9 replies to this topic

[Wolf O'Donnell]

Recently. BBC Horizon did a program about the mathematical concept of infinity.



I don't know about what you guys think, but something didn't seem quite right to me. What they said about the even set of infinite numbers and the fractional set of infinite numbers seemed outright contradictory.

If you can pair up every single even number with a whole number, then you can do the same with the fractional numbers. There's no shortage of whole numbers you could use to pair up the fractional numbers of 1.12456 and 2.1545645 with. If any of these infinite sets of numbers are larger or smaller than the other, then that should clearly imply that they were finite.

Also, what does that David Hilbert not understand about the concept of the word, full? If a hotel with an infinite number of rooms is full, then there is no room for extra guests, for even if you were to try to move all the guests one room along, you couldn't, because the next room along would ultimately be occupied. If you could move the guests along, then that would imply that the infinite hotel is not fully booked, as there would be rooms vacant.

[Showsni]

Hum, I don't have time to watch the whole programme, though we did quite a bit about infinity during my maths degree. Could you copy out the bits you want to talk about, or tell me where in the video to find them?

[GuardianNinja]

Thats right, if infinite rooms are full, then there is no more room

you can add, subtract and multiply infinities but the bigger number will always be equal to the smaller one as well.

[Wolf O'Donnell]

here. They've helpfully split the thing up into chapters. I'm specifically speaking about Chapter 3 and 5.

One's about David Hilbert's Hotel Paradox, which doesn't make any sense as far as I can tell because it's ignoring the definition of the word, full, and treating the guests as if they're a finite number. Chapter 5 contradicts the bit about how an infinite set of even numbers is equal in size to an infinite set of whole numbers.

[Showsni]

Okay, the hotel. By "full" we mean every room has a guest in it. That's a pretty good definition of full right? Now, we decide we need to make some more space. Let's take every guest out of their room and put them in the car park for a bit. Now, we tell them to go back to the room with twice the number of the one they've just come from. It's an infinite hotel, so all these numbers clearly exist; and every guest is going back to a different room... So we've just freed up half the space in our hotel. Tada! Does that make sense?

So, at the start:
Room 1 has Guest 1
Room 2 has Guest 2
Room 3 has Guest 3
Room 4 has Guest 4
Room 5 has Guest 5
Room 6 has Guest 6
Room 7 has Guest 7
and so on...

and afterwards:

Room 1 is empty
Room 2 has Guest 1
Room 3 is empty
Room 4 has Guest 2
Room 5 is empty
Room 6 has Guest 3
Room 7 is empty
and so on...

If you can pair up every single even number with a whole number, then you can do the same with the fractional numbers. There's no shortage of whole numbers you could use to pair up the fractional numbers of 1.12456 and 2.1545645 with. If any of these infinite sets of numbers are larger or smaller than the other, then that should clearly imply that they were finite.

Nope, doesn't work. You can't number the set of Reals with the set of Naturals. I mean, start numbering as follows:

1st 0.1
2nd 0.01
3rd 0.001
4th 0.0001
5th 0.00001

etc.
You can see that we're continuing on forever, and we're never going to reach 0.2, let alone anything else. (Obviously not a proof, but hopefully intuitive enough?)

With the even numbers, you can see how any even number would eventually be numbered if you went on long enough.

Or just use Cantor's diagonal argument as in the video to prove it. (The Real numbers are uncountable).

[Wolf O'Donnell]

Okay, the hotel. By "full" we mean every room has a guest in it. That's a pretty good definition of full right? Now, we decide we need to make some more space. Let's take every guest out of their room and put them in the car park for a bit. Now, we tell them to go back to the room with twice the number of the one they've just come from. It's an infinite hotel, so all these numbers clearly exist; and every guest is going back to a different room... So we've just freed up half the space in our hotel. Tada! Does that make sense?

So, at the start:
Room 1 has Guest 1
Room 2 has Guest 2
Room 3 has Guest 3
Room 4 has Guest 4
Room 5 has Guest 5
Room 6 has Guest 6
Room 7 has Guest 7
and so on...

and afterwards:

Room 1 is empty
Room 2 has Guest 1
Room 3 is empty
Room 4 has Guest 2
Room 5 is empty
Room 6 has Guest 3
Room 7 is empty
and so on...

But why do the two infinite sets change in size? If they were equal in size before, why should they change afterwards?

Nope, doesn't work. You can't number the set of Reals with the set of Naturals. I mean, start numbering as follows:

1st 0.1
2nd 0.01
3rd 0.001
4th 0.0001
5th 0.00001

etc.
You can see that we're continuing on forever, and we're never going to reach 0.2, let alone anything else. (Obviously not a proof, but hopefully intuitive enough?)

