07 Infinity
How big is infinity? The short answer is that ∞ (the symbol for infinity) is very big. Think
of a straight line with larger and larger numbers lying along it and the line stretching ‘off
to infinity’. For every huge number produced, say 10^1000 , there is always a bigger one,
such as 10^1000 + 1.
This is a traditional idea of infinity, with numbers marching on forever.
Mathematics uses infinity in any which way, but care has to be taken in treating
infinity like an ordinary number. It is not.
Counting
The German mathematician Georg Cantor gave us an entirely different concept
of infinity. In the process, he single-handedly created a theory which has driven
much of modern mathematics. The idea on which Cantor’s theory depends has to
do with a primitive notion of counting, simpler than the one we use in everyday
affairs.
Imagine a farmer who didn’t know about counting with numbers. How would
he know how many sheep he had? Simple – when he lets his sheep out in the
morning he can tell whether they are all back in the evening by pairing each
sheep with a stone from a pile at the gate of his field. If there is a sheep missing
there will be a stone left over. Even without using numbers, the farmer is being
very mathematical. He is using the idea of a one-to-one correspondence between
sheep and stones. This primitive idea has some surprising consequences.
Cantor’s theory involves sets (a set is simply a collection of objects). For
example N = {1, 2, 3, 4, 5, 6, 7, 8,.. .} means the set of (positive) whole
numbers. Once we have a set, we can talk about subsets, which are smaller sets
within the larger set. The most obvious subsets connected with our example N
are the subsets O = {1, 3, 5, 7,.. .} and E = {2, 4, 6, 8,.. .}, which are the
sets of the odd and even numbers respectively. If we were to ask ‘is there the
same number of odd numbers as even numbers?’ what would be our answer?
Though we cannot do this by counting the elements in each set and comparing
answers, the answer would still surely be ‘yes’. What is this confidence based on?