Science of Discworld III

‘umptyplex’ will mean 10 umpty , 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: π (‘pi’). To one decimal place, π 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 π. 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. 5
    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 6 that if umpty is any infinite cardinal, the umptyplex is a bigger one. So aleph-zeroplexplex isbigger 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 aleph-umptyplex 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.
    The foundations of mathematics are like that, which is why it’s best to construct the house first and put the foundations in later. That way, if you don’t like them, you can take them out again and put something else in instead. Without disturbing the house.
    This, then, is Cantor’s Paradise: an entirely new number system of alephs, of infinities beyond measure, never-ending – in a very strong sense of ‘never’. It arises entirely naturally from one simple principle: that the technique of ‘matching’ is all you need to set up the logical foundations of arithmetic. Most working mathematicians now agree with Hilbert, and Cantor’s initially astonishing ideas have been woven into the very fabric of mathematics.
    The wizards don’t just have the mathematics of infinity to contend with. They are also getting tangled up in the physics. Here, entirely new questions about the infinite arise. Is the universe finite or infinite? What kind of finite or infinite? And what about all those parallel universes that the cosmologists and quantum theorists are always talking about? Even if each universe is finite, could there be infinitely many parallel ones?
    According to current