Theistic Arguments Series: On the Impossibility of an Actual Infinite
One of the key concepts found in some theological arguments such as the Kalam Cosmological Argument is the assertion that the universe must have had a beginning, which is based on the notion that it is logically impossible for an "actual infinite" to exist. Theists have made numerous defenses of this assertion. It appeared in my previous post, where my interlocutor said:
This is a metaphysically untenable position. Why can't there be an eternal succession of people? Well, person one (p1) is going to have to give birth to them-self before they can give birth to p2. How can p1 give birth to them-self if they don't exist. Since contingent p1 can't be accounted for contingent p2...pn are not accounted for, and so the whole chain fails to exist.He seems to follow the logic of William Lane Craig, who has been one of the most prominent defenders of this position, as seen here. In this paper, Craig argues in support of the Kalam argument on two main fronts: that the existence of an actual infinite (whether it be a series of events, objects, or instances of time) is logically absurd, and also that physics does not allow for any kind of infinite succession of cosmological beginnings. It is my intention here to address only the first of these. Craig gives two main arguments against the existence of an actual infinite. One is that it is impossible for an actual infinite (and specifically, an infinite temporal regress) to exist, and the other is that an actual infinite cannot be formed by successively adding elements to a set.
Let me clarify a few term before I go on.
First, when speaking of a beginning of the universe, it is my contention that our known universe had a beginning (of sorts), with the big bang. But that doesn't mean that there could not be something (other than God) that preceded or caused it - some larger natural reality from which the known universe arises. Examples of this might be a multiverse, or it might be an infinite succession of cosmological epochs in which a universe is spawned, lives for some finite time, and then collapses, only to be spawned again. So when I postulate that there might be no beginning, I mean that there might be no beginning to this larger natural reality, not to our specific known instance of a universe. This is an idea that Craig specifically rejects. In his view the universe that we know is all there is in the natural world.
Second, the term "actual infinite" refers to a set or series of things that are instantiated, or that have actual physical existence. So while it may be acceptable to Craig to speak of an infinite set as a purely theoretical thing, there can be no real infinite set of existing things, including moments of time.
Craig's first argument against an infinite regress is based on the apparent absurdities that arise in the hypothetical Hilbert's Hotel.
What exactly is meant by "infinite"? In mathematics, there are different kinds or orders of infinity, and they are not the same size. For example, the set of all rational numbers is of a different (and larger) order of infinity than the set of all counting numbers. But with respect to arguments such as Craig's, we are talking about something called "countably infinite". This does not imply that it is possible for people to actually count all the elements of a countably infinite set. It does imply that there is a method of pairing all the elements of a countably infinite set in one-to-one correspondence with the set of counting numbers: [ 1, 2, 3, ... ]. Any such set is said to be the same size as any other countably infinite set. Even if you add or subtract elements from it, it is still possible to pair a set like this with the set of counting numbers in perfect one-to-one correspondence, as we see in this example where the first ten elements have been removed.
[11, 12, 13, ... ]
/ / /
[1, 2, 3, ... ]
This is a concept that Craig can't seem to grasp, as is evident in his discussion of Hilbert's Hotel. Unfortunately for Craig's analysis, the same arithmetic operations that are intuitively logical for finite numbers are not valid for infinite numbers. When working with infinite numbers, our intuition fails us. Infinity is not a definite quantity like the number 57, for example. It can't be treated as if it were a definite quantity. As mathematicians will attest, it is not logically valid to attempt to subtract some number of elements from an infinite set and obtain a definite difference. Attempting to do so will inevitably result in absurdities, as Craig notes:
But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.The absurdity is not due to the impossibility of the existence of infinite sets, it is due to Craig's faulty understanding of trans-finite arithmetic, and his faulty application of arithmetic logic.
Craig's second argument states that an actual infinity cannot be formed, because one cannot "reach actual infinity" by "successively adding one member after another." He further argues that this isn't dependent on the amount of time available. Craig refers to this as the impossibility of "traversing an infinite set".
At its heart, this boils down to Craig's self-contradictory use of the term 'infinite'. On the one hand, he acknowledges that an infinite series is an open-ended set - it lacks a bound, at least on one side. On the other hand, he speaks of "reaching" infinity, as if it were a definite value that corresponds to a bound at the far end of the set. The very term to 'traverse' a set implies that it is a closed set, bounded on both ends, with the implication that you can begin at starting point at one end, and then arrive at the other end of the set by traversing all the elements. Craig speaks about "counting down" moments of time from negative infinity to the present.
suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., -3, -2, -1, 0. We could ask, why did he not 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 by then.Notice that this implies both a starting and an ending point (even if Craig doesn't admit it) - a set of moments that is bounded on both ends. To see the problem more clearly, recall the definition of an "infinite" set, which is unbounded, at least on one end. Consider that in order to follow the correspondence of moments to the set of natural numbers, there is only one 'end' of the set at which you could begin enumerating this correspondence. Craig wants to start at the far 'end', even while acknowledging that there is no far end. But this violates the definition of infinity. It isn't good enough for him to simply say that the set contains an infinite number of elements. When he speaks of traversing the set from one end to the other, he is using contradictory terms. Again, he is treating infinity as if it were a definite quantity. Notice also that my interlocutor's argument, cited at the beginning of this post, follows this same pattern - speaking about an infinite series that begins at the far 'end'.
As an aside, one thing Craig never explains is why it must be necessary for an infinite set to be "traversed" at all. Or why it must be "formed" by successive additions. Why couldn't it simply exist?
So Craig concludes that something cannot exist for an eternity, because that would imply an infinite set of moments in time. It is metaphysically impossible for the universe to exist without a beginning, since that would imply the existence of an actual infinite. But if that were true, wouldn't the same logic apply to God, who is supposed to have no beginning in time? Why should it be the case that an actual infinite is impossible it it refers to the existence of the universe, but not impossible if it refers to the existence of God? Most theists agree that God can't do what is logically impossible. So if it is logically impossible for something to exist without beginning, that must apply to God as well.