The impossibility of an actual infinite has been defended in the last few decades by William Lane Craig, in his book The Kalām Cosmological Argument, his general apologetics book Reasonable Faith, and numerous philosophical articles, some of which were republished in Theism, Atheism, and Big Bang Cosmology, a book he co-wrote with atheist philosopher Quentin Smith. Most of this post is just paraphrases of Craig’s writings. In Reasonable Faith, Craig points out that there is a distinction between an actual infinite and a potential infinite.

A potential infinite is a collection that is increasing toward infinity as a limit but never gets there. Such a collection is really indefinite, not infinite. For example, any finite distance can be subdivided into potentially infinitely many parts. You can just keep on dividing parts in half forever, but you will never arrive at an actual ‘infinitieth’ division or come up with an actually infinite number of parts. By contrast, an actual infinite is a collection in which the number of members really is infinite. The collection is not growing toward infinity; it is infinite, it is ‘complete.’

In other words, an infinite amount of defined units is an actual infinite. An “infinitieth” of something is not a defined unit, so this is a potential infinite. An actual infinite is usually signified by aleph-null, but I can’t figure out how to type that in blogger, so I’ll just use ∞ instead, even though it usually represents indefiniteness rather than infinitude.

While actual infinites are used in conceptual mathematics they do not have any corresponding reality.

1. The most famous illustration of this is Hilbert’s Hotel (named after mathematician David Hilbert). Imagine a hotel with two wings that stretch out infinitely in opposite directions and which therefore contain an infinite number of rooms, and imagine that they are all occupied, that is, the hotel is completely full. Somebody shows up and asks for a room. In a finite hotel the proprietor would have to turn him away, but in an infinite hotel the proprietor could just move the person in room 1 to room 2, the person in room 2 to room 3, the person in room 3 to room 4, etc. Now room 1 is open and the person can check in. But the hotel was already full. Each room was occupied. Moreover, the same number of people are in the hotel even though no one has left and there is one more person than before. This is true for any finite number: if a million new people checked in, you could just move the person in room 1 to room 1,000,001, etc.

A friend of mine once suggested that if the hotel is infinite, then there would be no outside for someone to come in from. This is incorrect. We can easily imagine that the hotel extends infinitely in two directions along a street or something. Someone on the other side of the street (perhaps in the infinite restaurant) could then cross the street and check in.

2. But what if an infinite amount of people come to check in? Does the proprietor tell them that the hotel is full and turn them away? No, he just moves the person in room 1 to room 2, the person in room 2 to room 4, the person in room 3 to room 6, etc., moving each person to the room number double their previous number. He thus empties all the odd numbered rooms and the infinite number of new guests can check in. But before they came, each room was occupied. And again, there are the same number of guests as before, even though the proprietor just increased his occupancy by an infinite amount. And he can do this again and again, in fact infinitely many times, and there would never be one more person in the hotel.

3. Another illustration would be a bookcase with an infinite number of books. If you took seven books off the shelf there would be an empty space where they had been. But there’s still an infinite amount of books left, so therefore the bookshelf is still completely full with no empty spaces.

4. Conversely, if you took out all of the books but seven, you would have taken an infinite amount of books off the shelf. However, if you took an infinite amount of books off the shelf, it would be completely empty; but it’s not, there are seven left.

5. Now what if you took out every other book? That would leave a space between each remaining book. Although you took away an infinite number of books, there’s still an infinite number of books remaining, since there’s an infinite amount of odd numbers and an infinite amount of even numbers. The absurdity of this last example can be realized by imagining that you take the first book and push it up against the third book, so there isn’t a space between them any more. Then push these two books up against the fifth, and all of these against the next one, etc. Then there will be an infinite amount of empty space on the bookshelf (from the infinite number of books taken off it), so it’s necessarily empty. But there are still an infinite number of books left on the shelf. As such, it’s necessarily full. But if the bookshelf is full, the first book would be where it’s always been, even though you just moved it. The bookcase is simultaneously completely empty and completely full. When you look at this bookcase, what would you see?

Obviously, Hilbert’s Hotel and a bookcase like this could not exist in reality. Yet, if an actual infinite amount could exist, they could exist as well. Arguments 1-5 can be reduced to the following mathematical equations (where X and Y are actual amounts, that is, any number greater than zero):

1. X + Y ≠ X but ∞ + Y = ∞

2. X + X ≠ X but ∞ + ∞ = ∞

3. X - Y ≠ X but ∞ - Y = ∞

4. X - X = 0 but ∞ - ∞ = Y

5. X - X ≠ X but ∞ - ∞ = ∞

Note also that 4 and 5 contradict each other: infinity minus itself equals both infinity and an actual amount.

