The short answer is that the God of classical theism is infinite, and that it doesn't matter. The longer answer explains why it doesn't matter. Let's look at the longer answer first.
Metaphysical speculation is an infamously useless endeavour. But perhaps even more infamously, people still get paid to engage in it. (This does not include me.) Though genuinely confusing to those whose minds are more inclined to the natural sciences--what does it matter, after all, how many angels dance on the head of a needle?--it only takes a moment to realize that what metaphysicians are actually paid for is not the product of their speculation, but everything else that goes on around it. They teach students, for example, to 'critically' engage with the writings of others, an invaluable sort of persistent patience to engage with another person's arguments.
Such a 'critical' training may not, of course, serve an obvious utilitarian purpose, nor is it easily monetizable (for which reason it may be said to be invaluable). But it is, or at least it should be, conducive to the general leavening of an educated society with good manners and civil graces.
This is a tall claim. What exactly do I mean by it?
The common wisdom of today is that metaphysical speculation instills violent hatreds and provokes violence. Isn't metaphysical speculation what the resurgence of religious fundamentalism is all about?
I mean something like the following: Understanding what another person means is often tough going. Understanding what I myself think about this or that issue can often be even tougher going. And it does not take a speculative metaphysician of any advanced degree to realize other people are not likely to have a perfectly transparent understanding of things either. If I have difficultly understanding other persons and more difficulty understanding myself, so should other persons.
Metaphysical speculation, then, ought to be conducive to a general sense of intellectual modesty. Take, for example, the much maligned question about how many angels can dance on the head of a needle. I am not sure how many angels can dance on a needle head. I am also not sure why someone would think the question an important one to answer. But I can be very sure that I will never find out if I dismiss the question out of hand because it fails to conform to my idea of what can possibly be true. Hence, for the sake of understanding other person's better, I stifle my deep-seated need to dismiss such inquiries as manifest nonsense.*
These are valuable skills in their own rights to cultivate; and it is a good thing the institutional memory of higher academics continues to allow persons to pursue, if only for the sake of general literacy and mutual comprehension.
The question about whether God is infinite fits this same bill. How the question is answered will be more revealing about what they think about themselves than it ever will be about the speculative metaphysical entity that we have collectively agreed to name God. This is the first sense in which God is infinite, but, in itself, it does not matter.
My musings here are prompted by Stephen Webb's attempt to answer the question, Is God Really Infinite? over at First Things. Webb engages with recent and contemporary contributions to number theory. He wonders if there is any way to show that the classical theist God's infinity is bigger than Graham's number, a number so big that if each Planck volume in the universe contained a single digit, it could still not be contained. He comes around in the end, after invoking the authority of Anselm, Goeffrey Cantor, Aristotle, and much less well-known 20th century theologian Erich Przywara, to the conclusion that God might not be infinite, but our possible knowledge of him certainly is.
Webb's conclusion illustrates something of the point that I want to make; namely, that our answer to the question says more about ourselves than it ever does about a transcendent Divinity.
The idea of infinity is generally considered a numerical version of a limiting concept. The latter is any idea that we have in our heads that corresponds to nothing in reality (pace Immanuel Kant), and serves only to designate what cannot be thought. So, for example, I observe many things in the world, but I never observe world itself. The world is never something in the world. Hence, my idea of the world can only ever be a limiting concept.
Mathematicians can debate the finer points about whether it is possible to think the number infinity. The general wisdom is that you cannot actually think infinity. That is to say, it is not possible for me to hold in the finite 'space' of my attention the complete series of whole numbers (1, 2, 3, 4...n). Why not? Because my attention fastens itself to one number after another, and maybe a couple numbers at a time, but not the complete series of whole numbers simultaneously.
Mathematicians might propose a formula that allows you to think infinity, such that if you let your mind grasp the circularity of the formula, you can, so to speak, think infinity. But this is different than actually thinking infinity, which is to think complete series of whole numbers simultaneously. Since my mind must move ploddingly along from 1 to 2, and 2 to 3, and 3 to 4, there is just no way for me to do that.
Webb sees as much where he says, 'The problem with this judgment is that infinity—as in, God is infinitely unknowable—does not admit to degrees. An infinite God is not like an unimaginably large number that we could count to if only we had enough time.'
This is a promising line of inquiry, though Webb does not explore it in any great depth. If he had, he might have come to some surprising conclusions about himself.
It seems, on the one hand, that we can form the 'limiting concept' of infinity, the simultaneous possession of the complete set of whole numbers; but, on the other hand, we can't actually think the complete set of whole numbers simultaneously. The reason is ready-to-hand: our finite attention constrained to move from moment to moment through time. This movement of time is utterly indifferent to what I might want. I might want to speed it up, or slow it down; but while my perception of the rate of time's movement can vary by degrees (especially in cases of boredom, anxeity, utter serenity), I can still know that its rate of movement has not changed by reference to some external measuring device, like a clock. The movement of time constrains me to count numbers in sequence.
So let's grant that God is infinite in the sense that he possesses in his infinite attention the complete set of whole numbers. Does it matter if he is? Well no, but it does reveal something about myself/ourselves.
It draws attention to the fact that all my thinking is conditioned by the movement of time--or what we a materialist like Thomas Hobbes could call the 'trayne of thoughts'. I have an idea, or a limiting concept, that is analogous to the simultaneous possession of the complete set of whole numbers. But my attention is otherwise stretched out through a series of successive moments that extends backwards into the past and reaches forward into the future. These are counted in their slow succession through every single present moment.
So, God of classical theism is infinite, yes; but the only way this could mean anything to me is if I were not stuck counting moments as they pass me by.
And I am stuck counting moments as they pass me by.
*The question about how many angels can dance on the head of a needle regards how a purely intellectual being like an angel is said to exist. The question intends to ask are intellectual beings (like angels, the human mind, and God) extended in space like physical bodies are extended in space. A needle is spatially extended, while angels are not. So the question is nonsensical in the precise terms that it is stated. But far from being utterly silly, it anticipates what I have termed here a 'limiting concept.'