The Kalam Cosmologucal Argument and Infinite Regress by James Watson

Questions of whether an actual infinite exists, and in what capacity, have woven their way through the history of philosophy and mathematics. On the surface there do not seem to be many problems left. Few really believe in an actual infinite set of physical things or that such sets can be formed by successive addition. Transfinite mathematics has shown us that infinites, conceived as whole sets, can be attributed positive properties and formalized into consistent mathematical theories. Yet even Georg Cantor, the most sympathetic proponent of transfinite math, did not believe that any of these sets had ontologically instantiable import.[1] Here I will examine arguments for and against the existence of instantiated infinities in the debate surrounding the kalām cosmological argument for the existence of God. I hope to show, despite the continued hope for the infinite past of the universe or for the infinite nature of time, that the concept of infinity, its definition and implications, precludes any instantiation in or as space-time.

            When certain physical theories are developed, theories of cosmogony and cosmogeny, theories of time, etc., their immensity calls into question the common beliefs about infinite sets. Even Aristotle, though he thought infinities could only exist potentially, thought of time as entirely and ontologically infinite, and of time and space as infinite by division.[2] The problems become especially apparent in theories of cosmogony. Concepts such as ‘beginningless series,’ ‘a past point in time infinitely distant from the present,’ and ‘existing through an infinite amount of time until the present’ ask strange things of astrophysicists.

            Big bang theories of the universe escape most of these problems. An initial cosmological singularity, in which time and space began, without which there is no concept of ‘prior’ or ‘before’ (since these are temporal terms), supports the claim that an instantiated infinity does not exist with respect to the universe’s age. Medieval Islamic philosophers had no qualms with this type of theorizing. They had developed a philosophical theory that simultaneously proved the finitude of the universe and the existence of a creator.

The kalām cosmological argument, as opposed to the Thomistic and Leibnizian, is one of the better-respected arguments for the existence of God. Because its validity is not controversial,[3] because it aligns with the most prominent scientific theories of the universe, and because it agrees with general philosophical insight concerning properties of infinities,[4] it is one of the more interesting pieces of religious philosophy. It can be stated as follows:

(1) Whatever begins to exist has a cause of existence.

(2) The universe began to exist.

                (2.1) Argument based on the impossibility of an actual infinite:

                                (2.11) An actual infinite cannot exist.

                                (2.12) An infinite temporal regress of events is an actual infinite.

                                (2.13) Therefore, an infinite temporal regress of events is an actual infinite.

                (2.2) Argument based on the impossibility of the formation of an actual infinite by

         successive addition:

                                (2.21) A collection formed by successive addition cannot be actually infinite.

                                (2.22) The temporal series of past events is a collection formed by successive

                                           addition.

                                (2.23) Therefore, the temporal series of past events cannot be actually infinite.

(3) Therefore, the universe has a cause of its existence. [5]

 

There are dissenters. Many take the consistency of transfinite math to mean that it has ontological import, at least in some possible world.[6] Some think there is no problem with something coming to exist out of nothing.[7] Others attempt to formulate cosmological theories that agree with the evidence of a big bang singularity but also attempt to introduce something prior to the big bang.[8] For example, one theory posits a past infinity of universes coming into and going out of existence prior to what we think of as the singularity, as with the oscillating model. Few of these theories are physical; most are metaphysical and mathematical. But if the physical evidence points us only to the initial singularity,[9] why attempt to go beyond it? The kalām argument certainly points the same direction, both to a starting point and also beyond it. But its wording and intent give a clue as to why cosmologists feel a certain discomfort with an initial singularity. It is too much like creationism, since something must come from nothing.

J. L. Mackie and Graham Oppy have formulated several arguments to the effect that the consistency of transfinite numbers guarantees the logical possibility of an infinitely distant past. Oppy goes a step further and says that, even if this is not true, given the general acceptability of the big bang theory, continuous time would pose a significant difficulty for the kalām. I will walk through the major portions of a recent exchange between these two and William Lane Craig. After noting Craig’s responses I will attempt to strengthen his position for the intelligibility of the kalām and against an instantiated, that is, complete infinity with insights from Alvin Plantinga and A. W. Moore.

 

 

 

Objections to the Kalām Cosmological Argument

J. L. Mackie criticizes the kalām argument (KCA from here on) in his posthumous The Miracle of Theism.[10] Beginning with (2.2), Mackie says KCA proponents show a prejudice against actual infinities. In the medieval versions, this argument addressed the impossibility of traversing an actual infinite. “Since an infinite distance cannot be crossed, if the past were infinite, then today would never arrive. But this is obviously absurd, since today has arrived.”[11] Mackie claims that the arguers make the unwarranted assumption that the universe had an infinitely distant starting point and then claims that it is impossible to traverse that infinite. But, in taking infinity seriously, there would be no starting point at all – not even an infinitely distant one. So any point in past time is only finitely far from the present.

