Craig's Case Against the Existence of Actual Infinites, Part One*

(The rest of the posts in this series can be found here.)

William Lane Craig argues that the universe had a beginning as part of an argument that it had a divine "Beginner" or divine temporal First Cause of the universe. He offers four arguments for a finite past: two a priori and two a posteriori. The first is that an infinite past is impossible because infinites are impossible in general; the second is that even if concrete actual infinites are possible, they can't be traversed by successive addition; the third argues that the evidence for the Big Bang indicates an absolute beginning to the universe; and the fourth argues that the 2nd Law of Thermodynamics indicates that the universe is running down, in which case it must've been "wound up" with an initial input of matter-energy, and the latter was the beginning of the universe. In the current series of posts, I will consider Craig's versions of the first argument.

First Version: Parts as Large as the Whole
Infinite Library illustration: Imagine a library containing an actually infinite number of books, such that exactly one unique natural number is written on the spine of each book. Such a library would have just as many even-numbered books as it would have odd- and even-numbered books combined. This is because (i) the set of even numbers (2, 4, 6, …) can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, …), and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each set is just as large as the other. But this is absurd; the set of even-numbered books is a proper subset, or proper part, of the set of all the books in the library, and no proper part or subset of a set could be just as large as the set of which it is a part. Therefore, a library containing an actually infinite number of books is impossible. But this problem isn’t particular to just infinite libraries; it applies to any actually infinite collection of things whatsoever; therefore, concrete actual infinites are metaphysically impossible.

We can express the argument above as follows:

1. If concrete actual infinites are possible, then a library with an actually infinite number of books (each labeled with a unique natural number) is possible.
2. If a library with an actually infinite number of books (each labeled with a unique natural number) is possible, then there would be just as many even-numbered books as odd- and even-numbered books combined.
3. There couldn’t be just as many even-numbered books as odd- and even-numbered books combined.
4. Therefore, concrete actual infinites are impossible.

What to make of this argument? The first thing to notice is that there is an ambiguity in the notion of there being “just as many” books in (2) and (3).[1] On the one hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that the former has the same cardinal number as the latter. Then while (2) is true, it’s not obvious that (3) is true. Indeed, an actual infinite is standardly defined as a set that has the same cardinal number as some of its proper subsets. But, presumably, Craig's argument is supposed to persuade those who already know what an actual infinite is supposed to be, and yet remain at least agnostic as to whether actually infinite sets of things are metaphysically possible. It seems, then, that one wouldn’t accept (3) unless one already rejected the possibility of actual infinites. Now Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. We will consider those in later posts in this series. For now, though, we’ll just note that on the current interpretation of "just as many", (3) so far lacks the independent support it needs for the current argument to convince those who are "on the fence" about the possibility of actual infinites.

On the other hand, one might mean that there are just as many even-numbered books as odd- and even-numbered books together in the sense that neither set of books has members besides the ones they share in common. Then while (3) is true on that reading, (2) is not. For of course the set of odd- and even-numbered books has other books besides the set of books they have in common, viz., the even-numbered books.

The upshot is that the argument gets its intuitive appeal by equivocating on the notion of “just as many”. Thus, once the different readings of the expression are disambiguated, the argument loses its force.
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[*] Thanks to Mike Almeida for pointing out a problem in a previous draft of this post.
[1] At least two philosophers have raised this sort of criticism. See (e.g.) Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 154 (bottom)-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298; Draper, Paul. "A Critique of the Kalam Cosmological Argument", in Pojman, Louis and Rea, Michael (eds.), Philosophy of Religion, an Anthology, 5th ed. (Wadsworth, 2008). (Notes on Draper's paper can be found here and here.)