Craig’s List – Omniscience and actually existing infinities


William Lane Craig has famously argued for the ‘Kalam cosmological argument’ (in many places, but for example in Craig & Sinclair [2009]). Here is the argument:

  1. Everything that begins to exist has a cause.
  2. The universe began to exist.
  3. Therefore, the universe had a cause (Craig & Sinclair [2009], p 102).

The argument is clearly valid, as it is a version of modus ponens. Thus, in order to deny the conclusion, one must argue that the first or second premise is not justified.

Most people have argued against premise one, disputing whether all things which begin to exist have causes for their existence, or the fact that a fallacy of composition may be at play with the generalization from all things in the universe to the universe as a whole. I will not be pursuing this line of argument here, but will instead look at premise two.

Premise two seems to be supported by physics, specifically cosmogony, which some say indicates that the spacetime we exist within came into existence at the big bang. People who know more about this than I do tell me that this is actually a misconception of this theory, and that it is not really a theory about the origin of spacetime at all. However, we can avoid delving any further into the details of the physics, because Craig does not rest his argument on the interpretation of the big bang theory. There is a logical argument Craig spends time going into, according to which the universe must have had a beginning – that it is impossible for the universe to have always existed. Here is that argument:

2.1. An actual infinite cannot exist.

2.2. An infinite temporal regress of events is an actual infinite

2.3. Therefore, an infinite temporal regress of events cannot exist. (ibid, p 103)

It is on this supporting argument that I wish to focus. Specifically, it is the first premise of this argument that I will be spending time going into here. If we can knock this premise out, then it undermines the entire supporting argument, and with it the credibility of the main argument. If we can deny 2.1, we can avoid having to assent to 3.

Hilbert’s Hotel

In order to motivate 2.1 (that an actual infinite cannot exist), Craig uses the example of ‘Hilbert’s Hotel’. In this imagined hotel there is an infinite number of rooms. Infinity has a distinctive property, according to which a proper subset of it can be equal in cardinality to the whole, there are various counter-intuitive consequences, which Craig uses to motivate the idea that this could not actually exist. For example, if the hotel is full but a prospective guest arrives asking for a room, the hotel manager can simply ask each occupant to move into the next room, thereby making room number one free. Because there is an infinite number of rooms, there will be room for every occupant, thus making a newly free space for the new guest to stay in, even though the hotel was full. Even if infinite new guests turn up, the hotel manager can make room by getting everyone in the hotel to move into the room with the room number that is twice the number of their current room (so room number two gets room number four, room number four gets room number eight, etc.). This frees up an infinite number of rooms, even though the hotel was full. Craig comments:

“Can anyone believe that such a hotel could exist in reality? Hilbert’s hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible” (Craig & Sinclair [2009], p. 109-110).

If this is correct, then because a universe with no first moment would constitute an actually existing infinity, it follows that the universe had a first moment. Thus, the idea is that it is no objection to simply say that maybe the universe always existed. It couldn’t have always existed, says Craig.

However, it is not clear to me that his objection really applies to the universe, and I will spell this out in more detail now.

Pinning down the absurdity

One might wonder what specifically it is about Hilbert’s hotel that Craig finds absurd. It seems that the sheer scale of the hotel, the fact that it has infinite rooms, is not itself absurd to Craig. If it was, then the example would simply have been:

‘Imagine that there is a hotel with infinite rooms in – that’s absurd!’

Given that the example was more complex than this, it seems that just saying that the hotel is infinite is not enough for Craig to bring out the absurdity. Nor does simply adding that the hotel actually exists constitute the absurdity, otherwise the example would have been:

‘Imagine that there is a hotel with infinite rooms in, and that it actually exists – that’s absurd!’

Surely, when picturing Hilbert’s hotel, one pictures it as actually existing. Adding that it actually exists is somewhat empty as a property, and surely not enough on its own to make the difference between not absurd and absurd. So what is it that pushes us over this threshold?

