Craig's Critique of the Existence of Actual Infinites in His Blackwell Companion Article, Part I

Craig's recent defense of his philosophical arguments against the existence of actual infinites is unsuccessful. Start with his interaction with David Yandell's reply. Yandell correctly points out that no contradiction arises when one subtracts different infinite subsets from an actually infinite set. Thus: subtracting all the even numbers from all the natural numbers = all the odd numbers; subtracting all the natural numbers from all the natural numbers = 0. Now of course, if subtracting the very same subset yielded different answers on different occasions, then that would be a contradiction. But of course that doesn't happen when one subtracts from actually infinite sets.

In reply, Craig admits that no contradiction results from the infinite subtractions in Yandell's illustrations. Nevertheless, he asserts that a contradiction is yet to be found with another aspect of such subtractions. In particular, it's to be found in the fact that "an identical number of members have been subtracted from the identical number and yet did not arrive at an identical result." (p. 112) 

As those who follow the literature on the kalam argument know, however, Craig here fails to address a well-known, mainstream criticism from Wes Morriston[1] and Paul Draper[2] -- one that's been around for at least a decade. The criticism is that Craig's reply turns on an ambiguity in the notion of "an identical number" of members of a set. According to one construal, two sets fail to have an identical number of members just in case they can't be put into a one-to-one correspondence with one another. According to another construal, two sets fail to have an identical number of members just in case one of them has all the members of the other, and others besides. The problem is that Craig is relying on the former construal, while the latter construal entails that the two cases do not involve an identical number of members that are subtracted. Referring to the coarse-grained similarity of the two subtracted subsets (viz., correspondence) conceals this crucial difference. But once one shifts one's focus from sameness of cardinality to the finer-grained difference in the particular members of the two subtracted subsets (the set of all natural numbers has all the members of the set of all even numbers, plus all the odd numbers besides), the intuitiveness of Craig's assertion disappears. For then one would of course expect different results when the sets removed in the two scenarios are different in this crucial way. Craig has thus failed to show a contradiction or absurdity in the notion of an instantiated actual infinite. Here, as elsewhere, Craig is craigging.

(We've covered this point in more detail on other occasions (here, for example). An index of criticisms of virtually all of his other key a priori arguments against the existence and traversability of actual infinites can be found here. A critique of his reply to Mackie's criticisms can be found here.)
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[1] See, for example, Morriston, Wes."Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 153-155; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 299-300
[2] 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).