Real Mathematics : Infinities
I want to start by telling you about a very sad group of people: Aggies. An aggie is a student of or graduate of a college known as an agricultural college. There were many of these schools started in the less-settled parts of the US in the 1800s, with the goal of improving the business capabilities of the areas. The businesses were mostly farming and animal husbandry, so that's what the schools concentrated on. They were known as ag schools for short, and their students were aggies. The most prominent of the ag schools is Texas A&M University, which was founded as Texas Agricultural and Mechanical College in 1876. (At least, we Texas aggies think Texas A&M is the most prominent. Other aggies may have a different opinion.)
The thing you need to know about aggies is that we're sorta dumb. For one thing, we can count to 7, no higher. This causes problems because, as you know, everything's bigger in Texas. Listen in on Lester and Lowell, two aggie ranchers: "Les, ah reckon mah herd's biggern yers," drawls Lowell, looking over his cattle, which cover the plains to the north as far as the eye can see. "No Lowell, ah reckon yer wrong," drawls Lester, sitting on his horse on the other side of the fence, and looking at his herd, which covers the plains to the south as far as the eye can see.
Now since neither one of them can count any higher than 7, it would appear that they're doomed to argue forever with no resolution. But aggies, though lacking in arithmetic abilities, have a sort of primitive cunning, and Les and Lowell figured out a way to settle the argument. They'll set up a pair of chutes, side by side. Les will drive his cattle, one at a time, through his chute, as Lowell drives his cattle through his chute at the same rate. One of Lester's herd will be matched with one of Lowell's, and so on until one of them runs out of cattle. The one who still has cattle when the other has run out, has the larger herd.
The comparison of herds the size of Lester's and Lowell's takes a long time, so we'll just leave the two of them there and see what mathematicians can make of their idea of comparing sets of objects that are too large to count. Let's start with the natural numbers and the even natural numbers:
1 2 3 4 5 6 7 8... | | | | 2 4 6 8...
From those two lines, it's clear that there are more natural numbers than even natural numbers, since the even numbers match up with half of the natural numbers, and there is a bunch (an infinite bunch, in fact) of natural numbers left over.
Now let's do something alarming:
1 2 3 4 5 6 7 8... | | | | | | | | 2 4 6 8 10 12 14 16...
It's clear from this matchup that there are just as many even natural numbers as natural numbers. Using Lester and Lowell's method, we run the natural numbers through the chute one at a time, matching each one with the even natural numbers one at a time. We never run out of either set, of course, but it's clear that for every natural number n, there's a matching even natural number 2n, and vice versa.
One more time:
1 2 3 4... | | | | 2 4 6 8 10 12 14 16...
It's clear from this matchup that there are more even natural numbers than natural numbers! Because for every natural number n we found an even natural number 4n, and we have a bunch (again, an infinite bunch) of even natural numbers left over.
Now what are we supposed to make of this? Are there more natural numbers than even numbers, as in the first lineup? Or the same number of each, as in the second? Or more evens than natural numbers, as in the third? Or none of the above? Or what?
Well, it's clear that the intuitions developed for finite sets may not work well for us when we get to infinite sets. Les and Lowell will never find that the first time through, Les's herd is bigger, the second time they're the same size, and the third time, Lowell's herd is bigger. But with infinite sets, we have to face those possibilities, and the way mathematicians deal with them is this: If there is any way that the elements of set X can be matched one-to-one with the elements of set Y — one element of X matched with one element of Y, nothing left over on either side — then those two sets have the same number of elements. And if we can prove that there is no such matchup, then we know that one of the two sets is larger than the other.
You'll come back to that definition if you go on to compare more infinite sets, but meanwhile let's explore what can be done with infinite sets to get a feel for working with them. Let's take a vacation in a remarkable place called the Hilbert Hotel. The remarkable thing about this hotel is that it has an infinite number of rooms. You'd think that such a hotel would have no need for the "No Vacancy" sign in front, but it's a busy place, and one Labor Day the proprietor, David Hilbert, smiled at his wife Käthe after a busy day of checking in an infinite number of guests, and snapped on the No Vacancy sign. Hardly had they settled down to counting the infinite pile of receipts (which have to be shared among an infinite number of employees and pay off the debt of purchasing an infinite amount of furniture) when a car drove up and, ignoring the No Vacancy sign, the driver demanded a room.
Can the Hilberts accomodate him? With the cooperation of the present guests, they can. Käthe gets on the phone and starts leaving messages to all the guests, asking them to move to the next room please: The guest in room n moves to room n+1. Every guest which had a room still has one, and now room 1 is free for the new person.
So when the tour bus drives up and unloads 20 new arrivals, it's clear what to do. Käthe calls each room and asks the guest in room n to move to room n+20. And obviously any finite number of additional guests can be accomodated similarly.
Next, an infinite bus pulls up and unloads an infinite number of new prospective guests. What to do now? David can't just turn them away, there's an infinite amount of money at stake. If you remember the lineup above where we showed that the number of even natural numbers is the same as the number of all natural numbers, you can figure this out easily. Käthe gets busy, calling each room and asking the guest in room n to move to room 2n. Now all the odd rooms are empty, so David can assign the infinite number of new arrivals to those.
Now you may object that Käthe has an impossible job. How can she call an infinite number of rooms in a finite time? Even if it takes her only a hundredth of a second, or a billionth of a second, for each call, that's still an infinite amount of time.
Her secret is her remarkable efficiency. She doesn't start out very fast, but as she warms to her work, each call takes only half as much time as the previous call. Follow along on your watch as she makes an infinite number of calls in only one minute: The first call takes 30 seconds, half a minute. The next call takes a quarter of a minute. Next an eighth, then a sixteenth, and they keep coming, faster and faster, an infinite number of subdivisions as the second hand sweeps back toward the top — and when it hits 12 again, Käthe has completed her infinite task.
In arithmetic terms,
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1
That is, an infinite number of tasks, each taking a nonzero time, can be accomplished in a finite time, if the time needed for each task decreases quickly enough. (This fact explains the so-called paradoxes of the ancient philosopher Zeno of Elea, which you can read about on the net here and elsewhere.)
You can play around with these ideas. You might want to figure out what David and Käthe will do if an infinite number of infinite buses arrives at an already-full hotel. But don't get complacent about having this infinite set stuff figured out. There are plenty more surprises in wait. For example, from what we've seen about natural numbers and even natural numbers, it's tempting to conclude that infinite sets are all the same size — infinite, in fact. But in one of the most amazing developments of modern mathematics, Georg Cantor showed in the late 1800s that infinity comes in various sizes. What's even more amazing is that that result is accessible to any interested high-school student today. Get in touch with me if you'd like to explore that further.