It seems to me, given the examples used to illustrate the absurdity of Hilbert’s hotel, that Craig’s idea is as follows. The factor that gets us across the line is what we might call the behavior of the hotel. With an infinite hotel, given certain conditions obtaining, contradictions can be manifested, and contradictions are absurd. So it took the new guest to arrive, and for everyone to shuffle up one room, for an absurdity to become manifested; namely, the hotel is full, but also has a room available for a new guest. If the guest does not arrive, or arrives but is turned away by the manager, then where is the absurdity? How do we generate a contradiction without interacting with the hotel? It seems like the only way we could imply an absurdity in that case would be simply pointing out that the hotel has infinite rooms. But if this was on its own enough to constitute absurdity, why bother with the example of the guest arriving? Is it just for rhetorical effect? It seems to me that the answer is that without the guest arriving and the creation of the new free room, Craig thinks that nothing absurd is present.

If this right, then we could employ a distinction between active and passive infinities. An active infinity is one that manifests absurd behavior (like being full but also making room for a new guest), whereas a passive infinity is one that does not (like a Hilbert’s hotel which never admits new guests). Now, it should be noted that a passive infinite retains the potential to manifest absurdity; it is passive just so long as it doesn’t actually do so.

This makes the distinction between ‘actually existing’ and ‘not actually existing’ slightly wide of where the beef is here. It seems we could have an actually existing Hilbert’s hotel, which remains passive, and for all Craig has said, this would not be absurd. The absurdity only kicks in when an actually existing infinity becomes active.

The infinite universe is passive

The problem with Craig spelling out the nature of the absurdity associated with actually existing infinities like this, is that it doesn’t apply to the eternally existing universe. There are models where we could make his objection apply, but the most natural way of cashing it out avoids his problem, as I will explain.

Imagine a number line that contains all integers running from minus infinity, through 0 all the way up to positive infinity. Now think of 0 marking out this very moment now. This is a bit like the most natural way of thinking about the eternal universe; each moment has infinitely many earlier moments and infinitely many later moments. If this is how Craig is characterizing the eternally existing universe, then it is a passive infinity. There is no corresponding example to making a free room, or withdrawing a book. One cannot add a moment to time, nor take one away. It is a ‘closed’ infinity. In fact, it is arguably metaphysically impossible to add a time or take one away. Thus, Craig may be correct that active infinities are metaphysically impossible, but because the eternal universe is not one of these, then he has no objection to the eternal universe.

As I said, there are ways of cashing out the eternally existing nature of the universe according to which Craig’s point holds. For example, consider the ‘growing block’ theory of time. According to this theory, the past is a fixed set of facts, which is growing as time moves forwards. We continually add new truths to the stock of settled past truths. If this were the model, then we would have an infinite list of past truths, but we would be able to add to it. In a sense, this would resemble Hilbert’s Hotel and thus make the universe an active infinity.

It should be noted that even on this growing block theory, there is room to doubt whether this really counts as an absurdity. With the hotel example, we can derive a sort of contradiction, in the sense that the hotel was full, but had room for a new guest. If being full means that there is no room, then this is a contradiction. But it is not clear what is the contradictory sentence we are supposed to be able to make out of the growing block theory here. Sure, there are infinite past moments, and then a new one gets added to the pile as time moves forward. The only contradiction I can see here is that the cardinality of the past moments is the same, even after a new one is added to the block. If so, then we have our candidate.

It is a weak candidate, as it seems to me that we ought to simply accept that this is what an infinite block would be like. However, let’s assume that Craig has scored his point here, and that the growing block theory is absurd for that reason. No such account can be leveled at the eternal universe outlined above. It has an infinite number of moments, but there is no possibility of adding new moments or taking them away, so it is passive. It seems like we can block Craig’s argument by simply explaining clearly what an eternal universe looks like, and that while it is infinite in extent, it manifests no absurdity.

In fact, this will form one horn on a dilemma I wish to place Craig in. As we shall see, if there is a problem with the growing block theory, then it also affects Craig’s version of God. The dilemma will be that either the universe is infinite in temporal extension, or God doesn’t exist.

The Infinite God Objection

Craig’s God is omniscient. This means that ‘God knows only and all truths’. Watch him commit to this position here:

It is uncontroversial that there are mathematical truths, like that it is true that 2 + 2 = 4. God knows all these truths as well (Craig explicitly makes this point at 6:20 in the video above). To make the point as simple as possible, God knows the solution to every equation of the form x + y = z, where the variables are natural numbers. As there is an infinite number of such solutions (with a cardinality equal to the smallest infinity, ℵ0), it follows that God’s knowledge is correspondingly at least as infinite as the cardinality of the natural numbers (and obviously greater if he also knows all real number solutions as well).

