There are several points in the education of a mathematician where the student has to face the fact that mathematical reality is not always what one would expect, that the mathematical universe is, in fact, quite strange.

One of these points comes when one encounters Cantor and his famous “diagonal slash” proof. I’m not going to give the proof here (you can look it up if you wish, it is not a difficult proof to follow), but I do want to look at what that proof shows.

Cantor was interested in infinity. He knew, for instance, that the positive integers (counting numbers: 1,2,3…) are infinite in number. They do have a lower bound, 1, but then they go marching off over the horizon, never stopping, infinite. That’s how we tend to think of infinity, as something unbounded, going on forever. But mathematically that is not correct. Infinity is not tied to being unbounded. There are, for instance, an infinite number of real numbers (that is, numbers that can be expressed in decimal form, albeit of sometimes infinite length) between 0 and 1. Here we have a case of an infinite number of things in a finite, bounded, even small, space.

Fair enough, but still not too odd. Not, at least, until Cantor came along. He asked what, to most of us, is a stupid question: Are these two infinities the same size? Since both are infinite, we are tempted to say that of course they are the same size. Unfortunately, Cantor, being a great mathematician, wanted proof of that, but in trying to prove it he managed to prove instead that it is not true, that one of these infinities is, in fact, bigger than the other!

It gets worse. Given that one is bigger, most of us would probably choose the positive integers as the bigger set. After all, they go off out of sight while those reals are confined to that one small area. But Cantor’s proof is precisely that there are more reals between 0 and 1 than there are positive integers.

OK, so mathematics is weird—most of you probably thought so already—but so what?

Well, the “what” is that I’ve been thinking a lot about the incarnation, the amazing fact that the infinite God became a finite man. It is one of the great stumbling blocks to faith: how could the infinite God “squeeze himself” into a mere man?

The problem is precisely that we don’t really understand infinity. We say “God is infinite” and immediately we get a spatial conception. We imagine that God can only be infinite if he also goes off over the horizon like the positive integers. But why? What if God is a bigger sort of infinity, more like the reals between zero and one?

Our faulty conception of God’s infinitude has some serious consequences. We know that the universe is God’s creation and that it is vaster than we can imagine (though it is not, actually, infinite). We know that God is everywhere in the universe, upholding it all. But do we not also tend to imagine that God is therefore sort of spread out throughout the universe—perhaps even spread a bit thin? But that is pantheism. God is not spatial at all and his infinitude is such that wherever he is in the universe, there is all of him. This is the real meaning of immanence: not that there is a bit of God hidden in everything, but that God is there, all of him. Everywhere.

We are better at grasping this when it comes to time rather than space. We acknowledge that God is not bound to time as we are, that a thousand years are as a day to him and a day as a thousand years. We need to think of his relationship to space in the same way: A thousand light-years are as an inch and an inch as a thousand light-years.

This doesn’t, of course, make the incarnation any more comprehensible or prosaic, but it should let us dispense with that silly quibble. The universe is full of infinities residing in finitudes, so why shouldn’t the fullness of the godhead dwell in the man Jesus? God has a nature, so does man, just as the integers and the reals are all numbers. But God’s nature is different from man’s nature, just as the reals are different from the integers. And God’s infinite nature can fit in a man’s finite nature, just as the infinite set of reals between 0 and 1 fit into the short piece of the number line defined by those two integers.

We also need to remember that since wherever God is, all of God is, we don’t have just a little bit of God’s attention; We have it all. Each one of us. Just because God is giving you his whole attention doesn’t mean he is ignoring me, for I, too, have his complete attention!

Not only that, but you don’t just have a bit of God inside you either. In the Spirit, as in Jesus, is all of God. So in each one of us who have the Spirit is all of God. It is true that this union is subtly different from the union of man and God in Jesus, but it is still quite wonderful. It is also true that we don’t express this union as we should, but as we recognize it and accept it we will be able to yield to it more and thus express it better.

Tags: Cantor's diagonal argument, God, immanence, indwelling, Infinity, Jesus, Mathematics

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