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The Kalam Cosmological Argument Yet Again: The Question of the Metaphysical Possibility of an Infinite Temporal Series (2003)
(Revised 2014)

Arnold T. Guminski

ABSTRACT: In my first article, "The Kalam Cosmological Argument: The Question of the Metaphysical Possibility of an Actual Infinite of Real Entities" ("KCA" #1), I showed that the argument by William Lane Craig and others that real infinites are metaphysically impossible presupposes (what I call) the standard version (SV) of how Cantorian set theory presumably applies to the real world—that is, the world of concrete entities, whether natural or supernatural. This is the case because it is the application of SV to the real world which generates counterintuitive absurdities. However, I also showed in my previous article that there is an alternative version (AV) of applying Cantorian set theory to the real world. The application of this version does not generate counterintuitive absurdities. In the present article, I go on to show that given AV an infinite temporal series is metaphysically possible. In so doing, we reach a result that should be equally satisfying to those many theists and nontheists who are loath to believe that a beginningless temporal world is metaphysically impossible. However much these theists and nontheists may disagree about other issues, they are at least able to agree upon one important thing: the kalam cosmological argument fails insofar as it is grounded upon the alleged metaphysical impossibility of an infinite temporal series.[1]

A Necessary Discussion of the Standard and Alternative Versions of the Application of Cantorian Theory to the Real World

1. The argument by William Lane Craig that no actual infinite set of real entities is metaphysically possible depends upon his successfully maintaining that the application of Cantorian set theory to the real world generates counterintuitive absurdities. If this contention is sound, it follows that an infinite temporal series is metaphysically impossible since such a series constitutes an actual infinite set of real entities. Given this conclusion, Craig argues to his ultimate conclusion that "a personal Creator of the universe ... exists changelessly and independently prior to creation and in time subsequent to creation."[2] However, he acknowledges that the KCA does not show that the personal creator is all-good, omnipotent, omniscient, and so forth.[3]

2. In "KCA" #1[4], I acknowledged that the application of Cantorian set theory to the real world, according to what I call the standard version (SV), generates counterintuitive absurdities. This generation obtains due to the notion that what are commonly considered as sets in the real world are to be also deemed as sets within the meaning of Cantorian set theory. Given this posit, any two real infinites are equipollent (i.e., their respective members are in one-to-one correspondence) since each such infinite is equipollent to N (the set of all natural numbers). Thus a real infinite and any of its infinite proper subsets are also necessarily equipollent to each other, according to SV. Since SV generates counterintuitive absurdities, any real infinite is metaphysically impossible.[5] Therefore, since any infinite temporal series is a real infinite, any infinite temporal series is metaphysically impossible.

3. In "KCA" #1, I proposed that SV should be replaced by (what I call) the alternative version (AV). I argued that the application of Cantorian theory via AV to the real world does not generate counterintuitive absurdities such that every actual infinite set of real entities is metaphysically impossible.[6]

4. According to AV, a real infinite (although a set as commonly understood) is not a set as standardly understood within the context of Cantorian set theory, considered simply as pure mathematics. AV encompasses four principal propositions. AV1: Every real infinite and N (the set of all natural numbers) are equipollent because the members of the former correspond one-to-one with the members of the latter. AV2: The cardinal number of N (i.e., aleph-zero--אo) is the cardinal number of every real infinite because each such infinite is equipollent with N. AV3: Equipollence between two real infinites is a sufficient but not necessary condition for such two sets to have the same cardinality. AV4: No real infinite is equipollent with any of its infinite proper subsets, although both have the same cardinality (i.e., אo). Whether two real infinites are equipollent ultimately depends upon factually contingent or definitional matters pertaining to the real world (i.e., the domain of substances or continuants such as dogs, stars, gods, minds, molecules, governments, and the like, but not abstract entities such as numbers which subsist in another domain).[7]

A Review of the Structure of Craig's Argument that an Infinite Temporal Series is Metaphysically Impossible

5. In this article, I turn my attention to Craig's argument that any infinite temporal series of past temporal intervals[8] or events[9] is metaphysically impossible, even though it is to be assumed arguendo that some real infinites are metaphysically possible. However, I additionally presuppose that AV is the appropriate mode by which Cantorian theory is to be applied to the real world. Thus all those counterintuitive absurdities generated by the application of Cantorian set theory to the real world via SV, which (according to Craig) are actually intensified by the sequential nature of the temporal series, whether of events or temporal intervals, are necessarily obviated.[10]

6. Craig's first philosophical argument (FPA) is that any real infinite is metaphysically impossible. His second philosophical argument (SPA) is that any infinite temporal series is metaphysically impossible (without assuming the metaphysical impossibility of any real infinite). The two premises in the SPA—that is, (i) a temporal series is a collection formed by successive addition and (ii) a collection formed by successive addition cannot be an actual infinite—provide the sole grounds for concluding that no infinite temporal series is an actual infinite.[11] Craig expressly asserts that his SPA is independent of the FPA.[12] However, as I shall later argue, Craig in his post-Kalam Cosmological Argument writings nevertheless actually relies upon some FPA grounds in the exposition of what he purports to be his SPA.

7. Craig's assumption (made by him for argument's sake only) that an actual infinite in reality is metaphysically possible might well have come with some easy-to-overlook strings attached; a matter which would turn out to be very much to his polemical advantage were his SPA not to be understood as originally and strictly presented in his Kalam Cosmological Argument. Thus his discussion of the SPA might very well have expressly begun with a reproachful "So there!" approach: "All right, "he might well have written," since I assume (for argument's sake) that it is metaphysically possible for an actual infinite to exist in the real world, I also reluctantly assume that any two real infinites are necessarily equipollent to each other since each is equipollent with N and thus have the same cardinality."[13] "I must also," he might well have added, "assume arguendo that entities may be added to or removed from a real infinite."[14] The reason for these assumptions is that those indisposed to accept the KCA generally adhere to SV, and Craig himself thinks that SV is the only plausible version by which Cantorian theory can possibly be applied to the real world. It would seem, therefore, that Craig should have considered himself as estopped, in the interest of fairness, from advancing in support of the SPA all such grounds as had been especially advanced by him in support of the FPA.

8. A close reading of his Kalam Cosmological Argument discloses that Craig, with perhaps two rather de minimus exceptions, generally adhered to the policy of limiting his SPA as presented in that work upon grounds independently of those which purported to show that any real infinite is metaphysically impossible. Thus, for example, Craig in Kalam Cosmological Argument discussed the alleged impossibility of adding to a real infinite and those other counterintuitive absurdities generated by the Tristram Shandy paradox (as restated by him) in the course of his exposition of the FPA.[15] Thus the premises of the SPA were rather strictly construed in his book.[16] However, Craig's book also includes two appendices about "two other philosophical puzzles that embody issues closely related to the present argument [pertaining to the impossibility of a real infinite formed by successive addition]: Zeno's paradoxes of motion and the thesis of Kant's first antinomy of pure reason."[17] Indeed, he explicitly refers the principal theses of the two appendices as what "we have argued in support of our second premiss" (i.e., a collection formed by successive addition cannot be an actual infinite).[18]

9. Appendix 1 contends that "[t]he problem of whether an actual infinite could be formed by successive addition plays a central role in modern philosophical discussions of the Zeno paradoxes."[19] The burden of Appendix 1 is to show that it is impossible to form a real infinite by successive addition, specifically by the showing the absurdity of an "infinity machine" performing the super-task of transforming a potential infinite into an actual infinite. This is a conclusion that I am not at all disposed to contradict. It is important to remember that this conclusion rests upon the impossibility of "trying to transform a potential infinite into an actual infinite by counting."[20] And counting cannot do the trick, as Craig rightly comments, because there cannot be an "infinitieth" member of a set.[21] In other words, I would add, aleph-zero (אo) is not a number that can be counted up to or down from on the line of whole numbers. Rather, aleph-zero is the cardinal number of all whole numbers. It is in the concluding paragraph of Appendix 1 that we find that Craig incidentally mentions a second ground involving FPA considerations after stating another that does not do so.[22]

10. Appendix 2 pertains to Kant's first antimony that the world must be finite in time. Craig asserts that "Kant's proof that the world had a beginning in time is very much like ours."[23] According to Craig, "Kant's first antimony cogently argues that the present event could never arise if the temporal series of past events were infinite (Appendix 2)."[24] For Craig, "[t]he question is, how can an actual infinite come to be formed by successive addition? [T]his question ... lies at the heart of Kant's proof."[25] For Craig, Kant's assertion that an infinite series cannot be completed or enumerated by successive synthesis is equivalent to the assertion that it cannot be formed by successive addition.[26]

11. Craig understands Kant as having expressed in somewhat different words and from a somewhat different angle, as it were, what he himself urges in his SPA, i.e., that the formation of an infinite temporal series of events is metaphysically impossible because such a series cannot be formed (completed, enumerated) by successive addition (synthesis).[27]

Analysis of Craig's SPA in Kalam Cosmological Argument

12. So, we now turn to consider Craig's SPA insofar as it is understood not to presuppose grounds based upon SV considerations that support, or tend to support, the notion that any real infinite is metaphysically impossible.