A friend of mine once complained about these arguments by saying that when I add to or subtract from infinity I was treating it like an amount, and this is invalid. That is precisely the point: an amount, by definition, can be added to or subtracted from. Since we cannot do this with infinity, there cannot be an actual infinite amount of defined units. These arguments simply demonstrate this. As Craig writes,

There is simply no way to avoid these absurdities once we admit the possibility of the existence of an actual infinite. Students sometimes react to such absurdities as Hilbert’s Hotel by saying that we really don’t understand the nature of infinity and, hence, these absurdities result. But this attitude is simply mistaken. Infinite set theory is a highly developed and well-understood branch of mathematics, so that these absurdities result precisely because we do understand the notion of a collection with an actually infinite number of members.

Some might object that God is often referred to as infinite. Doesn’t this disprove God’s existence since an actual infinite can’t exist? It does not for the following reason: what makes an actual infinite impossible is that it consists of an infinite amount of units or members (like books or hotel rooms). God is not “made up” of any amount of units. In saying that God is infinite, we are saying he is unlimited by anything. Since God does not consist of a bunch of units, the argument against an actual infinite amount of units existing does not apply to him.

## 2 comments:

Excellent blog, added to my bookmarks list (came from Just Thomism). I have a concern about an argument against actual infinities made by William Lane Craig, whose work I admire generally, but whose specific argument in this instance is faulty, as far as I can tell, which argument you seem to using in your post. The problem is with the argument about infinity minus infinity leaving a leftover quantity, thereby leading to a contradiction.

Roughly, Craig seems to be confusing countability with quantity. The infinity of the natural number system in Cantor is aleph-null, the countable, not additive, infinity of the natural number system. The infinity of the even numbers is also aleph-null, hence the weirdness. But when you say the infinity of the natural numbers is the same quantity as the infinity of the even numbers, you simply mean they have the same number of members. You can't subtract members in one set from members in another simply because they might be the same number. You get nowhere by subtracting the evens from one another, with the odds somehow leftover. You simply tick them off, even to odd, next even to even, next even to next odd, next even to next next even, and so on, one-to-one.

Craig makes the mistake of lining up two sets of numbers so that like members are next to like members, (if I remember his example correctly, he has one set as 1,2,3,4, the second as 4,5,6,7, both of course heading to infinity). He then creates a specious subtraction relation between the numbers in each. He seems to think that he has accomplished something by subtracting 4 from 4, leaving 0, 5 from 5, leaving 0, etc, as if that somehow locks or freezes out 1, 2 and 3 like wallflowers at a dance when everyone else has partnered up. But that's just not what the infinity minus infinity thing is about. It's about counting out one-to-one. 4 is the first element in the second set, and is paired up with 1, the first element in the first set, 5 with 2, etc. You don't even have to pair them up in the order they occur, as long as for every one in one set, you can point to one in the other, ad infinitum.

To prove this point, just remake set one into an infinitely long list of something like names. So you've got Joe, Ron, Janet, Bill, Susan, etc., and 4, 5, 6, 7, etc. What's the quantity of each set? Aleph-null. But you won't make the mistake of thinking you're making progress by subtracting 4 from Bill just because Bill is the fourth item in the sequence, and 5 from Susie because she's number five, and so on. You'd realize right away it's all about matching up item with item in a counting exercise. If for every name, I can point to a number, and every number to a name, we've got a match.

I don't know how strong Craig's other arguments are, but that one just does not seem to work well at all. I say this as an admirer of his work. I think his position would be stronger if he removed this specific argument.

And this is a very intelligent blog. I look forward to any flaws in my point you can discover. If you can't, I'm going to write to Craig to advise him that his arguments against actual infinities would be strengthened with this particular weak link removed. My sympathies are with him enough that I would not want to see someone use this weakness in his argument against him. I think he's generally right about actual infinities, just not with his leftover quantity argument.

Hi Patrick, thanks for your comment, and sorry to take so long to respond. The problem you mention wouldn't just apply to Craig, but to David Hilbert as well (I think).

I don't get the distinction you're talking about. If we had an aleph-null number of people all lined up, and then had the first five people step out of the lineup, there would still be an aleph-null number of people left, as far as I can tell. If we reversed this so that everyone

exceptthe first five people stepped out of the line, then you would have an aleph-null amount minus an aleph-null amount equalling a finite amount.This is probably just me not getting it. I'm not a mathematician by any stretch. But as you suggested, I would encourage you to e-mail Craig and raise it with him. On his Reasonable Faith website (linked in the sidebar) he answers one e-mail a week, and I think your question is certainly worthy of further clarification on his part.

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