One upside to my recent convalescence has been that I have had plenty of time for reading. Currently I’m working my way through Graham Oppy’s book Philosophical Perspectives on Infinity, published by Cambridge University Press in 2006. Oppy is best known as a philosopher of religion, writing from a generally atheistic perspective. His book Arguing About Gods is really excellent, thought it definitely does not make for light reading.
As for the present book, I’m only through the first two chapters so far. I think I’m going to like the rest, though, since the preface contains remarks like this:
Part of my interest in philosophy of religion stems from the conviction that it must be possible to convince reasonable religious believers that traditional monotheistic arguments for the existence of God are worthless. Hence, not surprisingly, one of the subsidiary goals for the projected larger work is to make some contribution to the case for supposing that reasonable religious believers ought to recognise that the arguments for the existence of God provide no reason at all for reasonable nonbelievers to change their minds.
My kind of guy!
I’ve written about infinity a couple of times before (here and here.) But there is plenty more to say. So how about we take a quick look at one of the more intriguing thought experiments in mathematics. I refer, of course, to Hilbert’s Hotel.
Let’s begin with an imaginary hotel with one hundred rooms. Imagine further that all of them are occupied. If a new guest shows up in the lobby looking for a room, would we be able to accommodate him? Clearly not. At least. not without kicking out one of the guests already at the hotel.
In the context of a finite hotel the question is silly. If all the rooms are occupied and a new guest shows up, then that guest can only be accommodated at the cost of unaccommodating someone else. Seems obvious enough.
The cool part is that this is not true for an infinite hotel.
Let’s imagine that we have a hotel with infinitely many rooms. The rooms are numbered sequentially 1, 2, 3, 4, …, and for any natural number n there is a room with that number. Further assume that every one of those rooms is occupied.
If a new guest shows up wanting a room we can accommodate him by rearranging the guests who are already there. We shall simply tell the guest in room n to move to room n+1. The guest currently in room 1 will move to room 2. The guest currently in room 2 will move to room 3. The guest currently in room 3 will move to room 4.
What is the effect of this reshuffling? Well, everyone who previously had a room still has a room. No one has been thrown out of the hotel. The difference is that now there is no guest in room 1, and the new guy can be placed there.
Not a bad trick, and one that is easy to extend to any finite number of guests. If ten new guests show up, for example, then the guest in room 1 moves to room 11, the guest in room 2 moves to room 12, and more generally the guest in room n moves to room n+10. In this way everyone still has a room, but the first ten rooms have been freed up for the newbies.
We should pause for a moment to consider some practical difficulties. Obviously physics has something to say about the feasibility of such a hotel. If we’re thinking of actual, physical people, in an actual, physical hotel, then we might have a problem carrying out our reshuffling in a finite amount of time. How do we transmit the message, “Move over by one room! to everyone at once? The speed of light puts a limit on how quickly a signal can be transmitted, after all.
But then, that’s why we call it a thought experiment.
OK, I know what you’re thinking. But what happens if an infinite number of new guests turns up, one for each natural number. Incredibly, we can still accommodate them! The trick is easily done: we simply move the guest in room n to room 2n. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6 and so on. Once again, no one is thrown out of the hotel by this maneuver. Now, though, all the odd numbered rooms are now vacant, and the newbies have a place to say.
If the next night another infinite bus load shows up, we can accommodate them in exactly the same way. How about if infinitely many bus loads with infinitely many people show up all at once? No problem! We begin b vacating the odd numbered rooms as we did before. Then the first infinite bus load gets placed in the rooms whose numbers are powers of three. That is, the first new guest is placed in room 3, the second is placed in room 9, the third is placed in room 27 and so on. The people in the second bus are placed in rooms whose numbers are powers of five. In general, the guests on the n-th bus are placed in the room labeled with powers of the n odd prime. Since there are infinitely many odd primes, and since the fundamental theorem of arithmetic guarantees that no two odd prime powers can be equal to each other, we have now successfully accommodated all of the newbies.
Perhaps this seems paradoxical. We keep increasing the number of guests without making any change to the number of rooms. How can that be?
This requires some discussion, but the basic problem is in treating infinity as though it were a number. The phrase “increasing the number of guests,” which is crystal clear in the context of a finite number of guests, is not so easy to parse in the context of an infinite set.
This leads to subtle questions about how we compare the sizes of infinite sets, and to the distinction between cardinal numbers and ordinal numbers. Since this post is lone enough, we shall save that for another day. For now, let’s just agree that infinity can be both weird and fascinating!