The thing I want to tell you is that there is more than one infinity, and that some infinities are bigger than others. Isn't that a crazy idea? When Buzz Lightyear said to infinity and beyond, he wasn't talking nonsense, it's a real idea. And what is even more awesome, is that you dont have to take my word for it, I can prove it to you, and the ideas are really easy and lovely.

Surely there's only one infinity? What do I mean bigger? How could you tell one was bigger anyway, it's not like you can count them? Let me take you back in time, to a hunter-gatherer tribe in Brazil. Specifically to hunter-gatherers from the PirahÃ£ tribe whose language only contains words for the numbers one, two and many. Suppose two people want to exchange some goats for pots … or something. And they agreed that one pot was worth one goat, so there would be the same number of goats and pots, how would they do this? They cant just count out 10 pots and 10 goats and they dont want to just give roughly the right number, someone might cheat and not give enough. Well it's easy if you think about it, just line them up, if you put a pot next to each goat then you know that every pot has a goat, every goat has a pot and you can both go away happy that you've done a fair deal. This idea is a very important one. We can say that two sets of objects have the same size (the same number of objects I mean) if you can pair up the objects from one and the objects from another. And if after you've done the pairing there's some left over in one set then the sets aren't the same size. If there is a pot at the end that doesn't have a goat to pair with then one of the guys knows he's been conned.

How does this help us with infinity? Well what I propose to do is name two sets of things that are infinite and which I claim have different size. If I'm wrong then there should be some clever way to line up all the goats from one with all the pots in the other so that nobody is left without a partner. If I'm right, then no matter how cleverly you line things up there's always going to be an extra goat.

Enough beating about the bush, let's name these sets then: First off the pots, the pots are all the whole numbers 1,2,3,4 etc., there are an infinite number of these, you can always invent a new one by saying “plus one”. Next, the goats, these are all the decimal numbers, no matter how complicated, between 0 and 1. So these are numbers like 0.5, 0.3333333..., 0.173658365345... to simplify things we write them all as a list of digits that goes on forever so that a half becomes 0.50000.... there are an infinite number of these. Ok, so let's say I'm wrong and that someone clever comes along and says he can line up the pots and goats. So he makes a table like this:

Pot --------- Goat

1 --------- 0.

**7**8234694378967389456....

2 --------- 0.7

**8**937463496836542525....

3 --------- 0.23

**4**52653589456411420....

4 --------- 0.663

**9**6745877967897665....

5 --------- 0.5464

**3**461321555554555....

6 --------- 0.84516

**4**54894542888257....

and so on.

Now it's important to remember to remember that no matter who proposes whatever way of lining up the pots that what I'm about to say works just the same. So the list I've made above is totally random. But let's imagine there's some complicated pattern to what this guy has said and that it carries on forever and all the pots are there on the left, and all the goats on the right, … there's a problem though. He's missed one of the goats. And if he tries to put this goat in there's still going to be a missing goat, no matter how he tries it. And here's how you find the goat:

Look down the diagonal I've put in bold. Now think about the number 0.895045... What I've done is looked along the diagonal and added one to whatever digit I see (9 becomes 0). This goat is missing. Think for a second about where in the list this goat could be. It's not the first place, because it's first digit is one more than the first digit of that one, because of how I've built it. No matter how far down you put this goat, somewhere it will cross this diagonal line, and where it crosses the diagonal line it will be wrong at that digit. So, no matter how you line the pots up you will always have an extra goat. So the set of these pots must be smaller than the set of these goats.

So we have found that there are two sets of things, both of them infinite, but one is bigger. It's a totally mental idea, it's so utterly profound and wonderful, it blows a lot of works of art out of the water. This is what real maths is, it's learning about wonderful profound ideas. It's really really sad when people say that they dont like maths because of school. Because the crap you do at school isn't maths's fault. I just hope that even if I dont convince you to love maths that you'll at least know that there's some stuff in maths that's not nasty or boring or hard.

I don't want to be mean/stupid, but surely that doesn't work?

ReplyDeleteYour theory works, but only if you assume that an a decimal will be left over. And since both the integers and decimals are infinite, there should be none left over.

All you've done is say "There's two infinite sets, but one has more in. Therefore, there are two kinds of infinity." It is only true because you set up the senario that way

I would agree with the statement that there are different sized infinities, but I don't agree with the proof.