            Addressing (2.1), Mackie says that transfinite mathematics refute the premise that an actual infinity cannot exist. Just because properties differ from infinite sets to finite ones does not entail any contradiction or absurdity with regard to the actualization of those infinities.

…our normal criteria for smaller than and equal to fail to be mutually exclusive for infinite groups. For finite groups to be smaller than means that the members of one group can be correlated one to one with a proper part of another group; to be equal to means that the members of the two groups can be exactly matched in a one to one correlation. These two criteria are mutually exclusive for all finite groups, but not for infinite groups. Once we understand this relation between the two criteria, we see that there is no real contradiction.[12]

 

Mackie further rejects (1) on the grounds that there is no a priori reason to accept it. Craig quotes: “…there is a priori no good reason why a sheer origination of things, not determined by anything, should be unacceptable, whereas the existence of a god [sic] with the power to create something out of nothing is acceptable.”[13] Mackie claims that, despite the fact that astronomy and physics support a finite past for the universe, given Cantorian formulations with infinite numbers and that there are, a priori, no good reasons to accept (1), the KCA is a flawed argument.

Graham Oppy agrees with Mackie and claims that the KCA is full of unwarranted assumptions.[14] Proponents of the KCA generally accept the validity of transfinite mathematics, but they also ask whether it is possible for these infinities to be instantiated in the actual world. Oppy admits that he does not understand exactly what is meant by this question. The logical possibility of infinities referred to in transfinite math entails their possibility in some possible world. So what is the question meant to address? Oppy offers three ways of interpreting the question,

One suggestion is that the question is whether there are any infinities in the actual world. Another suggestion is that the question is whether it is possible for there to be any infinities in the actual world. And a third suggestion is that the question is whether it is possible for there to be any infinities in any world.[15]

 

Oppy takes the third suggestion to be out of the question since proponents of the KCA accept the logical consistency of transfinite infinities. The KCA takes it for granted that the first suggestion is not true, stating it without argument. Oppy concedes for the sake of argument, but points out that Mackie does not.[16] Therefore, Oppy says a stalemate exists between those who accept the existence of actual infinities, like Mackie, and those who do not.

            In considering the second suggestion, Oppy says, “in order to distinguish this claim from the third suggestion, it seems that we shall need to interpret it to be asking whether the existence of infinities is compatible with the actual laws of physics…”[17] But since this ceases to be an a priori question and thus an a posteriori one, and since “we do not yet know what are the laws of nature,”[18] we are in no position to judge the question.

            Oppy finds the arguments for (2.2) much more interesting. Oppy does not seem to agree with Mackie’s claim that KCA proponents misunderstand infinities by claiming an infinitely distant starting point. Proponents have clarified that the KCA refers to sets of the type *ω, when referring to the past regress of an infinite amount of time. William Lane Craig explains,

For in this case the past would be like the second version of Zeno’s Dichotomy paradox, in which Achilles to reach a certain point must have traveled across an infinite series of intervals from the beginningless and open end, with this exception: in the case of the past, unlike the case of the stadium, the intervals are actual and equal. The fact that there is no beginning at all, not even an infinitely distant one makes the difficulty worse, not better… For the past to have been ‘traversed’, would be equivalent to saying someone has just succeeded in enumerating all the negative numbers ending at 0. But this seems to be inconceivable; as G. J. Whitrow urges, a collection of order type *ω is simply not constructible.[19]

 

            But Oppy says that this is merely one way to construe a beginningless set. He gives a prima facie tip-o-the-hat to the idea that *ω sets cannot be constructed. But goes on to say that there are other types of beginningless sets that admit of some ability to be traversed. For example, {1, 2, 3,…3, 2, 1}. Given other forms of consistent, beginningless sets the second sub-argument is unsound.

            However, since this new set has a starting point we must examine the case further. If we simply raise the objection based on this set, it would only prove that, whether time was infinite or finite, it would have a starting point; but this is what the KCA wants to prove. The KCA needs a starting point because a beginningless set, one that has no first member, cannot be traversed. Oppy claims this is tautologous. “But what does this mean? Well, as far as I can see, it means that it is a legitimate objection to infinities which have no first member that they have no first member!”[20] He says that it is no argument against the actualization of infinities that they have no first member, but merely more prejudice against actual infinities. “Once we grant…that Cantorian set theory reveals that worlds with actual infinities are logically possible, there can be no good a priori argument against actual infinite temporal sequences.”[21]

Support for the Kalām Argument

 

