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Infinity: Big and Bigger


On the Internet, you can never go wrong by quoting the The Hitchhiker's Guide to the Galaxy.

Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space.

Now, it's kind of a cheat, because i'm not going to talk about that kind of space, I'm going to talk about spaces in a mathematical sense. But I'm offering something in exchange, because I'm going to talk about spaces that are much bigger than mere physical space.

The point of this is really to talk about (echo effect) infinity. And beyond.

Mathematically, space is much simpler than the thing in which your coffee cup is located just out of reach and that keeps your cat from being exactly where you're sitting, no matter how much he tries. In mathematics, a space is simply a set of some sort with some kind of additional structure. (A set is just some collection of things with no duplicates, like {1, 2, 3, 4, 5}. By convention, we put sets into braces like that example.)

So far, that's not a space -- we haven't said anything further about it than there is a bag full of things. But -- since I've chosen a set we conveniently already know a lot about -- we know that the set is ordered because we agree that 5 is bigger than 4. And we have a space.

Okay, it's a pretty boring space, but it's a space.

There are some other rules we think we know, like addition -- 1+2=3. But in our little space, we immediately run into trouble, because 3+4 equals what? Oh, 7, but 7 isn't in the set. To take care of 3+4, we need to expand the set to be at least {1,2,3,4,5,6,7} and then we're immediately going to have the problem of 4+5, or for that matter, 7+1.

Now, with nothing more than the idea of addition (we talked about ordering, but we can define an order in terms of addition) we've run into our first experience with infinity. There is a set N that we can define like this:

  • 0 is part of N
  • For anything that is part of N, which we'll call n, n+1 is also in N.

We call N the natural numbers.

Now, N is pretty big. After all, no matter what n we pick, there's always something bigger. This is what we call infinite. And all is well, until we think about subtraction: we know 3-1=2, and we know 2-1=1, and we know 1-1=0, but 0-1 isn't in our set. So we define a new set called the integers which has new elements -1, -2, -3, and so on. We can throw in multiplication now, and all is good, but when we put in division we're in trouble again: 2÷3 and 1÷2 aren't in there. So we define another set called the rational numbers, Q.

Now, we've pretty much defined all the numbers anyone had any use for until the Greeks and Egyptians screwed it all up by trying to measure fields and distances.