Science of Discworld III

the symbol for 7 is different. So what is seven? It’s easy to say what seven days, or seven sheep, or seven colours of the spectrum are … but what about the number itself? You never encounter a naked ‘seven’, it always seems to be attached to some collection of things.
    Cantor decided to make a virtue of necessity, and declared that a number was something associated with a set, or collection, of things. You can put together a set from any collection of things whatsoever. Intuitively, the number you get by counting tells you how many things belong to that set. The set of days of the week determines the number ‘seven’. The wonderful feature of Cantor’s approach is this: you can decide whether any other set has seven members without counting anything. To do this, you just have to try to match the members of the sets, so that each member of one set is matched to precisely one of the other. If, for instance, the second set is the set of colours of the spectrum, then you might match the sets like this:
Sunday
Red
Monday
Orange
Tuesday
Yellow
Wednesday
Green
Thursday
Blue
Friday
Violet 3
Saturday
Octarine
    The order in which the items are listed does not matter. But you’re not allowed to match Tuesday with both Violet and Green, or Green with both Tuesday and Sunday, in the same matching. Or to miss any members of the sets out.
    In contrast, if you try to match the days of the week with the elephants that support the Disc, you run into trouble:
Sunday
Berilia
Monday
Tubul
Tuesday
Great T’Phon
Wednesday
Jerakeen
Thursday
?
    More precisely, you run out of elephants. Even the legendary fifth elephant fails to take you past Thursday.
    Why the difference? Well, there are seven days in the week, and seven colours of the spectrum, so you can match those sets. But there are only four (perhaps once five) elephants, and you can’t match four or five with seven.
    The deep philosophical point here is that you don’t need to know about the numbers four, five or seven, to discover that there’s no way to match the sets up. Talking about the numbers amounts to being wise after the event. Matching is logically primaryto counting. 4 But now, all sets that match each other can be assigned a common symbol, or ‘cardinal’, which effectively is the corresponding number. The cardinal of the set of days of the week is the symbol 7, for instance, and the same symbol applies to any set that matches the days of the week. So we can base our concept of number on the simpler one of matching.
    So far, then, nothing new. But ‘matching’ makes sense for infinite sets, not just finite ones. You can match the even numbers with all numbers:
2
1
4
2
6
3
8
4
10
5
…
 
    and so on. Matchings like this explain the goings-on in Hilbert’s Hotel. That’s where Hilbert got the idea (roof before foundations, remember).
    What is the cardinal of the set of all whole numbers (and hence of any set that can be matched to it)? The traditional name is ‘infinity’. Cantor, being cautious, preferred something with fewer mental associations, and in 1883 he named it ‘aleph’, the first letter of the Hebrew alphabet. And he put a small zero underneath it, for reasons that will shortly transpire: aleph-zero.
    He knew what he was starting: ‘I am well aware that by adoptingsuch a procedure I am putting myself in opposition to widespread views regarding infinity in mathematics and to current opinions on the nature of number.’ He got what he expected: a lot of hostility, especially from Leopold Kronecker. ‘God created the integers: all else is the work of Man,’ Kronecker declared.
    Nowadays, most of us think that Man created the integers too.
    Why introduce a new symbol (and Hebrew at that?). If there had been only one infinity in Cantor’s sense, he might as well have named it ‘infinity’ like everyone else, and used the traditional symbol of a figure 8 lying on its side. But he quickly saw that from his point of view, there might well be other infinities, and he was reserving the right to name those aleph-one, aleph-two, aleph-three, and so on.
    How can there be other infinities? This was the big unexpected consequence of that simple, childish idea of matching. To describe how it comes about, we need some way to talk about really big numbers. Finite ones and infinite ones. To lull you into the belief that everything is warm and friendly, we’ll introduce a simple convention.
    If ‘umpty’ is any number, of whatever size, then