1 year's time, then the most recent day he could have recorded is 1 year ago. But if he has been writing for 2 years, then that same day could not have been recorded by him. For since his intention is to record consecutive days of his life, the most recent day he could have recorded is the day immediately after a day at least 2 years ago. This is because it takes a year to record a day, so that to record 2 days he must have 2 years. Similarly, if he has been writing 3 years, then the most recent day recorded could be no more recent than 3 years and 2 days ago. In other words, the longer he has written the further behind he has fallen. In fact, the recession into the past of the most recent recordable day can be plotted according to the formula (present date - n years of writing) + n - 1 days. But what happens if Tristram Shandy has, ex hypothesi, been writing for an infinite number of years? The most recent day of his autobiography recedes to infinity, that is to say, to a day infinitely distant from the present. Nowhere in the past at a finite distance from the present can we find a recorded day, for by now Tristram Shandy is infinitely far behind. The beginningless, infinite series of days which he has recorded are days which lie at an infinite temporal distance from the present. This is not in itself a contradiction. The infinite past must have in this case, not the order type of the negative numbers w*, but the order type w* + w*, the order type of the series . . . , -3, -2, -1, . . . , -3, -2, -1. 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.19

But suppose that such an infinite task could be completed by the present day. Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . . , -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the

19. Oppy rightly observes that it is the whole scenario that is impossible, which includes the requirement that consecutive days be recorded (Oppy 2006a, p. 57, n. 3). But given that the task of writing one's autobiography at the rate of 1 day per year seems obviously coherent, it seem to us that the blame can be placed on the infinity of the past.

day before or the year before? By then an infinite time had already elapsed, so that he has had ample time to finish. Thus, at no point in the infinite past should we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity.

Conway and Sorabji have responded that there is no reason to think that the man would at any point have already finished (Sorabji 1983, pp. 219-22; Conway 1984). Sorabji thinks the argument confuses counting an infinity of numbers with counting all the numbers. At any given point in the past, the man will have already counted an infinity of negative numbers, but that does not entail that he will have counted all the negative numbers. Similarly, in Conway's analysis, the nub of the argument lies in the conditional

(*) If an infinite number of numbers had been counted by yesterday, then the man will have finished by yesterday.

But Conway's conditional is quite ambiguous, and the arguments that he suggests in support of it have no apparent relevance to the reasoning behind the paradox. The mutakal-lim is not making the obviously false claim that to count infinitely many negative numbers is to count all the negative numbers! Rather, the conditional at the heart of the paradox is a counterfactual conditional like:

(**) If the man would have finished his countdown by today, then he would have finished it by yesterday, and the truth of this conditional seems plausible in light of Hume's Principle. It is on the basis of this principle that the defender of the infinite past seeks to justify the intuitively impossible feat of someone's counting down all the negative numbers and ending at 0. Since the negative numbers can be put into a one-to-one correspondence with the series of, say, past hours, someone counting from eternity would have completed his countdown. But by the same token, the man at any point in the past should have already completed his countdown, since by then a one-to-one correspondence exists between each negative number and a past hour. In this case, having infinite time does seem to be a sufficient condition of finishing the job. Having had infinite time, the man should have already completed his task.

Such reasoning in support of the finitude of the past and the beginning of the universe is not mere armchair cosmology. P. C. W. Davies, for example, utilizes this reasoning in explaining two profound implications of the thermodynamic properties of the universe:

The first is that the universe will eventually die, wallowing, as it were, in its own entropy. This is known among physicists as the 'heat death' of the universe. The second is that the universe cannot have existed for ever, otherwise it would have reached its equilibrium end state an infinite time ago. Conclusion: the universe did not always exist. (Davies 1983, p. 11)

The second of these implications is a clear application of the reasoning that underlies the current paradox: even if the universe had infinite energy, it would in infinite time come to an equilibrium since at any point in the past infinite time has elapsed, a beginningless universe would have already reached an equilibrium, or as Davies puts it, it would have reached an equilibrium an infinite time ago. Therefore, the universe began to exist, quod erat demonstrandum.20

