Counting - continued.

We saw last time that the idea of putting sets into one-to-one correspondence leads to counter-intuitive ideas like that there are as many even numbers as numbers - and of course there's nothing special about even numbers, you could just as easily show that there are as many century numbers (100, 200, 300...) as numbers. Indeed providing you have a set of things where you can organise them in some systematic way so that you can point to each element in turn then they can be put into one-to-one correspondence with the set of numbers.  Of course you could say we are cheating here, because we are dealing with infinity.  And indeed for some time mathematicians treated the concept of infinity as an embarrassing irrelevance. But then a bloke called Cantor back in the 1870s showed that infinity was an important mathematical tool, and in fact that there was more than one sort of infinity - that there were sets which were too big to be put into one-to-one correspondence with the set of numbers - were in fact uncountable.  But just what do we mean by counting?  We'll look at that next time.