            William Lane Craig is one of the staunchest supporters of the KCA and rebuts Oppy’s claims of prejudice and unwarranted assumptions in a series of articles. Craig does not feel that Oppy has accomplished anything toward supporting Mackie’s objections. That Cantorian set theory is logically consistent does not imply that it is possible in some worlds. There is a subtle distinction that Oppy fails to address:           

But how does Cantorian set theory show that there are possible worlds in which there are actual infinites? And even if there are, how does that show that an actual infinite is ontologically possible? The issues involved here are more subtle than Oppy seems to realize. He states, "[Craig] concedes that infinite set theory is a logically consistent system; consequently, it seems that he concedes that there are logically possible worlds in which various 'infinites' obtain." But it is by no means obvious that this second alleged concession follows from the first. The validity of this inference depends on how broadly one construes the logical modality involved in one's possible world semantics.[22]

 

            According to Craig, Oppy fails to take into account the distinction between ‘strictly’ logical and ‘broadly’ logical. To say that a system’s logical consistency in first-order logic indicates that it is true in some possible world is to construe the system much more broadly than is normally considered in possible world semantics. Alvin Plantinga, in The Nature of Necessity, provides some background on modal construal.

            Plantinga explains,

Kareem Abdul-Jabbar’s being more than seven feet tall is a state of affairs, as is Spiro Agnew’s being President of Yale University. Although each of these is a state of affairs, the former but not the latter obtains, or is actual. And although the latter is not actual, it is a possible state of affairs; in this regard it differs from David’s having traveled faster than the speed of light and Paul’s having squared the circle. The former of these last two items is causally or naturally impossible; the latter is impossible in that broadly logical sense.

A possible world, then, is a possible state of affairs—one that is possible in the broadly logical sense. But not every possible state of affairs is a possible world.[23]

 

And, also important to the distinction,

Objects or individuals exist in possible worlds, some like Socrates existing in only some but not all possible worlds, and others, like the number seven, existing in every world. To say that an object x exists in a world W is to say that if W had been actual, x would have existed; more exactly, x exists in W if it is impossible that W obtain and x fail to exist.[24]

 

Notice that it is “objects or individuals” that exist in possible worlds. Given these distinctions, actual infinities and properties of transfinite math do not necessarily exist in some possible world. Their independent coherence determines that. ‘Socrates’ and ‘Spiro Agnew’ are independently coherent objects described in a situation as possible or impossible, but with a symbol or a vague term it is unclear how to determine possibility without reverting to mere definitions. To attribute properties to a symbol and place it in a situation where contradictions between the properties and the situation cannot be derived is to create the possibility of instantiation. Despite the fact that infinities are capable of being introduced in complex theories and attributed properties that limit their functions (such as subtraction and division in cardinal arithmetic), it is the individuals themselves, the infinities, that are defined in such a way as to reject instantiation, not the consistent mathematical systems. Transfinite arithmetic is not an ontological argument for infinities. For Oppy to rely on transfinite arithmetic in support of the possibility of an actual infinity is to miss the point of what is required for modal instantiation.

Craig points out that Plantinga has criticized Mackie on just this point. “…for the resources of first order logic do not permit us to deduce a contradiction from propositions like ‘2+1=7’ or ‘Some prime numbers weigh more than Jackie Gleason,’ but we should not regard such propositions as therefore possible.”[25] ‘Broadly logical modality’ typically indicates a concept of possibility that is “narrower than that of strictly logical possibility (which characterizes a proposition just in case it is not the negation of a thesis of first-order logic, for example) but broader than physical possibility (which characterizes a proposition just in case it does not violate a law of nature)…”[26] Therefore, in failing to note this distinction, Oppy has failed to provide any evidence that the logical consistency of transfinite numbers necessarily lends to their instantiation in some possible world. And even if we define a “sphere of accessibility containing strictly logical possible worlds,” as does Oppy, and concede the possibility of strictly logically possible worlds, a logically consistent system such as infinite set theory does not imply that an infinite is ontologically possible.[27]

A Note On Instantiation

D. M. Armstrong sets out a criterion for instantiation:

C1: “The Possible is restricted by the actual in the sense that all possibilities are composed from actually existing elements, but actually existing combinations form a subset of possible combinations.”[28]

 

Armstrong is defending his version of naturalism against primitive modality. He is basically saying, “…the only universals which can exist are those which are instantiated in the natural world…”.[29] If infinity is taken to have the predicate un-completeable totality, we certainly have record of objects that, given their form constitute ‘always incomplete,’ the natural numbers being the most obvious. However, predicates like ‘the cardinal number of N’ do not seem to exist by any empirical method. Therefore if one is a naturalist, one can accept Armstrong’s thesis concerning the possibility of instantiating a modal predicate for infinity, and thus reject the possibility of an actually completed infinity.

