Wednesday, March 07, 2007

Maths part 6 the nature of infinity

So i'm here. Finally. Remember all our components so far? We have our types of numbers, each set bigger than the other, remembering that a set is just a collection of objects, in this case numbers, and functions, which given a value from one set, give you a value from the other set.

And we now know functions can be "bijective". The actual meaning is not important now- although it's nice if you understand it. All you need to know that if there exists an injection from A to B then there are at least as many objects in B as there are in A- so the number of objects in B is greater or equal to that in A, and with a surjection, the opposite is true- the number of objects in A is greater or equal to that in B. So if we have both, the number in both is equal.

The great thing about the way we have set up our definition is that we don't need to go through assigning a number to each element in a set- we just have to show it's possible. So it's actually possible to apply these definitions to infinite sets.

A simple example, the natural numbers (1,2,3,4....) to the even numbers (2,4,6,8,....) Now f(x)=2x will easily give us a bijection here- we can assign a natural number to every even number. So while both sets have an infinite amount of objects, we can see that they (sort of), have the same amount. Because amounts don't necessarily mean much when we are talking about infinity, mathematicians say "cardinality" instead. But it's reasonable for you to say they have the same amount of objects.

Now this an interesting result, because one would expect the number of natural numbers to be double that of the even numbers- but is not true, at least not according to the version of counting we are using.

To give us our idea of infinity, we tend to split infinite sets into broad categories, countable and uncountable. Countable sets are ones which have a bijection to the natural numbers, that is we can assign a natural number to each member of the set. If you imagine an infinite cars driving by on the road, they will never stop, but you can certainly count them as they go.

Uncountable sets are bigger sets than the natural numbers- that is there is an injection from the natural numbers to the bigger set, but no surjection. So we cannot assign a natural number to each object. Imagine yourself plunked in the middle of an infinite desert and trying to count all the sand... there is no real order you can count them, no sensible pattern which will allow you to count them. Uncountable sets are huge, ridiculously huge in fact, which we shall talk about a little later.

Now the integers (all the whole numbers, including the negative ones) are countable- assign 1 to 0, 2 to 1, 3 to -1, 4 to 2, 5 to -2.... and so on, and you can describe each one.

Whats more, it is possible to show that even the rational numbers (all the fractions) are countable- after all they are just integers on top of natural numbers.

So where do we get uncountable numbers from? Well the real numbers are uncountable- there are a multitude of proofs for this, but all of them will involve mathematics a little too hard for here, and also I can't remember the easier ones (easier conceptually anyway. The mathematics involved is quite hard...). Bizzarely, or probably not, only the transcendental numbers (remember them- they are the extremely weird numers like pi and e) are uncountable.

Now whats odd about this is it's possible that compartively, the countable numbers take up a very tiny space in infinity, so if you were to pick out a number at random you'd always get an uncountable one- which is why, incidentally, all those physical constants are so inconviniently long..... The weird thing is that as a fourth year mathematician I know of very few examples of transcendental numbers... but they make up almost all of the numbers... crazy no?

Finally we have a notion of size in infinity, with uncountable sets sitting at the top of the roost. I have some more things to say about mathematics, and a little bit more about infinity, but for now... fare thee well.



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