We know that God created the universe a finite amount of time ago, but defenders of the Kalaam cosmological argument say that God had to do it this way. He had no other choice.
William Lane Craig has proposed the following argument to support this claim:
1) An actually infinite number of things cannot exist.
2) A beginningless series of events in time entails an actually infinite number of things.
3) Therefore, a beginningless series of events in time cannot exist.
Depending on your view of time, there are potential problems with the second premise—as I’ve written elsewhere.
However, here I’d like to consider the first premise.
Is it true that an actually infinite number of things can’t exist?
What’s an “Actual” Infinite?
We need to be aware of the difference between what philosophers and mathematicians call “potential” infinities and “actual” infinities.
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- Something is potentially infinite if it goes on endlessly, but there is no frame of reference in which all of its elements exist.
- Something is actually infinite if it goes on endlessly and there is a frame of reference in which all its elements exist.
For example, suppose that you have a machine that makes cubes. Today it makes a cube, tomorrow it makes a cube, and it keeps on like that endlessly. No matter how many days you go into the future, there will still be a finite (limited) number of cubes.
From within the perspective of time, there is no day where the “infinity-eth” cube pops out of the machine, because “infinity” is not a number on the number line. There is no “number just before infinity,” and so you can’t count to infinity.
Yet the series of cubes that the machine will make is endless, which is what “infinite” means—unlimited or unending (Latin, in- “not” + finis = “limit, end”).
So, while the number of cubes you have grows toward infinity, it never gets there. That’s why the cubes would be said to be potentially infinite rather than actually infinite—because they don’t all exist at the same time.
But suppose you didn’t have a cube-making machine. Suppose instead that God decided to create an infinite number of cubes all at once. Bam! It’s done. All in a flash.
In this case, you would have an infinite number of cubes—that all exist at once—and so that would be an actually infinite set of cubes.
The key point to remember is that both potential infinites and actual infinites involve limitless numbers of things. The difference is that in a potential infinity these elements don’t all exist at once, while in an actual infinity, they do.
Can Actual Infinities Exist?
While many authors assert that actual infinities can’t exist, we should test this. Can we think of anything actually infinite?
How about numbers?
We often represent the set of natural numbers like this: {0, 1, 2, 3 . . . }. The reason we put the ellipsis (the three dots) at the end is to say that this series of numbers goes on forever in the same way it began, with one number after another, with no end to them.
That’s an infinite set!
And all those numbers already exist. It’s not like there’s a mathematician somewhere inventing new numbers in his workshop.
We may make up names for new numbers—like the number googol, which is 10-to-the-power-of-100, or googolplex, which is 10-to-the-power-of-googol—but we didn’t create these numbers. We only named them.
Even before they were thought of or named, it would remain true that googol multiplied by 2 is 2 googol, that googol minus google is 0, and that googol divided by google is 1.
So, it appears that an infinite quantity of natural numbers exists, whether we’ve thought of or named them or not.
Since the set of numbers does not grow with time—only our knowledge of them does—the set of natural numbers is an actually infinite set, as mathematicians commonly acknowledge.
The Truth of the Matter
Numbers are not the only actual infinity we can think of. There also are truths (facts), as we can easily see:
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- It is true that 1 + 1 = 2.
- It is true that 2 + 1 = 3.
- It is true that 3 + 1 = 4.
- And so on.
So, not only does an actually infinite quantity of numbers exist, an actually infinite quantity of truths also does.
How might a defender of the Kalaam argument respond to this?
The Nature of Numbers and Truths
It’s easy to point out that there is a difference between numbers and truths and the kind of objects we see in the world around us.
For example, if I have two cubes that I’m holding in my hands, I’m physically touching them. But I don’t seem to be touching the number 2. Numbers aren’t physical objects you can see, hear, or touch.
Neither are truths. It may be a truth that Abraham Lincoln died in 1865, but I can’t hold this truth in my hands like an apple or put a ruler beside it and measure how long it is.
Things like numbers and truths seem to be in a different category than things like cubes and apples. Things in the first category are often called abstract objects, while those in the latter are often called concrete objects. We also might call them physical objects.
The sciences have given us a lot of information about how physical objects work, but we can’t use science to investigate abstract ones. They lie in the realm of philosophy, and among philosophers there are different views about their nature.
