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 primary

[1]Yes, traditionally `Indigo' goes here, but that's silly - Indigo is just another shade of blue. You could equally well insert `Turquoise' between Green and Blue. Indigo was just included because seven is more mystical than six. Rewriting history, we find that we have left a place for Octarine, the Discworld's eighth colour. Well, seventh, actually. Septarine, anyone?

to counting.[1] 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 adopting

[1] This is why, even today when the lustre of `the new mathematics' has all but worn to dust, small children in mathematics classes spend hours drawing squiggly lines between circles containing pictures of cats to circles containing pictures of flowers, busily `matching' the two sets. Neither the children nor their teachers have the foggiest idea why they are doing this. In fact they re doing it because, decades ago, a bunch of demented educators couldn't understand that just because something is logically prior to another, it may not be sensible to teach them in that order. Real mathematicians, who knew that you always put the roof on the house before you dug the foundation trench, looked on in bemused horror.

such 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 'umptyplex' will mean 10umpty, which is 1 followed by umpty zeros. So 2plex is 100, a hundred; 6plex is 1000000, a million; 9plex is a billion. When umpty = 100 we get a googol, so googol = 100plex. A googolplex is therefore also describable as 100plexplex.

In Cantorian mode, we idly start to muse about infinityplex. But let's be precise: what about aleph-zeroplex? What is 10^aleph-zero?

Remarkably, it has an entirely sensible meaning. It is the cardinal of the set of all real numbers - all numbers that can be represented as an infinitely long decimal. Recall the Ephebian philosopher Pthagonal, who is recorded as saying, `The diameter divides into the circumference ... It ought to be three times. But does it? No. Three point one four and lots of other figures. There's no end to the buggers.' This, of course, is a reference to the most famous real number, one that really does need infinitely many decimal places to capture it exactly: n ('pi'). To one decimal place, n is 3.1. To two places, it is 3.14. To three places, it is 3.141. And so on, ad infinitum.

There are plenty of real numbers other than n. How big is the phase space of all real numbers?

Think about the bit after the decimal point. If we work to one decimal place, there are 10 possibilities: any of the digits 0, 1, 2, ... , 9. If we work to two decimal places, there are 100 possibilities: 00 up to 99. If we work to three decimal places, there are 1000 possibilities: 000 up to 999.

The pattern is clear. If we work to umpty decimal places, there are 10^umpty possibilities. That is, umptyplex.

If the decimal places go on `for ever', we first must ask `what kind of for ever?' And the answer is `Cantor's aleph-zero', because there is a first decimal place, a second, a third ... the places match the whole numbers. So if we set 'umpty' equal to 'aleph-zero', we find that the cardinal of the set of all real numbers (ignoring anything before the decimal point) is aleph-zeroplex. The same is true, for slightly more complicated reasons, if we include the bit before the decimal point.' [1]

All very well, but presumably aleph-zeroplex is going to turn out to be aleph-zero in heavy disguise, since all infinities surely must be equal? No. They're not. Cantor proved that you can't match the real numbers with the whole numbers. So aleph-zeroplex is a bigger infinity than aleph-zero.

He went further. Much further. He proved [2] that if umpty is any infinite cardinal, the umptyplex is a bigger one. So aleph-zeroplexplex is [1] Briefly: since the bit before the decimal point is a whole number, taking that into account multiplies the answer by aleph-zero. Now aleph-zero x aleph-zeroplex is less than or equal to aleph-zeroplex x aleph-zeroplex, which is (2 x aleph-zeroplex, which is aleph-zeroplex. OK?

[2] The proof isn't hard, but it's sophisticated. If you want to see it, consult a textbook on the foundations of mathematics.

bigger still, and aleph-zeroplexplexplex is bigger than that, and ...

There is no end to the list of Cantorian infinities. There is no 'hyperinfinity' that is bigger than all other infinities.

The idea of infinity as `the biggest possible number' is taking some hard knocks here. And this is the sensible way to set up infinite arithmetic.

If you start with any infinite cardinal aleph-umpty, then alephumptyplex is bigger. It is natural to suppose that what you get must be aleph-(umpty+1), a statement dubbed the Generalised Continuum Hypothesis. In 1963 Paul Cohen (no known relation either to Jack or the Barbarian) proved that ... well, it depends. In some versions of set theory it's true, in others it's false.