Let’s consider Craig’s God’s knowledge of these arithmetic solutions as a list of truths, which we could call ‘Craig’s List’. It would be an infinitely long list. So Craig’s God’s knowledge is infinite.

But, according to the Hilbert’s Hotel argument from above, the infinite cannot actually exist. Therefore, an omniscient God cannot actually exist. Craig’s God is omniscient. Therefore, by his own argument, Craig’s God cannot exist.

Call this the ‘Infinite God Objection’.

God’s knowledge is of induction schemas

It could be objected here that God does not need to know every arithmetic truth, such as 2 + 2 = 4, because as long as he knows the base case and all relevant induction schema, he would know enough to deduce the answer to any similar equation. If this were the case, then it would drastically limit the amount of propositions God would need to know, from infinite to a mere handful.

My response to this is that if this were all that were required to know all mathematical truths, then I know all mathematical truths. After all, I know the base case (that 0 is a number) and the relevant induction schema. God and I both have the same resources at hand, and if this is all it takes to know all mathematical truths, then we both know all mathematical truths. This is an awkward consequence, to say the least.

But this consequence is not just awkward. It is intuitively true that there are lots of arithmetical equations that I do not know the answer to, even though I could work them out given my knowledge of the induction schema. It seems more natural to say that I do not know the answers to these questions, but I know how to work out the answers. This makes the response in the God case inadequate though. To concede that God does not know the answer to any mathematical question, but knows how to work out the answer, is just to concede that there are things he does not know. The fact that he could work it out it not a defeater to the claim that he does not know it.

On the other hand, perhaps the similarity is only apparent, and that due to my limited nature, as compared to God’s unlimited all-powerful nature, there is a meaningful difference between the two cases. Perhaps it is the case that I slowly lumber through, applying the schema to the case at hand to derive the answer, and with the possibility that I could always go wrong on the way. In contrast, God applies it at lightening speed, without the possibility of getting it wrong on the way. In this case, there is no arithmetic question you could ask God to which the answer would be ‘I don’t know, but I will work it out for you’; as soon as you have asked the question he has already worked it out. Therefore it is never true that there is something he does not know.

But I could just stipulate an equation, without asking God directly. Even though, were he to think about it he would get the answer immediately, given that he is not currently applying the schema to the case, it is not true that he knows it. So there is something he doesn’t know. So he is not omniscient.

And if we avoid this by saying that he is constantly applying the schema to all cases, then we are right back to the original case, where he knows an infinite number of truths.

Thus this escape route will not help.

God’s knowledge is non-propositional

Craig could say that God’s knowledge is non-propositional, as in the Thomist conception. On this idea, God does not know lots of individual propositions, but rather has one unified knowledge of himself, which is perfectly simple.

To begin with, this contradicts his statements in the video above, where Craig explicitly states that God knows all propositions. Perhaps we can let this slide, as it is him talking somewhat informally.

In a paper entitled ‘A Swift and Simple Refutation of the “Kalam” Cosmological  Argument?‘ (1999), Craig considers a very similar objection, namely that if mathematical truths are just divine ideas, then God’s mind has infinitely many ideas. In defense of the divine conceptualist, Craig offers the following reply:

“[T]he conceptualist may avail himself of the theological tradition that in God there are not, in fact, a plurality of divine ideas; rather God’s knowledge is simple and is merely represented by us finite knowers as broken up into knowledge of discrete propositions and a plurality of divine ideas.” (Craig, (1999), p 61 – 62).

This theological tradition goes back to Thomas Aquinas, and as an explanation of this, Craig cites William Alston’s paper ‘Does God have beliefs?’ (1986). In that paper, Alston says the following:

“[C]onsider the position that God’s knowledge is not propositional. St Thomas Aquinas provides a paradigmatic exposition of this view. According to Aquinas, God is pure act and absolutely simple. Hence there is no real distinction in God between his knowledge and its object. Thus what God knows is simply His knowledge, which itself is not really distinct from Himself. This is not incompatible with God’s knowing everything. Since the divine essence contains the likenesses of all things, God, in knowing Himself perfectly, thereby knows everything. Now since God is absolutely simple, His knowledge cannot involve any diversity. Of course what God knows in creation is diverse, but this diversity is not paralleled in the intrinsic being of His knowledge of it. Therefore ‘God does not understand by composing and dividing’. His knowledge does not involve the complexity involved in propositional structure any more than it involves any other kind of complexity” (Alston, (1986), p. 288).

