Sunday, April 11, 2010

William Lane Craig’s Argument Against Actual Infinities

William Lane Craig is a prominent defender of the Kalam Cosmological Argument (KCM) for the existence of God. (Nowadays philosophers tend to discuss “arguments for” – rather than “proofs of” – the existence of God.) The basic outline of the KCM is as follows:

1. Whatever begins to exist has a cause of
   its existence.

2. The universe began to exist.

3. Therefore, the universe has a cause of its

Now, there are a lot of points about this argument that could be questioned, and many philosophers have objected to various points of Craig’s argument. But one part that stands out for me, and that seems to have been missed by those attempting to answer Craig, is his discussion of the nature of infinity.

As part of his version of the KCM, he asserts that

2.11. “An actual infinity cannot exist.”

To support this assertion, he offers the example of Hilbert’s Hotel.

Perhaps the best way to bring home the truth of (2.11) is by means of an illustration. Let me use one of my favorites, Hilbert's Hotel, a product of the mind of the great German mathematician, David Hilbert. Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many times and yet there would never be one single person more in the hotel than before.

But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. 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.

He ends with a rhetorical question. As Daniel Dennett reminds us, (Freedom Evolves (2003) p. 8) the rhetorical question usually marks the weakest point in an argument. It certainly does so here, for Craig has offered no proof that the features of Hilbert’s Hotel cause any real problems. In fact, in other writings, Craig has admitted that these counterintuitive properties of infinity do not lead to any logical contradiction. However, he insists that they nevertheless constitute a proof of metaphysical impossibility. But all he offers in support of this claim is the counterintuitive (but not self-contradictory) features of Hilbert’s Hotel.

In fact, Craig has done nothing more than appeal to his own personal incredulity: “I can’t imagine how such a thing could be true, therefore it’s impossible.” This type of appeal is, in fact, a well-known logical fallacy, and as such this part of Craig’s argument falls apart. That is really all that needs to be said to demolish Craig’s argument for the impossibility of an actual infinity; nonetheless, let’s proceed and show how a little imagination can help sort out Craig’s difficulties. Here, then, is a different version of Hilbert’s Hotel, one which will show that, while these unusual features of infinity may seem awkward at first, there is nothing metaphysically impossible about them.

Rather than an infinite hotel, suppose we have an infinite universe containing an infinite number of galaxies. From our location here on Earth, we choose a particular direction in space and imagine extending a ray infinitely outward in that direction. (Note I am talking about a conceptual ray, not anything physical, like a light beam or a particle beam.) Now, for a reasonably dense and random arrangement of galaxies in the universe, this ray will intersect an infinite number of galaxies. Let us label the galaxies by calling the one closest to us along the ray #1, the next galaxy #2, and so forth.

As an aside, note that I am assuming that the property of “intersecting the ray” is a binary one: each galaxy either intersects or doesn’t intersect, and there is a clear dividing line when a galaxy goes from non-intersecting to intersecting.  

Now let us assume that the galaxies of the universe are in motion, in such a manner that on a particular day (say Monday), all the galaxies that were intersecting on Sunday are still intersecting, but one new galaxy has moved into intersection with the ray, in a position closer to us than galaxy #1. When this happens, the new galaxy immediately becomes galaxy #1, the old galaxy #1 becomes galaxy #2, and so forth. We have added one to infinity, and still remain with an infinite number of galaxies.

Now, I don’t think any of what I suggested above is in any way metaphysically impossible. (Apart, possibly, from the assumption of an infinite number of galaxies – but Craig can’t object to that because the impossibility of an actual infinite is what he’s trying to prove.) The process can be repeated infinitely many times: just let all the galaxies drift away from us, so that there is room for another galaxy to slip into the closest position. But this situation is exactly equivalent to Hilbert’s Hotel: the numbers are the “rooms” and the galaxies are the “guests.” So Craig’s first objection to Hilbert’s Hotel is swept away.

Craig also seems disturbed by the idea that, even though we have added one galaxy/guest, we still have “the same number” of guests as before, namely, an infinite quantity. Here Craig seems to have confused the ideas of number and cardinality. He expects that infinite cardinalities can be dealt with in the same way as natural numbers, so that (infinity + 1) is somehow a bigger “number” than infinity. But in this Craig is simply demonstrating his mathematical ignorance: no mathematician thinks that infinite cardinalities obey the rules of addition for finite numbers. (It is well known that there are, in fact, some infinite cardinalities that are bigger than others. You can’t obtain a bigger cardinality simply by adding a few more elements to your set, however. The correct way to get a bigger infinity is to take the “power set” of an infinite set: the set of all subsets of the original set.)