            Craig does not get off so easily. He is an anti-naturalist and holds a theory of universals much closer to the classic version: namely, there are uninstantiated modal properties. This means that the logical possibility of an infinity’s being complete, would require the acceptance of the existence of an infinity in some possible world. Craig wishes not to accept this and must argue as to why it is unreasonable to do so. He gives us little or no reason to this effect in the exchange with Oppy, though he relies on the reader’s prior knowledge of an article by Plantinga concerning essential properties, which we will examine later. For now, let us see why Craig might be right.

Peter Suber provides some interesting insights into the way infinite set theory is conceived among mathematicians. In, “A Crash Course in Mathematics of Infinite Sets,” Suber highlights something to which we cling intuitively, “Now you know how many natural number there are: א0. But this is not profound. So far we’ve only invented a name (numeral) for the number of natural numbers.”[30] The “so far” seems an obvious indication that he will flesh out later why we can do so and what this magnificent number could be. But he never does this except in relation to larger infinities. The only understanding he or anyone else has concerning a completed infinity is the א0’s relationship to exponentially higher infinities. The argument is that there must be a higher infinity than cardinal because between any two points there is necessarily a third. But, given that we know this, why postulate the cardinal number in the first place. All the other numbers existed all along, every point between 1 and 2 is an infinity equal to the natural numbers. But the same goes for counting by tens (10, 20, 30, 40…), all these are equal to the infinite number of cardinals; just as an infinite number of cardinal infinites is equal to the exponential increase of one cardinal. But this is merely a mathematical way of saying there are infinite numerals and the natural numbers are a smaller subset, however, they are also infinite, but a smaller infinite. The question remains how could there be a ‘smaller’ infinite if a basic definition of infinite is that it is never complete?

Also, the use of infinity in Calculus is on par with that of Aristotle because, with the infinitely small, things ‘tend toward’ zero, which means that zero is the stopping point and any non-extended point on your trajectory from a starting point to zero is perfectly willing to act as an indication of the direction you are heading: toward zero. There is nothing that calculates the cardinal number of points between your starting point and zero; it is merely potential.  The same goes for the infinitely large. Numbers, propositions, possibilities tend toward infinity, but are inexhaustible.

Therefore, if one is a naturalist, if you accept Armstrong’s theory of universals you are forced to accept the infinity statements of the KCA. On the other hand, if you take Craig’s position, you need to find something between cogent talk of infinities in mathematics and logical possibility. From the classic point of view, after defining the terms appropriately, it may be possible to speak of infinities, yet contradictory to treat them as completed.

An Aside Concerning Modal Possibility

            Craig briefly describes properties generally attributed to ‘broadly logical possibility.’ Continuing from the previous quote, he says:

Actualists like Plantinga and Stalnaker construe the possibility of the abstract objects which are possible worlds to consist in their instantiability and hold that the framework of possible worlds is grounded in these abstract objects’ possessing the modal property of being possibly instantiated. Broadly logical possibility/necessity is therefore frequently identified with metaphysical possibility/necessity. A state of affairs which is strictly logically possible may, in fact, be metaphysically impossible, incapable of being instantiated. [31]

           

This is the crux of Craig’s argument against the actualization of infinities, but it involves some intricacy. Craig says that broadly logical modality is typically left undefined, except as construed above, and then examples are given to show their uniqueness. However, no examples are given for the impossibility of ontological instantiability of logically consistent states of affairs. This leads us to question our typical notion of taking ‘logically consistent’ to mean ‘logically possible.’

            Craig seems to be relying on Plantinga’s argument for the ability to identify modality de re via modality de dicto, though he does not indicate this directly. Roughly the argument states that existence need not be considered a property for determining essential properties of an object, and that if it can be shown that “de dicto modal properties determine whether x has P essentially.”[32] Given the complexity of this argument, we may substitute a simpler argument along the same lines.

We will take it that infinity (i) has the property of (P) incompleteness (u). (This is incompleteness in the quantitative sense, rather than the Anaxagorian ‘unboundedness,’ or qualitatively incomplete.) If we do not take it this way, it would be finite, the word ‘infinite’ would not exist, and discussion would be over. Therefore we must investigate to what degree infinity has the property of incompleteness, that is, we must determine the sense of our predicate. Our range includes the traditional categories of necessity, possibility, essentially, and contingently. Some, like Armstrong, say that no object has a property necessarily but that all objects necessarily have some property or another. This means that it could not have the property essentially. Therefore, if it is the case that [Piu], then it is contingently. This means that it could have been otherwise and that in some possible world infinity does not have the property incompleteness, and it may be the actual world. But if infinity does not have the property of incompleteness in this world, we have merely chosen the wrong word to define it. We are actually working with a finite sum. We are merely noting that in some possible world there exists a number of objects that has no coherent ending point, but we are not saying that it is the case in this world. But this is senseless; even the mathematicians who chose to use the word infinite claim that it is not characterized by any known numeral.

