An infinite series is a familiar mathematical concept, where ‘…’ effectively indicates ‘don’t ever stop’ – for example 1 + ½ + ¼ + ⅛… an infinite series totalling 2. This, though is the last of a short series of posts about infinity.
Sets
Cantor’s first great contribution was to formalise the mathematics of sets. Sets had been around really as long as people conceptualized – but Cantor embedded them firmly into mathematics. A set is really just a group of things. They could have something in common – like the set of things that look like an orange, or the set of people with the name Brian – or they could be as disparate as the set of things you thought about today. Cantor built on work of other mathematicians to pull together a picture of how sets operate that lets us do everything from define the numbers to establishing simple mathematical operations. But for our purposes I just want to pick out one aspect of set theory because it is going to be very relevant to the jump from sets to infinity. It’s a property of a set called its cardinality.
Let’s think of a two very simple sets – the first is the set of legs on my daughter’s dog. The second is the horsemen of the apocalypse. These two sets are said to have the same cardinality if I can pair of the members of the set on a one-to-one basis. Let’s try that.
Front right, for example could pair up with Death, Front left with Famine, Back right with Pestilence and Back left with War. I’ve exhausted both legs and horsemen, so they have the same cardinality. But – and here’s the clever thing – I needn’t have known how many legs the dog has or how many horsemen there are. Okay, in this example it’s no secret, but it wasn’t necessary to know. (can I just say that no animals were hurt in the making of that graphic).
With set theory in place, Cantor was ready to build on Galileo’s observations about the way that with infinity you could get one-to-one correspondence with a ‘smaller’ infinite set. Remember Galileo’s description of the integers, the counting numbers, and the squares. We can say that the infinite set of counting numbers has the same cardinality as the set of squares, because we can do that one-to-one pairing off, even though we don’t know how many we are dealing with.
And we know that the squares are a subset of the integers. Subset’s a spot of set theory that has escaped into the world, but specifically we mean that all the squares are members of the set integers, but it isn’t the full set – there are members that are squares. Cantor, cunningly, used this strange behaviour to define an infinite set. An infinite set is one that has a one-to-one correspondence with a subset.
Aleph null
Once we’re taking a set theory approach to infinity – and dealing with the true infinity, not Aristotle’s potential infinity – it’s useful to have a different symbol for it. Cantor chose aleph ℵ, the first letter of the Hebrew alphabet, and specifically he called the infinity of the counting numbers aleph zero or aleph null because we can’t assume that all infinities are same, so this is the basic infinity, the infinity of the counting numbers. Aleph null has some pretty strange qualities. We can add one to it and still end up with the same number. You can see why if you imagine doing the one-to-one correspondence starting with an extra value called x, then go on with the rest of the numbers.
What’s more, add aleph null to itself and you get aleph null. Again, we can imagine this happening by setting up a correspondence alternately with two lots of counting numbers. And for that matter you can multiply aleph null by itself – and still get aleph null. Imagine matching each of the numbers with its square. There are enough gaps left to take up all the remaining sets of aleph null.
Mind boggling though this behaviour is, we are actually used to special cases in arithmetic. After all, 0 + 0 = 0 and 1 x 1 = 1. And really we should hardly be surprised if the answer never changed, because surely there can be nothing bigger than infinity. Only that’s exactly what Cantor set out to prove wasn’t true.
The cardinality of the rational fractions
We’ve looked at aleph null squared, but Cantor wanted to check how flexible aleph null really was. He’d checked out the counting numbers, but how about fractions? He started with rational fractions, fractions made out of the ratio of whole numbers. Cantor was to prove that there were also aleph null of these using a delightfully neat proof that requires no maths. Imagine laying out every fraction there is in a huge table. In fact, to make it simple we’re actually repeating many of the fractions. If you look down the diagonal coloured in yellow in the image below, they’re all 1. It doesn’t matter that we’ve got redundancy, the table, continued for ever in both directions, has every single ratio in it.
Now what Cantor did was to set up a repeating path through the table. A simple, repeatable rule that enables us to work through the whole table. Finally, he puts each jump in one to one correspondence with a counting number. We’ve just proved that aleph null applies to the rational fractions as well – they have the same cardinality as the integers because we can match them one-to-one by going through this sequence. Incidentally, though the first one in the video below is the path Cantor used, it’s not the only one that exists. He could just as well have used the second one, for instance… the point is that there is a mechanism to set up a one-to-one correspondence.
What about irrational fractions?
Yet these aren’t the only sort of fractions. There are also the irrationals, the numbers that written as decimals go on for ever and ever – presumably to infinity. Do these also squeeze into aleph null? With another blindingly simple proof, Cantor was to show that this wasn’t the case. (I ought to make it clear that what comes next is not the actual proof, but illustrates how it works.)
Let’s imagine putting every single fraction, rational and irrational, between 0 and 1 into a list. If I could really achieve that, then I could use exactly the same one-to-one matching proof, matching each fraction against its position in the list, an so would prove this was another aleph null set. Now to work this proof I need to be able to study sequential numbers in the list. If I have them in order I can’t do that, as two numbers in order only differ right at the end of an infinity long set of digits – so I’m going to scramble the list and pick out the first few randomly selected numbers.
Now let’s look at the diagonal set of digits in those numbers. You could imagine this as a number in its own right: 0.220709 and so on. Now let’s add one to each digit. So instead of reading 0.220709, the diagonal is 0.331810 (9 flips over to zero).
Finally, let’s compare this new number with our original table. It’s not the first number, because they differ in the first digit. It’s not the second number, because they differ in the second digit. It’s not the third number. And so on. We have generated a number that doesn’t appear in our list. Cantor had proved with beautiful simplicity that you can’t cram all the fractions between 0 and 1 into a list with cardinality aleph null. The count of these fractions was something bigger, something bigger than infinity. Take a moment to think about that.
