Monday, March 28, 2011

The Kalam and the Infinite

The kalam cosmological argument’s second premise is “the universe began to exist.” While modern science is virtually united in this premise’s affirmation, some philosophers of science have taken to denying this premise. Philosophically, there are arguments demonstrating an actually infinite number of things cannot be instantiated in reality. In this case, we will examine the case for an infinite amount of time. If time is not infinite, then the universe is not infinite.

William Lane Craig puts the argument this way:

1. The series of events in time is a collection formed by adding one member after another.

2. A collection formed by adding one member after another cannot be actually infinite.

3. Therefore, the series of events in time cannot be actually infinite.[1]

(1) is easily defensible on an A-theory of time.[2] Given that, we should turn to (2). The idea is that if time is infinite, then an infinite number of moments have already passed. If that is the case, then the present moment should have already arrived. Think about that for a moment.

In fact, one may inquire as to why every moment has not already arrived, since an infinite number of moments encompasses any moment. From any moment in the past that is chosen, an actually infinite number of moments would have already passed, so that every moment should have already come and gone. But this leads to an even greater absurdity: if every moment were added to every other moment to form the successive collection of an actually infinite number of moments, no moment could ever arrive! The present moment, however, has indeed arrived. Therefore, the forming of the set of each successive moment to the other cannot be actually infinite.

Some counter with the objection that this ignores Cantor’s set theory, in which it is possible, as a whole, to have a set with one-to-one correspondence which is actually infinite. Craig dispels this well when he writes,

But on an A-theory of time the universe does endure through successive intervals of time. It arrives at its present event-state only by enduring through a series of prior event-states. So before the present event could occur, the event immediately prior to it would have to occur . . . and so on ad infinitum. . . . Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd.[3]

A somewhat less-abstract example could commend itself in the following example. Suppose you had a shovel and you wanted to fill a hole. Now this particular hole is a bottomless pit. Fortunately, you are supplied with an infinite amount of time and an infinite amount of dirt. Were you to start filling the hole, when would your task be complete? The answer is obviously “never.” For no matter how much dirt was successively added, you would never fill the hole completely because the hole is bottomless; it doesn’t matter if you filled the hole with two tons of dirt per day every day for a trillion years; you’d never be any closer to completion than you were the moment you started! This is highly absurd.[4]

Finally, consider a final example involving the orbits of two planets. Planet A completes 2.5 orbits for every orbit completed by Planet B. It takes B one year to complete this orbit. If they both start at the same time, how many orbits will A and B have after ten years? The mathematical answer is A=25 and B=10. Extrapolating to 100 years, we know that A=250 and B=100. If these two planets have been orbiting for an infinite but equal amount of time in the past, what are the values of the orbits for each planet? The correct answer is somehow the number of orbits are the same! Even stranger, because of an actual infinite, “the number of completed orbits is always the same.”[5]

How do we apply this apologetically? First, we must use this philosophical defense to show that an actually infinite time has not passed. This means time had a beginning. Second, we must show if time had a beginning, then it is impossible that the universe is beginningless. Third, we must draw our listeners back in to the kalam itself. Pointing out, then, that we must accept “the universe began to exist,” we can couple that with the intuitive idea that “whatever begins to exist had a cause,” and thus conclude the universe had a cause. It is at that point we may discuss God and evidences for Christianity!

                [1] William Lane Craig, Reasonable Faith (Wheaton, IL: Crossway, 2008), 120.

                [2] A discussion of the A-theory vs. B-theory is not in view here, but one should recognize an entirely different argument would commend itself were the B-theory to be true.

                [3] Craig, 122.

                [4] Thanks to a discussion with Max Andrews in which he presented the basic idea of this illustration.

                [5] Craig, 123.

All posts, and the blog Possible Worlds, are the sole intellectual property of Randy Everist. One may reprint part or all of this post so long as: a) full attribution is given (Randy Everist, Possible Worlds), b) all use is non-commercial, and c) one is in compliance with the Creative Commons license at the bottom on the main page of this blog.


  1. Thanks for this!

    I think when discussing this issue it is important to define what we mean by "the universe". Do we mean this, i.e., our universe? Do we mean The Universe of universes, encompassing multiverses (if they exist)? Or, do we mean, "physical reality"? If we mean the latter, then isn't there some wiggle room where the anti-theist can argue, "well, time as such may not have existed, but something physical may well have, which at some point begat time?". Just a thought.

    In my experience, all arguments for theism, while reasonable and persuasive even to the reasonable person, leave some wiggle room for the atheist.

  2. Thanks for commenting Rk! Typically, what is meant just is space and time in reality when we say "the universe." Because of how time is construed both philosophically and physically, "space" as such could not exist without time; space just being that physical material reality we're discussing. But yes, there is always "wiggle room" in any argument of which we're not 100% certain; but certainty is largely a red herring. If anyone is looking for "wiggle room" while accepting the basic idea of premises, they're typically trying to avoid the conclusion more than anything else! :)

  3. Have you seen the articles on the Kalam and the infinite in the last issue of Faith and Philosophy by Craig and Wes Morriston? Very interesting.

  4. I have only had the opportunity of perusing those articles. I find Wes to be quite interesting, as it seems he is always working on another objection to the kalam. For the life of me, I can never figure out why. :) Thanks for commenting!

  5. Hey Randy,
    What do you make of objections from Supersets? I've looked around on the internet and haven't been able to find much material on philosophical debate about supersets disproving the conception of the impossibility of an actual infinite. Though the argument from supersets seems intuitively faulty to me, since when one speaks about "added events" in time, he speaks of equal intervals being added, not each interval being half the other.

    Also, what is your opinion about the infinite divisibility of time?

    Thank you brother!

  6. Hi Evan thanks so much for your comment! To be honest, I don't do much with set theory myself. Are you referring to work done by or based on Cantor's set theory? As I understand it, it is quite intuitively faulty. It works in theory, given its own axioms, but what's to say these axioms would work in reality? They are not logically necessary. Hence, these axioms, which we can employ to stop absurdities like Hilbert's Hotel, or the orbits of two planets rotating at differing speeds from all eternity equaling each other, cannot in reality be utilized. There's just no stopping the absurdities an actual infinite in reality engenders.

    AS to the infinite divisibility of time (or indeed, anything else for that matter), I think we can use both intuition and a thought experiment or two. First, the intuitive: with an object, such as the handle on a baseball bat, we would not say that it is made up of an actually infinite number of, say, wood particles. We wouldn't say this even if an objector responded that for any and every wood particle we produce, we may simply divide it in half and have another. The reason? It would then follow everything was actually infinite! Similarly, with time, it seems absurdly counterintuitive that any second that passes is actually infinite, even though that second can be divided over and over, ad infinitum.

    Second, the thought experiment. Really, it is simply analysis. Infinite divisibility does not get one to traverse an actually infinite number of moments. For in order to do this, the intervals themselves must be progressively smaller as they approach an absolute zero (or Planck time, if you wish). But when added they can only form a distinct and whole interval of time that is not actually infinite!

    Further, let us suppose we grant them an actual infinite in this way (that is, even given what is above, we admit it is an infinite). In this case, however, it does not at all follow the universe did not have a beginning; for in this case an actually infinite number of divided moments passed to form the whole traversed sequence, which, as far as anyone can tell from scientific models, had an absolute beginning! It won't do to claim the argument above as still demonstrating an actual infinite and then abandon the argument when it comes to the beginning of the universe. Great question my friend and brother!


Please remember to see the comment guidelines if you are unfamiliar with them. God bless and thanks for dropping by!