13. The premises in the SPA, as stated in the Kalam Cosmological Argument, assert that a temporal series of events is a collection formed by successive addition and that a collection formed by successive addition cannot be an actual infinite. But as Craig explained: "Here we do not assume that an actual infinite cannot exist. Even if an actual infinite can exist, the temporal series of events cannot be one, since an actual infinite cannot be formed by successive addition, as the temporal series is."[28]

14. It is essential for the reader to notice that Craig's position is that "[t]he only way in which an actual infinite could come to exist in the real world would be to be instantiated in reality all at once, simply given in a moment."[29] Thus when Craig speaks of a collection being formed or completed he is thus speaking of how a collection "could come to exist," rather than of simply defining or describing the structure or nature of its composition. However, he acknowledges that "[t]he only way a collection to which members are being successively added could be actually infinite would be for it to have an infinite 'core' to which additions are made."[30] "But then," he adds, "it would not be a collection formed by successive addition." He explains that the surd infinite (to which additions are made) is but simply given.[31]

15. Thus whenever Craig writes in his Kalam Cosmological Argument of a collection having been successively formed he is always referring to the case of one that must have a first member. Any doubt concerning his meaning is dispelled by his declaration:

It is important to understand exactly why it is impossible to form an actual infinite by successive addition. The reason is that for every element one adds, one can always add one more. Therefore, one can never arrive at infinity. What one constructs is a potential infinite only, an indefinite collection that grows and grows as each element is added. Another way of seeing the point is by recalling that אo has no immediate predecessor. Therefore one can never reach אo by successive addition or counting, since this would involve passing through an immediate predecessor to אo.[32]

16. Unless one correctly understands Craig's use of the term "formed by successive addition," it is unlikely that the objector to his SPA will notice that it does not do his/her cause any good to respond that, although an infinite series of events cannot be "formed by successive addition" in a finite amount of time, it is possible for it to have been so formed in an infinite amount of time. Craig rightly responds, in a later writing, that his SPA "can be restated in time itself," by replacing the term "temporal series of events" with "temporal segments of equal duration, say hours," since "it would be nonsensical to reply that it is only impossible for them [i.e., an infinite number of previous hours] to elapse in a finite time, for the argument concerns time itself."[33] The purported basis for this contention is the first premise of the SPA, i.e., the impossibility of an actual infinite formed by successive addition.[34]

17. Given the way Craig uses the term "formed by successive addition," it is indeed impossible for any real infinite to be formed unless by instantaneous instantiation of coexisting members. This is the case because Craig's use of "formed by successive addition" in his Kalam Cosmological Argument presupposes a finite or empty set of events as being temporarily prior to the existence of any real infinite. Accordingly, I readily agree that Craig is quite right when he asserts that a collection formed by successive addition cannot be an actual infinite.

18. However, Craig overlooks the possibility that a collection itself as a whole need not be one formed by successive addition although every finite segment thereof has been so formed. Thus every so-called surd infinite of past events, to which additional members are added, itself consists of finite segments each of which has been successively formed.[35] This is the case because every past event in every temporal series has been preceded by another past event. Therefore the so-called surd infinite, in an augmented infinite temporal series, is not but simply given in a moment in the sense that the surd infinite consists of coexisting events but rather it (considered as a whole) is a never-formed collection of past events which has been successively or sequentially instantiated.[36]

19. Hence it does not follow that it is metaphysically impossible for there to be a real infinite that has neither been formed by successive addition nor one that has been instantaneously and simultaneously instantiated (whether or not new members are subsequently added to a surd infinite). Accordingly, it is coherent to speak of an infinite temporal series, every finite segment thereof consisting of immediately consecutive events that have been successively formed, even though it considered as a whole has not been successively formed. Such a collection can be said to be successively formed throughout only in a rather Pickwickian sense that every finite segment has been so formed. But this is a quite a different matter from saying that a collection as a whole has itself been formed by successive addition, since such a collection so formed necessarily has a first finite segment. Hence an infinite temporal series as a whole up to any given event or moment has not at all been formed or completed, in the sense of Craig's SPA.[37]

20. Commenting upon some remarks by J. L. Mackie about the SPA,[38] Craig has written:

For the issue is how the whole infinite past can be formed by successive addition, not merely some finite portion of it. Does Mackie think that because every finite segment of the past can be formed by successive addition, the whole infinite past can be formed by successive addition? That is as logically fallacious as saying that because every part of an elephant is light in weight, therefore the whole elephant is light in weight.[39]

Craig is correct in this matter: one may not validly infer that the whole infinite past has been formed by successive addition just because every finite segment of the past has been so formed. This would indeed be an instance of the fallacy of composition. But it is also logically fallacious to conclude that there cannot be an infinite series of past events itself not formed by successive addition simply upon the ground that every finite segment thereof has been formed by successive addition. This would indeed be an instance of the fallacy of division.

21. Given the foregoing, it appears fairly obvious why something is radically misleading with the first premise in Craig's SPA: i.e., that every temporal series of events is a collection formed by successive addition. The premise is necessarily true of every finite segment of a temporal series of events. But it is simply begging the question to assert that this applies to every temporal series.[40] Craig has fallaciously concluded that having been formed by successive addition is an essential property of every temporal series because it is the common (indeed essential) property of every finite temporal series. What is the essential property of every temporal series of past events is that every event in the series took place after one or more events in the series; it being the case that a temporal series is not one "whose members all coexist [but] [r]ather it is a collection that is instantiated sequentially or successively in time, one event following upon the heels of another."[41] Every temporal series may be said to be characterized by successive or sequential instantiation, whether or not it has been formed by successive addition. But a temporal series which as a whole has been formed by successive addition is one that necessarily has a first segment.[42] Such a temporal series must be finite because it not only has a first but also a last segment which terminates in the present or some specific past event or temporal interval.[43]

22. Having examined Craig's SPA, as expounded in his Kalam Cosmological Argument, I find it to be fundamentally flawed. Its first premise (i.e., a temporal series is a collection formed by successive addition) is more likely an idée fixe on Craig's part rather than being an intuition of a metaphysical truth. However, Craig's book discloses that he, when prescinding from grounds supportive of the FPA, believed that an infinite temporal series is metaphysically impossible if and only if a temporal series is one necessarily formed by successive addition. Consider first the following passage, bearing in mind that the SPA in Kalam Cosmological Argument assumes for argument's sake that a real infinite is metaphysically possible:

[S]uppose 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 on 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 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. The impossibility of such a traversal has nothing at all to do with the amount of time available: it is of the essence of the infinite that it cannot be completed by successive addition.[44]

23. Note that in this passage the impossibility of traversal pertains to an infinite yet to be crossed. Thus, of course, it does not disclose Craig's opinion about the metaphysical possibility of an infinite temporal series (assuming a real infinite is possible) if and only if a temporal series is necessarily formed by successive addition. But now consider Craig's note accompanying the just quoted text:

But suppose someone objects that the man in question has been running from eternity past: if his foot strikes a stone every second and there are in eternity past an infinite number of seconds, will he not have completed his course successfully? In one sense, yes: if an infinite number of seconds could elapse, then an infinite number of stones could be traversed. But this only pushes the issue one step backward: how can an infinite number of seconds elapse? One does not eliminate the problem of forming an infinite collection by successive addition by superimposing another collection on top of the first; for if one is possible, both are possible, and if one is absurd, both are absurd. Since any collection formed by successive addition cannot be infinite, an infinite number of seconds cannot have elapsed. This means that time had a beginning or that measured time was preceded by an undifferentiated time.[45]

24. This passage pertains to an infinite which has been crossed. In this passage Craig distinguishes between the elapsing of infinitely many moments or events and the traversing of a real infinite. Craig appears to acknowledge that an infinite number of stones could have been traversed if an infinite number of seconds had elapsed. So one can here agree with Craig that although a traversal is commonly understood to involve a beginning and a finishing terminus, the presence of the former is not an essential property of traversal of past moments or events.[46] But he nevertheless concludes that an infinite series of seconds cannot elapse because that series was totally formed by successive addition. Well, it is rather easy for me to willingly concede that "[f]or the set of all past events to have been both formed by successive addition and to be an actual infinite is ... absurd."[47] However, this concession does not entail that an infinite temporal series of past events is metaphysically impossible because not every temporal series, whether of moments or events, need be a collection formed by successive addition.

25. This is perhaps the place to repeat that given AV it is neither the case that a real infinite is metaphysically impossible because the application of Cantorian set theory to the real world generates counterintuitive absurdities; nor is it the case that an infinite temporal series is metaphysically impossible because not only "the absurdities pertaining to the existence of an actual infinite apply to it, but also that these absurdities are actually heightened because of the sequential character of the series."[48] I do not propose to show here in detail how Craig in fact quite remarkably failed in some of his later writings to adhere to the policy (generally followed in Kalam Cosmological Argument), of treating the SPA as one to be considered independently of FPA considerations.[49] Actually, whether or not he has done so is not particularly critical to our present inquiry. Since we presuppose AV, it follows that those counterintuitive absurdities generated by the attempted application of Cantorian set theory to the real world, as especially intensified with respect to infinite temporal series, do not obtain whatsoever. So it seems to me that it behooves Craig to either refute AV or to show (assuming AV for argument's sake) that an infinite temporal series is nevertheless impossible for reasons not designed to show the impossibility of its formation by successive addition.