            If infinity possibly has the property of incompleteness we are stuck with the same quandary. If it is possible that [Piu] then it is not the case in every possible world, but need only be so in one possible world. If it were the case that [Piu] in every possible world, then it would be so necessarily and therefore ‘i’ would have ‘u’ essentially. If then, [Piu] in this, the actual world, then in all other possible worlds it may be the case that [~Piu] and the word would then be meaningless in those possible worlds. This is the most likely alternative to essentialism for infinity. But if it is the case then in the actual world, [Piu] necessarily in this world and any additional predicate that contradicts this predicate makes the term incoherent.

            Given that infinity is properly a predicate of mathematical sets and we have dispensed with the use of ‘infinite’ as a qualitative whole, this argument involves the invocation of second order properties. Despite the skepticism concerning their efficacy, they serve to indicate the sense in which the property ‘infinite’ is used in various contexts. Here especially, if it is the case that it has some property or other essentially, then a contradictory attribution is unacceptable.

            So where does this leave us with Craig’s argument? He says:

Oppy, like Mackie, seems to take a proposition’s freedom from inconsistency in first-order logic to be indicative of that proposition’s being true in some possible world. But this involves a notion of possibility which is much broader than that normally countenanced in possible world semantics.[33]

 

The propositions of infinite set theory that claim equal properties for infinities of different sizes serve as an example here. One infinite can be larger, even substantially larger than another infinite, yet they are considered to be of equal value. For example, in Hilbert’s Hotel with its infinite number of rooms, if there were no vacancies but an infinite number of guests arrived needing rooms, the accommodation would be effortless. Though this poses no problems in infinite set theory, given that the sets to which it adds possess amorphous maximums, but it creates some headaches for mathematics of distinct objects. Terms apply in different ways in different types of mathematics. “This is because our normal criteria for smaller than and equal to fail to be mutually exclusive for infinite groups.”[34] However, in our world they are mutually exclusive, even contradictory. Does this transfer to broadly logical standards? In an interesting sense, smaller than = equal to on Cantor’s scheme. What is needed is to show that there are necessarily no possible worlds in which smaller than and equal to are not mutually exclusive.

            At minimum, it seems obvious that a world where smaller than and equal to are not mutually exclusive could not be a world that contained even one physical object. Smaller than and equal to are conventions of language that describe physical things in addition to conceptual ones, so they share a similarity with married and bachelor. To say that one physical object is smaller than and equal to another physical object simultaneously, and in the same sense without qualification, is a misuse of language and a contradiction.

It might be objected that we merely do not know of any objects on which this contradiction would not rest. But is it because we do not know of any ‘married bachelors’ that we call it a contradiction? If we devised a theory in which a creature was so extravagantly virile or his marital status was so surreally conceived as to consider him a ‘married bachelor,’ would that lend any weight to its instantiability in a possible world? Perhaps, though it does not seem possible with other contradictions like round-squares. In the case of the bachelor, we would have then altered the definitions such that they reject exclusivity. But that seems to be what is done with infinite set theory. The infinite is discussed in finite terms: an unlimited set, א0 is now limited to a symbolized finite sum smaller than a larger sum, א1. But in doing so, the necessarily inexhaustible nature of infinity has led to theories of talking about it that have nothing to do with infinity. We have already seen that to add a predicate that contradicts ‘incompleteness’ makes the term incoherent. We must remove the predicate altogether to add another, but then we are left with a finite sum, rather than an infinite one. 

To flesh out the coherence of the objects of infinite set theory we have to reduce them to definitional situations and determine their consistency. The natural numbers are considered to be infinite because for any finite natural number, another follows it. The process itself is infinite. The natural numbers are, by definition, incomplete on any rendering. They cannot be rendered in their entirety, regardless of human capacity. We could then define an infinity as follows: a set of objects, conceptual or physical, the number of which cannot be conceived (1) by continuous process of accumulation, even granting the possibility of super-tasks and indefinite time, or (2) as complete in its entirety as defined by a single cardinality, belonging to that set, yet succeeding all other members to the point that none follow it. Neither א nor ω indicate a completed set, for doing so violates the very definition of ‘infinite.’ Therefore, “An actual infinite [in the sense of complete whole] cannot exist.” Therefore the infinities in set theory are merely symbols representing logical potentialities (in Aristotelian terms), not logical completions.

