The blog of Dr Glenn Andrew Peoples on Theology, Philosophy, and Social Issues

Turnabout on the "actual infinite" argument?


In an interesting turnabout, Fancois Tremblay has sought to turn William Lane Craig’s use of the kalam Cosmological argument against him.

Part of the argument, namely premise 2: “The universe began to exist,” draws on, among other things, Craig’s argument against the possibility of an actual infinite (it also draws on the empirical scientific evidence that the universe did begin to exist). Basically, the argument is that if the past is infinite in duration, then an infinitely large number of days (or months, or years, take your pick) has been traversed. But since an infinite number of things cannot actually be traversed, the past cannot be infinite.

Temblay has accepted this premise, but he thinks it is damaging to the Christian faith traditionally expressed, or to a number of faiths, I suppose. Why? Because, says Francois,

This is where I must now part ways with William Craig. While his argument against infinite regress is reasonable, we also have to contend with his belief in an infinite god. And we have to ask, what does it mean for a god’s knowledge, power, benevolence, and presence to be qualified by “infinite”?

Since an actual infinite cannot exist, says Francois, an infinite God cannot exist either, and so if there is a God, he must be finite.

The main question I would want to ask is who the argument is supposed to be directed against. Are there any philosophers of religion who defend a concept of Theism whereby God’s knowledge is said to consist of an infinite number of propositions? And if there are, does it do any harm to traditional Christian theism to show that they are wrong? After all, all it takes is a simple internet search for the words “God,” “knowledge,” “propositional” and “intuitive” to find out that omniscience doesn’t have to be viewed as having an infinitely large collection of true beliefs. God’s knowledge, as Aquinas noted, can be (and should be) construed as intuitive, rather than propositional or discursive. Likewise, omnipresence need not be defined in terms of being present in an infinite number of places (in fact if God is literally not extended in space, it is just obvious that this is not what it means), and so forth.

Turnabout is fair play, but in this case it did not pay off.

Glenn Peoples


James, the brother of Jesus and son of Joseph


Ed hits it on the head


  1. Matthew Flannagan

    There is an exchange between Alvin Plantinga and Patrick Grim on this matter that was published in a journal (it slips my mind) but was republished on the web.

    From memory, it evolved around the point Grim made that an infinite set of proposition is impossible and so omniscience is impossible. Plantinga responded by asking why omniscience requires an infinite set of propositions to exist. The discussion proceeds from there.


  2. Derek

    I get your point, but I think Tremblay’s argument falls down even before that. Craig’s argument simply says that “time” cannot be an actual infinite, as it is traversed, and an actual infinite cannot be traversed. It doesn’t seem to say anything about the existence of an actual infinite that is not traversed. Unless Tremblay could show that God Himself is traversed, I don’t see him as having much of a foothold. It’s pretty clear that the “actual infinite” argument only applies to things that began to exist.

  3. Derek, I think Tremblay’s argument falls down because of his equivocation on the word “infinite.” Craig does argue against the possibility of an actual infitine, and not merely an infinite time. It’s because there can’t be an actually infinite number. But when Tremblay uses this to argue against an “infinite God,” he’s no longer talking about an infinite number of things – or at least if he is (e.g. an infinite number of beliefs that God holds), then he just doesn’t understand a non-platonic concept of God.

  4. Ilíon

    Propositions are not physical or spatial entities.

    There is no problem with having infinities (numbers, for instance), per se. It is only when we try to translate infinity into time/space/material entities that the problem arises.

  5. Ilíon

    At the same time, there is nothing in Christianity that says or requires God to hold an infinite number of propositions.

  6. Ilion, the problem of infinity doesn’t exist because of the physical nature of the things in question. It’s a logical problem. It’s a problem for numbers for exactly the same reason that it’s a problem for apples.

  7. Joshua

    Craig has explained this objection before. He does not object to something being infinite, he objects to an actual infinite being reached by adding individual components up progressively. If the claim was that God acquired infinite power by successive periods of bench-pressing or acquire infinite knowledge by reading an infinite number of books, then he would never actually have acquired this infinite power or knowledge. But that is not the claim of classical theism, nor Christianity. God has always been omniscient, omnipotent, etc…

  8. Matt

    Very simply put Trimblay has confused the quantitative with the qualitative. An infinite number of particulars hence an infinite regress of time is quantitative (impossible for a infinite) while the attributes of God are infinite in a qualitative (possible) sense.

  9. Andrew

    Bill Craig has a great response to this argument. Glenn is right to point out that there’s a not so subtle equivocation going on here, but I don’t think he’s stated it clearly.

    Craig simply make a distinction between qualitative and quantitative infinity. Tremblay’s equivocation lies in the fact that he fails to distinguish these two points and assumes that God has constituent parts (something which no theist worth his salt is going to accept (since God is a non-object).

    Also, @ Matt,
    I should regard it as remarkably silly of Brim to make the assumption that God could only ever have propositional knowledge. Why could it not be the case that God’s knowledge is simply a holistic intuition of reality?

  10. “Craig simply make a distinction between qualitative and quantitative infinity.”

    Andrew, that’s a different response than the one I gave (which may explain why you think I didn’t state Craig’s response clearly – I didn’t state it at all!) My response was that omniscience does not consist of an actually infinite number of propositions, ergo it is not subject to the objections associated with an actual infinite. The distinction isn’t between qualitative infinity vs quantitative infinity (the latter of which I think is a terrible unclear concept), but rather between propositional or discursive knowledge vs intuitive knowledge.

    Similarly, my response about omnipresence does not depend on the quantitative vs qualitative distinction. Instead, it was that God is not spatial, it’s mistaken to think of him being physically present in all of physical space.

    Truth be told, I don’t think it’s helpful to speak about qualitative “infinity,” for infinity really just refers to the idea of numbers never ending. I think we should instead speak simply of God being “perfectly X” rather than “infinitely X.” I don’t think using the concept of infinity here brings clarity at all.

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