Analysis of Craig's Second Argument in post-Kalam Cosmological Argument writings

26. Craig, given his view that SV is the only way in which Cantorian set theory can arguably apply to the real world, confuses the issue somewhat when he declares in Kalam Cosmological Argument:

[W]e argued that it would be impossible to add to a really existent actual infinite, but the series of past events is being increased daily. Or so it appears. For if Cantor's system were descriptive of reality, the number of events that have occurred up to the present is no greater than the number that have occurred at any point in the past.[50]

But one should not be confused by this statement. Craig is not merely saying that the set of events that have occurred up to the present has the same cardinality as any of its infinite proper subsets. He also necessarily means that, given his acceptance of the SV, both are necessarily equipollent to each other. And since (per Craig) each is equipollent to the other, it would make sense to ask if up to and including but not after the year 1000 such-and-such occurs throughout every preceding year of a beginningless temporal series, then why did not such-and-such stop occurring in any year before 1000? or why does not such-and-such still occur every year up to 2014? These questions in a way make sense provided SV is the only possibly arguable way of applying Cantorian set theory to the real world. But it does not make sense to pose these questions as AV provides a coherent way in which to apply Cantorian set theory to the real world.

27. Our reasoning is as follows: According to AV, assuming arguendo that the present year has been preceded by infinitely many years, the number of the years up to the present year, i.e., 2014, is indeed identical to that of the years ending with any year before 2014. And this number is אo (aleph-zero) because every real infinite is equipollent with N (the set of all natural numbers). All such real infinites are thus equinumerous, and only in that sense can it be said any real infinite has as many members as any other. But, according to AV, the set of years up to 2014 is not equipollent to any of its infinite proper subsets ending in some earlier past year.

28. We now proceed to run through and briefly discuss several other issues concerning the metaphysical possibility of an infinite temporal series, thereby analyzing some First Argument grounds that Craig smuggled into the SPA as expounded in post-Kalam Cosmological Argument writings.

29. Does the assertion that an infinite temporal series is metaphysically possible entail that it is possible to count from infinity? No, not any more than it is possible to count to infinity. Craig rightly declares, in explaining why counting to infinity is impossible: o has no immediate predecessor. Therefore one can never reach אo by successive addition or counting, since this would involve passing through an immediate predecessor to אo."[51] A fortiori, one cannot count from infinity. That is, one cannot count from אo because אo has no immediate successor. Responding to those who so unwisely "say that while an infinite collection cannot be formed by beginning at a point and adding members, nevertheless an infinite collection could be formed by never beginning but ending at a point after having added one member after another from eternity," Craig rhetorically exclaims: "But this method seems more unbelievable than the first method. 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 simply by moving in the opposite direction?"[52] True, one cannot count down from infinity because אo has no immediate successor. As to the second point, I shall have more to say about traversal of an infinite, but it suffices to remark that it is impossible to traverse an infinite past by moving in the opposite direction from now; and that any supposed traversal to the present moment through the infinite past is not a traversal by moving in the opposite direction. In any event, the impossibility of counting down from infinity does not entail the metaphysical impossibility of an infinite temporal series or that of a real infinite consisting of coexisting elements. In any event, Craig is quite right: there is no moment or event at which a beginningless temporal series becomes infinite with the addition of a supposedly last member in the series. But then, any moment in the series has indeed been preceded by infinitely many moments as the case may be.

30. Well what about counting down or up through a real infinite? The concept of such an operation is quite different from that of counting to or from infinity.[53] In the latter situation, one is allegedly attempting the impossible task of arriving or departing from אo. In the latter situation, we are not so engaged. But neither are we talking of אo as a number that can be counted up to or down from within an infinite series. Indeed we are not even trying to determine what is the cardinal number of a real infinite. We already know what is the cardinal number of any real infinite. It is אo, the cardinal number of N (the set of all natural numbers). So the question must be whether there can be a counting up or down through an infinite in the sense of successively pairing in appropriate order the elements of that infinite with those of N (or some other appropriate denumerable mathematical set (such as -N). And here the metaphysical impossibility of either counting up or down through an infinite can be readily admitted. We cannot possibly count up through an infinite (in the sense just indicated), although we do have a starting element, because the operation can never be completed in the absence of a last or ending element. We cannot possibly count down through an infinity because there is no starting member. And we obviously cannot successively count backwards through an infinite from now; since the infinite past would not have an ending moment even if now is deemed to be the starting moment. In any event, the metaphysical impossibility of an infinite temporal series is not entailed by the impossibility of counting up or down through such a series, or of any real infinite.

31. So the next matter we should consider is whether there is something somewhat similar to but not the same as counting down (or up) through an infinite. So let us call this operation as quasi-counting down (or up) though an infinity. Before we can meaningfully speak of a quasi-countdown (or countup) through an infinity, we must first consider another notion of what I shall call s-enumeration.[54]

32. Now there is an equipollence between any real infinite and N. The one-to-one correspondence between N (or -N {. . . . -5, -4, -3, -2, -1, 0}) and any infinite temporal series is one that intrinsically obtains and does not entail any process or operation. Now what I mean by s-enumeration is that of an intuitive, all-at-once apprehension by Someone (e.g., an omniscient God) of each and every one-to-one correspondence between any moment or event in an infinite temporal series with any appropriately ordered negative number.[55] And, according to some theological theories, an omniscient God would know what is the exact numerical correspondence between any one event in an infinite temporal series terminating is some specific future event (marked, as it were, as the zero event) and any appropriately ordered number.[56]

33. Richard Sorabji has very well pointed out that "we cannot imagine that the beginningless series of past years has been subjected to counting in any straightforward way; for it has no first member to match the first number used in counting."[57] So also there cannot be counting in a backwards direction because there is no starting number. However, he has explained that there is something akin to backwards counting as follows:

Ought it not to be possible for a beginningless being to count off the years, descending from the higher numbers and finishing, say, in this century with the years four, three, two, [one] and zero. A backwards-counting angel might then sigh with relief in 1982 [the year before the publication date of Sorabji's book] and say, "Thank heavens, I have reached the year zero; I have just finished counting infinity." If this is not possible, then how can the traversal of an infinity of years be possible?

My answer to this is that something like the backwards count would indeed be possible in principle. I am not at all sure it would to be called counting, but it is conceptually possible that God should have included a beginningless meter in his beginningless universe, to record how many years remained until some important event, say, until the incarnation of his Son. At zero B.C., the meter would register zero, but the counting would never have begun. Rather, for every earlier year, the meter would have displayed a higher number. Whether or not this should be called counting, there is no logical barrier to it, I believe; and therefore no logical barrier has been exhibited to the traversal of an infinity of past years.[58]

34. The important thing to notice is that this so-called countdown (or countoff) through the infinite past presupposes a simultaneous apprehension of all future events up to some zero or target event. This allows, as it were, the s-enumeration, i.e., the intuitive all-at-once intuitive apprehension by Someone of the all the one-to-one correspondences between the infinitely many years up to and including the target-date with all the negative numbers. The Someone appropriately programs the meter or counter (whether a person or not) according to the given s-enumeration, which presupposes some specific future event as the zero (or target) event.[59]

35. I think that Craig egregiously mishandles the issue of a quasi-countdown through an infinity. In his review of Sorabji's book Craig considers the former's thesis that the traversal of the infinite past is possible. He first refers to (what he understands to be) Sorabji's opinion that an infinite lapse of years is possible--and notes that Sorabji had dismissed the possibility of counting down through an infinity and finishing at the present moment because a starting number is lacking.[60] Craig continues as follows:

But this response is clearly inadequate, for the years of an infinite past could be enumerated by the negative numbers, in which case a completed infinity of years would, indeed, entail a countdown from infinity [i.e., countdown through an infinity]. Sorabji anticipates this rebuttal, however, and claims that such a backwards countdown is possible in principle and therefore no logical barrier has been exhibited to the elapsing of an infinity of past years.... [T]he question I am posing is ... whether such a countdown is not metaphysically absurd. For such a countdown should at any point already been completed, since an infinity of time had already elapsed.[61]

36. First, the reader will notice Craig's obfuscating use of "countdown from infinity," instead of using some expression such as "quasi-countdown through an infinity." Secondly, the phrase "could be enumerated" suggests some kind of temporal process different from an all-at-once intuitive s-enumeration of the elements of the infinite past by that Someone who programs the countdown meter. Next Craig, having first noted Sorabji had rejected the possibility of a countdown through an infinity because of the lack of a starting number, recites that Sorabji admits the possibility in principle of a backwards countdown. To the contrary, what Sorabji favored is "something like the backwards count"--which presupposes the all-at-once intuitive apprehension by Someone of the one-to-one correspondence between the infinite temporal series terminating in the year of "some important event" and some denumerable mathematical set (e.g., -N). Moreover, Craig implicitly waives any objection to the factual possibility of an eternal counter or meter always having sufficient time/space to display the requisite number (however large it may be) of the moments/events before the target event. He does so because he wants to use FPA grounds to show the metaphysical impossibility of an infinite temporal series.