Some philosophers (known as anti-realists) deny that abstract objects are real and propose other ways of understanding them. Other philosophers (known as realists) hold that they do exist, but again there are different understandings (e.g., do abstract objects exist in an abstract realm of some kind? do they only exist in physical objects? are they based in the mind of God?).
Options for the Kalaam Defender
A Kalaam defender could adopt an anti-realist position and say that things like numbers and truths simply do not exist, which would mean that there aren’t actual infinities of these—because numbers and truths don’t exist in the first place!
But this seems hard for many to imagine, including various supporters of the Kalaam argument.
There is another option, which is to draw a line between the two classes of objects and say something like, “Look, whether or not actual infinities of abstract objects exist (maybe they do; maybe they don’t), that’s not what I’m talking about. When I say that actual infinities can’t exist, I mean that they can’t exist concretely, in the physical world.”
A person taking this position could acknowledge that actual infinities of abstract objects can exist. What he disputes is that actual infinities of concrete, physical ones can.
Why Not?
The question would be: Why not? Why can’t actual infinities of physical objects be real?
This question is particularly acute from a Christian perspective, since the Christian faith holds that God exists and that he is omnipotent, which means that he can create anything that does not involve a logical contradiction.
So, let’s do a thought experiment:
Imagine a cubic foot of empty space that has a single hydrogen atom in it. If we can imagine that, so can God.
Now imagine a second cubic foot of empty space sitting right next to it, also with a hydrogen atom in it. God can imagine that, too.
Now put a third cubic foot of space next to that, also with a hydrogen atom, so that we have a row of three.
Then imagine a fourth, a fifth, a sixth, and so on. God can imagine each of these as the line of cubic feet extends off into the distance.
In fact—due to his omniscience—God can imagine any number of such units. Unless there is a logical contradiction involved, God could imagine a cubic foot of space with a hydrogen atom for every natural number.
And so, God can imagine an actually infinite volume of space, with a single hydrogen atom in each cubic foot.
Either this scenario involves a logical contradiction or it doesn’t. If it doesn’t, then God can imagine it, since God can do anything logically possible.
And it does not appear to involve a contradiction. As even many defenders of the Kalaam argument admit, the mathematics of infinity are consistent and do not contain logical contradictions.
This situation isn’t like saying, “Suppose God imagines a four-sided triangle.” Not even God can visualize that, because “four-sided triangle” is a contradiction in terms.
“Four-sided” and “triangular” mean different and contradictory things. This expression is just word salad—not something that is actually meaningful.
But our volume of space isn’t like that. Even our puny minds can imagine a row of cubic feet with hydrogen atoms in them. God’s mind is infinite, and so he can imagine the line of cubic feet extending on endlessly—and there doesn’t seem to be a logical contradiction involved in him doing that.
If there’s not, then we’re ready to add a new element to our thought experiment.
“Let There Be Space!”
Because of his omnipotence, God can do anything that doesn’t involve a logical contradiction. So, if God can imagine something, he can also create it.
As a result, if the idea of an infinite row of cubic feet of space—each with a hydrogen atom—doesn’t involve a logical contradiction, then God can make it real.
God thus could create an infinite number of hydrogen atoms and an infinite volume of space to contain them.
If God chooses, actual infinities of physical things could exist!
The only way to avoid this would be to say that, even though the idea of finite space with finite atoms is logically coherent, a contradiction in terms is generated if we extend this to infinity.
In that case, God couldn’t imagine or create this any more than he could a four-sided triangle.
But in that case, I want to know: What’s the contradiction?
As we’ve discussed, a Christian who understands God’s omnipotence should affirm that God can make something unless it is shown to involve a logical contradiction.
The Kalaam defender thus needs to name the contradiction: Which are the specific terms that contradict?
And it won’t do to change the scenario and pose some other one where a new logically contradictory entity is subtly introduced. That happens all too often.
Changing the scenario is what you do when you can’t deal with the current one.
Neither is it sufficient to say, “Hey, infinities have weird properties.” Yes, they do. That doesn’t mean they’re beyond the reach of God’s omnipotence.
So please, deal with the thought experiment I’ve described, flesh out its terms in detail, and name which terms contradict each other—just like how I pointed out that “four-sided” and “triangular” contradict.
Either that or be prepared to acknowledge that God has the power to create actual infinities of physical objects.
