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#197 Does God Know an Actually Infinite Number of Things?

January 24, 2011

Dr. Craig,

I can't tell you what your work has meant to my life. Part of who I am today is due majorly to your obedience to God in teaching and researching. For this I thank you deeply.

I do have a question concerning the idea of actual infinites not existing. As I can see all the arguments supporting this, a possible "actual" infinite has come to mind. You make the point that since God has started time rolling at creation, time could never cease. Therefore, the future is potentially infinite. My question concerns God's knowledge. If God knows all true propositions, including future tensed propositions, and he doesn't learn them as time goes on (which is a thought I highly support) then doesn't God know truth statements about all future moments in time at once. Meaning, isn't God's knowledge about the future an actual infinite, since he currently knows everything about every point in the potentially infinite future? Wouldn't the number of things God knows, be an actual infinite in quantity? I certainly buy his knowledge being infinite in quality, but this seems to be an actual infinite in quantity.

Since God is omniscient, it seems that the potentially infinite future, becomes actually infinite in God's knowledge.

By no means am I trying to unravel your arguments. This has just been on my mind for a bit and I'd like to see where my thinking is off... since this question could actually be asked of me in the potentially infinite future.

Thank you for all you do,




Dear Dr Craig,

I have a few questions regarding Omniscience and Actual Infinity

1) There is potentially an infinite amount of numbers, you can keep counting forever and you will never reach the last number. Now since God is omniscient does that mean he knows all possible numbers?

a) If so, how is that possible if there is potentially an infinite amount and you can always add more and never reach the last number?

b) If not, how would you reply to someone if he thought this counted as evidence against omniscience?

c) Also, if you say that God knows all possible numbers, and there is an infinite amount of numbers, doesn't that mean than actual infinity does exist (namely, numbers)?

Are your answers to these questions based on your view of God being timeless sans creation and temporal since creation?

Would you answer differently if you held the view that God is timeless and knows all truth in a single intuition at a single moment? If so, how would you answer the above questions if you held the view that God is timeless and knows all truth in a single intuition at a single moment?

Please give two answers to each of the above questions (one based on the view that God is timeless sans creation and temporal since creation and one based on the view that God is timeless and knows all truth in a single intuition at a single moment), unless one answer applies to both views.

2) In Question 108 "Omniscience and Actual Infinity" you stated:

"Perhaps what Sotnak is implicitly assuming is that if Peter will live forever, then his afterlife will be actually infinite. But then he's begging the question against Peter's afterlife's being potentially infinite. What is wanted is some argument for that assumption. Moreover, all the talk about omniscience becomes in that case just superfluous window-dressing. The argument should just be that immortality implies the existence of an actual infinite. Omniscience doesn't even enter the picture. Whether immortality implies the existence of an actual infinite was a central issue in my dialogue with Wes Morriston on the kalam cosmological argument, which I commend to you. (see "Debate on the Kalam Argument" Reasonable Faith Podcast)"

However, would you answer differently if you held the view that God is timeless and knows all truth in a single intuition at a single moment? If so, how would you answer Sotnak on this view?

3) Is there anything that God knows an infinite amount of? For example, is there an infinite amount of possible worlds or possibilities? Wouldn't this imply that an actual infinity exists (namely, possible worlds or possibilities)?

Is there no such thing as an actual infinity? Not even in the mind of God?



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Dr. craig’s response


In my opinion, Paul, an actually infinite number of things does not exist, even in the mind of God. In Philosophical Foundations for a Christian Worldview, I write:

some thinkers such as William Alston, while rejecting complete [divine] simplicity, have advocated that God's knowledge be construed as simple. On Alston's view God has a simple intuition of all of reality, which we human cognizers represent to ourselves propositionally. Such a view is in line with Aquinas's adaptation of the Augustinian notion of the Divine Ideas. In order to preserve divine aseity in the face of Platonism, Augustine located the Platonic forms in God's mind as the Divine Ideas. Aquinas went further by contending that God does not, strictly speaking, have a plurality of Divine Ideas but rather an undifferentiated knowledge of truth. We finite knowers break up God's undivided intuition into separate ideas. Similarly, Alston maintains that God's knowledge is strictly non-propositional, though we represent it to ourselves as knowledge of distinct propositions. Thus, we say, for example, that God knows that Mars has two moons, and He does indeed, know that, but the representation of His knowing this proposition is a merely human way of stating what God knows in a non-propositional manner. Such a conception of divine knowledge has the advantage that it enables us to embrace conceptualism without committing us to an actual infinite of divine cognitions or Divine Ideas.

A nice analogy of God’s cognition on the above view would be your visual field, which you see as an undifferentiated whole, but which could be analyzed as composed of pixels.

So, Gordon, when we say that God knows an infinite number of propositions, we are speaking of the extent of His knowledge, not the mode of His knowledge. There’s no reason, by the way, to think that God’s cognitive state can’t be changing, just as my visual field changes. Indeed, if God is in time, His cognitive state must be constantly changing or He would be in error, still thinking, for example, that it is 3:00 o’clock.

Now to your specific questions, Paul, the answers to which are, in my opinion, independent of God’s relationship to time:

1. On my view there is not a potentially infinite number of numbers. Rather there are no numbers at all! Numbers, if they exist, are abstract objects, and I’m strongly inclined to say that only concrete objects exist. What is true to say and what God knows is that according to standard number theory, there is an actually infinite number of numbers. Moreover, God knows that an actually infinite number of arithmetic truths follow from the axioms of standard arithmetic, like 1+1=2, 2+1=3, 3+1=4, . . . . But, as explained above, He does not know these truths propositionally, as I have just expressed them, but non-propositionally. Therefore, there is literally neither an infinite number of numbers nor an infinite number of propositions.

2. As indicated, my view of the mode of God’s cognition is independent of issues of time. The talk of “a single intuition of reality” was meant to express the non-propositional mode of God’s knowledge and thereby to eliminate an actually infinite number of divine ideas or items of knowledge. Unfortunately, the recording of my dialogue with Morriston I referred to was so poor that we couldn’t market it. But the current issue of Faith and Philosophy contains a similar exchange with Morriston (“Taking Tense Seriously in Differentiating Past and Future: A Response to Wes Morriston,” Faith and Philosophy 27 (2010): 451-6). As I explain there,

Morriston needs to find something that is part of reality which is actually infinite in quantity in order to make an analogy with a beginningless series of past events. Morriston later returns to his suggestion that rather than future events, which on an A-Theory of time are not part of reality, we consider future-tensed truths or corresponding tensed facts. But this move makes two unjustified assumptions: first, Platonism with respect to propositions and, second, the actual infinitude of propositions or facts. If we accept these assumptions, there is no need for appeal to future-tensed truths in order to designate an actual infinitude of propositions, since for every proposition p there is the further proposition that Tp, or that it is true that p. The finitist will therefore either deny Platonism with respect to propositions, taking them to be useful fictions perhaps, or deny that there are an infinite number of propositions, since, God’s knowledge being non-propositional, propositions are the byproduct of human intellection and so merely potentially infinite in number, as we come to express propositionally what God knows in a non-propositional way.

3. Not on my view! Possible worlds don’t really exist. They’re just a heuristic device for illustrating primitive modal expressions like “Necessarily, God is omnipotent” or “Possibly, God is timeless.” In short: no room for actual infinites!

- William Lane Craig