 

Continuing…

Further, Craig finds the paradoxes of the infinite themselves to be a decisive reminder that infinities cannot be instantiated. Any infinity, if instantiated, would create a conundrum of contradictory situations. The opponent of the KCA must show there to be a reason or possible situation in which these problems would not ensue. Craig responds to Mackie, “Rather than alleviating the difficulties entailed therein, Mackie has merely specified an aspect of that system which supplies the conditions which, if instantiated in the real world, would spawn the absurdities like Hilbert’s Hotel or Russell’s Tristram Shandy paradox.”[35]

            Except for some quibbling, Craig’s attack on Oppy’s view of successive addition (2.2) mainly concerns Oppy’s introduction of different types of infinite sets, especially {1, 2, 3,…3, 2, 1}. Craig calls this “bizarre.” Is this really a completable set? “If I started counting now, when would I arrive at that second 3? Let us have no fictional suggestions about counting progressively faster so that the infinite super-task is completed in a finite time, for such scenarios are wholly unrealistic.”[36] Such a set is necessarily incompletable in the case of adding members. But Mackie does not dispute that a series type *ω is unformable. He argues that the KCA proponents mistakenly assume that the present exists as a point succeeding the completion of a ω-type series, that is, an infinite series that had a definite beginning point, though infinitely far from the present. But Craig denies that the KCA makes any such assumption and argues that no ω-type series can be formed by successive addition.  

Comments

            It is obvious that Oppy’s central task is to undermine the KCA’s strict finitist metaphysics. He claims that, if we get rid of this unwarranted notion, then the KCA falls apart. It is imperative that we examine such claims, because if they undermine the KCA, they also undermine several hundred years of physics and metaphysics.

            First in this final section, I will address Oppy’s objection to the claims concerning actual infinities and then close with comments concerning the relevance of these claims to the KCA.

Oppy focuses his sights on Craig’s response to transfinite math and its broadly logical possibility of instantiation of infinities. Oppy merely states that he is not willing to concede the point. His claims that the arguments are question-begging seem to hit all around the point. He wants to say that the finitist metaphysic is a presupposition, grounded only in prejudice. With regard to inverse operations with transfinite cardinals, Oppy says, “so why should one who thinks that Cantorian infinities might be physically instantiated lose any sleep over these operations?”[37] The most obvious reasons are the fact that the Cantorian system works on a different conceptual plane of operation, that is, one in which subtraction and division “cannot be defined,” and smaller than and equal to mean the same thing, and that incompleteness seems an essential property of infinity at least in this world.

Oppy’s defense of the ω+ω* set is baffling. The only answer he can offer conjures the notion of topological space and then invokes “supertasks”[38] to complete the traversal of an infinite in a finite amount of time. Oppy rejected this series the first time because it had a beginning, and that was what the KCA was trying to prove. How does re-invoking it help his case? Where are the ‘other’ infinite types that do not fit the description of an untraversable infinite, that he said existed in the first argument? Again, Craig is charged with question-begging. “…[I]t seems to me that all we have is the expression of a question-begging intuition. Certainly, I have not been able to find anything in Craig’s writings which give one who believes that an ω*-series can be formed by successive addition a reason to change her mind.”[39] But really, what more does he want? Infinities are, by definition, untraversable. Both Aristotle and Cantor admit as much.

Oppy continually criticizes the KCA proponents for holding a ‘strict finitist metaphysic.’ He assumes that there is some reason to doubt that the universe, time, sets, events,… whatever, can exist in complete, though infinite packages. But is this the case? Certainly the advancements in transfinite mathematics make room for these questions and demand a certain amount of precision in our answers to them. To begin, we have seen that a consistent system can be said to possess ontological import in some possible world as long as its individual components possess real possibility (though ‘real possibility’ tends to remain a fuzzy concept). The individuals in our case are sets that contain an infinite number of members. Though we have never discovered an actual infinite, Oppy would say this is merely a posteriori and does not address the modal possibility of these individuals. Granted. Also, to say that we do not have a conception of what an infinite number of things is like is merely untrue as far as Cantorian formulas are concerned; there are a number of positive properties of infinities that abide by certain fixed laws of transfinite math.

So what we need is an understanding of infinity that resolves the problems of both temporal and ontological construction implicitly in its definition. Whether such beasts can exist at all is our question, but, as A. W. Moore explains, intuitionists take it to be merely a matter of chronographic ineptitude that we cannot construct an infinity. They invoke ‘super-tasks’ (sound familiar?) to explain away the impossibility.

Let us call any story in which infinitely many tasks are performed in a finite time a ‘super-task story’. Then one thing is surely beyond dispute: that logically consistent super-task stories are there for the telling. In one such story someone constructs all the natural numbers in a minute. They spend half a minute constructing 0, a quarter of a minute constructing 1, and so on ad infinitum. [40]

 

So, the question is, does this make any sense? Moore says no. “It literally makes no sense to describe anything as infinite in this or that respect. We can only use ‘infinity’ to describe the endlessly nested possibilities that (finite) things afford.”[41] This still sounds like prejudice. However, Moore does not think that much else is available to refute the intuitionist.

