## Abstract

A popular critique of the kalām cosmological argument is that one argument for its second premise (what I call the Impossible Formation Argument, IFA) illicitly assumes a finite starting point for the series of past temporal events, thereby begging the question against opponents. Rejecting this assumption, opponents say, eliminates any objections to the possibility that the past is infinitely old and undermines the IFA’s ability to support premise 2. I contend that the plausibility of this objection depends on ambiguities in extant formulations of the IFA and that we may resolve these ambiguities in a way that does not presuppose a finite staring point. I also argue that this disambiguation allows us to construct an argument demonstrating that the concept of an infinite past entails a contradiction.

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## Notes

I retain the language of ‘possibility’ here to make room for discussing fictional or hypothetical temporal events.

Also for the sake of simplicity, I am ignoring the relativity of simultaneity. It may be that not all events are happening ‘simultaneously’ in the pre-relativistic sense. Nevertheless, my account might be updated to accommodate the relativity of temporal reference frames. For instance, we might map all coincident and non-coincident space-time events within the scope of the present light cone onto a set,

*t*, and then map each set of previous non-overlapping space-time events to sets corresponding to denumerable functions of −1, and map each set of successive non-overlapping space-time events to sets corresponding to denumerable functions of +1.I take it that more than one event may occur simultaneously (in some accepted sense of simultaneous), perhaps even an infinite number of them, but if the temporal event containing those states of affairs is finite, then each temporal moment occurs one-by-one, in finite succession. This is consistent with what I will call ‘successive addition of finite temporal events’ because it does not refer to a finite number of states of affairs. Alternatively, one might add, not just a single temporal event, but an infinite series of temporal events (events that stand in before and after relations to one another). That would be an instance of ‘successive addition of infinite temporal events.’ I take it that, if an infinite set of anything is possible, it is not incoherent to imagine adding an infinite set to a finite set even if the IFA defended in this paper is sound. For instance, the set of natural numbers, N, has cardinality ℵ

_{0}. And if we already have a finite set of negative integers, say, { −3, −2, −1}, it is not impossible to construct an infinite set by adding N to this set. This would be an example of constructing an infinite set by successive addition of infinite members. The question for this paper, then, is not whether an infinite number of states of affairs could be added (as a single temporal event) to any previous temporal event, or whether an infinite series of events or temporal events could be added all at once, but whether an infinite number of temporal events (regardless of how many members each contains), added one-by-one, in finite succession could ever constitute an infinite set.As they may be quite different depending on which theory of time we adopt.

There is some concern—rightly, I think—about the notion of ‘existence’ and ‘coming-to-be’ implied these discussions. When we are speaking purely in terms of mathematical entities, we need not be assuming the actual existence of anything at all, even when we say, for example, ‘there is or is not a first member of set X’ (barring Platonist arguments to the contrary). We are simply expressing certain types of relations between hypothetical referents. Thus, we have reason to be careful when using transfinite mathematics to draw inferences about actual events. However, we must also avoid using this concern as a means of dismissing the significance of the debate. See my reply to Craig on this point at the end of § 1. Thanks to an anonymous referee for raising this concern.

The challenge here might be that, if time is dense, there is no first moment, and therefore, the notion of a successor is meaningless. In addition, one might object that, even if we could make sense of (i), (ii) isn’t true because every dense duration is infinitely dense such that any temporal duration constitutes an infinitely ‘old’ universe. I take it that both objections are trivially true but irrelevant, since regardless of whether time is discrete or dense, the relevant notions of ‘successor’ and ‘actual infinite’ could be interpreted to accommodate the difference, e.g., ‘The temporally dense series of events is a series formed by continuous addition.’ For simplicity, I will continue to use the language of discrete time, comfortable that the differences of expression needed for dense time can be accommodated.

The objection here might be that, if time is static, all temporal moments persist such that the notion of ‘addition’ must be interpreted as metaphorical, and, therefore, misleading. If the argument for premise 2 depends on this dynamic interpretation of ‘addition,’ that would be correct, and it would remain an open question whether the universe includes an infinite number of temporal events. But even on a status theory of time, the metaphor of addition need not be construed merely as a rhetorical fiction. Most versions of static theory retain substantive senses of addition, causation, and succession that render the IFA meaningful and, therefore, worth taking seriously. (See Oaklander and Smith 1994).

