Infinite musings

Infinity, as no end of people keep telling me since I wrote A Brief History of Infinity is a big subject, so I like to revisit it now and again. One of the joys of doing my talk on infinity, a real favourite of mine, is the way people's minds are duly boggled by the idea that there can be something bigger than infinity. And what's more, you can prove it without a single equation.

Thanks to the great German mathematician Georg Cantor we can establish this painlessly. The first step is to discover the concept of cardinality in set theory. A set is just a collection of things, and set theory is the maths that describes the workings of such collections, and from which all the basics of arithmetic can be derived. Cardinality is a measure of the size of the set, and the important thing to be aware of is that if we can pair off items in two sets so they are in one-to-one correspondence, those sets have the same cardinality - they are the same size.

Take a simple example - legs on my dog, Goldie, and the horsemen of the Apocalypse. I can pair of one leg with each of the horsemen. At the end of the process there are no horsemen or legs unpaired. So I can say the legs and the horsemen have the same cardinality. The clever thing about this is I can do it even if I have no idea how many legs or horsemen there are. (I do. It's four. But I don't have to know.)

So let's move on to infinity. The simplest infinite set is just the integers - the counting numbers. Now if I have another infinite set - for instance all the rational fractions - and I can pair off the items in the set with the integers then the two sets have the same cardinality - they are the same 'size' of infinity. And you can do this with the rational fractions. But not all sets of numbers are this accommodating.

Think of the set of every number between 0 and 1. Every single decimal number. This is also an infinite set, but has it the same cardinality as the integers? If I can set up a simple list of the numbers, it has.

So lets imagine that list, going:

0
...
0.5
...
1

and let's imagine writing out the first few numbers. I can't really do this in order, as after zero we will have

0.000000.... all the way to infinity 1
0.000000.... all the way to infinity 2
...

which is rather difficult to write out, so we'll scramble the list and come up with something like this:

0.21584032048303404035930…
0.92939212493373921239446…
0.52030202578403024842231…
0.65873032294825193482488…
0.15740302049304958102733…
0.33335939320293919290111…

Now I'm going to add 1 to the first decimal place of the first number, the second decimal place of the second number and so on all the way through. (If it's 9, I'll change it to 0.) So we go from

0.21584032048303404035930…
0.92939212493373921239446…
0.52030202578403024842231…
0.65873032294825193482488…
0.15740302049304958102733…
0.33335939320293919290111…

to

0.31584032048303404035930…
0.93939212493373921239446…
0.52130202578403024842231…
0.65883032294825193482488…
0.15741302049304958102733…
0.33335039320293919290111…

Now let's pull out those incremented values to see this number:

0.331810...

This is a very interesting number. It's not the first number in the original list, as it differs in the first decimal place. It's not the second number in the list, as it differs in the second decimal place. And so on, all the way through. What we've done here is produce a number that isn't in the list. It actually isn't possible to have a simple list of every number between 0 and 1 - we can't match all the numbers off one-by-one with the integers. The set of numbers between 0 and 1 has a different cardinality to the infinity of the integers - it's bigger.

Enjoy letting your mind boggle at that. This is why infinity is such fun...