37. Now in footnote 29 to the quoted passage by Sorabji, he (Sorabji) unfortunately comments that "however far back you go, a backwards counter would already have counted an infinity of numbers; but he [Fred I. Dretske] wrongly infers that the counter would have finished. That is to move illegitimately from 'an infinity of numbers' to 'all the numbers.'"[62] The choice of words was somewhat improvident because of its equivocation, but quite understandable because a fair reading of the accompanying text should have indicated to Craig that although a so-called backwards counter (i.e., what Craig termed the "eternal counter") would have quasi-counted down through an infinity of years up to any given past year (other than the zero or target year) in the series of years terminating in "some important event," there would have still been a finite number of years yet to elapse before the occurrence of the "some important event." Craig failed to realize that what-was-something-like-the-backwards-count is similar to a countdown or checking off until the occurrence of a target event (i.e., like the countdown at the Space Center until the firing of a missile, e.g., {...3, 2, 1, blast-off!}. But the problem with Sorabji's proposal is that it is infected with the SV virus. Sorabji's proposal considered in the light of SV permits a critic such as Craig to exploit the former's footnote 29.

38. Craig, citing the principle of correspondence in Cantorian set theory as applied by SV (i.e., that any two real infinites are equivalent if and only if they are equipollent),[63] points out that "on this account the counter should at any point in the past have already finished counting [down] all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers."[64] Triumphantly, he proclaims that "a deeper absurdity bursts into view: for suppose there was another counter who counted at a rate of one negative number per day. According to the Principle of Correspondence, which underlies infinite set theory and transfinite arithmetic, both of our eternal counters will finish their countdowns at the same moment, even though one is counting at a rate of 365 times faster than the other!"[65]

39. Pace Craig, but his argument in this matter depends not upon Cantorian set theory but rather SV, which specifies how such theory is to apply to the real world. So it does not follow that the eternal counter should have finished at any moment before the target-event. Nor does it follow that there is anything strange in that the eternal-per-year counter and the eternal-per-day counter would finish at the same moment because the set of infinitely many years and that of infinitely many days terminating at the same moment are not equipollent. If the zero or target date involving "some important event" is, say, 4 July 2014, then the eternal meter which quasi-countdowns on a one-per-year rate and the other meter at a one-per-day rate will have read -1 and -365 respectively on 4 July 2013.

40. So, after all is said and done, it has not been shown that it is metaphysically impossible for there to be a quasi-countdown through an infinite temporal series of past events to some past moment, or the present moment, by some eternal meter or counter. So, finally we come to consider the issue of the metaphysical possibility of the traversal of an infinite temporal series of past years by one and the some entity.

41. With respect to any set of infinitely past years terminating in the present or some past year, it is always true that there was never a time that there was an infinite to be traversed in order to reach the present or any past year. In other words, up to the present or to any given past year there has already been infinitely many years that have elapsed or have been traversed before the present of any given past year, as the case may be.[66] As so succinctly expressed by William of Ockham (in his refutation of the argument that the world necessarily had a beginning because an infinite cannot be traversed):

[I]t is true in general that an infinite that at some time was to be gone through never can be actually gone through; nor can there ever be a last [element] of such an infinite.... But an infinite that at no time was to be gone through but always had been gone through can be gone through despite its infinity. This is the reason why in virtue of the very fact that something has been gone through which at some time was to be gone through, it is finite. But if anything has been gone through which never was to be gone through, it need not be finite but can be infinite. Now, however, if the world existed from eternity, it never was the case that all past years were to be gone through, because at no temporal instant would this proposition have been true; "All these years (indicating all those [now] past) are to be gone through." Therefore, the conclusion does not follow.[67]

42. The proposition that it is metaphysically impossible to traverse an infinite where at every moment there is yet an infinite to be traversed fits very well with the A-theory of time and the notion that a concrete entity exists sequentially from moment to moment, or event to event. Hence it is metaphysically impossible to traverse infinitely many but progressively shorter distances which nevertheless sum to a merely finite distance. And similarly, it is metaphysically impossible for there to be a universe of finite duration but such that its history is constituted, in whole or in part, by the elapsing of infinitely many temporal intervals which are of progressively shorter duration in the earlier direction. Hence it must be the case that motion over a finite distance involves the successive occupation of finitely many but overlapping subdistances and, necessarily, the elapsing of finitely many but not overlapping temporal intervals of non-zero duration. But we have seen that the traversal of denumerably many moments of infinite duration that terminates in the present moment (or at some past moment) does not involve the traversal of an infinite yet to be traversed (or to have elapsed).[68]

43. Here we may take notice of some relevant remarks by Craig about the foregoing matter. In his essay (coauthored with James D. Sinclair), Craig remarked:

It is surprising that a number of critics, such as Mackie and Sobel, have objected that the argument [i.e., that the traversal of infinitely many equal, actual temporal intervals is metaphysically impossible] 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" ([citations]). But, in fact, no proponent of the kalam argument of whom we are aware has assumed that there is 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.[69]

The reader may readily see that Craig is quite misleading here because an infinite temporal series (considered as a whole) is not a collection formed by successive addition, but rather a collection characterized by being instantiated sequentially or successively, one event or moment immediately followed by another.

J. P. Moreland's Argument from the Nature of Causal Sequences

44. I do not think there are fairly plausible philosophical arguments in support of KCA other than those already noted in this article and in my previous article with one exception. J. P. Moreland, Craig's colleague, friend, and ally, advanced one additional argument worthy of mention in his Scaling the Secular City: A Defense of Christianity.[70] The argument is allegedly based upon the nature of causal sequences. Moreland explains:

Consider any event.... This event is caused by another event which preceded it in time.... In order for any event to take place, the entire chain of its causal antecedents must have already occurred and be actual. Otherwise, a necessary condition for the last member of the chain (the event under consideration) would not have occurred and the rest of the chain would not have occurred either (since its existence depends upon this necessary precondition).[71]

45. Moreland concludes that any chain of events leading up to the present must have a first member "since the entire sequence is already actual."[72] With all due respect, this does not impress me as a good argument. It may be worthy of mention but it is not worthy of acceptance because it is based upon an equivocation. Yes, all past events can be said in one sense to be actual. If there are infinitely many past events in one temporal series, the series can be deemed to be an actual infinite for our purposes. As Craig explains: "The fact that the events [in a temporal series] do not exist simultaneously is wholly irrelevant to the issue at hand; the fact remains that since past events, as determinate parts of reality, are definite and distinct and can be numbered, they can be conceptually collected into a totality."[73] He further writes: "[A]n important feature of past events that is not shared by future events [is] their actuality. For past events have really existed; they have taken place in the real world, while future events have not, since they have not occurred."[74] Thus Craig further asserts: "past events are actual in a way in which future events are not."[75]

46. The point is that both Craig and Moreland adhere (as do I) to the dynamic or tensed A-theory of time, according to which every event is not only earlier than, simultaneous with, or later than some other event, but it is also either past, present, or future. Within this context, past events are not actual. As Craig and Moreland declare in their Philosophical Foundations For A Christian Worldview:

On an A-theory of time these different attitudes [pertaining to how we regard an event depending on its pastness or futurity] are grounded in the reality of temporal becoming. A future event has yet to exist and will be present; but a past event no longer exists and was present.[76]

47. Craig and Moreland quite obviously endorse the view inasmuch as they state:

[A] tensed or A-theory of time implies a commitment to presentism, the doctrine that the only temporal entities that exist are present entities. According to presentism, past and future entities do not exist. Thus there really are no past or future events, except in the sense that there have been certain events and there will be certain others; the only real events are present events.[77]

48. So, Moreland's argument from the nature of causal sequences collapses because only among those copresent events within one and the same moment immediately prior to the next succeeding event is the actual cause of the latter-prescinding from issues pertaining to simultaneous or divine causation.[78] Thus the previous members of the causal series terminating in the event within the moment immediately before the event-effect should not be deemed to be actual in a relevant sense.[79]

Final Considerations

49. This essay only concerns philosophical arguments about the metaphysical possibility of infinite temporal series, and therefore does not pertain to the real or supposed deliverances of the natural sciences—or, should I say, the deliverances of some natural scientists. But I should tell the reader that I hold that the real deliverances of the natural sciences have for now and for the reasonably foreseeable future little probative value as to whether the temporal series that constitutes or includes the history of this universe has a temporal beginning with respect to so-called metaphysical or real time, as distinguished from empirical or clock time. There is much to say about this matter, but this essay is not the place. It suffices to point out that if someone thinks that the kalam philosophical cosmological arguments fail then he should consider himself estopped from relying upon the conclusions of these arguments in expounding upon the nature of the cause of the beginning of the universe (now based solely upon the real or suppose deliverances of the natural sciences)—such as to hypothesize that the creator is a personal being of at least great power and knowledge, and who is timeless sans creation but who, according to Craig, enters time with the creation of the universe. That is to say: the deliverances of the natural sciences are insufficient to warrant the conclusion that the universe had a beginning; or, if it had a beginning, that it was caused by only one supernatural being, itself of infinite duration or atemporal, or whose mental life does not totally consist of a succession of infinitely many event-filled moments.

