### 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.

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[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.