Wednesday, January 11, 2012

God and Infinity

We who believe in God often call him infinite. Even those who do not believe often accept the infiniteness of God as part of the necessary definition of God. I’d like to examine this usage in more detail, and suggest that using “infinite” in referring to God needs to be done carefully.

The basic problem is how one defines “infinite”. In common usage it means “immeasurably great” or “unbounded”. There is nothing wrong with these ideas in and of themselves. The supreme being should be “immeasurably great” and “unbounded”. A bounded being certainly is less than whatever binds it and so can no longer be considered supreme. As a believer in the Bible, I also find that God is there often described as having immeasurably great power and unbounded mercy and love.

The problem is not with these concepts, but trying to extend beyond these to draw logical conclusions. As a mathematician I’ve learned that ideas about infinity do not follow our normal intuition.

Mathematical Infinity

For many centuries mathematicians were aware of infinity, but never really defined it. Some considered it so vague that it should not be part of proper mathematics. Even during my time at UCLA, long after infinity had been precisely defined, the professor who taught me the most about infinity and its definition personally disliked the idea of infinity and told me he thought we should avoid it. When mathematicians, who excel at precise definitions avoid something, we should take notice.

Sets and Size

Mathematicians have defined infinity by using the notion of a set. A set is a group of things. If a set contains the numbers 1, 2, 3, and 4 it would be shown like this: {1, 2, 3, 4}. A set can contain anything. For example, I could also have the set {cat, apple, computer, moon}.

To define infinity, we first address the question of how big a set is. More specifically, how do we tell if two sets are the same size? The obvious answer is to count. The set {1, 2, 3, 4} has four elements, as does the set {cat, apple, computer, moon}.

However, there is another way to compare sizes. We can tell if sets are the same size by matching the members. For example, we could make this match: 1 – cat, 2 – apple, 3 – computer, 4 – moon. The match shows the sets are the same size. Mathematicians call this a mapping, specifically a one-to-one mapping. Now say we have another set, {dog, fish, pen}. We could try the same sort of mapping: 1 – dog, 2 – fish, 3 – pen, 4 - ?. However there is a problem. The 4 has nothing left to match. This means the sets are different sizes. There is no one-to-one mapping between them.

Determining a set size by matching is useful when counting is hard. For example, if I have a few hundred pennies and want to count them, I might put them in stacks of ten. Rather than count each stack, I make one stack of ten, then just compare each new stack with the first stack to make it the right size. Doing all those comparisons is a lot faster and less prone to error than counting every penny. To get the final count I just count the stacks and multiply by ten.

This brings us to infinite sets and Georg Cantor. Infinite sets are not only hard to count, they are impossible to count. Cantor asked whether all infinite sets are the same size. This seems like a ridiculous question. Of course they’re all the same size – they’re infinite! But what about when you have an infinite set that contains another infinite set. Is the contained set smaller or the same size?

Take the set of all the counting number, the positive integers: {1, 2, 3, …}. (The “…” means they just keep going.) Also look at the set of even numbers: {2, 4, 6, …}. Which is bigger? One obvious answer is the counting number are bigger because even numbers are contained in (are a subset of) the counting numbers. However, another obvious answer is that they are the same size, because both are infinite. Which is correct?

Consider a mapping between the sets where the counting numbers are multiplied by two. So counting number “1” maps to even number “2”, counting number “2” maps to even number “4”, and so on. Every counting number can be mapped to an even number. In other words, given any counting number, just multiply by two to get its corresponding even number. If you have 324, its corresponding even number is 648. Likewise, given any even number, you can find its associated counting number by dividing by two. If you have the even number 486, its corresponding counting number is 243. The point is every counting number has a matching even number and every even number has a matching counting number. Thus the sets have a one-to-one mapping and they are the same size.

Infinity Defined

At this point many might be thinking, “It’s a trick. The even numbers must be smaller. They are a subset of the counting numbers.” All I can say to that is that it’s not a trick, but follows directly from the idea of a one-to-one mapping. In fact, it leads to the following definition of an infinite set.

An infinite set is a set which has subsets that are not equal to it but are the same size.

That definition probably does not make anyone happy at first glance. It seems again like a trick. Shouldn’t the definition be more like “An infinite set is one that goes on forever”? Trouble is, words like “forever” contain the idea of infinity, so such a definition is circular. The mathematical definition recognizes an infinite set as the only kind of set which can have elements removed and still remain the same size. As such it supports the notion that infinity remains infinite even tweaked a little. Removing some stuff does not reduce its size. Conversely, adding some stuff does not increase its size.