50. The metaphysical possibility of the lapse of infinitely many past moments or events does not entail that one and the same entity has traversed them all. Conversely, the metaphysical impossibility of one and the same entity having traversed any infinite temporal series of past moments or events does not entail the metaphysical impossibility of a temporal series of past moments or events. However, what the metaphysical possibility of a temporal infinite series does entail is the existence of someone or other entity or of one and the same entity during every finite segment of the series. Of course, anyone who denies the metaphysical possibility of one and the same entity having traversed an infinite temporal series implicitly denies the existence of an omnitemporally everlasting God. On the other hand, a naturalist could with perfect consistency admit the metaphysical possibility of an infinite temporal series of past events, but nevertheless deny, doubt, question, or suspend judgment as to metaphysical possibility of one and the same entity having traversed an infinite temporal series of past moments or events.[80] He might very well accept with equanimity that there has been infinitely many entities, each with finite duration, which have progressed one-by-one throughout an infinite past consisting of equal temporal intervals. Thanks to AV, the viability of the thesis of the metaphysical possibility of an infinite temporal series should also please those naturalists and theists who share my opinion of the metaphysical possibility of the history of the natural world, namely that it is constituted by multiple temporal series of causally related events which had no beginning and is of infinite duration.

51. On the other hand, acceptance of the KCA as further developed by Craig entails that an everlasting God (i.e., the supreme being, but whose mental life consists of infinitely many successive events without beginning or end) does not and cannot exist.[81] The theologically conservative theist (including perhaps the atemporal eternalist) who is an adherent of the KCA does not enjoy an enviable status. If the KCA is sound, he is then obliged to concede that an allegedly omnipotent and omniscient God is without power to create a beginningless temporal world (in the sense of having no first moment), or a world consisting of infinitely many entities, spatially and causally interrelated, or of a superworld consisting of infinitely many worlds, none of which is spatially and causally interrelated with another. Not only that, he is obliged to concede that God cannot severally know or apprehend infinitely many things. Furthermore, he is obliged to concede that God is without power to create a world with a future consisting of infinitely many moments or events.[82] Finally, he cannot consider as an open question of natural theology of whether God is atemporally eternal or omnitemporally everlasting since he cannot possibly be the latter. Alas! The supporter of the KCA, who thinks of himself as a theologically conservative theist, has succeeded in defining with great rigor what are the precise limits of divine omnipotence and omniscience.[83] Either that, or he has painted himself into a quite a corner by limiting, by virtue of his natural theology, the possible content of divine revelation. So it might well behoove such theists, to quote C. D. Broad and to mix metaphors, "[to] appreciate the concluding lines of Mr. [Hilaire] Belloc's Cautionary Tale about the boy who ran away from his nurse in the Zoo and was eaten by a lion. 'Always keep hold of Nurse, for fear of finding Something Worse.'"[84]

52. I am what I am pleased to call a commonsensible naturalist—and therefore believe that it is much more probable than not that it is a metaphysically necessary fact that the temporal series of events which constitutes or includes the history of the natural world is denumerably infinite and of infinite duration. Surely, it seems at least prima facie more probable than not that before any moment during which a natural entity exists, there was an immediately preceding moment during which the same or some other natural entity existed. Merely to understand the assertion that before any moment pertaining to the natural world there was another such moment is to understand the proposition that the temporal series of moments pertaining to the natural world is denumerably infinite and of infinite duration.[85] I have not otherwise mentioned this thesis in this essay because I want to focus only on the question of the metaphysical possibility of an infinite temporal series. Acceptance, therefore, of the metaphysical possibility of an infinite temporal series is in this context equally possible for the naturalist and the theologically orthodox theist.

53. So what is the present state of the controversy? We are, I think, at a point much further along than St. Thomas Aquinas' conclusion in De aeternitate mundi contra murmurantes: "[N]o demonstration has as yet been forthcoming that God cannot produce a multitude that is actually infinite."[86] On the other hand, it is probably still somewhat rash to predict that some future ecumenical council will define:

If anyone says that the one, true God, our omnipotent and omniscient creator and lord, lacks the power to create a temporal world without a beginning; or that He is without power to create infinitely many entities in one and the same world, or to create infinitely many worlds none of the same being spatially related to any other but each with finitely many entities therein: let him be anathema.

Appendix: Metaphysical Possibility, the Causal Principle, and Other Matters

54. In this paper I use the terms metaphysically possible, metaphysically necessary, and metaphysically impossible as being much stronger than the terms broadly logically possible, broadly logically necessary, and broadly logically impossible. In possible worlds semantics there are maximal descriptions of reality or of how reality might conceivably be. Each maximal description is both strictly logical and semantically significant and consistent, and exhaustively includes positive and negative propositions (each being formally true or false). A state of affairs, although broadly logically possible in the sense just indicated, is not metaphysically possible unless it must also be actual or actualizable. Thus what may be broadly logically possible may not be metaphysically possible in the foregoing senses of those terms. I shall assume that, for argument's sake, Craig would accept the distinction between the metaphysically possible/necessary/impossible and the broadly logically possible/necessary/impossible as explained above.[87]

55. If so, Craig would surely agree (as I do in fact hold) that the proposition that not everything (i.e., every concrete entity) that begins to exist has a cause of (the beginning of) its existence is a broadly logically possible truth. However, he actually holds (as do I) that the just-stated causal principle (i.e., that everything that begins to exist has a cause for its existence) is or closely approximates a metaphysically necessary synthetic truth which states a first principle that must apply to every broadly possible world in which there are things that begin to exist.[88] As Craig explains: "That some synthetic truths are intuited to be metaphysically necessary a posteriori is evident from such examples as 'Gold has atomic number 79' and 'This table could not have been made of ice.' It could well be that only logically posterior to our experience of reality do we intuitively grasp the necessary truth of the causal principle."[89]

56. A first philosophy involves an acceptance of synthetic principles, which are neither self-evident nor a priori in the usual strict sense; nor are they falsifiable empirical generalizations. However, not all metaphysically necessary truths are components of a first philosophy. First principles are presupposed and implicit in our making of our properly basic and incorrigible commonsensible beliefs (such as the existence of the external world or of other minds), and in the making of generalizations and inferences in ordinary life, as well as in historical, scientific, and forensic inquiry. These first principles are what some call the fundamental deliverances of natural reason.[90] C. D. Broad, for example, preferred to speak of postulates, describing them in his Mind and its Place in Nature[91], but doing so more succinctly in his essay "Critical and Speculative Philosophy."[92] There he wrote:

I do not find [postulates] self-evident. I do not know of any self-evident premises from which they could be deduced by any known logical principles. Hence I cannot group them as a priori propositions. If they are to be grouped as empirical propositions they would have to come under the head of inferred empirical propositions. And this seems impossible for most of them. All inductions make some assumption about the structure of nature, which may be called the 'Uniformity of Nature,' for want of a better name. It would evidently be circular to try to prove such a proposition inductively. Again, any particular perceptual judgment may be defended by argument if we grant the general principle that all sensa are appearances of physical objects. But I can see no possibility of inferring this principle either inductively or deductively from the existence and correlations of sensa. On the other hand, it is equally impossible to refute these propositions by argument. And in practice everyone assumes them, and it is difficult to see that we could possibly unify our experience or that we should have any motive for carrying our researches further if we did not assume them to be true.

57. Even Bertrand Russell, in his Human Knowledge: Its Scope and Limits,[93], found himself required to formulate postulates required to validate scientific method. As he put it: "we may be said to 'know' what is necessary for scientific inference, given that it fulfills the following conditions: (1) it is true, (2) we believe it, (3) it leads to no conclusions which experience confutes, (4) it is logically necessary if any occurrence or set of occurrences is ever to afford evidence in favor of any other occurrence."[94]

58. I ask those readers who do not accept the notion of a first philosophy to assume the truth of the causal principle embodied in the first premise of the KCA for purposes of argument. It is very important, however, to understand what the causal principle does not assert. It does not assert that whatever exists has a cause. Neither does it assert that every event is caused if and only if its occurrence is fully determined. In other words, the causal principle is not that of ontic determinism; nor does the principle entail that whatever constitutes the cause of the beginning of the existence of an entity is, under the circumstances, a set of necessary and sufficient conditions. However, I should like to remark that although I am working with the causal principle as stated in the first premise of Craig's KCA, I believe that it is only approximately true. What, in my opinion, best captures what should be understood as the causal premise was that stated by C. D. Broad:

I suspect that my difficulty about a first event or phase in the world's history is due that the fact that, whatever I may say when I am trying to give Hume a run for his money, I cannot really believe in anything beginning to exist without being caused (in the old-fashioned sense of produced or generated) by something else which existed before and up to the moment when the entity in question began to exist. That this principle has no trace of self-evidence when 'cause' is interpreted in terms of law, and not in terms of generation, is, of course, irrelevant. When 'cause' is interpreted in terms of generation I do find it impossible to give up the principle...[95]

59. In this paper I ask the reader to also assume (again if only for argument's sake) the A-theory of time (i.e., the objective reality of temporal becoming and therefore the objective reality of past, present, and future). According to Craig, the KCA presupposes the A-theory of time.[96] As Craig further explains it: "From start to finish, the kalam cosmological argument is predicated on the A-theory of time. On a B-theory of time, the universe does not in fact come into being or become actual at the time of the Big Bang, it just exists tenselessly as a four-dimensional space-time block that is finitely extended in the earlier than direction."[97]


NOTES

[1] Unless otherwise indicated, bracketed matter in quotations has been supplied or changed by me, and emphasis or parentheses in quotations appears in the original passages.

