Earlier this week, my brother Chris sent me a Facebook question posed by my preschool-aged nephew: "What is infinity minus infinity? What is zero minus infinity?" I'm sure most of us at some point engaged in the one-upmanship game of making bigger numbers than our sibling or friend.
And then your brother makes the leap!
Brian: Infinity + 1!
Chris: That's just infinity. There's nothing bigger than infinity. I win!
No fair! What's up with this? Up to the point where we leap into the infinite, we are dealing with numbers. As it happens, infinity is not a number. Some people like to say, "Infinity is not a number, it's a concept." But this isn't very helpful at all. What good does it do to say it's a concept? What is a concept anyway? Or for that matter, what is a number?
Another of my Facebook friends relatively recently introduced me to a TED video that explores the cardinality idea of infinity. That is, cardinality is about sizes of sets, and natural numbers (positive integers) or counting numbers are exactly the type of numbers that measure cardinality. The idea of infinity in relation to cardinality corresponds to the idea that you have an unending number of elements in a set.
Bizarre things happen in the cardinality sense of infinity. For example, if you take all of the positive integers (1, 2, 3, 4, ...) and double each number (2, 4, 6, 8, ...), we still have the same number of elements in the set (since we just manipulated each object). But our new set happens to be a subset of the original. When dealing with finite sets, a proper subset always has fewer objects than the original set. But we just saw an example where a proper subset has an equal number of objects (infinitely many). So here, the phrase "number of objects" does not represent an actual number, but represents the concept of cardinality.
Alternatively, we could have started with all of the positive integers (1, 2, 3, 4, ...) and then just deleted every other number in the list. This also gives us (2, 4, 6, 8, ...) since the odd numbers were all removed. This is one way of showing that an infinite set taken away from an infinite set can still be infinite. This is an example where ∞ - ∞ = ∞. On the other hand, if we start with (1, 2, 3, 4, ...) and then take away the infinite collection of numbers (11, 12, 13, 14, ....), we are left with (1, 2, 3, 4, 5, 6, 7, 8, 9, 10). This would be an example where ∞ - ∞ = 10. In fact, by choosing how many numbers we want to leave and then just deleting all of the rest, it is easy to create examples where ∞ - ∞ = n for any integer represented by n.
The point here is that a cardinality interpretation of subtraction (removing elements from a set, like taking candy pieces out of a pile) reveals that the cardinality of ∞ (not a number) does not follow ordinary arithmetic rules. The infinite does that; it breaks the ordinary rules we are comfortable with when dealing with finite things.
In mathematics, we say ∞ - ∞ is indeterminate because the result actually depends on how the subtraction takes place.
A cardinality interpretation of 0 - ∞ does not actually make sense. Interpreting 0 - ∞ first requires an extension of the idea of numbers to negative numbers. For example, my children want to tell me that 2 - 5 = 0 because if you start with 2 candies and try to take 5, you don't have any left. But at first they don't realize that there are 3 candies that you never got to take.
We could introduce the idea of borrowing (loans). Suppose that I have 2 candies in my bowl and my daughter wants to eat 5 candies. If I want to give her 5 candies, then I can give her my 2 candies but then I'll need to get 3 more candies from someone else. This puts me in debt for 3 candies (-3) which I might represent by little paper IOUs. I have 3 IOUs. I can pay off the IOUs when I obtain candies. For each candy I receive, I pay off an IOU.
This is the idea of addition extended to all integers. -5 + 2 corresponds to have 5 IOUs and 2 candies. The 2 candies pay off 2 IOUs, leaving me the same as if I had only 3 IOUs to begin with. So -5 + 2 = -3. Subtraction is normally thought of as actually taking candies away. That is 5 - 3 corresponds to having 5 candies and taking 3 away, leaving only 2. The extended idea of subtraction is to think of having 5 candies and using 3 IOUs: 5 - 3 = 5 + -3. Taking away the candies is equivalent to redeeming IOUs.
That is, once we start dealing with negative numbers, subtraction is really about adding negatives (redeeming IOUs). The value -∞ simply means that we have an infinite number (there's that cardinality idea again) of IOUs. So 0 - ∞ is really the idea 0 + -∞, which means we start with a pile of 0 candies and an infinite number of IOUs. Since we can't redeem any of the IOUs, we still have infinitely many. That is, 0 - ∞ = -∞. By the same argument, 5 - ∞ = -∞ and ∞ - 5 = ∞.
In addition, this gives us a way of extending our earlier idea of subtraction for ∞ - ∞ to end with any possible answer from -∞ to ∞. The extended cardinality approach to ∞ - ∞ would mean that we have infinitely many candies and infinitely many IOUs. If we put the candies and IOUs each in some order (an interesting philosophical question is if this is always possible), then we can make a choice on how we redeem our candies.
We might redeem every IOU but occasionally (or regularly) skip some of the candies. For example, the first IOU could take the 1st candy, the 2nd IOU takes the 3rd candy, the 3rd IOU takes the 5th candy, and so on, leaving candies 2, 4, 6, .... This would correspond to ∞ - ∞ = ∞. Or we could redeem only some of the IOUs but use all of the candies. For example, the 1st candy redeems the 1st IOU, the second candy redeems the 3rd IOU, and so on, leaving infinitely many IOUs unredeemed. This corresponds to ∞ - ∞ = -∞. Or we could leave the first 12 IOUs unredeemed and then redeem each subsequent IOU by the subsequent candies, corresponding to an example of ∞ - ∞ = -12.
It is essential to see that when we deal with arithmetic involving ∞ or -∞, we are not dealing with numbers. For cardinality, we are actually dealing with correspondences between sets. ∞ - ∞ does not adequately describe the correspondence. But 12 - 5 is adequate because every finite correspondence results in the same final answer. But for infinite correspondences, the nature of the correspondence itself makes a difference. Indeterminate really is indeterminate.
Edit: I had a colleague point out to me that there is a second sense of infinite using ordinal numbers. In this setting, it is possible to talk about "Infinity plus one" as having meaning that is actually distinct from "Infinity". For now, you might want to look at the wikipedia page, but I'm working on my own response as well. (October 8, 2013)