But the possibility that super-task stories are not coherent will not—yet—help the intuitionist. For there can be no non-question-begging way of explaining the incoherence if all we have to appeal to is pure temporal structure and the fact that we are immersed in time. We need some independent leverage.[42]

 

Moore suggests that we conceive of infinites as Wittgenstein: “as abuses of grammar, misappropriations of the language.”[43] Now we are back to needing an understanding that resolves the issue of time.

Without answering again the argument from transfinite math, the best suggestion that I can offer is that, implicit in the definition of ‘the infinite’, is the notion of inexhaustibility. If an actual infinity exists, instantiated in some objects or other, the whole of them would remain untraverseable. As in one of Moore’s arguments, inexhaustibility is, “a matter of there coming a point—some point or other—beyond which we cannot exert ourselves…[and that would] remain beyond us however far we extended our powers and abilities,”[44] including ‘super-task’ speed. If this understanding is correct, then the possibility of an actual infinite being formed by successive addition (or subtraction), whether in ω or ω* or ω+ω* set-types, or even existing at all as completed or whole, is incoherent. And this leaves untouched transfinite math’s claim to consistency, as Moore points out, “…[L]ogical consistency does not guarantee coherence—a point that intuitionists are especially keen to emphasize.”[45] Yet at the same time it rejects ontological instantiability in the sense of completing an infinite set, by construction or numeration.

Closing

            There are many objections to the KCA,[46] some concern the coherence of cosmological singularity,[47] some concern the warrant to believe the valid arguments of the KCA,[48] some concern the coherence of God’s existence sans the big bang, that is without temporal succession of, say, intrinsic divine thoughts leading to the act of creation,[49] and some which primarily question the intuitive nature of the concept ‘ex nihilo, nihil fit.’[50] But whatever the outcomes of those debates, though I think they are very positive for the KCA, the issues of infinity involved in the argument seem to present no problems for the mechanics of cosmology. No one is having problems making their theories work because an actual infinity must be constructed, and we are just waiting on the tools to do so. In fact, many physicists, mathematicians, and philosophers agree to the claims made by the KCA, given their continuous attempts to debunk their consequences. It is their application that makes these theoreticians uneasy. Given their validity, they must be addressed in one form or another, say, in the construction of a consistent theory of a beginningless universe. But no one is running around trying to find things that come into being out of nothing (except perhaps Quentin Smith), and the evidence continually points to an initial cosmological singularity.

 

 



[1] A. W. Moore, The Infinite (London: Routledge/Taylor and Francis, 2001), “[Cantor] was adament throughout his life that the whole idea of an infinitesimal was demonstrably inconsistent,” p. 117; concerning the paradoxes of the infinitely big, Cantor claimed some totalities “…were too big to be regarded as genuine sets at all… There was no such set as Ω,” p. 127; “There could not be any sets that were genuinely infinite, p. 128, italics his; “…the truly infinite is that which resists mathematical investigation,” p. 198.

[2] Ibid., pp. 36-44.

[3] Its logic seems the least problematic of the theistic proofs. The ontological argument suffers the pain of explaining “existence” as a coherent predicate and the teleological argument has trouble avoiding the naturalistic fallacy.

[4] In that it is a logical argument supporting the intuition that an infinity cannot be instantiated.

[5] W. L. Craig schematizes the argument this way in “Professor Mackie and the Kalām Cosmological Argument,” Religious Studies, No. 20 (1985),  p. 367.

[6] As Mackie and Oppy will argue here.

[7] Quentin Smith has argued this in “The Uncaused Beginning of the Universe,” Philosophy of Science, Vol. 55, No. 1, (1988), pp. 39-57. 

[8] Paul Davies, God And The New Physics (New York: Simon And Schuster, 1983); Adolf Grünbaum, "The Pseudo-Problem Of Creation In Physical Cosmology" in Leslie, J. (ed.) Physical Cosmology And Philosophy (New York: MacMillan, 1990), pp.92-112.Grünbaum, "Creation As A Pseudo-Explanation In Current Physical Cosmology" Erkenntnis 35, (1991), pp.233-254. Steven Hawking, A Brief History Of Time (New York: Bantam Books,1988).

[9] By ‘physical evidence’ I am referring to those several independent confirmations of an inflationary universe, including red shift and background microwave radiation. See Alan H. Guth, The Inflationary Universe (Reading, MA, Perseus Books, 1997).

[10] J. L. Mackie, The Miracle of Theism: Arguments For and Against the Existence of God (Oxford: Clarendon Press, 1982).

[11] Craig, “Professor Mackie and the KCA,” p. 368.