See this paper's Appendix, "Metaphysical Possibility, the Causal Principle, and Other Matters" (paragraphs 54-59), for my explanation of such terms as "broadly logical possibility," "metaphysical possibility," and kindred expressions, as well as some philosophical principles that I ask the reader to assume if only for argument's sake.

Here I wish to gratefully acknowledge the helpful comments and/or encouragement by Stephen T. Davis, Eyal Gerecht, Stanley Mullen, Quentin Smith, and Tom Masterson about earlier versions of this paper. I am also thankful to the anonymous referee of this version of my paper and to Keith Augustine, Scholarly Paper Editor for the Secular Web, for their careful editorial review.

[2] Kalam Cosmological Argument (New York, NY: Barnes & Noble, 1979), p. 152. The KCA asserts: "1. Everything that begins to exist has a cause for its existence. 2. The universe began to exist. 3. Therefore the universe has a cause for its existence" (Ibid., p. 63).

[3] Kalam Cosmological Argument, p. 63, and p. 172n171.

[4] Philo 5 (2002): 196-215. My article is also available online on the Secular Web (2006; updated 2014).

[5] Neither Craig nor any typical supporter of the KCA has grounded his/her objection to the metaphysical possibility of real infinites upon the physical impossibility of real infinites in this world of spatially and causally related entities or in any other world composed of entities having physical properties more-or-less similar to those in this world. See Guminski "Kalam" (2002), at p. 210n11. But consider, for example, that there are infinitely many worlds each one more or less similar to this world and with each such world containing finitely many humans each with exactly two hands. There would then be infinitely many humans and infinitely many human hands. According to SV, there is a one-to-one correspondence between the infinite sets of humans and of their hands, and a one-to-one correspondence between the infinite set of hands and each of its subsets of left and right hands.

[6] Guminski "Kalam" (2002) and the present article pertain, for the sake of argument, only to denumerable infinite sets. See ibid., p. 199.

[7] Readers interested in more completely understanding the propositions asserted in the accompanying paragraph, as well as their defense, are invited to read the most recent electronic version of "The Kalam Cosmological Argument: The Question of the Metaphysical Possibility of an Infinite Set of Real Entities," updated on the Secular Web in 2014. (Those who choose to read the original version in Philo should note that the Secular Web version is canonical.)

[8] According to William Lane Craig in his first essay—"The Finitude of the Past and the Existence of God"—in Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (Oxford: Clarendon Press, 1993), p. 29: "Specified instants are not temporal intervals, but merely the boundary points of intervals, which are always non-zero in duration." Craig holds (as do I) that "the notion that the present is a solitary instant ... seems to pose insuperable problems for the reality of temporal becoming, since instants have no immediate successors, so that one after another cannot elapse" ("A Critique of the Cosmological Argument for God's Non-existence" in Theism, Atheism, and Big Bang Cosmology, p. 260). As Craig further explains: "[O]n the A-theory of time, according to which temporal becoming is an objective feature of reality, treating time as composed of instants (degenerate temporal intervals of zero duration) seems to land one in Zeno's clutches since temporal becoming would require the actualization of consecutive instants, which is inconclusive" (Craig & Sinclair, 2002, p. 113n13). Although I use "temporal intervals" in this article to ordinarily refer to only intervals of equal duration, I agree with Craig that whether or not a real infinite is metaphysically possible does not turn upon whether the intervals are of equal duration. "The Finitude of the Past," pp. 29-30. Hence I would agree with Craig that the natural world (prescinding from issues pertaining to God's eternity) exists from temporal interval to the next—or, more homely said, from moment to moment.

[9] See Craig, Kalam Cosmological Argument, p. 95: "By 'event' we mean 'that which happens.' Thus, the second premiss is concerned with change." In ibid., p. 103, and Theism, Atheism, and Big Bang Cosmology, p. 30, Craig's SPA refers to events. In some other writings, e.g., Time and Eternity (Wheaton, IL, Crossway Books, 2001), p. 221, he uses "equal past intervals of 'time' instead of events" in his second premise and the conclusion of his SPA. I shall use the term "moment" in this essay to refer (unless otherwise indicated) to the shortest factually possible temporal interval (whatever that may be) that could conceivably encompass one or more events or parts of events (i.e., changes) or one or more eventless states or parts of the same pertaining to a concrete entity. By "eventless states" I mean states that are temporally bounded in both the earlier and later directions.

[10] See Kalam Cosmological Argument, p. 97, where Craig asserts that "the sequential nature of the temporal series of events actually intensifies [all the absurdities attending the real existence of an actual infinite]."

[11] Ibid., pp. 102-103; "The Finitude of the Past and the Existence of God," p. 30.

[12] "Here we do not assume that an actual infinite cannot exist. Even if an actual infinite can exist, the temporal series of events cannot be one...." Kalam Cosmological Argument, p. 103. Cf. Craig's Time and Eternity, p. 229, where he refers to the FPA and SPA as "two quite plausible, independent arguments for the finitude of time." Cf. also ibid., p. 226: "But suppose the above [his argument that an actual infinite cannot exist in reality] is altogether wrong. Suppose that an actual infinite can exist. That still does not imply that the past can be actually infinite." (Bracketed matter added.)

[13] See, e.g., "Time and Infinity" in Theism, Atheism, and Big Bang Cosmology, p. 100, and "A swift and simple refutation of the Kalam cosmological argument," Religious Studies 35 (1999), p. 64, for evidence as to this point.

[14] See, e.g., Kalam Cosmological Argument, pp. 85, 105-106, for evidence as to this point.

[15] Ibid., pp. 97-99. However, his subsequently written essays, "The Finitude of the Past and the Existence of God" and "Time and Infinity" in Theism, Atheism, and Big Bang Cosmology, pp. 33-34 and pp. 99-106 respectively, disclose that Craig transported discussion of Tristram Shandy from the rubric of his first philosophical argument to that of his second.

[16] Kalam Cosmological Argument, pp. 102-105.

[17] Ibid., p. 105.

[18] Ibid. pp. 109-110.

[19] Ibid., p. 175.

[20] Ibid., p. 184.

[21] Ibid., pp. 182-184.

[22] This paragraph reads: "Thus, if we were to come upon a man who tells us that he has been transferring marbles from one tray to another from eternity past and that he is now finishing this super-task, we will know him to be a liar. For (1) then he would have completed an actual infinite by successive addition, which is impossible because you can always add one more, and (2) as we saw with Tristram Shandy, the man's task would have always been done, and hence never been done, which is contrary to our seeing him doing it now. This goes to show that the temporal series of past events cannot be an actual infinite. If it were, it would correspond to the second version of the Dichotomy paradox, where to be at any point, Achilles must ... 'have traveled along the series of points from the infinitely distant and open "end" .... This is an even more astounding feat than the one he accomplishes in winning the race against the tortoise.' For the set of all past events to have been formed by successive addition and to be an actual infinite is thus absurd." Ibid. p. 185 (note omitted). Ground (1) simply states in other words that a potential infinite cannot be transformed into an actual infinite; but it misapplies this rule in a situation where with respect to any past event there has already been infinitely many prior events in the same temporal series. Ground (2) presupposes grounds purporting to show that an infinite temporal series of events is impossible because of the supposed equipollence between the set of years ending, say, in 2004 and that of any subset ending in any earlier year. In any event, the notion of an infinite temporal series of past events should not lead us to worry about the second version of the Dichotomy because this paradox involves the question whether Achilles can traverse any distance from a starting point. But an infinite temporal series does not have any starting point. The other possible deviation from Craig's general policy is his statement in note 21 of Appendix 1 that counting from infinity entails that one will have always been finished, which means that he has never been actually counting. This, says Craig, shows the absurdity of an actually infinite series of events. Ibid., pp. 187-188.

[23] Ibid., p. 189.

[24] Ibid., pp. 109-110.

[25] Ibid., p. 199.

[26] See, e.g., ibid., pp. 104, 190, and 198-199, showing that Craig uses "formation" and "completion" synonymously, and does the same with "successive addition" and "succession synthesis."

[27] Alvin Plantinga, in his Warranted Christian Belief (New York, NY: Oxford University Press, 2000), concludes that Kant's argument with respect to his first antinomy "has no force at all [because] this transition to the conclusion completely begs the question by assuming what was to be proved: that the series in question has a beginning." Ibid, p. 25. Plantinga declares that he is "sorry to say that it is hard to take the argument seriously." Ibid., p. 24.

[28] Kalam Cosmological Argument, p. 103.

