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

Third Version: Making Room in a Fully-Occupied Region Without Removing Occupants; Filling Up an Unoccupied Region Without Adding Occupants

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

Hilbert's Hotel Illustration:
Imagine a hotel with an actually infinite number of rooms. If such a hotel were possible, then it would have some remarkable features. For example, suppose every one of its rooms were full. Then one could create vacancies in some of the rooms even if none of the guests left. So, for example, the hotel manager could have the guests in room 1 move to room 2, the guests in room 2 move to room 3, and so on for all the rest of the rooms. In this way, room 1 would could become vacant for a new guest, even though not a single guest left the hotel.

In fact, one could create infinitely many vacant rooms without removing a single guest. For one could move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and in general, for each room n, move the guest in n to 2n. The result would be that an infinite number of odd-numbered rooms would become vacant for more guests to occupy.

Furthermore, one could also remove any vacancies -- i.e., fill all the rooms - without adding a single new guest (again, assuming one has an infinite number of guests to move around). So, for example, one could just reverse the movings of guests stated in the previous examples, and thereby go from an infinite number of vacancies to a fully-occupied hotel.

These features of an infinite hotel illustrate that no such hotel could exist -- not even an omnipotent God could create one. For it is metaphysically impossible for a spatial region to go from full to not-full merely by moving its occupants around; without removing at least some of what occupies a region, that region will remain occupied by something or other. Similarly, it is metaphysically impossible for a spatial region to go from not-full to full merely by moving its occupants around; without adding at least some new occupants to a given region, that region cannot become fully occupied.

We can express the argument in the illustration above as follows:

1. If concrete actual infinites are possible, then a hotel with an infinite number of rooms and guests is possible.
2. If a hotel with an infinite number of rooms and guests is possible, then a spatial region can go from full to not-full without removing any of its occupants, and it can go from not-full to full without adding occupants (i.e., merely moving occupants around is sufficient).
3. No spatial region can go from full to not-full without removing occupants, and no spatial region can go from not-full to full without adding occupants (i.e., merely moving occupants around isn't sufficient).
4. Therefore, concrete actual infinites are impossible.

What to make of this argument? One might have doubts about premise (3): Granted, no fully-occupied finite region of space, containing a finite number of occupants, could go from full to not-full, or from not-full to full, by just moving some of its occupants around. But are these things obviously true for infinitely many elements within infinite regions of space?[1]

In any case, I'd like to set aside concerns about (3) to focus on a concern about the justification for (1). For the supposed absurdities of Hilbert's Hotel don't obviously generalize to all actual infinites.[2] More to the point, the supposedly problematic features of Hilbert's Hotel aren't a part of sets of past events – whether infinite or not. For the supposed problems only arise when one combines infinites with co-present regions, and only when the occupants of such regions can be rearranged. But of course, temporal regions are not co-present (at least not according to Craig’s preferred A-theory of time). Nor can the events that occupy the temporal regions be rearranged or moved around. Therefore, even if the combination of features in Hilbert’s Hotel and Craig’s infinite library give rise to absurdities, such a combination of features is not present in an infinite past. But if not, then Craig's justification for premise (1) is undercut.

[1] Here are some considerations that make me doubt premise (3). What is the source of the intuition that seems to support it? Perhaps a simpler case than Hilbert's Hotel will help us get clear on this. Thus, imagine three people standing in a line, each holding an iPhone. So everyone's "phone hand" is "fully occupied". Now imagine that they want to free up some space -- free up a "phone hand" -- for a new iPhone by having each person hand their phone to the person to the right. Now let us stipulate that if a person in the line has no one to the right to hand their phone to, then they must hold onto it. Given these conditions, they won't be able to free up a "phone hand" and make space for a new phone. Why not? Because the person on the right end has no one to the right to which they could hand their phone. And since that person can't get rid of their phone and free up their phone hand, no one else can, either.

So far, we have failed to free up a phone hand for a new phone just by moving phones around. And the reason seems to be that at least one person can't move their phone down the line, since they have no one to hand it to. But now suppose we change the scenario a bit. Thus, imagine that the line of people holding phones extends (literally) without end to your right. Finally, suppose their plan to free up space for a new phone, and their rules for permissible passing of phones, are the same as before. The result this time is that the first person in the line is now without a phone! What changed to make this possible? This: Despite the fact that, as with the first scenario, everyone started out holding phones -- all the "space" was occupied -- in the new scenario, every person in the line has someone to hand their phone to (since the line of people has no end). And because of this, a hand was freed up to receive a new phone. It appears, then, that the factors that generated our intuition against freeing up space to add a new occupant just by moving things around aren't present in the second case. But that case is relevantly similar to Hilbert's Hotel. Therefore, it's no longer clear that it's impossible to make room for new occupants in a fully occupied region by moving things around when infinite regions are involved.

Here's another way to come at the point. Perhaps the concern stems from a sense that filling a region of space without adding new occupants sounds like "magic", in the sense that we're getting something from nothing. Relatedly, one might have the sense that making room in a fully-occupied region of space without taking something out sounds like "magic", in the sense that it would require an occupant to pop out of existence. However, remember that in the original example of Hilbert's Hotel, we were dealing with a limitless, inexhaustible supply of rooms and a limitless, inexhaustible supply of guests. If so, then it's not clear why we'd need more guests to fill any unoccupied rooms, and it's not clear why we would need less guests to make a new vacancy.

[2] Wes Morriston has made this sort of criticism in (e.g.), "Must Metaphysical Time Have a Beginning?", Faith and Philosophy 20:3 (July 2003), pp. 288-306, esp. 295-298.


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