ReplyDeleteAlthough by the method you specified you can always come up with a new number goat set, you can also always come up with a number in the pot set to match it. Also if you take the infinite set of all whole numbers and add "0." to the front of all its elements, you will have created the second set. By this method it could be argued that these sets are in fact the same size.

However I think that the original statement is true and will try to create an example that shows it(I'm just doing this quickly so I might mess up). Consider the set of all non-negative integers(let this be the pot set) and the set of all non-negative real numbers(let this be the goat set).

All the integers are real numbers, however not all the real numbers are integers. That is the integers are a subset of the real numbers, but the real numbers are not a subset of the integers.

Now for every element of the integers you associated with its equivalent number in the real numbers (1 with 1, 243124 with 243124, etc.). This will cause all elements of the integers set to be paired with an element of the real numbers set. However, numbers which are not integers in the real numbers set(0.5, pi, 12.1, etc.) have no pair in the integers set and there are no elements available to pair with. You have goats left over.

Therefore it can be said that the set of all non-negative real numbers is "bigger" than the set of all non-negative integers, despite the fact that they are both infinite.

AH. I see what you did. You are confusing the density of an infite set and the length.

ReplyDeleteSee, both sets are the exact same length, but between 2 intergers exist an infite amount of real numbers.

Yet you can pair every real number with an interger. Take the next number up from an interger and pair that with 2, then the next with three and so forth.

This is similar to fitting two infinite sets within a single one, by alternating the placement.

However, infinity has some weirder properties such as that adding anything to it results in itself.

If you want amazing math, do calculus with natural logs and the relationship with all other powers of constants.

ThreeWords: I'm not sure quite what you mean sorry, basically what I tried to show is that whatever pairing up you do at all there will always be one missing, and even if you put that one back in you will be able the use the same argument, no matter what pairing you imagine there will always be something left out. Sorry, if I've misunderstood your point. Try again if you think I've made a mistake, I may get it the second time.

ReplyDeleteAnonymous: Your proof is nice and simple, but it does lead to a problem, that you may get people saying "well of course you cant pair up all the elements if you do it like that, but there might be some other way of doing it". It may be that there's a way to answer this objection, but at half midnight i cant work it out lol, but your method does have the advantage of being simple to understand.

I think I may have failed to explain an important point properly that may be the cause of your objection at the start though. You say that this extra goat I've found can always be paired up by finding another pot. The point I should have stressed is that the way I found the extra goat works for absolutely any pairing you may care to imagine.

So suppose we add this extra goat into the line up, then the pairing we have imagined is still missing one (that I haven't taken out by the way, so it wasn't there in the first pairing either) so it's still the case that there are more goats. I should have explained more clearly that adding the extra goat doesn't get you out of the mess. Hope that clears things up, if not I'll try again.

In fact, I'll add in the second beautiful wonderful thing (tm). This might help explain the difficulty of your proof:

Theorem: The size of the set of rational numbers (for people who dont like technical terms, i mean fractions with whole numbers top and bottom) is the same as the size of the set of whole numbers.

Proof: I can set up a pairing, (which I cannot do for the real numbers with infinite decimal expansions) which goes like this:

1: 1/1

2: 1/2

3: 2/1

4: 1/3

5: 2/2

6: 3/1

7: 1/4

8: 2/3

9: 3/2

etc

If it's not clear how I got that, picture a grid with the numerator from 1 to infinity on the bottom axis, and the denominator from 1 to infinity on the vertical axis, then take diagonal slices like:

.\.\.\.\.\.

.\.\.\.\.\. (sorry for the rubbish

.\.\.\.\.\. drawing, it's hard to

.\.\.\.\.\. do in ascii)

and work up the diagonals in order. Every possible pair of numbers will eventually be reached this way, no line is infinitely long, so how ever large the fraction it matches to some number. So a pairing exists, which I said was our measure of the size of the set of things.

Why cant I do something like this for all the real numbers? Because pi and root two and an infinite (of the bigger kind) amount of numbers like them can never be written as a fraction and so will never be reached by this.

All three of you, if anything has confused you or you think I've made a mistake (I probably have it's very late) please comment again. Thanks for your responses, it was nice to get a response for once lol.