Oppy's response to the problem at hand is to say that the man's finishing his countdown when he does rather than earlier is just "a brute feature of the scenario, that is, a feature that has no explanation" (Oppy 2006a, p. 59; cf. p. 63; Oppy 2006b, pp. 141-2). It has always been the case that he will finish when he does, but why the man finishes when he does rather than at some other time is just inexplicable. Resting with inexplicability may seem unsatisfactory, however, especially in light of the respectable role such reasoning plays in scientific cosmological discussions. Oppy justifies his response on the basis that principles of sufficient reason requiring that there be an explanation in such a case are highly contentious. Oppy presents the typical objections to various versions of the Principle of Sufficient Reason such as the impossibility of providing an explanation of what has been called the "Big Contingent Conjunctive Fact" (BCCF), which is the conjunction of all the contingent facts there are, or of libertarian free choices (Oppy 2006a, pp. 279-80). The problem with this justification, however, is twofold. First, plausible defenses of the Principle of Sufficient Reason can be given.21 Second, and more to the point, there is no reason to think that requiring the need for an explanation in the present case demands for its acceptability or plausibility the enunciation and defense of some general Principle of Sufficient Reason. Indeed, any such principle is apt to be tested inductively for its adequacy by whether cases like this constitute plausible counterexamples. The exceptions offered by Oppy, such as the inexplicability of the BCCF and libertarian choices, are simply irrelevant to the present case, for the BCCF is not at stake nor can a person counting from eternity at a constant rate choose arbitrarily when to finish his countdown. In the case under discussion, we have a good reason to think that the man should have finished his countdown earlier than any

20. See the similar reasoning of Barrow and Tipler (1986, p. 601-8) against inflationary steady-state cosmologies on the ground that any event which would have happened by now would have already happened before now if the past were infinite.

21. See Pruss' article in this volume. We shall leave to him the defense of principles of sufficient reason. Oppy himself thinks that it is "very plausible" that there are acceptable instances of the following schema for a Principle of Sufficient Reason:

O (for every FG of kind K, there is an F'G' that partly explains why the GFs rather than Q possible alternatives), where O is an operator like "necessarily," "it is knowable a priori," etc., G is an ontological category such as a proposition, state of affairs, etc., F is a restriction such as true, contingent, etc., and Q is a quantifier like "any," "every," etc. (Oppy 2006a, p. 285, cf. pp. 275-6). But he thinks that it is not at all clear that there are acceptable instances of this schema that can be used to rule out scenarios like counting down from infinity. Although it is not clear what Oppy means by "GFs," the following principle would seem to be an instance of his schema: Necessarily, for any contingent state of affairs involving concrete objects there is a contingent state of affairs that partly explains why that state of affairs obtains rather than any other. Such a principle would require that there be some partial explanation for why the man finishes his countdown today rather than at some other time. But not even a partial explanation can be given, for regardless of how we vary such factors as the rate of counting, they will be the same regardless of the time that he finishes and so do not furnish even a partial explanation of why he finishes today. So why is this instance of the schema not acceptable?

time that he does, namely, he has already had infinite time to get the job done.22 If we deny that infinite time is sufficient for completing the task, then we shall wonder why he is finishing today rather than tomorrow or the day after tomorrow, or, indeed, at any time in the potentially infinite future. It is not unreasonable to demand some sort of explanation for why, if he finishes today, he did not already finish yesterday. By contrast, if such a countdown is metaphysically impossible, then no such conundrum can arise. But clearly, there is no metaphysical impossibility in counting backward for all time, unless time is past eternal. It follows that the past cannot be infinite.

For all of these reasons, the formation of an actual infinite by successive addition is a notoriously difficult notion, even more so than the static existence of an actual infinite.