[29] Ibid., p. 104. Craig adds: "To try to progressively instantiate an actual infinite in the real world would be hopeless, for one could always add one more element." Ibid.

[30] Ibid., p. 105.

[31] Ibid.

[32] Ibid., p. 104. Cf. ibid., p. 198: "the temporal series is by nature successively formed and cannot therefore be infinite."

[33] "Time and Infinity," in Theism, Atheism, and Big Bang Cosmology, p. 105.

[34] Ibid.

[35] Cf. Kalam Cosmological Argument, pp. 182-183: "[It] is a common tenet of set theory, that a denumerable number of finite sets forms a set which is itself denumerable. But the question is, could such a set be formed by successive addition? And here, set theory contains another tenet which states that the answer is no: a finite set added to another finite set is always a finite set. An actually infinite set is made up of an infinite number of finite sets, but is it never formed by adding these finite sets together." (Note omitted.) The term "forms" in the first sentence surely means the same as "constitutes." Thus to say that denumerably many finite sets forms a set which is itself denumerable is equivalent to saying that a set which is itself denumerable consists of denumerably many finite sets.

[36] As Craig says, ibid., p. 103: "[T]he collection of all past events prior to any given point is not a collection whose members all coexist. Rather it is a collection that is instantiated sequentially or successively in time, one event following upon the heels of another." (Emphasis in original.)

[37] We can speak of an infinite temporal series ending with the start of the year 2000 as having been completed with the start of 2000. But this is true of any given year, and so "completed" is here used as a façon de parler. An infinite temporal series, just as any other real infinite, is never completed but always complete, in the sense that it does not involve a necessarily-always-finite set which never can be completed so as to be an infinite set by the addition of members. As Craig puts it: "A potential infinite is a collection that is increasing toward infinity as a limit but never gets there....By contrast, an actual infinite is a collection in which the number of members really is infinite. The collection is not growing toward infinity; it is infinite, it is 'complete.'" Time and Eternity, p. 221 (emphasis in original).

[38] J.L. Mackie, The Miracle of Theism: Arguments for and against the Existence of God (Oxford: Clarendon Press, 1982), pp. 93-95.

[39] Time and Eternity, p. 229.

[40] In his debate with Professor Michael Tooley ("A Classic Debate on the Existence of God," November 1994, University of Colorado at Boulder), responding to Tooley's argument "that probably there is no mind that exists independently of some associated arrangement of matter that is either identical or at least causally dependent upon," Craig replied: "[A]ll the evidence shows is that being embodied is a common property of minds, but that doesn't show that it is an essential property of minds. To draw the conclusion that there is no unembodied mind you'd have to show that this is an essential property...." (Dr. Craig's First Rebuttal). Thus Craig very well knows the difference between a merely common property and one that is essential for entities of a particular kind.

[41] Kalam Cosmological Argument, p. 103.

[42] Paul Draper, in his "A Critique of the Kalam Cosmological Argument," in Louis P. Pojman, ed., Philosophy of Religion: An Anthology (Belmont, CA: Wadsworth, 3rd ed. 1998), p. 43, writes: "[I]f the temporal regress of events is infinite, then the temporal series of events is not an infinite collection formed by successively adding to a finite collection. Rather, it is a collection formed by successively adding to an infinite collection....[I]t has always been the case that the collection of past events is infinite."

[43] See Craig's discussion in Kalam Cosmological Argument, pp. 96-97. As he says: "In the real sense, the set of all events from any point into the future is not an actual infinite at all, but a potential infinite. It is an indefinite collection of events, always finite and always increasing." Ibid., p. 97.

[44] Ibid., p. 104.

[45] Ibid., p. 159n47. See also Theism, Atheism, and Big Bang Cosmology, p. 32, where Craig incorporates both passages in his abridged version of his argument presented in his book.

[46] As Craig puts it: "To traverse a series means just to cross it or pass through it one member at a time." "Graham Oppy on the Kalam Cosmological Argument," Sophia 32 (1993), p. 5. But he also denies that "the notion of traversal entails a beginning point, so that a series with no beginning cannot be traversed." Ibid.

[47] Kalam Cosmological Argument, p. 185.

[48] Ibid., p. 99.

[49] Craig, in his purported discussion of his SPA in his later writings, can be seen to retain the form but not the substance of that argument since he depends upon several considerations which presuppose some of the grounds he advanced to support his FPA. See, e.g., his essays "The Finitude of the Past and the Existence of God" and "Time and Infinity" in Theism, Atheism, and Big Bang Cosmology, pp. 30-35 and pp. 99-106 respectively; his Apologetics An Introduction (Chicago: Moody Press, 1984) pp. 79-81; and Time and Eternity, pp. 227-228.

[50] Kalam Cosmological Argument, a 97. (Emphasis in original.) The quoted declaration appears in Craig's exposition of the FPA.

[51] Ibid., p. 104.

[52] Time and Eternity, p. 227. In this, and some other writings, Craig refers to the impossibility of forming an actual infinite as sometimes being called the impossibility of counting to infinity or traversing the infinite. Ibid. Surely, these three notions (however closely related) are conceptually distinguishable.

[53] Unfortunately, Craig sometimes synonymously uses such expressions as counting from (eternity) (infinity) and counting down (or countdown) from (eternity) (infinity). See, e.g., ibid., pp. 227-228. I owe much in my discussion on counting down or up through an infinite to Richard Sorabji's Time, Creation and the Continuum: Theories in Antiquity and the Early Middle Ages (Ithaca, NY: Cornell University Press, 1983), p. 219-220.

[54] The enumeration in s-enumeration is suggested by the process of listing, identifying, or designating members of a collection. The s in the expression is suggested by such simultaneous or quasi-simultaneous enumeration were it done all at once by a temporally everlasting God or however it is that an atemporal God accomplishes an enumeration of a collection. Again, I acknowledge heavily borrowing from Sorabji, Time, Creation and the Continuum, pp. 219-220, for the substance of the discussion of the subject in question.

[55] An omniscient God would, of course, be capable of such all-at-once intuitive apprehension of these infinitely many correspondences between past events and negative numbers because (we hypothetically assume at least) that a real infinite is metaphysically possible. God, according to St. Thomas Aquinas (in his Summa Theologiae I-II, q.14, a. 12), knows all things simultaneously, and not successively.

[56] Craig in his Time and Eternity, 243-265, and in his The Only Wise God (Grand Rapids, Mich., Baker, 1987), concludes that God's foreknowledge extends to all future events, including all future contingents. According to Craig, God is atemporal sans creation; but that he becomes temporal beginning with creation or with the initiation of a series of moments in metaphysical time.

[57] Time, Creation and the Continuum, p. 219.

[58] Ibid., pp. 219-220. (Emphasis in original; bracketed matter added; note omitted.)

[59] Of course, what is presupposed in this scenario is that the counter or meter has sufficient time and/or space to display (utter, record) the requisite number, however large it may be, of any corresponding moment/event in a specific infinite temporal series.

[60] "Time, Creation, and the Continuum--Richard Sorabji," International Philosophical Quarterly 25 (1985) p. 323. Craig uses "counting down from infinity" instead of "counting down through an infinity." But he clearly means the latter.

[61] Ibid. (Bracketed matter added.)

[62] Time, Creation and the Continuum, p. 220n29.

[63] Craig slips in his review of Sorabji's book by omitting "and only if." Op. cit., p. 323.

[64] Ibid. Essentially the same argument is repeated by Craig in other writings. See, e.g., Time and Eternity, pp. 227-228. Our discussion in this paragraph assumes, for convenience sake, an exact 365-to-1 ratio between days and years.

[65] Ibid.

[66] For excellent discussions of the metaphysical possibility of traversal of an infinite temporal series to the present (or any past) year and of the impossibility of the traversal of an infinitely many years yet to elapse or to be traversed, see: Norman Kretzmann, "Ockham and the Creation of the Beginningless World," Franciscan Studies 54: (1985), pp. 12-22; and his The Metaphysics of Creation: Aquinas's Natural Theology in Summa Contra Gentiles, II, (Oxford: Clarendon Press, 1999), pp. 175-180; J. J. MacIntosh, "St. Thomas and the Traversal of the Infinite," American Catholic Philosophical Quarterly, 68 (1994): 157-177; and Richard Sorabji, Time, Creation, and the Continuum, pp. 221-222.

[67] William of Ockham, Quaest. Variae, q. 3 (Oth VIII, 81, lin. 382-82, lin. 400), as quoted (with emphasis added) in Kretzmann, "Ockham and the Creation," pp. 20-21.

[68] As I show in my third KCA essay, "The Kalam Cosmological Argument as Amended: The Question of the Metaphysical Possibility of an Infinite Temporal Series of Finite Duration" (2004, updated 2005; revision forthcoming), Craig's kalam cosmological argument based upon the real or supposed deliverances of the natural sciences involves his ostensible acceptance of (1) the metaphysical possibility of the elapsing of a denumerably infinite series of progressively shorter temporal intervals in the earlier direction and (2) the metaphysical impossibility of the elapsing of a denumerably infinite series of temporal intervals of equal duration.

