Skip to main content

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

Fourth Version: Subtracting the Same Amount Yields Contradictory Results

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

Hilbert's Hotel Illustration:
Imagine a hotel with an infinite number of rooms and an infinite number of guests. Such a hotel would have some surprising features. So, for example, suppose all the guests in the even-numbered rooms check out. Then the hotel would still have infinitely many occupied rooms, viz., the odd-numbered rooms. On the other hand, suppose all the guests in rooms 4, 5, 6, ... check out of the hotel. Then the hotel will no longer have infinitely many occupied rooms -- it will have three. But how can this be? Just as many guests checked out in the first scenario as in the second scenario. This is because (i) the infinite set of even-numbered rooms (2, 4, 6, …) can be put into a one-to-one correspondence with the set of rooms numbered 4 and higher (4, 5, 6, …), and (ii) any two sets that can be put into a one-to-one correspondence with one another are the same “size” – each set has just as many members as the other. But this is absurd; subtracting the same amount cannot yield different amounts. Therefore, an infinite hotel is metaphysically impossible. But the absurdity here isn't particular to just infinite hotels. Rather, they generalize to every denumerably infinite set of concrete entities whatsover. Therefore, concrete actual infinites are metaphysically impossible.

We can express the argument illustrated above as follows:

1. If concrete actual infinites are possible, then a hotel with infinitely many rooms and infinitely many guests is possible.
2. If a hotel with infinitely many rooms and infinitely many guests is possible, then it’s possible to remove just as many guests from the hotel and get different amounts of remaining guests.
3. It’s impossible to remove just as many guests from the hotel and get different amounts of remaining guests.
4. Therefore, concrete actual infinites are impossible.

What to make of this argument?[1] By now the problem with Craig's argument is a familiar one: there is an equivocation on the locution "just as many" in (2) and (3). On the one hand, it might mean that "just as many" guests were removed in the two scenarios in the sense that the two sets of rooms have the same cardinal number. Then while (2) is true, it’s not obvious that (3) is true. For while the two sets of rooms have the same cardinal number, one set has all the members of the other set and then some, viz., the odd-numbered rooms. Referring to the coarse-grained similarity of the two sets (viz., sameness of cardinal number) conceals this crucial difference. But once one shifts one's focus from the sameness of cardinality to the finer-grained difference in the particular members of the two sets, the intuitiveness of premise (3) disappears. For then one would expect different results when the sets removed in the two scenarios are different in this crucial way.

On the other hand, one might mean that "just as many" guests were removed in the two scenarios in the sense that neither set of guests 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 guests removed in the second scenario has all the guests that were removed in the first scenario and others besides.

Yet again, 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] For more careful and rigorous versions of the following sort of criticism of the argument at hand, see the following articles: Morriston, Wes. "Craig on the Actual Infinite", Religious Studies 38 (2002), pp. 147-166, esp. pp. 151-152; Morriston, "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 300-301. See also Dever, Josh. "Worlds Apart: On the Possibility of an Actual Infinity", esp. pp. 8-10.

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…