### Craig's Case Against the Existence of Actual Infinites, Part Six

Craig’s Diagnostic Argument: "The Deeper Problem"

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

According to Craig, the supposedly counter-intuitive implications we've discussed arise in virtue of the following set of claims:

1. A set is larger than any of its proper subsets. (a version of Euclid’s Maxim)
2. If two sets can be put into a one-to-one correspondence with one another, then neither set is larger than the other -- i.e., each set is just as large as the other. (The Principle of Correspondence)
3. Actually infinite sets are possible.

Craig points out that (1)-(3) jointly entail a contradiction.[1] Which proposition should go? Craig thinks that it should be (3). But why should we follow him in thinking this? Well, it's not clear that we should. For yet again, Craig equivocates on a key term.[2] For there are at least two interpretations of the notion of a set A being "larger than" one of its subsets B. On one interpretation, A is larger than B in the sense that A has a larger cardinal number than B. But if this is how "larger than" is to be understood, then while (2) seems true, it's not clear that (1) is true. For if actual infinites are possible, then they have the same cardinal number as some of their proper subsets.

According to another interpretation, A could be "larger than" its proper subset B in the sense that A has all the elements of B and others besides. But if this is how "larger than" is to be understood, then while (1) seems true, it's not clear that (2) is true. For actually A has all the members of B, and others besides. So, for example, consider the set A of natural numbers: {1, 2, 3, ...}. This set has all the members had by its proper subset B of all the even numbers: {2, 4, 6, ....}. And while the two sets can be put into a one-to-one correspondence with one another, it remains true that A has all the members had by the set of even numbers and more besides, viz., the set of all the odd numbers: {1, 3, 5, ...}. And since this is so, it looks as though (2) is unjustified.

Now Craig appeals to the supposed absurdities we've discussed in previous posts as a basis for rejecting (3) over (1) and (2). The idea is that the absurdities arise because Euclid's Maxim and The Principle of Correspondence -- statements (1) and (2) above, respectively -- come into conflict when we apply them to infinite sets. Therefore, given the intuitive plausibility of (1) and (2), the best explanation is that (1) and (2) are true for all possible concrete sets, and that therefore actual infinites are impossible, i.e., (3) is false. This is Craig's proposed diagnosis of the cause of the supposed absurdities that arise in the cases we've discussed in previous posts. But the problem is that, as we've seen in other posts, all of the apparent absurdities are generated by an illicit equivocation on the notion of two sets having "just as many" members; once the equivocation is noted and the different senses of "just as many" are disambiguated, the apparent absurdities disappear. It looks, then, that Craig's diagnostic argument, like the others, fails to support Craig's claim that actually infinite sets of concrete entities are impossible.

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[1] To see this, suppose an actual infinite set S is possible. Then by the definition of denumerable infinites, S can be put into a one-to-one correspondence with one of its proper subsets. For example, the set of natural numbers {1, 2, 3, ...} can be put into a one-to-one correspondence with its proper subset of the even numbers {2, 4, 6, ...}. Then by (2), S's proper subset of the evens is just as large as S itself. But by (1), S is larger than its proper subset of the evens. So S both is and isn't just as large as one of its proper subsets, which is a contradiction.

[2] The following sort of criticism can be found in the following articles: 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; 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.)