Now imagine that every other galaxy along the ray is moving in such a manner that, on Tuesday, all the even numbered galaxies become non-intersecting.  This is exceedingly unlikely in any actual physical arrangement of galaxies, but not, surely, metaphysically impossible. This is equivalent to having an infinite number of guests check out of the hotel. Clearly, there will still be an infinite number of galaxies that intersect the ray. So we have subtracted infinity from infinity and are left with infinity! So we have disposed of Craig’s second “absurdity.”

Finally, we imagine all the galaxies numbered 4 and higher become non-intersecting (on Wednesday, say) and we are left with just three galaxies that intersect the ray. Here Craig’s objection is about the “number” of guests: he complains that “the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out.” His objection, it seems, is that in one instance (infinity – infinity = infinity) and in another instance (infinity – infinity = 3). Once again, it seems Craig has confused regular addition of integer numbers with operations on infinite cardinalities. It is simply not legitimate, mathematically speaking, to assume that the arithmetic rules for ordinary integers can be extended to rules for infinite cardinalities. As we all learned in math class (or should have learned, anyway), the expression (infinity – infinity) is “undefined.” That’s for a very good reason: the same reason Craig pointed out, that (infinity – infinity) can have different values depending on how the operation is conceived.

In fact, there is one definition of infinite cardinality that runs like this: “A set of elements S is said to be infinite if the elements of a proper subset S' can be put into one-to-one correspondence with the elements of S.” So Craig’s complaint – that the infinite subset of even-numbered galaxies is of the same size as the infinite set of the galaxies themselves – is actually the very definition of infinity.

To Craig’s rhetorical question, “Can anyone sincerely believe that such a hotel could exist in reality?” there is a clear answer: “Yes!” I suspect that Hilbert originally chose the hotel example specifically to emphasize the counterintuitive nature of infinity. But when we change the context, what once seemed absurd now seems perfectly natural. The claimed absurdities of Hilbert’s Hotel are not, in fact, absurd after all. 

(A more recent, and more technical, paper by Craig and Sinclair actually takes note of the definition of an infinite set given above, so it seems he has learned a little more math. But not enough: Craig continues to use the subtraction-of-infinities argument and the Hilbert Hotel "absurdities" argument.)


  1. Nice blog Robert! Keep up the good work.

    I feel like the confusion between number and cardinality is key to this issue. But I need to read a lot more before I feel comfortable enough to comment. Wes Morriston has some of the better articles on this topic. Available at his website.

    I also tend to think that the debates about infinity are a red herring. I find the jump from "there is a cause" to "the cause is God" to be the weakest part of the argument. That's probably because I am most familiar with the free will/agency debate that Craig glosses over in making his argument for a personal cause.


  2. Hi, John D. Thanks for the comment! (I love your blog, by the way.)

    I agree that the jump you mention is the weakest part of the argument. With my math background, I felt more comfortable attacking the mathematical part of the argument. And apart from the existence of God, or even a first cause, I think it's an interesting question whether the universe had a beginning or not. I doubt such things can be settled by pure reason, however. (Or science either, for that matter.)

    Even supposing the jump to God to be justifiable, you still have the question of what sort of a God is implied by the argument. Is this first cause God good, loving, caring, involved with the world? Or cold, distant, removed? There doesn't seem to be anything in the argument to imply one or the other.

  3. Here an argument that does show that the actual infinite leads to a logical contradiction by philosopher Casper Storm Hansen of the University of Aberdeen called "New Zeno and Actual Infinity" which can be found here:


    In 1964 José Benardete invented the “New Zeno Paradox” about an infinity of gods trying to prevent a traveler from reaching his destination. In this paper it is argued,contra Priest and Yablo, that the paradox must be re-solved by rejecting the possibility of actual infinity. Further, it is shown that this paradox has the same logical
    form as Yablo’s Paradox. It is suggested that constructivism can serve as the basis of a common solution to New Zeno and the paradoxes of truth, and a constructivist interpretation of Kripke’s theory of truth is given.