Skip to main content

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...

Comments

Popular posts from this blog

Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's recent gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some ex

Which idiot came up with percentage-based gradient signs

Rant warning: the contents of this post could sound like something produced by UKIP. I wish to make it clear that I do not in any way support or endorse that political party. In fact it gives me the creeps. Once upon a time, the signs for a steep hill on British roads displayed the gradient in a simple, easy-to-understand form. If the hill went up, say, one yard for every three yards forward it said '1 in 3'. Then some bureaucrat came along and decided that it would be a good idea to state the slope as a percentage. So now the sign for (say) a 1 in 10 slope says 10% (I think). That 'I think' is because the percentage-based slope is so unnatural. There are two ways we conventionally measure slopes. Either on X/Y coordiates (as in 1 in 4) or using degrees - say at a 15° angle. We don't measure them in percentages. It's easy to visualize a 1 in 3 slope, or a 30 degree angle. Much less obvious what a 33.333 recurring percent slope is. And what's a 100% slope