However, see Oppy (2006), §4.5 for an interesting discussion.

In his (1984), Craig explicitly treats ‘formed by successive addition’ and ‘traversed’ synonymously (p. 369).

Since we are talking about the coming-to-be of ordered temporal events, ‘simultaneity’ here must be understood in a meta-temporal sense. I am considering the possibility that temporal events are co-instantiated, and therefore the notion of simultaneity refers to this co-coming-to-be rather than to simultaneity at a temporal event in the series.

Morriston (2010) attempts to undermine the kalām argument by showing that proponents of the IFA committed to the omnipotence of God are compelled to admit the existence of an actually infinite future. Morriston’s argument presupposes a static, eternalist view of time on which any infinite series, such as counting the natural numbers, would tend toward infinity, and thus, if God commands that an infinite series begin, it follows that it will tend toward infinity, and since God is omniscient, the end of such a series is not only foreseeable, but known. The problem is that Morriston conflates the existence of an infinite set with the formation of one. If God foresees the completion of an infinite set on a static view, then an infinite amount of time and all the counting instances exist simultaneously. But that is question-begging. We are asking if an infinitely large set can be formed by anyone, God included. If God creates, all at once, an infinite series of temporal events, each of which includes counting one natural number, then such a set may exist. If an infinite number of coin flippers flip a coin simultaneously, then an infinite number of coins have been flipped. But the question is whether such a series of coin flips could be completed one-by-one in finite succession. My argument shows that not even God could complete an actually infinite set by finite addition of finite members.

There is also some concern over Craig’s use of the term ‘collection’ rather than ‘set,’ since it may be that collections have different implications than sets. Nevertheless, since Craig evaluates the plausibility of his argument and objections in terms of set theory, I assume he is treating collections and sets synonymously. To avoid confusion, and because Craig seems to treat collections and sets synonymously, I will write only in terms of sets.

More specifically, a set is determined by which members it has, that is, by its membership rule. This may imply that two sets with the same number of members, even zero members, may be different sets, e.g., the (empty) set of married bachelors and the (empty) set of round squares.

Among some philosophers, even this is controversial. If an infinite number of additions or divisions can be described across a finite length of time, then it seems, according to Cantorian Transfinite Mathematics, logically possible to complete an infinite number of finite tasks. For example, in Zeno’s paradox of Achilles and the Tortoise, Achilles should never be able to overtake the Tortoise in a race because we can describe a function according to which Achilles must first cross half the distance the Tortoise moved since both began moving, and then half that distance, and half that, and so on, such that he is committed to an endless task. However, since we know that Achilles

*can*overtake the Tortoise, Zeno is wrong. Graham Oppy (2006) offers a striking argument for this, showing that, ‘If we sum the series [of steps between Achilles and the Tortoise], we find that the limit of the series of distances that Achilles travels is [equal to] the limit of the series of times that the Tortoise travels…. There is nothing even*prima facie*puzzling about this’ (p. 96).But on the assumptions I have set out, the very possibility of summing to infinity is what’s at issue. The paradox presupposes that an infinite number of time is

*available*in which to complete tasks (according to the limit function applied), and, therefore, the question of how a set of finite tasks could be infinite is moot. But if this supposition is not yet established (namely, that such a function is being ‘applied’ in some active sense), the paradox does not arise. Or, in response to Oppy: Who is doing the summing? If it is Achilles or even God, he will never finish.I begin this argument with the number (4) to avoid confusion with the premises of the kalām argument I presented at the beginning.

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Watson, J.C. The End of Eternity.
*SOPHIA* **56**, 147–162 (2017). https://doi.org/10.1007/s11841-017-0590-0

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DOI: https://doi.org/10.1007/s11841-017-0590-0

### Keywords

- Kalām
- Cosmological argument
- Infinity
- Eternity