[12] Ibid, italics his.

[13] Ibid, pp. 368-69, quoting Mackie, The Miracle of Theism, p. 94. “[sic]” belongs to Craig.

[14] Graham Oppy, “Craig, Mackie, and the Kalām Cosmological Argument,” Religious Studies, Vol. 27, No. 2 (June 1991) pp. 189-197. 

[15] Ibid., p. 193, italics mine.

[16] “[Craig] tells us that the proponent of the kalam argument is committed to the claim that there are no infinities in the actual world; however—at this point in his paper—he provides no further evidence for the truth of the claim that there are no infinities in the actual world beyond the thought that it would be absurd to suppose otherwise. Since Mackie does not share this intuition…; at best we have a stalemate,” Ibid., pp. 193-94.

[17] Ibid., p. 194.

[18] “Perhaps, on the basis of our current knowledge of the actual laws of nature, we can judge that it is fairly likely that there are no actual infinities; however, it is hard to see that we have much reason to be very confident about this,” Ibid.

[19] Craig, “Prof. Mackie and the KCA”, pp. 369-70. “Construction” here is taken in the normal sense, affirming that the concept of infinity implies the inability to achieve it by finite process.

[20] Ibid.

[21] Ibid., p. 195.

[22] Craig, “Graham Oppy on the Kalām Cosmological Argument,” Sophia 32 (1993), p. 2, citations omitted.

[23] Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), p. 44, italics his.

[24] Ibid., p. 46, italics his.

[25] Craig, “Graham Oppy on the KCA,” p. 2. Craig directs readers to Plantinga’s “Is Theism Really a Miracle?” Faith and Philosophy 3 (1986): 117.

[26] Ibid.

[27] Ibid.

[28] Steven K. McLeod, Modality and Anti-Metaphysics (Burlington, VT: Ashgate, 2001), p. 79.

[29] Ibid, p. 84.

[30] Peter Suber, “A Crash Course in Mathematics of Infinite Sets,” St. John’s Review, XLIV, 2 (1998), pp. 35-59. 

[31] Ibid., italics mine.

[32] Alvin Plantinga, “De Re et De Dicto,” Nous, S. 69, V. 3, pp. 235-258).

[33] Craig, “Graham Oppy on the KCA,” p. 2.

[34] Craig, “Prof Mackie and the KCA,” p. 368, italics his.

[35] Craig, “Prof Mackie and the KCA,” p. 371.

[36] Craig, “Graham Oppy on the KCA,” pp. 3.

[37] Ibid., p. 17.

[38] ‘Super tasks’ are problematic in themselves, in that they presuppose performing an infinite amount of tasks in a finite amount of time. But given that the possibility of an actual infinity is at stake it seems to beg the question to introduce them.

[39] Oppy, “Reply to Prof. Craig,” p. 18.

[40] Moore, The Infinite, pp. 213-14.

[41] Ibid. p. 215.

[42] Ibid,. p. 214.

[43] Ibid.

[44] Ibid., p. 213.

[45] Ibid., p. 214.

[46] Three prominent objections are brought out in Graham Oppy’s article, “Professor William Craig’s Criticisms of Kalam Cosmological Arguments by Paul Davies, Steven Hawking, and Adolf Gruenbaum,” Faith and Philosophy, Vol. 12 (1995), pp. 237-50.

[47] Quentin Smith questions the nature of the initial cosmological singularity traditionally postulated by physics and claims that, given the unanswered problems, big bang cosmology actually contradicts theism, “Atheism, Theism, and Big Bang Cosmology,” Australasian Journal of Philosophy, Vol. 69, No. 1 (March 1991) pp. 48-66.

[48] As we have seen Oppy takes a shot at this. John Taylor does also in “Kalam: A Swift Argument from Origins to a First Cause?” Religious Studies 33 (1997) pp. 167-179.

[49] see Adolf Grünbaum, "The Pseudo-Problem Of Creation In Physical Cosmology" in Leslie, J. (ed.) Physical Cosmology And Philosophy  (New York: MacMillan, 1990) pp.92-112, and "Creation As A Pseudo-Explanation In Current Physical Cosmology" Erkenntnis 35,(1991) pp.233-254.

[50] In addition to Graham Oppy, Quentin Smith has criticized the rejection of something coming into existence out of nothing. See "The Uncaused Beginning of the Universe," in Theism, Atheism, and Big Bang Cosmology, by William Lane Craig and Quentin Smith (Oxford: Clarendon Press, 1993). Though in 1999 he changed his contention from the universe coming into existence out of nothing to the universe coming into existence as its own self-causal agent, “The Reason the Universe Exists is that it Caused Itself to Exist,” Philosophy 74 (1999), pp. 579-86.