[69] W. C. Craig & James D. Sinclair, "The Kalam Cosmological Argument," The Blackwell Companion to Natural Theology (eds. W. L. Craig & J. P. Moreland) (Wiley-Blackwell, 2012), pp. 119-120.

[70] (Grand Rapids, MI: Baker Book House, 1987). Some readers may protest that I have not even mentioned the many versions of the Grim Reaper paradox and kindred paradoxes posed by some supporters of the KCA. Their many Gedankenexperimente involve bizarre assumptions about highly improbable (to say the least) agents and their alarming operations, which are deemed broadly logically possible, and arguments designed to show that there cannot be an infinite temporal series given these outlandish scenarios. But these arguments invariably suffer from the defect that all that they do is to show, or attempt to show, that it is metaphysically impossible for an infinite temporal series to obtain given the assumptions pertaining in any particular scenario involving Grim Reapers and the like. In other words, just because an infinite temporal series of events that includes the operations of strange agents is metaphysically impossible, it does not follow that it is metaphysically impossible for the temporal series of events constituting or including the history of our universe (in which Grim Reapers and friends are absent) to consist of denumerably infinite moments and be of infinite duration. Thus in this domain what is one man's modus ponens (if P is true then Q is true, and P is true; therefore Q is true) is another's modus tollens (if P is true then Q is true, but Q is false; therefore P is false). I am pleased to see that Craig seems to have no sympathy whatsoever for Grim Reaper and similar paradoxes. See Craig & Sinclair, 2012, p. 113n63: "The most plausible way to avert such paradoxes [involved in the many Grim Reaper scenarios] is by denying the time and space are constructions out of an actually infinite number of points." Another plausible way to avert paradoxes with respect to some preposterous scenarios is by denying that there can be denumerably infinite series of progressively shorter temporal intervals in the earlier-than direction.

[71] Scaling the Secular City, p. 28.

[72] Ibid.

[73] Kalam Cosmological Argument, p. 96.

[74] Ibid., pp. 96-97.

[75] Ibid., p. 97.

[76] (Downers Grove, IL: InterVarsity Press, 2003), p. 384.

[77] Ibid., p. 388.

[78] Cf. Richard Swinburne, The Existence of God (Oxford: Clarendon Press, 1991), pp. 23-25, 72-77.

[79] I am unaware of Craig ever having endorsed Moreland's argument. In any event, it does not appear in their jointly authored book, Philosophical Foundations for a Christian Worldview (see pp. 465, 468-580).

[80] The "third way" of St. Thomas Aquinas purports to prove God's existence based upon that of contingent things and that all such things will eventually fail to exist. Hence there must be a necessary being which either has its necessity caused by another or has of itself its own necessity, and so forth. Summa Theologiae, I-I, q.3, a.3. The argument is thoroughly criticized by Alvin Plantinga, God and Other Minds (Ithaca, NY: Cornell University Press, 1967), 4-25, and Anthony Kenny, The Five Ways: Saint Thomas Aquinas' Proofs of God's Existence (Notre Dame, IN: University of Notre Dame Press, 1980), p. 46-69. That the argument implicitly presupposes that all contingent beings must simultaneously fail to exist is among the difficulties of the "third way."

[81] See: Steven B. Cowan, "A Reductio Ad Absurdum of Divine Temporality," Religious Studies 32 (1996) 371-378; Craig, Kalam Cosmological Argument, pp. 151-153; Time and Eternity, pp. 233-236.

[82] Wes Morriston, in his "Craig and the Actual Infinite," Religious Studies 38 (2002), pp. 156-161, very persuasively argues that the KCA requires the theist to conclude that God neither can severally apprehend infinitely many abstract entities, nor know infinitely many truths, including truths about a future consisting of infinitely many moments or events. He has also very ably argued in "Beginningless Past, Endless Future, and the Actual Infinite" (Faith and Philosophy 27 (2010): 439-450) that even if the temporal series of future events is only a potential infinite, the general arguments for the KCA provide parallel considerations for concluding not only that the temporal series of past events must have a first member, but that there must be a final member of the temporal series of future members. However, Craig argues that the objectivity of temporal becoming makes the series of future events a potential infinite only, and therefore not an actual infinite; and therefore its cardinality cannot be (אo) aleph-zero. See Craig, "Taking Tense Seriously in Differentiating Past and Future: A Response to Wes Morriston." Faith and Philosophy 27 (2010): 451-456, and Craig & Sinclair (2012), p. 116. Although I agree with Craig that the series of future events is a potential infinite, nevertheless the true propositions (i.e., the facts) about future events of an endless future (say in Heaven or Hell) are infinitely many, and hence an actual infinite. Of course, given SV, were the infinitely many facts about potentially infinite future events deemed an actual infinite, then it would follow that the cardinality of the set of infinitely many true propositions about future events as of this moment (whatever that may be for the reader) would be identical to that of the set of infinitely many true propositions about future events after additional n moments had elapsed; but this would be absurd since (according to SV) the two sets would necessarily be equipollent. So if it is AV that actually applies to the real world, then the cardinality of the two sets would be identical (i.e., aleph-zero), but the sets would not be equipollent.

[83] Cf. Edmund Gibbon, The Decline and Fall of the Roman Empire (New York, NY: Modern Library) vol. 1, p. 689n69, where he rather sarcastically notes that one Estius, believing that there is one power (that of creation) which God cannot communicate to a creature, "so accurately defined the limits of Omnipotence." Gibbon remarked that the said Estius "was a Dutchman by birth, and by trade a scholastic divine." Thank goodness that Estius was not a Frisian!

[84] "Relations of Science and Religion," in Reason, Philosophy and Psychical Research (New York, NY: Harcourt, Brace & Co., 1953), p. 243.

[85] Cf. Some Main Problems of Philosophy (New York, NY: Collier Books, 1962), p. 211: "There is no obscurity, or inconceivability, or lack of clearness and distinctness about this notion. Merely to understand the assertion that before any minute there was another minute is to understand the proposition that past time was infinite in extent." However, I should point out that Moore held that "though it does not seem to be quite certain, but only exceedingly probable, that there cannot have been moment which was the first when anything existed, it does seem to [him] quite certain that there cannot have been a first moment of time itself" (Ibid., p. 194).

[86] Cyril Vollert, Lottie H. Kendrzierski, Paul M. Byrne, ed., St. Thomas Aquinas, Siger of Brabant, St. Bonaventure: On the Eternity of the World (Milwaukee, WI: Marquette University Press, 1984 2d ed.), p. 24. Perhaps some readers will not agree that I have succeeded in showing that a denumerably infinite temporal series of infinite duration is metaphysically possible. But I trust that they will at least agree that Craig and his allies have not succeeded in showing that a denumerably infinite temporal series is metaphysically impossible.

[87] The reader should bear in mind that Craig generally (but not perhaps without exception) synonymously uses the terms metaphysically and broadly logically possible (or necessary and its derivatives). However, he is very well aware that some writers distinguish between the different senses of broadly logically possible (and its derivatives) and metaphysically possible (and its derivatives). See "The Kalam Cosmological Argument" by Craig and James D. Sinclair in The Blackwell Companion to Natural Theology (eds. William Lane Craig & J. P. Moreland) (Wiley-Blackwell, 2012), pp. 101, 105-106, as well as W. L. Craig & J. P. Moreland, Philosophical Foundations of a Christian Worldview (InterVarsity Press, 2003), pp. 49-50.

[88] Cf. William Lane Craig, "Must the Beginning of the Universe Have a Personal Cause? A Rejoinder." Faith and Philosophy 19 (2002), p. 94, 97.

[89] Ibid., p. 99.

[90] See inter alia: Thomas Reid, Essays on the Intellectual Powers of Man (Cambridge, MA: MIT Press, 1969), pp. 593-644 and W. L. Craig & J. P. Moreland, Philosophical Foundations, p. 348: "Science cannot be practiced in thin air. It fact, science itself presupposes a number of substantive philosophical theses which must be assumed if science is even going to get off the runway."

[91] (1925) (reprint: Paterson, NJ: Littlefield, Adams, & Co., 1960), p. 217.

[92] Contemporary British Philosophy: Personal Statements (First series) (ed. J. H. Muirhead) (London, UK: G. Allen and Unwin, 1924), pp. 77-100. See also his Kant: An Introduction (ed. C. Lewy) (Cambridge University Press, 1978), pp. 175-177.

[93] (New York, NY: Simon and Schuster, 1948), pp. 487-496.

[94] Ibid., p. 496. See also the chapter "Non-Demonstrative Inference" in Russell's My Philosophical Development (London, UK: George Allen & Unwin Ltd, 1959), pp. 190-207.

[95] Kant: An Introduction, p. 218. The suppressed end of the quoted passage reads: "...and with that confession of the intellectual impotence of old age, I must leave this topic." Broad's mention in the penultimate phrase of "that confession of the impotence of old age" is quite evidently a sign of his humility because, surely, it very frequently happens, as in his case, that with advancing years even into old age there are many who acquire great maturity of intellectual judgment.

[96] Craig & J. P. Moreland, Philosophical Foundations for a Christian Worldview (InterVarsity Press, 2003), pp. 469-470.

[97] Craig & Sinclair (2012), pp. 101, 183-184.


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