Friday, 19 February 2010

Jupiter and beyond the infinite - Part 2

defining infinity, sets and counting

Plato: So lets recap what we worked out last time. If I remember right we realised that if our definitions aren't sorted out we can end up confusing and contradicting ourselves. And as we dont find definitions set in stone, we have to make them to suit our needs. (Though I'm not sure I agree with you about that).

Adam: Yeh that's right, basically today I'm going to suggest that the idea of a set is a useful one for discussing infinity. … hang on a second … why am I talking to you?

Plato: Oh it's called a Socratic dialogue. You talk to fictional student,in this case me, then in arguing with me you find a solution to the problem.

Adam: Bit weird though. Isn't it big headed of me thinking I can act as a teacher to Plato?

Plato: You've got to remember that you are from over 2 and a half thousand years after me. I only know one thing about infinity, that you get paradoxes if you try and think about it.

Adam: What sort of paradoxes?

Plato: We talked about Zeno's paradoxes last time. But I've come up with a better problem that I think makes the idea of defining infinity nonsense. Think of a square. Now the number of points in the square is infinite right? All the points in the square make up an infinite number. But they make up one. One square. And what's more, each point has no area, it covers no space at all, it's a sizeless point. But together they make up a shape with area. How can it make sense that this square is infinity and one, and that it has area, but it is made of things with no area?

Adam: These questions are hard, but if we try hard, I think we can find a solution. Now my first candidate for a definition is from my dictionary:

1.immeasurably great: an infinite capacity for forgiveness.
2.indefinitely or exceedingly great: infinite sums of money.

Plato: … these are rubbish definitions. We cant really say anything with these. This is why the idea of mathematicians just proving things from definitions is too simple. If mathematicians just went along blindly proving things from any definition they were given they would produce a lot of nonsense that didn't really tie in with what we expect. These definitions aren't helpful, if infinity cant be measured then how can we talk about it being big? And “exceedingly great” is just hopelessly vague.

Adam: Agreed. What we need to do then is to make (or find, sorry Plato) some definition of infinity that will allow us to study this idea. Infinity can be very interesting, but if we dont know what it is we may as well be discussing angels on pinheads.

Plato: What kind of definition do we want?

Adam: The definition we want must fit with our intuition of what infinity should act like, and it should allow us to deduce things about it mathematically. So for example, “really big” fits with our intuitive idea, but we cant really talk about what it is and how it works. And we can get a lot of theorems out of “all those functions that transform a plane onto itself” … but that's not what we think of when we think about infinity.

Plato: That's not what I think of for sure. I dont even know what that is. I'll have to ask my friend Eudoxus, he's much better at maths than I am. … Sorry, carry on Adam.

Adam: Let's think about some of the things we would like to call infinite. Lets make it clear that we dont just mean big. The planet Jupiter is huge, unimaginable enormous, two whole Earths could fit on the red spot.

Plato: But it's not actually infinite, right. So we dont want to say that the number of grains of sand on all the beaches on earth is infinite. So we need to distinguish large but finite and infinite. We also want to be able to distinguish between the two ways of looking a square that I talked about before: We want a square to be one square, containing an infinite number of points, but not contradict ourselves.

Adam: Right, a lesson that the 19th and 20th centuries teaches us is that a lot can be got out of the idea of a set. I'll explain now what a set is and why it is useful.

Adam: A set is a collection of some number of objects or ideas. To distinguish this from anything else we write a set with curly bracket. So if I have four apples on a table the set {things on the table} can also be written as {apple 1, apple 2, apple 3, apple 4}. We are allowed to have sets that are infinite, for example {1,2,3,4,5...} or {all the points on the surface of the earth}.

Plato: So a set is like a big box that we can put as many things into as we like?

Adam: Yes. And we can do something else that's very useful. Think about a box. If I have two small boxes, I can put them inside a bigger box. So that now, the bigger box has exactly two things in it – two boxes.

a box in a box
A two boxs in a box.

