Skip to main content

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

Second Version: Adding Without Making Larger

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

Library Illustration: Imagine a library containing an actually infinite number of books. In such a library one could add one, two, three, or indeed denumerably infinitely many more books to the shelves without making the library any larger in terms of how many volumes it contains. This is because (i) the original infinite set of books can be put into a one-to-one correspondence with any set of books that could result from such additions, and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each is just as large as the other. But this is absurd; the original set of books is a proper subset, or proper part, of each set of books that would result from such additions, 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 is possible.
2. If a library with an actually infinite number of books is possible, then adding any finite or denumerably infinite amount of books to the library would result in it having just as many books as before.
3. Adding any finite or denumerably infinite amount of books to the library couldn't result in it having just as many books as before.
4. Therefore, concrete actual infinites are impossible.

What to make of this argument? As with the argument in the previous post, the notion of there being “just as many” books is ambiguous in (2) and (3).[1] On the one hand, one might mean that there are "just as many" books before the additions as after 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 as noted in the previous post, Craig thinks he can provide independent support for (3) via other thought experiments besides the one under consideration. Again, we'll consider those in later posts in this series. For now, though, we’ll just note again 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" books before the additions as after 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 resultant collection of books after each addition is one that has other books besides the original set, viz., the ones added.

As with the argument discussed in the previous post, 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.
-------------------------------
[1] As noted in the previous post, 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.)

Comments

Popular posts from this blog

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 Mackie’s reply:
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…