I think most of us have had this conversation. Two kids keep trying to one-up each other with bigger and bigger numbers, until one inevitably pulls out the trump number, infinity.
The problem is, Superman here is wrong.
Infinity plus one is totally a thing.
To really understand infinity, we need to think about about it in two different ways. The first is to ask, “What is the number infinity?” The second is to ask, “What does it mean to have infinitely many?” The first we’ll answer in this post, the second we’ll answer next week.
So, what is infinity?
All of this depends on what precise definitions you choose to use, but the school child’s idea is still about right. What number occurs after all the other numbers? Well, infinity, of course.
Think about it this way: two is the number after one. Three is the number after two, and so on. Each counting number (also called natural number) is followed by another. Then, we define a number , the Greek letter omega, to be the number just after all of the counting numbers^{1}. This clearly has to be infinity!
But what’s the number after ? Well, of course. Then . And, after all of those, . It goes on. You can go on for as many layers as you wish.
Numbers like 1, 2 or are called ordinals since they define the order of things. You have the first thing, the second thing and the -th thing.
While isn’t useful in a lot of places in math, it does provide one resolution of the famous Zeno’s paradox. (Actually, Zeno has six surviving paradoxes, but let’s talk about the most famous one.) Achillles, the fastest of the Greek warriors, is chasing a tortoise. However, he can’t ever catch the tortoise. Why not?
In order to catch the tortoise, Achilles must first reach the place where the tortoise began. But by the time he arrives, the tortoise has moved farther away. Thus, Achilles has to run to that spot, but, again, the tortoise has moved on. Since Achilles must repeatedly reach the spot where the tortoise was, he can never catch the tortoise.
Now, of course this sounds silly. Many of these philosophical paradoxes do. Zeno tried to use this to show that all movement is an illusion, despite our experience to the contrary.
But let’s talk about this using our new idea.
Let’s say Achilles runs 10 times faster than the tortoise^{2}, but starts 100 meters behind him. After Achilles has gone 100 meters, the tortoise is still 10 meters ahead. After Achilles has gone that next 10 meters (110 meters) total), the tortoise is yet another meter ahead. And after Achilles has gone that additional meter (111 meters total), the dang tortoise is already ahead by another tenth of a meter. And so on and so forth, ad nauseum.
After 20 cycles of this, Achilles has still not caught up, though he is closer than the diameter of an electron.
Achilles is still behind the tortoise, though.
But the -th cycle is, by definition, the cycle after all of those cycles. And since he’s getting infinitely close during the first finitely many cycles, at the -th cycle, Achilles has exactly caught up to the tortoise.
How long does it take for Achilles to catch the tortoise?
If he is traveling at 10 m/s (Olympic sprinter pace), the first cycle―100 meters―takes ten seconds, the first two cycles―110 meters total―takes 11 seconds, while the first cycles takes 11.1111… seconds (or 100/9, to be exact).
And so, Achilles can quickly catch the tortoise.
Using this idea, we’ve come up with a number representing infinity. We also have numbers that are after infinity. But while is after , is it bigger than ? How about or ?
Before you decide, listen to this story.
The Hitchhiker’s Hotel is the largest hotel in the galaxy. In fact, it has infinitely many rooms. One night, every room was filled, when an unfortunate traveler came in begging for a room for the night. Fortunately for her, the hotel has a policy for such circumstances. They called each room and told them to pack up and move to the room with the next higher number on it. Each guest did so and the hotel worker put the new guest up in room number one.
When we started, we had infinitely many guests in infinitely many rooms. After our lucky guest came, we had one more guest and the same number of rooms, but each guest still had their own room. So they must be the same size, right?
In the next post, I’ll talk about how to carefully tell which set of things is bigger, and then show that some surprisingly different things are the same size.
- Really, to define this carefully, we need set theory. The ordinal zero is the empty set. The ordinal one is the set of the empty set, i.e., the set of the ordinal zero. The ordinal two is the set of the ordinals zero and one. The ordinal three is the set of the ordinals zero, one and two. This goes on infinitely. The ordinal is then defined to be the limit of this process over all the counting numbers. Following this same pattern, is union of with the set , i.e., the set of all the finite ordinals along with . For more details, see Wikipedia, that endless source of knowledge. ↩
- The land speed record for a tortoise is 5 miles per hour, so Achilles is going cheetah speed! ↩
I am definitely going to have to have my kids read this. As an engineer, I have a concept of “close enough”. So, while Achilles might not catch up by the 20th iteration, he is close enough.
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This brings back fond memories of my first university mathematics class. The professor shared the same paradoxes. I remember feeling delighted and overwhelmed.
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“What number occurs after all the other numbers? Well, infinity, of course.”
It feels like there’s an unstated premise underlying this part. Each non-infinity number comes after some other specific non-infinity number. Then infinity is defined as coming after a set?
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Yeah, that’s exactly what’s happening, and that’s exactly where the new idea comes in. We usually think of order as “the number after the last number,” but if we only did that, we’d be stuck counting forever. Cantor’s idea, here, was to say, well, we need to break out of this rut, so let’s put the ordinal omega after ALL the counting numbers. This is vaguely similar to saying “One is the number that comes after all the numbers from 0 to 1 (exclusive.)”
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To further add to this, you can describe fractions as “the number between these two numbers”. So 1/2 is “the number between 0 and 1”, 1/4 is “the number between 0 and 1/2”, and 1/3 is “the number between {0, 1/4, 5/16, 21/64, …} and {1/2, 3/8, 11/32, 43/128, …}.
You can further extend this construction to the whole of the real numbers, and if you go further you get what are known as the surreal numbers, of which all the ordinals are a subset.
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I actually had an argument with my husband a month ago about this very topic… and now I know that I won! Ha!
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Good to know that we’re helping marital peace! 😉
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