Jupiter and beyond the infinite - Part 3

Counting the infinite.

Adam: Hello Plato, sorry I was busy in Friday, care for another chat about infinity?

Plato: Definitely, let's see. Last time we decided that a set is a collection of some number of objects, that can be thought of as a box. We also said it makes sense to say that an infinite set cannot be given a number as its size. But I said that we had missed something and wanted to know how we could say the size of something infinite.

Adam: Indeed. So we want to work out some way of talking about size of a set of objects that can be extended to infinity. First, I'd like to think about a hotel.

Plato: Why a hotel, what has that got to do with infinity?

Adam: This hotel is infinite. It has one room for every single whole number that exists.

Plato: I'm guessing this is an imaginary hotel.

Adam: Indeed, and it's full. Now suppose you arrive at this hotel and that it is completely full, there is someone in every room. Will you be turned away?

Plato: Yes, there is someone in every room so there's no empty room for me.

Adam: Ah, but, what if you were to go to room 1, and very kindly ask if the people there wouldn't mind you having their room, whist they go into room 2?

Plato: But then where would the people from room 2 go? Room 3 I suppose?

Adam: Indeed, and so on forever.

Plato: But the person in the last room would have nowhere to go.

Adam: Ah that's just the thing, there is no last room, this hotel is infinite. Every single person can go to the room above them.

Plato: I see. So even if the hotel is full you can always add one more person. And I guess you could do this twice if you wanted two extra rooms?

Adam: You can even do this if a coach party with infinitely many people in it turns up.

Plato: Hang on, how does that work? You'd have to move the people in the hotel infinitely many rooms away, the person in room 1 can hardly move to room infinity+1 can they?

Adam: That's true yeh, but what happens if the person in room 1 moves to room 2, the person in room 2 moves to room 4, the person in room 3 moves to room 6, and so on?

Plato: So everyone doubles their room number? Then everyone in the hotel has somewhere to go. But then all the odd rooms are free. And we can move the people from the coach into the odd rooms. I see.

Adam: So we can see that we can add a finite or infinite number of people into an infinite hotel without changing it's size.

Plato: Yes, I see. And we can take out an infinite number of people too by reversing the process. So we want to define counting so that the size of an infinite set is the same if you add a finite set to it or double it?

Adam: Right. Now lets think about counting. What do you do if you're counting very slowly and deliberately?

Plato: I point to each thing in turn and say the next number, and make sure I haven't missed anything out.

Adam: Right. Now there are some tribes of people with no words for numbers. They cant just say the next number because they dont have the word for it. How do you think they count?

Plato: Point to each one and count on their fingers?

Adam: Good idea, but what about if they want to count more than 20? Suppose they want to do something easier, they have some goats and some sheep in a field, they want to know if there are more goats than sheep, how can they work this out?

Plato: If they cant count them I dont think they could. They've got no way of knowing how many of each there are.

Adam: Suppose that each sheep had a number written on its side. Could they do it then?

Plato: Well they could use the sheep to count to goats, so yes.

Adam: How?

Plato: They would point to each goat in turn, but instead of saying the next number they would go to the next sheep.

Adam: So what's happened is you have paired up the goats and the sheep?

Plato: Basically yes, you put a goat next to sheep 1, sheep 2 and so on.

Adam: And how do you tell if there are more sheep than goats?

Plato: You see if the number of goats you count is smaller than the number of sheep. If this happened then at some point in the counting you would run out of goats.

Adam: Now can you see a way of solving the problem if we dont have numbers painted on the side of the goats?

Plato: Yes, we can just line the goats and the sheep up. If there are more goats then one of them wont have a partner.

Adam: Exactly right. So we can say two sets are the same size if they can be paired up together with nothing being left out.

Plato: But what has this got to do with infinity?

Adam: Well remember that very odd property of infinity, that adding one to it didn't matter? We can explain that now. Think about the sets {1,2,3,4,....} and the set {0,1,2,3...}. Are these the same size by our definition?

Plato: Well the second one clearly has an extra element, the zero. But we can line them up next to eachother. We can pair 1-0, 2-1, 3-2 etc. So yes, they must be the same.

Adam: So there are as many numbers above 1 as there are about 0? That's quite impressive. What about {1,2,3,4...} and {2,4,6,8...}?

Plato: We can do the same. Pairing 1-2, 2-4, 3-6, 4-8 and so on. So these are the same size.

Adam: But every single thing in the second set is in the first set, but not everything in the first is in the second. We called this a subset last time. Now does it seem possible that this could happen with a set that wasn't finite?

Plato: No, if you tried comparing a finite set with a set that had some elements removed you would end up with one set being smaller.

Adam: So, how's this for a definition of infinity? An infinite set is one which is the same size if you remove one of its elements.

Plato: Yes, that seems sensible. What can we say with this definition though? We've already worked out there are as many odd numbers as there are numbers altogether. But what else can we say?

Adam: Well next week I'll argue that there are different sizes of infinity, that we can say that some infinite things are bigger than others.