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To Infinity and Beyond: Contemplating the Infinite

By Pastor Doug Van Dorn

Nov 2005

(Note: If it reads "Pastor Doug Van Dorn" then this was an article I wrote for our church newsletter. I'll be posting some of the more timeless and, to me, interesting of them on the Og Blog. Might as well start with an article on time and infinity.)

I have recently been reading a book called “The Infinite Book.” The book’s purpose, of course, is to help us think about infinity. One of the chapters is called “The Hotel Infinity.” Let me tell you about this otherworldly hotel.

In a conventional hotel there are a finite number of single rooms. If they are all taken, then there is no way you can be accommodated at the hotel without evicting one of the existing guests from their room. When it’s full it’s full.

At the Hotel Infinity things are different. Suppose that you turn up at the check-in counter of the Hotel Infinity only to find that the infinite number of rooms (numbered 1, 2, 3, 4, … and so on, forever) are all occupied. The receptionist is perplexed – the Hotel is full – but the manager is unperturbed. No problem, he says: move the guest in room 1 to room 2, the guest in room 3 to room 3, and so on, forever. This leaves room 1 vacant for you to take and everyone still has a room!

You are so pleased with this service that you return to the Hotel Infinity on the next occasion that you are in town, this time with an infinite number of friends for the ultimate reunion. Again, this popular hotel is full, but again, the manager is unperturbed. We can easily accommodate an unexpected party of infinity, he explains to the nervous receptionist. And so he does, by moving 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 so on, forever. This leaves all the odd-numbered rooms empty. There are in infinite number of them and they are free to accommodate you and your infinitely numerous companions without leaving anyone out in the cold. Needless to say room service is a little slow at times to some of the high-numbered rooms.

The day after the infinite contingent of unexpected guests have been accommodated, the disgruntled guests in the even-numbered rooms all decide to leave. They are fed up with being constantly moved around by the crazy manager and spending all their time queuing for everything.

The manager is very upset that half of the hotel’s rooms (all the even numbers) are now empty. He has to supply statistics on the occupancy of the Hotel and 50 per cent occupancy is a failure. Unless things look up he is facing closure. As a demanding traveler you have begun to get the idea how things work at this hotel now. You don’t want to see such a flexible establishment close down, so on hearing of the manager’s problem you suggest that the remaining guests just be moved closer together to get rid of the unoccupied rooms. You propose that they leave the guest in room 1 alone, move the guest in room 3 to room 2, 5 into 3, 7 into 4 etc. In the end all the rooms are filled again even though no new guests have arrived. The manager is delighted.[1]

These are the sorts of games one can play with infinity. An Infinite “number” is not just quantitatively different from a finite number. Infinity is not the biggest number conceivable, because if you conceive of that number, just add one to it. Rather, infinity is qualitatively different. When dealing with the infinite, you have moved into a different realm altogether.

Now, when you have a group of numbers (1, 2, 3, 4, … and so on, forever), you have what is called an infinite “set”[2] of numbers. In the hotel infinity, we were really only playing around with this one infinite set. This might surprise some, who think that we were playing with more than one set. On the other hand, if you understand this, you might suppose that there is only one possible infinite set. However, one would suppose wrongly.

At first you might think that making bigger infinites is child’s play. Suppose you have an infinite collection of numbers 1, 2, 3, … Just add one more thing to it – say the object *. Isn’t that bigger? Unfortunately not… Adding one, or two, or even all the whole numbers to a countable infinity still leaves a countable infinity. [This would make it the same infinite size]. In order to jump up a level to a new order of infinity something different is required.

If you have any infinite set, then you can generate one that is infinitely bigger by considering the set that contains all its subsets. This is called a power set. As a finite example consider the set of three objects {A,B,C}. (These could be people and the ‘sets’ groups of friends, families, or secret societies.) It contains subsets containing the following members:

{0}, {A},{B},{C},{A,B},{A,C},{B,C},{A,B,C},

There are 8 = 2x2x2 = 2³ subsets. In general, if the original set has A members then there are 2ª = 2x2x2x2x... (A times) possible subsets and members of its power set.

Thus from an infinite set like א0 we can create an infinitely larger set (by which we mean one that cannot be put in one-to-one correspondence with it) by forming its power set P[א0]. Now we can do the same again by forming the power set of P[א0]. That will be infinitely bigger than P[א0]. And so on, without end.[3]

Admittedly, talking about power sets is complicated and extremely difficult to comprehend. In reality actually trying to contemplate the hotel infinity is extremely difficult! Perhaps power sets are infinitely more difficult to understand?

If you have managed to stick with this until now, here is what I believe is a theologically profound point. Scripture says of God “I am the first and I am the last” (cf. Isaiah 48:12). In reality, this does not mean that God has a beginning or an ending, but that he is before all things and the purpose of all things. Jesus calls himself both the “beginning and the end” (Rev 22:13).

Theologians really have no better word for this than do scientists or mathematicians. The point is God is infinite. After what we have just said, you might think (as many Christian mathematicians prior to 1900 thought) that if you can have an infinite number of infinites, that this somehow diminishes God by reducing him to just one among an infinite of others. But this is not true when we consider two things. First, God created all things. Therefore, even all mathematically potential infinites derive their source in him (so do any real infinities that may occur in the universe). They are not greater than He. Second, these things we have been dwelling on are dim reflections of God himself, who is beyond even absolute mathematical infinity.

Here is, what I believe to be, the crux and excitement of this matter. If infinity is not just quantitatively different but qualitatively different from the finite, we might suppose that it is impossible to know anything about it. In the same way, if God is infinitely greater than us finite creatures, we might suppose that it is impossible to know anything true about him. This would be very bad, of course.

Yet, we can see from this discussion that we do know something about infinity, even though we cannot fully comprehend it. We know it is logically possible. We can mathematically prove its existence. We can see and work with it up to a point. And this is tremendously heartening. For it means that God has created in man the ability to understand something quantitatively different from himself, even if he cannot fully comprehend it. This in turn would mean that true communication with God really is possible.

Now, I’m not using math to prove that we can know about God. It is the word that tells us we can know God truly. What we see in the world of math or science is only what we should expect to find knowing what God has said is true in his word. And this makes the creation ordinance that God gave to us to go into the world and be free to study all kinds of things a very enriching and exciting task. We don’t have to be “in ministry” to do God’s work!

But here is a caveat; a great lesson from infinity. When we think about God, we are not to think that what we know about him is true univocally (that is identically). For he is not like us. This must humble is. Rather, all we know about God – ALL WE KNOW – is true analogically (that is, by way of analogous human experience). In this sense, everything that Scripture tells us about God is an anthropomorphism, a way of communicating his attributes by way of our human experience.

Nevertheless, all Scripture tells us about God is both correspondingly true (just like when Scripture talks about God covering us with his wings we know intuitively both that God doesn’t have literal wings and that this means that God protects and cares for us) and knowable, even though he is infinite. How exciting is this! Because of this, we may be sure that when the Infinite One talks to us, that his finite creatures can hear. Praise God that he allows us to know more about him in everything he has made, both in the Good Book, and in the book of nature that testifies to his Greatness.

[1] John Barrow, The Infinite Book, p. 44-45. [2] A set is a Many that allows itself to be thought of as One. [3] Ibid., p. 74-75.

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