Different Sized Infinities

Despite what has been shown so far, despite the definition of an infinite set that allows some stuff to be added or subtracted from a set and that set remains the same size, there are different sizes of infinity. I feel I’ve already provided enough confusion. In exchange for not confusing you further, I ask that you accept the possibility that infinite sets can be different sizes. In fact, Cantor showed that not only are there there infinite sets of different sizes, he showed that there are an infinite number of different sizes of infinite sets! For those that wish a better discussion I recommend reading A Brief Introduction to Infinity.

Are These Infinities Real?

Do mathematical infinities correspond to God or any other reality? Are there physical (or spiritual) infinities of different sizes? I don’t know. I do know that the mathematical definition of infinity is the only rigorous one I know. It also seems to be about as simple as possible. Despite its relative simplicity, it leads to bizarre consequences like subsets that are the same size as their contained set and infinities of different sizes. Can we expect real-life infinities, if they exist, to be any less bizarre?

Infinity in Theology

Now that we’ve looked at a rigorous definition of infinity, let’s examine some theological concepts in light of the mathematical definition of infinity.

First, consider a common argument against the idea of God existing. “If God exists and is all-powerful, can he make a rock that he can’t lift?” If he can’t make the rock, the argument says, then he is limited and not all powerful. If, on the other hand, God makes such a rock, then the existence of the rock is a limitation of God’s power. Again we have a God who is limited. Quite some time ago I came to the conclusion that the problem with this argument is defining “a rock God cannot lift.” It is a nonsensical statement, like saying can God make a true lie? The sentence is correct grammar but a nonsensical combination words, as nonsensical as Jabberwocky.

However, now that I look at this argument from an infinity point-of-view, how do I know it contains a nonsensical definition? I can only know that if I understand infinity. Perhaps the real problem with this argument is assuming too much knowledge about “all-powerful”. Perhaps my objection to it and the argument itself both assume too much about infinity. Given that a mathematically infinite set can lose something and still be infinite, maybe God can lose part of his power and still be infinitely powerful. So maybe a rock God cannot lift is not a nonsensical concept. But by the same token, if God does create such a rock, maybe he is still all-powerful.

A related concept has to do with God being all-powerful and people having free will. Maybe an infinitely powerful God can give people some of his power and still remain infinitely powerful. Some say that God cannot allow free will in people because doing so reduces his sovereignty. Maybe these people can now relax because in the world of infinite it is possible to give power away and still retain the same amount of power.

We’ve talked about God’s infinite power, what about his boundless love and mercy? People often say that God cannot be loving because there is so much evil in the world, or so much evil in their own lives. But given how counter-intuitive even relatively simple mathematically infinity is, how can we expect to understand God’s infinite love? Are questions of God’s love valid? Certainly. Will giving a mathematical proof of his love comfort someone who is hurting? Certainly not. However, in our times of calm reflection and rational thought, perhaps we can at least realize how limited our understanding is of an infinite God.

Finally, let’s consider the Christian doctrine of the trinity, or triune nature of God. How can God be one and three at the same time? Again, I don’t know. But perhaps the mathematically concept of infinity can be illustrative. Earlier we saw how the even numbers are the same size as the counting numbers. Likewise the odd numbers are also the same size. We are able to split the counting numbers into two sets, odds and evens. Those two sets are not half as big, but are each as big as the original. A three-way split could just as easily be accomplished (multiples of three, numbers that divide by three with remainder of one, numbers that divide by three with remainder of two). Could not an infinite God also be divided into three parts, each of which is as big and the original? Does this prove or even fully explain the trinity? Of course not. But at least it is food for thought and opens new avenues of thinking.

Infinite Possibilities

As I said above I make no claims that mathematical infinity matches any reality. In fact, making such a claim can lead to other problems. For example, as we mentioned above mathematical infinities have infinitely many different sizes. If we claim God fits the mathematical definition of infinite, we must immediately ask what size is God’s infinity. Or is God somehow more infinite than all mathematical infinities (which, I think, was roughly Cantor’s view)? Rather than making such a strong claim, what I hope I’ve done is convince you to be careful in your handling of concepts that include infinity and to open you up to new thought possibilities in your search for truth and contemplation of the infinite.