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#12 Forming an Actual Infinite by Successive Addition

July 09, 2007

Dear Dr. Craig,

I have a question concerning one of the philosophical arguments you offer in support of the view that the universe began to exist, namely the argument from the impossibility of forming an actual infinite by successive addition. You set up the argument as follows:

1. A collection formed by successive addition cannot be actually infinite.
2. The temporal series of past events is a collection formed by successive addition.
3. Therefore, the temporal series of past events cannot be actually infinite.

This argument exposes a feature in the notion of an infinite series of events that I find bewildering. To set the situation up, we’ll assume the past is infinite. By virtue of a tensed conception of time, every event in the infinite past up to the present was a real event that had to be “lived” through. But, if that’s the case, how could all those events have been lived through, one by one, up till now? Just how, exactly, could we reach the end of that beginningless series? How could the present event arrive if, before it could arrive, an infinite number of previous events had to arrive first?

Like I said, this seems very puzzling. But I can’t quite put my finger on why. Is it simply that, on an intuitive level, I find the idea of traversing a beginningless series absurd? As you wrote in your reply to John Taylor, “The question is whether an infinite series of events, having no beginning and having an ending in the present, is metaphysically possible given a tensed view of time. Intuitively, this does not seem possible, for it seems that the present event could not arrive if its arrival had to be preceded by the successive arrival of an infinite number of prior events.” [“A Swift and Simple Refutation of the Kalam Cosmological Argument?” Religious Studies 35 (1999): 57-72. Footnote 26] This is exactly what impels me to accept the argument. But is there a way to analyze our intuition more deeply to find out exactly why such a traversal is impossible? Or is it somehow non-analyzable?

The “traditional objection” to this argument is that it is only impossible to traverse infinity if one begins at some point. But, whatever this reply manages to do, it doesn’t seem to rebut our intuition or reduce the apparent absurdity engendered by the situation; after considering the objection, I’m still genuinely perplexed as to how such a traversal could happen.

What underlies our intuition isn’t the argument that, for every number one counts, another number can always be counted before reaching infinity--is it? For this does seem to be susceptible to the traditional objection. Since, as Wes Morriston [“Must the Past Have a Beginning?” Philo, Vol. 2 (1999) no. 1, 5-19.] et al have pointed out, this observation seems only to involve counts that begin at some point, it doesn’t seem like it could be effectively used to support the notion that a beginningless series of events is impossible. Is our intuition dependent on or independent of this observation?

(Note: I’ve been considering this argument in its “bare bones” form only, leaving separate, e.g., discussion of the Tristram Shandy paradox(es). I wanted to see if the argument could be defended without the use of such puzzles and thought experiments.)


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Dr. craig’s response


Well, Michael, I obviously share your intuition!  Basically, you’re asking about the warrant for premiss (1).  You find that as you reflect on the idea of forming an actually infinite collection of things by successive addition the task seems impossible.  You want to know if we can unpack this intuition a little more to see why such a task is impossible.

In the case of beginning with some finite quantity and adding finite quantities to it we can pinpoint the problem clearly:  since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever.  Infinity in this case serves merely as a limit which we never attain.

What becomes truly puzzling, even mind-boggling, is the suggestion that we can, by adding only finite quantities, form an infinite quantity or collection--say, a certain collection of baseball cards--by  never beginning but ending at some time!  Here the impossibility cannot be analyzed as due to the impossibility of adding finite quantities to finite quantities and getting an infinite quantity, for in this case the quantity to which finite additions are being made is always and already infinite.  We are successively adding finite quantities to an already infinite quantity, so of course the sum is an infinite quantity.  Here infinity is not functioning as a mere limit but as a collection of concrete elements.

Now notice that one still hasn’t explained how we are able to form our infinite collection of baseball cards by successive addition.  For at any time in the past the collection is already infinite, and yet the total collection has not yet been formed.  The total collection will not be formed until the last card is added.  From any point in the past one need add only a finite number of cards to complete the collection.  But that leaves unsolved the problem of how the entire infinite collection could have been formed by successive addition.

Here’s the problem, it seems to me:  in order for the collection to be completed, we must have already enumerated, one at a time, an infinite number of previous cards. But before the final card could be added, the card immediately prior to it would have to be added; and before that card could be added, the card immediately prior to it would have to be added; and so on ad infinitum. So one gets driven back and back into the infinite past, making it impossible for any card to be added to the collection.

This way of putting the argument is somewhat akin to Zeno’s argument that before Achilles could cross the stadium, he would have to cross half-way; but before he could cross half-way, he would have to cross a quarter of the way; but before he could cross a quarter of the way, he would have to cross an eighth of the way, and so on to infinity.  Therefore, Achilles could not arrive at any point.  Zeno’s paradox is resolved by noting that the intervals traversed by Achilles are potential and unequal.   Zeno gratuitously assumes that any finite interval is composed of an infinite number of points, whereas Zeno’s opponents, like Aristotle, take the interval as a whole to be conceptually prior to any divisions which we might make in it.  Moreover, Zeno’s intervals, being unequal, add up to a merely finite distance.  By contrast, in the case of an infinite past the intervals are actual and equal and add up to an infinite distance.

About the best that the critic of the argument can do at this point, I think, is to say that if one adds cards at a rate of, say, one card per second, then the collection can be completed because there has been an infinite number of seconds in the beginningless past.  But clearly this response only pushes the problem back a notch:  for the question then is, how can the infinite collection of past seconds be formed by successive addition?  For before the present second could elapse, the one before it would have to elapse, and so on, as before.  Because the problem is applicable to time itself, it cannot be resolved by appealing to infinite past time.

Of course, proponents of a static or tenseless theory of time will deny that moments of time really do elapse, but then their objection is actually to premiss (2), not premiss (1).

If one is not yet convinced by this argument, then I would offer a further defense of premiss (1) by arguing that if an actual infinite could be formed by succesive addition, then various absurditites would result.  Consider the scenario imagined by al-Ghazali of our solar system’s existing from eternity past, with the orbital periods of the planets being so co-ordinated that for every one orbit which Saturn completes Jupiter completes 2.5 times as many.  If they have been orbiting from eternity, which planet has completed the most orbits?  The correct mathematical answer is that they have completed precisely the same number of orbits.  But this seems absurd.  Think about it:  the longer Jupiter and Saturn revolve, the greater becomes the disparity between them, so that they progressively approach a limit at which Jupiter has fallen infinitely far behind Saturn.  Yet, being now actually infinite, the number of their respective completed orbits is somehow magically identical.  Indeed, they will have “attained” infinity from eternity past:  the number of completed orbits is always the same. So Jupiter and Saturn have each completed an infinite number of orbits, and that number has remained equal and unchanged from all eternity, despite their ongoing revolutions and the growing disparity between them over any finite interval of time.  This strikes me as nuts.

It gets even worse.  Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0.  We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity.

So in defense of premiss (1), I offer both the direct argument against the possibility of forming an actual infinite by never beginning but ending at a point and indirect, reductio arguments that if premiss (1) is denied, then various absurdities follow.

- William Lane Craig