2.22. Successive formation of the series of past events

Premise (2.22) may seem rather obvious. The past did not spring into being whole and entire but was formed sequentially, one event occurring after another. Notice, too, that the direction of this formation is "forward," in the sense that the collection grows with time. Although we sometimes speak of an "infinite regress" of events, in reality an infinite past would be an "infinite progress" of events with no beginning and its end in the present.

As obvious as this premise may seem at first blush, it is, in fact, a matter of great controversy. It presupposes once again an A-Theory of time. On such a theory, the collection of all past events prior to any given event is not a collection whose members all tenselessly coexist. Rather it is a collection that is instantiated sequentially or successively in time, one event coming to pass on the heels of another. Since temporal becoming is an objective feature of the physical world, the series of past events is not a tenselessly existing manifold, all of whose members are equally real. Rather the members of the series come to be and pass away one after another.

Space does not permit a review of the arguments for and against the A- and B-Theories of time respectively. But on the basis of a case such as is presented by Craig (2000a,b), we take ourselves to be justified in affirming the objective reality of temporal becoming and, hence, the formation of the series of temporal events by successive addition. It is noteworthy that contemporary opponents of Zenonian arguments such as Grunbaum resolve those puzzles only by denying the objective reality of temporal becoming and treating time as a continuum of tenselessly existing point-instants. If moments of time and, hence, events really do come to be and elapse, then it remains mysterious how an infinite number of such event-intervals can be traversed or manage successively to elapse.

2.23. Conclusion

It follows, then, that the temporal series of events cannot be actually infinite. The only way a collection to which members are being successively added could be actually infinite would

22. Notice, too, that if there is any probability of his finishing in infinite time, then he will have already finished.

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.

2.3. Scientific confirmation

The sort of philosophical problems with the infinity of the past, which have been the object of our discussion, are now being recognized in scientific papers by leading cosmologists and philosophers of science.23 For example, Ellis, Kirchner, and Stoeger ask, "Can there be an infinite set of really existing universes? We suggest that, on the basis of well-known philosophical arguments, the answer is No" (Ellis, Kirchner, & Stoeger 2003, p. 14; emphasis added). Similarly, noting that an actual infinite is not constructible and, therefore, not actualizable, they assert, "This is precisely why a realized past infinity in time is not considered possible from this standpoint - since it involves an infinite set of completed events or moments" (Ellis, Kirchner, & Stoeger 2003, p. 14). These misgivings represent endorsements of both the kalam arguments defended earlier. Ellis and his colleagues conclude, "The arguments against an infinite past time are strong - it's simply not constructible in terms of events or instants of time, besides being conceptually indefinite" (Ellis, Kirchner, & Stoeger 2003, p. 14).

Apart from these philosophical arguments, there has emerged during the course of the twentieth century provocative empirical evidence that the universe is not past eternal. This physical evidence for the beginning of the universe comes from what is undoubtedly one of the most exciting and rapidly developing fields of science today: astronomy and astrophysics. Prior to the 1920s, scientists had always assumed that the universe was stationary and eternal. Tremors of the impending earthquake that would topple this traditional cosmology were first felt in 1917, when Albert Einstein made a cosmological application of his newly discovered gravitational theory, the General Theory of Relativity (Einstein 1917, pp. 177-88). In so doing, he assumed that the universe is homogeneous and isotropic and that it exists in a steady state, with a constant mean mass density and a constant curvature of space. To his chagrin, however, he found that General Relativity (GR) would not permit such a model of the universe unless he introduced into his gravitational field equations a certain "fudge factor" A in order to counterbalance the gravitational effect of matter and so ensure a static universe. Einstein's universe was balanced on a razor's edge, however, and the least perturbation - even the transport of matter from one part of the universe to another - would cause the universe either to implode or to expand. By taking this feature of Einstein's model seriously, the Russian mathematician Alexander Friedmann and the Belgian astronomer Georges Lemaitre were able to formulate independently in the 1920s solutions to the field equations which predicted an expanding universe (Friedmann 1922; Lemaitre 1927).

23. Besides the paper by Ellis et al., see Vaas (2004).

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