Plato: Hang on, does that mean that if I had a box with two smaller boxes inside it. Then the bigger box would have two things in it, even in the smaller boxes had something in them too?

The outer box has only two things in it.

Adam: Exactly right. If I walk into a room and see two boxes on the floor I would say there are two objects in the room. Even if those boxes were full to the top of loads of different things. And in the same way we can have sets inside sets. So {{1,2},{3,4}} is a box that has two smaller boxes inside it, in one box there is a 1 and a 2, and in the other there are 3 and 4. Now this is quite a different thing to the set {1,2,3,4}, which is a box with just 1,2,3,4, in it, without any other smaller boxes, and it is not the same as {{1,3},{2,4}}, because the numbers are in different boxes. And we can talk about one box's realationship with another in terms of what things they both have in them. So you can have a box that's only filled with things from another box and so on.

Plato: It would be easier if we had a name for the things in the sets, and some way to describe the different sets instead of saying “thing” and “bigger box” all the time.

Adam: You're right. How about we call the things inside the box “elements” of the set. And how about we say that if one box is made up of some of the elements from another box then we call the smaller box a “subset”.

Plato: Let me make sure I've got this right. So if I think about the set {1,{2,3},4} then I can say that 1 is an element, and so is the set {2,3} because these are both things in the set? And I can say that {1,4} is a subset because it's made of things that are in the first set?

The set {muffin, apple, creepy starfish},

is a subset of the set {muffin, apple, creepy starfish, shoe, dove}.

Adam: That's right yeh. This idea of sets is a useful one because we can think about almost anything mathematical in terms of sets. We can think about the number 3 as a set with 3 elements, we can think about a square as {all points inside of 4 lines} and say that this set is contained in the set {all point in the page}. This relationship of being a part of something else is useful too. We can say that a square is a subset of the plane, so we know that if we can say something about the whole plane, we can something about the square too. If we can say something about the set of all numbers {1,2,3,4,5.....} we can say something about some specific numbers {1,2,7,51,90} too.

Plato: How does this help my square problem?

Adam: Like this: Now you remember how I said that a square is a set of all the points inside 4 lines? Now we can say that this is one set, which is why you said it was one. But that it contains infinitely many points. So to say a square is both one and infinite is no more a contradiction than to say that a bag is still only one bag because it has lots of things in it.

Plato: I think I understand that. But I dont see how we can define infinity with this. So far all we've done is said some things can be infinite without knowing what we're talking about. What about infinite sets, sets with an infinite amount of things in them? What do we want that idea to mean?

Adam: Well, can we agree that {1,2,3} isn't infinite.

Plato: No, it has 3 elements in it.

Adam: And the set {apple, cabbage, grapefruit, your mum} isn't infinite either. Your mum's not that fat.

Plato: Oy! Watch it you. No that's not infinite. It has 4 things in it.

Adam: And in general if there are some definite number of things in the set we dont want to say that is infinite either?

Plato: Right.

Adam: Even if the amount of things in the set is huge, say if the set is {all the 6.7 billion people on earth}?

Plato: No, we dont want to say that is infinite, even if it is huge.

Adam: So maybe our definition of infinite might be a set where there is no number that is equal to the amount of elements in the set. I think this is a good definition, it fits with what we think infinite means, and it mean there are useful things we can work out.

Plato: Hang on Adam. You've missed a step here. We know easily what it means to say that {1,2,3,4} has 4 elements in it, but what does it mean to say that an infinite set has size? I dont think that makes sense.

Adam: You're right. And I do have a way of answering you. But it's late and I'm getting tried. How about we carry this on same time next week?

Plato: I'll hold you to that.

Adam: So to summarise: A set is a collection of some number of objects, that can be thought of as a box, it makes sense to say that an infinite set cannot be given a number as its size. We will talk about what size means in infinite sets next week.

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