### 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.)

### Epicurean Cosmological Arguments for Matter's Necessity

One can find, through the writings of Lucretius, a powerful yet simple Epicurean argument for matter's (factual or metaphysical) necessity. In simplest terms, the argument is that since matter exists, and since nothing can come from nothing, matter is eternal and uncreated, and is therefore at least a factually necessary being.
A stronger version of Epicurus' core argument can be developed by adding an appeal to something in the neighborhood of origin essentialism. The basic line of reasoning here is that being uncreated is an essential property of matter, and thus that the matter at the actual world is essentially uncreated.
Yet stronger versions of the argument could go on from there by appealing to the principle of sufficient reason to argue that whatever plays the role of being eternal and essentially uncreated does not vary from world to world, and thus that matter is a metaphysically necessary being.
It seems to me that this broadly Epicurean line of reasoning is a co…

### Notes on Mackie's "Evil and Omnipotence"

0. Introduction
0.1 Mackie argues that the problem of evil proves that either no god exists, or at least that the god of Orthodox Judaism, Christianity, and Islam, does not exist. His argument is roughly the same version of the problem of evil that we’ve been considering.
0.2 Mackie thinks that one can avoid the conclusion that God does not exist only if one admits that either God is not omnipotent (i.e., not all-powerful), or that God is not perfectly good. 0.3 However, he thinks that hardly anyone will be willing to take this route. For doing so leaves one with a conception of a god that isn’t worthy of worship, and therefore not religiously significant.
0.4 After his brief discussion of his version of the problem of evil, he considers most of the main responses to the problem of evil, and concludes that none of them work.

1. First Response and Mackie's Reply
1.1 Response: Good can’t exist without evil; evil is a necessary counterpart to good.
1.2.1 this see…

### Notes on Swinburne, "On Why God Allows Evil"

Notes on Swinburne’s “Why God Allows Evil”

1. The kinds of goods a theistic god would provide: deeper goods than just “thrills of pleasure and times of contentment” (p. 90). For example:
1.1 Significant freedom and responsibility
1.1.1 for ourselves
1.1.2 for others
1.1.3 for the world in which they live
1.2 Valuable lives
1.2.1 being of significant use to ourselves
1.2.2 being of significant use to each other

2. Kinds of evil
2.1 Moral evil: all the evil caused or permitted by human beings, whether intentionally or through negligence (e.g., murder, theft, etc.)
2.2 Natural evil: all the rest: evil not caused or permitted by human beings (e.g., suffering caused by hurricanes, forest fires, diseases, animal suffering, etc.)

3. The gist of Swinburne’s answer to the problem of evil: God cannot – logically cannot -- give us the goods of significant freedom, responsibility and usefulness without thereby allowing for the possibility of lots of moral and natural evil. This is why he has al…