Hilbert's Hotel Is Just Fine
8:07 AM

I'm taking a philosophy of religion class this semester where we explore the question of whether such a thing as god exists. Having been an agnostic atheist for most of my life, I am not religious, but also do not claim to deny the existence of god. I am, however, deeply interested in learning about the arguments for and against god's existence, and am ready to concede his existence if a sound argument is presented to me to prove it. The arguments and reasons I have encountered so far, however, are unconvincing and often fallacious. To my great surprise, it seems that the fallacious nature of many such arguments is often not addressed clearly and in detail, or if it is, that such responses have not been popularized enough. I have decided, therefore, to publish my comments on theistic arguments which I believe to have flaws that I feel deserve being written about and being refuted. I expect to write more of my thoughts on philosophy of religion as I encounter more arguments in my class, so I suppose my posts could be read as my journey to find out whether god's existence is something that can be rationally believed in.

    The first argument that I encountered and that immediately struck me as fallacious is a part of the Kalam Cosmological argument, nowadays most strongly represented by William Lane Craig. I will not go into details of it here, but a critical part of the argument rests on the impossibility of infinite things existing in reality. That particular section of the argument goes as follows:
  1. If we look back in time, we can see that there's a regress of events, i.e. one event precedes another and so on.
  2. This cannot go on infinitely because an infinite number of things cannot exist in reality.
  3. Therefore, this regress of events must end somewhere.
  4. This end represents the beginning of the universe.
  5. If the universe began to exist, there must be a reason why it began existing.
  6. That reason is God. 
    In this post, I want to address point 2. To my mind, it seems perfectly plausible that the regress of events be infinite, and that it never end. After all, you can imagine an infinite regress of events in your mind: one event preceding another, and so on ad infinitum. It would seem bizarre to me, if the regress suddenly stopped and suddenly a god appeared. Craig, however, says that it must necessarily end because infinity, though it can exist in mathematics, cannot exist in reality. This is proved, he believes, by a seeming logical paradox called the Hilbert's Hotel.

Why Hilbert's Hotel is not Paradoxical
Hilbert's Hotel. It’s always booked solid, yet there’s always a vacancy. For the Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).
Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests. Later that night, an endless convoy of buses rumbles up to reception. There are infinitely many buses, and worse still, each one is loaded with an infinity of crabby people demanding that the hotel live up to its motto, “There’s always room at the Hilbert Hotel.”

    (See a sixty-second explanation of the problem here and the way Craig present it here.) According to Craig (and many other thinkers throughout the history), the absurdities in the example of Hilbert's Hotel demonstrate that infinity cannot exist in reality. However, there is nothing absurd or logically impossible about the hotel, and it does not by any means constitue proof that an infinite number of things could not exist in reality. Let's take into consideration one by one the two key points which seem to puzzle Craig:

"As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. Before he [the new guest] arrived, all the rooms were full!" (198)
    Now Craig is bothered by the seeming paradox above because he does not understand that, normally, when a hotel is said to be "full", it means two things:

  1. All the rooms have guests.
  2. No more guests can be accommodated (because no more new rooms can be added to the hotel)
    These two conclusion apply only to hotels that have a finite number of rooms, because if they were full, and no more rooms could be added to the hotel, obviously no more people could check in. However, in an infinite hotel, you can add as many rooms as you like, because per definition it is an infinite hotel, and adding more rooms does not change the fact that it still the same hotel because it still would have an infinite number of rooms. Suppose for example, that instead of taking all the pains to shift an infinite number of guests whenever a new guest arrived, we just built a new room in front of room #1. Then we could shift down the numberings of all the rooms by one, i.e. calling the new room #1 and the old #1 room #2, and so on. The hotel has one more room, and one more guest, but it still has an infinite number of rooms, so it still is the same Hilbert's hotel. In this way, we can see that there's nothing paradoxical about accommodating additional new guests in an infinite hotel. (An alternative scenario would be that no new rooms are added, but that the room shifting would go on to infinity and forever. As there's no reason why the shifting should ever end, there's nothing paradoxical in this scenario either.)

    Here's the second main seeming paradox Craig bases his arguments on.

“Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be?.... How can there not be one more person in the hotel than before?.... And, again, strangely enough, [after infinity of additional guests is accommodated] the number of guests in the hotel is the same after the infinity of new guests checks in as before, even though there were as many new guests as old guests…. And yet there would never be one single person more in the hotel than before.” (198)

    All the muddle that Craig has gotten himself into seems to stem from the fact that he does not understand what 'infinity' means. He thinks that it would be paradoxical if a number was added to infinity but the sum would nevertheless equal infinity. I believe that this is mainly because Craig seems to think that infinity is some specific number, although he doesn't know what it is. But infinity is not any specific number. When we say that something is 'infinite' we simply mean to say that it does not have a boundary or limit, that if we tried to measure it, we would never finish, and that if we tried to count an infinite number of things, we would also never finish. We do not mean to say that that something is extremely large, but that we don't know how large, or that an infinite number of rooms is very many, but we don't know how many. Therefore, when we say that Hilbert's Hotel has an infinite number of rooms, we mean to say that there are countless number of rooms, that if we tried to count the number of rooms, we would never finish. Because even if new rooms are added to the hotel, the hotel, although it would have additional rooms, would still have countless number of rooms, we understand that the hotel still has an infinite number of rooms. So because adding a room to a countless number of rooms does not change the fact that the number of rooms is countless, we say that the hotel still has an infinite number of rooms or guests within it. It does not mean that an additional guests or rooms were not checked in or that "there would never be one single person more in the hotel than before” as more guests were added.

   Therefore, as we can see from above, Hilbert's Hotel is not a paradox, and fails to prove that infinity cannot exist in reality.

Citations from:
Peterson, Michael, William Hasker, Bruce Reichenbach, and David Basinger. Philosophy of Religion: Selected Readings. New York: Oxford UP, 2010. Print.
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