With the even numbers, you can see how any even number would eventually be numbered if you went on long enough.

Or just use Cantor's diagonal argument as in the video to prove it. (The Real numbers are uncountable).

But surely you can still number all the... did you call them real numbers? Anyway, surely you can number all of them too, because there's no shortage of whole numbers with which you can pair them with. I suppose the direction in which you head towards infinity is different. Whereas with the whole numbers, you can add incrementally, with these others you can't add incrementally. But surely that means infinity has a directional quality to it? The size should be the same, but the direction is different.

Edited by Wolf O'Donnell, 16 February 2010 - 04:47 PM.

[Showsni]

But why do the two infinite sets change in size? If they were equal in size before, why should they change afterwards?

They don't; both have the same "size" as the natural numbers. (That is, the counting numbers, 1, 2, 3, 4, etc.) That is, they're what's known as "countably infinite;" you can count them. The rational numbers (all numbers that can be written as a/b with a and b natural and b not 0) have the same size, the natural numbers including negatives have the same size, and so on. Intuitively these feel very different; it seems as though the rational numbers should be much, much bigger than the natural numbers. But that's just how infinity is. (Which is why it's called a paradox.)

But surely you can still number all the... did you call them real numbers?

The real numbers (all decimal numbers, if you like) are a different kettle of fish. Here's the proof:
Let's just look at all real numbers between 0 and 1. Our assumption is that you can number them, 1, 2, 3, 4, and so on. So, we can write them down in the order given by that numbering; for example, it might look like this:

1st 0.47204493026718...
2nd 0.56202675518356...
3rd 0.03127456299110...
4th 0.04724842728499...
5th 0.11261111721181...
etc.

Now, we constuct another real number as follows. The first digit after the decimal point is any digit that's not the first digit in the first number on our list. That digit is a 4; so, let's pick 1, say.
0.1
The second digit is any digit that's not the second digit of the second number on our list. That digit is 6; so let's pick 1, say.
0.11
The third digit is any digit that's not the third digit in the third number on our list. That digit is 1; so let;s pick 2, say.
0.112
and we repeat...
0.11212...

Now, as you can see, the number we end up constructing is a real number between 0 and 1; so it must appear somewhere on out list, since the list contains all real numbers between 0 and 1. However, by the way we've constructed it, it's impossible to be on our list, since the nth digit will always be different to the nth digit of the nth number on the list. In other words, we've reached a contradiction. The only way to solve our contradiction is to realise that our initial assumption must have been false; in other words, it's impossible to number all of the real numbers between 0 and 1. (Or, indeed, between any two distinct real numbers; there are "more" real numbers between 0.001 and 0.002 than there are natural numbers if you could count forever!)

Which brings us to the idea of orders of infinity; infinities of different sizes. You can't think of infinity as a number in the traditional sense.

[Wolf O'Donnell]

But why do the two infinite sets change in size? If they were equal in size before, why should they change afterwards?

They don't; both have the same "size" as the natural numbers. (That is, the counting numbers, 1, 2, 3, 4, etc.) That is, they're what's known as "countably infinite;" you can count them. The rational numbers (all numbers that can be written as a/b with a and b natural and b not 0) have the same size, the natural numbers including negatives have the same size, and so on. Intuitively these feel very different; it seems as though the rational numbers should be much, much bigger than the natural numbers. But that's just how infinity is. (Which is why it's called a paradox.)

I'm sorry, but this isn't making it clearer.

If these infinities are the same size, then there shouldn't be any chance of extra rooms appearing. That would mean the room set is bigger than the guest set, in the same way the rational set is "bigger" than the natural set.

[Showsni]

Ah, but the rational set isn't bigger than the natural set. With infinite sets, we can't really talk about bigger or smaller very well (except when we make the jump to uncountably infinite sets); all countably infinite sets have the same size. The natural numbers, the prime numbers, the even numbers, the rational numbers... they're all the same size. Let's just look at the evens as opposed to the natural numbers.

Now, obviously there are twice as many natural numbers as there are even numbers; right? That's just common sense. But no, that's wrong. We've got infinite numbers off to the right of our number line. We can pair up every natural number with a different even number, and we'll never run out of numbers as we can always find more. So the set of even numbers is actually the same size as the set of natural numbers; in a way, it's the same set. (There's a bijection between the two sets, a one to one and onto function mapping from one to the other). Mathematically, if we can find such a bijection (in this case, double the natural number to get the even number, or half the even number to get back to the natural number) then we can pretty much view the sets as being identical for most purposes.

Infinity can "stretch," as it were. So long as we're dealing with countable things, you can always stretch the hotel out to fit in more guests.

It's pretty hard to get your head around, maybe...

[GuardianNinja]