William Lane Craig’s kalam cosmological argument (henceforth KCA) has proved to be a vastly influential natural theological argument for the existence of God. It is simple to state, and inquiry into whether it is sound has opened up new avenues of research in the philosophy of religion and science. In this essay I will argue that a crucial premise in defense of his KCA, that an actual infinite cannot exist, is false if God exists.
The KCA consists of three basic steps:
(1) Whatever begins to exist has a cause.
(2) The universe began to exist.
(3) Therefore, the universe has a cause.
(1) and (2) entail (3), so whether or not the KCA is a good argument hinges on its soundness, i.e., whether the premises are true or not. Craig offers the following argument to demonstrate the second premise:
(2.1) An infinite temporal regress of events would constitute an actual infinite.
(2.2) An actual infinite cannot exist.
(2.3) Therefore an infinite temporal regress of events cannot exist.
Judeo-Christian theology maintains that God is omniscient, that is, has knowledge of all things logically possible for Him to know. What philosophers usually have in mind is that God knows things like the age of the universe, that His Son, Jesus Christ, was crucified and revivified three days after, and so on. That is, God possesses complete irreflexive knowledge—total knowledge that is not about God Himself. God’s reflexive knowledge is equally uncontroversial. There is nothing paradoxical, for instance, in the assertion that God knows He has the power to do anything he pleases (so long as it does not entail a logical contradiction). If He is omniscient, God presumably knows, since it entails no logical inconsistency, that the following proposition is true:
(4) Craig says that an actual infinite cannot exist.
Now consider God’s knowledge of His own knowledge of something. Since there is no logical inconsistency in asserting that God knows that He knows (4) is true, it follows from his omniscience that God knows that He knows (4) is true. Given this, God must also know that He knows that He knows (4), and so on ad infinitum, so long as God’s omniscience entails complete reflexive knowledge.
In short, there exist an actual infinite number of propositions that God knows to be true, namely, propositions about His own knowledge. This follows from the very definition of God, but outright conflicts with premise (2.2) of his defense of the KCA, that “An actual infinite cannot exist.” Independent of the KCA and premise (2.2) of the argument defending it, it seems perfectly reasonable that an omniscient God should have complete reflexive knowledge, leading to a dilemma: either God doesn’t exist (and thus the KCA loses all utility), or an actual infinite can exist (and thus the conclusion that the universe began to exist rests on a false premise, that an actual infinite cannot exist).
One apparently simple solution is to limit the domain or range of (2.2) to events occurring within the physical universe. This would exclude any nonphysical objects, such as propositions that are the objects of God’s knowledge (including His reflexive knowledge). So might (2.2) might be revamped to read:
(2.2′) An actual infinite consisting of physical objects cannot exist.
But if it is acceptable to limit the range of premise (2.2) to avoid an undesirable conclusion for the theist (that God does not exist), it is equally acceptable to limit the range of premise (1) to avoid an undesirable conclusion for the atheist (that God exists). Since the point here is epistemological, this move will not give the KCA proponent what he wants.
But there is another way to defend (2.2′): show that it is true apart from the KCA. Thought experiments such as Hilbert’s Hotel have putatively established independent arguments for (2.2), but (2.2) implies that God does not exist; so KCA proponents need independently justified supplementary premises that show that Hilbert’s Hotel, or some other argument, establishes the stronger premise (2.2′). This has yet to be done.
The latter solution is perhaps more promising. Craig has argued that God’s knowledge is not propositional; that is, God does not know any propositions are true. God’s knowledge is better expressed:
as an undivided intuition of reality, which we finite knowers represent to ourselves in terms of propositions. We express propositionally what God knows non-propositionally.
Construing omniscience nonpropositionally in terms of “an undivided intuition of reality” (whatever that means) entails that there are not an infinite number of propositions; rather, there are only as many propositions as human beings have cognized. Theists are thereby freed from countenancing an actual infinite. On this view, God does not know all true propositions; rather, he knows all truths, which are acquired through “an undivided intuition of reality.” Craig further suggests that propositions need not play any role in defining omniscience, citing Charles Taliaferro’s characterization:
[He] proposes … that omniscience be understood in terms of maximal cognitive power, to wit, a person S is omniscient iff it is metaphysically impossible for there to be a being with greater cognitive power than S and this power is fully exercised.
This approach resonates well with Craig’s idea that the modal terms in (2.2) should be understood in terms of metaphysical necessity, a stronger form of necessity than logical necessity—so long as “necessity” and “metaphysical necessity” are intelligible.
Propositions may very well be “useful fictions.” But there is a friction between (2.2) and omniscience whether or not we countenance propositions. Let us countenance truths, then, as Craig suggests: “God knows every truth.” It is still self-contradictory to suppose that (a) God knows whatever it is logically possible for Him to know and (b) an actual infinite cannot exist. For suppose that God knows (4), where (4) is taken as a truth rather than a proposition. If we replace “proposition” with “truth,” it still follows that an actual infinite number of truths exist. Moreover, it is logically possible for God to simultaneously have and hold an infinite number of divided beliefs, just as we can think of human knowledge as including truths that may be individuated from each other, yet held all at once. The only apparent reason to resist this is to save the soundness of the KCA. Thus the objection that God actually knows things in virtue of an “undivided intuition of reality” is extremely weak, for it does nothing to prevent the individuation of the truths of God’s knowledge as something “actual” and “infinite.”
Thus whether we countenance propositions or truths when defining omniscience, if (2.2) is true, then God cannot exist. If God knows all true propositions and believes no false ones, then God must know an infinite number of propositions, as there are an infinite number of the objects of his self-knowledge. And since it is logically possible for God to know an actual infinite number of truths about Himself, if He knows all truths, then He does know an actual infinite number of truths about Himself. So Craig must either independently justify limiting the range of the tacit universal quantifier in (2.2), or else concede that either (2.2) is false or that God does not exist.
 I am grateful to Robert Greg Cavin for suggesting this specific example. Other examples are equally suitable, such as the original one I used to derive the paradox: that God knows that he knows everything.
(1′) Whatever begins to exist that is a physical object within spacetime has a cause.
If (1′) took the place of (1) in the KCA, then (2) would no longer follow from (1), since the universe is not a physical object within spacetime.
 For an explanation of Hilbert’s Hotel, see William Lane Craig, Reasonable Faith (3rd ed.) (Wheaton, IL: Crossway Books, 2008), p. 118.
 William Lane Craig, “Theistic Critiques of Atheism” in The Cambridge Companion to Atheism, ed. Michael Martin (New York, NY: Cambridge University Press, 2007): 69-85. <http://www.reasonablefaith.org/site/News2?page=NewsArticle&id=6645>.
 Or propositions might even be useless fictions, for that matter. See W. V. O. Quine, “Propositional Objects” in Ontological Relativity and Other Essays, ed. W. V. O. Quine (Columbia, NY: Columbia University Press, 1969).
 Craig could also argue, perhaps, that God is “omniscient” in a way which would make logically possible the existence of a more knowledgeable being than God, i.e., a being that knows everything that it is logically possible to know.
Copyright ©2009 Jeffrey T. Allen. The electronic version is copyright ©2009 by Internet Infidels, Inc. with the written permission of Jeffrey T. Allen. All rights reserved.