No doubt, William Lane Craig's kalam Cosmological Argument (henceforth KCA)[2] has proved a vastly influential natural theological argument for the existence of God. It is simple to state, and its soundness has opened new avenues of research in the philosophy of religion and science. The criticism I wish to focus on in this essay is that the KCA's premise "An actual infinite cannot exist" is false if God exists. I propose a seemingly harmless solution to this paradox but only at the cost of drawing doubt on the truth of the first premise, "Whatever begins to exist has a cause."

The KCA consists of three general 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, the question of whether the KCA is a good argument or not hinges on the question of its soundness, i.e., whether the premises are true or not. To demonstrate the second premise, in particular, Craig offers the following argument:

(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.

It is customary in Judeo-Christian theology to (partially) define God as omniscient, i.e., having knowledge of all things logically possible for Him to know. Usually, what philosophers have in mind is that God knows things like "The Universe is 13.4 billion years old," "Jesus was crucified and revivified three days later," etc., i.e., God possess complete irreflexive knowledge--that is to say, knowledge that is not about God Himself. God?s knowledge about Himself, though, is presumably just as uncontroversial as His irreflexive knowledge. There is nothing paradoxical, for instance, in supposing God knows He has the power to do anything he pleases (so long as it does not entail a logical contradiction). God is also aware of His creating the world and sending his Son, Jesus Christ, to die for all of humans' sins; and, again, there is presumably nothing paradoxical or controversial in supposing these things are true. However, what if we turn God's knowledge on His own knowledge of something? If God knows everything that is logically possible to know, including the fact that Craig says an actual infinite cannot exist, then it follows that God knows that He knows that Craig says an actual infinite cannot exist (since this fact entails no self-contradiction)[3]. Given this, God must also know that He knows that He knows Craig says an actual infinite cannot exist, and so on ad infinitum. In short, there exist an actual infinite number of propositions which God knows to be true, viz. propositions about his own knowledge. Assuming this, however, which follows from the very definition of God, outright conflicts with premise (2.2), "An actual infinite cannot exist," of the KCA. It seems perfectly reasonable, independent of the KCA and (2.2), that God should have full reflexive knowledge, so it seems we are in a bind: either God doesn't exist (in which case, the KCA loses all utility) or an actual infinite can exist (in which case, the conclusion that the universe began to exist rests on a false premise, viz. something cannot, in principle, consist of an actual infinite of things).

There is an apparently simple solution to this paradox, however. It is to limit (2.2)'s domain of discourse (or range) to only include physical objects, e.g. events occurring within the universe, so as to exclude nonphysical objects, e.g. propositions which are the objects of God's knowledge, including reflexive knowledge. Thus, (2.2) may be revamped to read:

(2.2') An actual infinite consisting of physical objects cannot exist.

To this solution, my question is: since the range has been limited on a premise because it leads to an undesireable conclusion, viz. God doesn't exist, for the theist, can't we limit the range of the first premise so as not to lead to an undesirable conclusion, viz. God exists, for the atheist?[4] The point here is epistemological. If we consider it acceptable for the theist to revise (2.2) on the basis that, with the larger range, it implies an undesirable conclusion for them, it is equally acceptable for the atheist to revise (1) on the basis that, with the larger range, it implies an undesirable conclusion for them, as well. Though this is clearly a useless way for a proponent of the KCA to argue, there is another way to defend (2.2'): independently establish it as true; i.e., show that it's true apart from any relation to the KCA. Indeed, thought experiments such as Hilbert's Hotel have putatively provided for Craig and KCA proponents alike the independently established arguments for (2.2); but, since (2.2) implies God does not exist, the premise requires independently justified supplementary arguments to show how either Hilbert's Hotel or another argument altogether only establishes the stronger premise, (2.2'), which has yet to be done.

Notes

[1] For Nelson Pike, who has greatly inspired my work against Craig's menacing kalam Cosmological Argument.

[2] William Lane Craig, The Kalam Cosmological Argument (London: Macmillan & Co., 1979).

[3] I am gratefully indebted to Robert Greg Cavin for this specific example. Others are equally suitable, e.g. God knows he knows everything, which was the original example I used to derive the paradox.

[4] The change in range that I have in mind for the first premise would be something like the following:

(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 within spacetime.