Thus, if the divine conceptualist can avail himself of this Thomistic tradition of God having non-propositional knowledge, then Craig himself could make the same move to avoid the charge that God knows an infinitely long list of arithmetical truths.

There is a problem of going the Thomist route here, as Aquinas himself is very explicit about whether God knows infinite things:

“Since God knows not only things actual but also things possible to Himself or to created things, as shown above, and as these must be infinite, it must be held that He knows infinite things” (Aquinas, Summae Theologica, Q14, A12).

Alston is perhaps trying to spell out a Thomist inspired view, rather than a Aquinas’ actual views. Even if Aquinas insisted that God knows an infinity of things, perhaps a non-propositional knowledge model can be adopted whereby God knows all mathematical truths without knowing an infinite list of truths. Indeed, Alston turns to F. H. Bradley’s idealism to spell out this possible model. Aston says that on Bradley’s view, the ‘base of our cognition is a condition of pure immediacy’, in which there is no distinction between different objects of knowledge. It is like taking in a painting as a whole, without focusing on any one particular bit of the painting. We can ‘shatter this primeval unity and build up ever more complex systems of propositional knowledge’, which would be like focusing on a particular brush stroke rather than the scene as a whole. This second mode of understanding is more discursively useful, but lacks the ‘felt oneness’ of the primeval apprehension. In contrast to these modes is the nature of the ‘Absolute’ itself – the world beyond our comprehension, which ‘includes all the richness and articulation of the discursive stage in a unity that is as tight and satisfying as the initial stage’. God’s knowledge, says Alston, could be modelled like this.

Wes Morriston, in his paper ‘Craig on the actual infinite’ (2002) considers this move by Craig, and concludes that Alston’s idea is of no help here:

“On Alston’s proposal, then, God’s knowledge is certainly not chopped up into a plurality of propositional states. On the other hand, it is said to have ‘all the richness and articulation’ of discursive thought. Even if this ‘richness and articulation’ does not consist in a multiplicity of propositional beliefs, it must surely involve some sort of distinction and variation and multiplicity within the divine intellect. However ‘tight and satisfying’ the unity of God’s knowledge, it must be thought of as a unity within a multiplicity – a one in a many” (Morriston, (2002), p. 159).

Ultimately, Alston’s idea is just that a God’s knowledge is a sort of synthesis of multiplicity and unity, and Morriston’s reply is that this does not eliminate the multiplicity. So it is not really any help to Craig.

Thus it seems that the non-propositional nature of God’s knowledge is not really a way of getting out of the claim that God is infinite.

Craig’s God is a passive infinity

Given that we now have the distinction between the active and passive infinity at hand, it could be that Craig’s reply would just be that God’s knowledge of arithmetic truths is a ‘closed totality’ of knowledge, and as such is passive. Just as no new moments can be added to the timeline, no new arithmetic truths can be added or subtracted from the totality of mathematical truths. As such it is infinite, but can never manifest absurdities as a result. As such, God can be infinite in this regard and not get chewed up in the teeth of Craig’s argument.

This would be a satisfactory response by Craig, but for one thing. Craig’s God has a very distinctive relationship to time, because Craig has a very particular theory of time. This makes Craig’s God particularly vulnerable to the actively infinite God objection.

Craig’s God and Time

Craig has a fairly nuanced view about God’s relationship to time. Roughly, God existed in an atemporal manner before he created the universe, but then entered into time and became temporal.

“God exists changelessly and timelessly prior to creation and in time after creation” (Craig [1978], p 503).

Craig also believes that the correct theory of time is the ‘A-theory’, according to which the fundamental temporal relations are tensed (like ‘it is now raining’, or ‘it will be sunny’, etc), rather than tenseless (like ‘raining at t1 is earlier than sunny at t2’, etc). For Craig, there is a fact about what is happening now which is metaphysically basic, and continually changing as time rolls forwards. God, being a temporal entity in time, has knowledge of this now, of ‘where he is’ on the timeline so to speak, and consequently what is presently happening:

“As an omniscient being, God cannot be ignorant of tensed facts. He must know not only the tenseless facts about the universe, but He must also know tensed facts about the world. Otherwise, God would be literally ignorant of what is going on now in the universe. He wouldn’t have any idea of what is now happening in the universe because that is a tensed fact. He would be like a movie director who has a knowledge of a movie film lying in the canister; he knows what picture is on every frame of the film lying in the can, but he has no idea of which frame is now being projected on the screen in the theater downtown. Similarly, God would be ignorant of what is now happening in the universe. That is surely incompatible with a robust doctrine of divine omniscience. Therefore I am persuaded that if God is omniscient, He must know tensed facts” (taken from, which is a transcript of a paper given in Cambridge in July 23rd 2002)

This makes Craig’s God an ‘temporal epistemic agent’, that is one who is continually updating his knowledge set with new facts about reality as time passes; namely what is presently true. He doesn’t just know that at t1 it is raining – he knows that it is now raining.

Craig’s God is an active actually existing infinity

According to Craig then, God comes to know new things as time moves forwards. But he already knows an infinite number of truths, all the mathematical truths etc, and then he adds to his knowledge as time passes. However, the cardinality of his knowledge, how many truths he knows, stays the same – it is still infinite. So he knows more things, but also the same number of things. This is a manifestation of absurdity, just like Craig complained about with Hilbert’s Hotel, and at least as convincing as the growing block problem. Thus, by his own arguments, Craig’s God cannot exist.


It could be that Craig objects to the distinction between active and passive infinities. Perhaps it was made for rhetorical force only. If so, then his objection should be characterized as:

‘Imagine a hotel with infinite rooms, that’s absurd, therefore it couldn’t actually exist’.

If so, then I find it very implausible. In order to accept it, we would need to have something to justify it, and all Craig offers is that one can derive ‘absurd’ consequences from it, by which he means something contradictory. I agree that if we can derive contradictions from something, then it is to be rejected. However, we have seen that the only way we can get anything absurd from Craig’s examples is if we interact with the infinity, by getting the manager to free up a room for us, etc. Craig has never offered an example of any absurd consequences from thinking of actually existing infinities that are passive. Thus, if he wants to take this option, he still has all his work ahead of him for motivating the first premise of his supporting argument. Until he has provided this motivation, we are free to refrain from assenting to it, and consequently refrain from assenting to the conclusion of the Kalam argument.

But then if Craig accepts the active/passive distinction, then he has a pair of serious problems. Given the eternal universe model, it is infinite but passive. So not absurd. So it can exist. In addition, Craig’s A-theoretic nature of God means that God manifests absurd behavior. Therefore, he cannot exist.

The conclusion, then, is that either Craig has a lot of work to do explaining why actually existing infinities cannot exist, or he has in fact argued himself into a corner where an eternal universe could exist and God cannot. It seems there are big problems for Craig’s God.


8 thoughts on “Craig’s List – Omniscience and actually existing infinities

  1. Here’s another aspect of this debate on actual infinities. In this video, Craig gives his Aristotelian (and Leibnizian) whole-prior-to-the-parts reply to Swinburne’s question. The question is: do the ‘nows’ traverse an actual infinity of points even if the past is finite? This Aristotelian reply is supposed to avoid actual infinities without having to commit to discrete time. The problem is that it’s hard to make sense of how this could work on his presentism, since we would need to mentally divide up the past 13.7 billion years, whereas the only thing that exists to divide up is the present.


  2. Yes, I think this points to a part of my argument I need to expand on; namely the underlying theories of time in his conception of the universe and the eternalist model I proposed. That whole problem of the moving now having to traverse an infinite series doesn’t come up on the B-theory model I have in mind, so I should be clearer about that.


    1. The B-theorist who happens to reject actual infinities can at least get the Aristotelian argument off the ground because, unlike on A-theory, there actually exists a whole 13.7 billion years to divide up mentally. The other alternative for either the A or B theorist, who rejects actual infinities, is to say time is discrete, but Craig doesn’t go that route.


  3. Even better, the universe appears spatially flat as best we can measure, which predicts an infinite universe under a big bang model.

    That is, the universe was spatially infinite at the big bang which was more like an “everywhere” stretch, but like Hilbert’s Hotel rooms continued to make *more space* by expanding.

    The expanding spatially infinite universe, which may in fact be what we live in, would be an example of an “active” infinity.

    Hilbert’s Hotel seems absurd, but may indeed entail no more contradiction than the expansion of space in an already-infinite universe.


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