Although infinity doesn’t come up much at GCSE, it’s alluded to from time to time. When talking about experimental vs theoretical probability for example, or where the line ends for a function without limits. recurring decimals, the asymptote of the tan lines, or even the nth term of a sequence. So it seems apt that we look a little deeper into the rather difficult concepts behind infinity.

What is infinity?

Well, that depends. There’s a few kinds for starters. Let’s go with ‘traditional’ infinity, known sometimes as ‘potential infinity’. This is the infinity most of us know and probably don’t particularly love. Real numbers keep going, and going, and going, and going. They never end, they are endless, they are infinite. There is no ‘upper bound’. It is therefore bound*less*.

The symbol for (positive) infinity is the Lemniscate symbol, which is essentially a mobius strip, itself a never ending one-sided shape. ∞

Negative infinity is just -∞ and an infinitely small number, an *infinitesimal* number would be something like 1 / ∞

If we stop there, infinity is quite a manageable thing. However, if we delve a little deeper, it’s a hideously bizarre anomaly that is jam packed with counter intuitive mind blowing ridiculousness. So let’s do that!

Let’s compare odd and even numbers… to infinity!

There are infinitely many even numbers, but they can be put in ascending order. It’s fair to assume that each even number is ‘every other number’, as we’ve neglected to think about the odd numbers. So therefore, there are n even numbers for 2n numbers, where 2n must be infinity. Still with me? But we know there are in fact infinite even numbers, then there are 2 x infinite real numbers, but that can’t be right, because there’s no such thing as 2 x infinity. So… umm… now what?

Well, I guess we’ll have to accept what Galileo wrote about infinity:

“The attributes ‘equal’, ‘greater’ and ‘less’ are not applicable to infinite, but only to finite quantities’.

So maybe infinity is finite after all? Only a few nutters think like that!

But then we get ‘actual infinity’ which is kind of infinity, but with a defined limit. A box to put it all in if you will. Mathematicians like boxes to put things in.

Consider 1/2, 1/4, 1/8, 1/16, 1/32 … and so on. This is an infinite sequence, but you know it’s never going to get past 1. You could bound it using notation like this

{1/2, 1/4, 1/8, 1/16, …}

So now it’s an example of ‘actual infinity’, which causes umpteen paradoxes. So is there an ‘actual infinity’? Who the hell knows. Some say yes, some say no. Maths says yes so that you can manipulate things easier. Brain says no.

Another example of a sort-of-bounded infinity is the Sierpinski Triangle. Each triangle gets subdivided into smaller triangles, then they get subdivided, then they get… but it all stays inside the original triangle.

I’ll do a post all about fractals on another day, but they’re a great exploration into the infinite.

We’ve also stumbled across different *sizes* of infinity. Our fraction sequence is infinitely long, but in terms of SIZE, it’s nowhere near as big as an infinite sequence of real integers. It never gets past ‘1’ for starters.

So now we get to Cantor, who in 1891 started us all using the term ‘cardinalities’ to describe the different sizes of infinity.

He ended up constructing a whole series of sizes of infinity. The smallest of which were the natural numbers. He showed that there were more real numbers than natural numbers (infinitely more?!), even though there are infinite real and infinite natural numbers. What the hell is going on? Maybe the diagram below will help. If there’s infinite natural numbers, and infinite whole numbers, and infinite integers, and infinite rational numbers, and infinite real numbers, you can see that each ‘infinite’ is of a different size, as there’s simply more numbers to use!

So if there’s more of one than the other, are they in some way countable then? Countable infinites?! Well it turns out there’s plenty of those. Dr James Grime prefers the term ‘listable’ to countable. I’m inclined to agree.

Let’s see…

Listing (“counting”) **Natural Numbers:**

1, 2, 3, 4, 5, 6,… simple. I’m clearly not going to miss any of these out if I keep going…forever.

Listing **Integers:**

0, 1, -1, 2, -2, 3, -3 … simple… but arguably *twice as big* as the infinite natural numbers…?! So now we have a bigger countable infinity than the natural number infinity. But aren’t we *not *comparing sizes? Maybe we are now. I’m having trouble keeping up with myself here.

Listing **Rational Numbers:**

Hmm… trickier.

You have to be clever here, but it’s doable. If you just add 1 to the denominator of a fraction each time, you’ll never get to a whole number. Doh! There must be some kind of clever way of doing it… right? Right!

Here’s a way:

Listing *all* the** Real Numbers**? Well that includes **Irrational Numbers**. That causes a bit of a problem. How can I count the irrational numbers? They’re not fractions so… do I just randomly pick out a bunch of digits…to infinity…? Clearly these guys are a bit of a headache to be considered ‘countably infinite’.

Turns out we can’t count them… so they’re the biggest… infinity.

Are we finished? I didn’t mention any paradoxes yet! Here are some of the more famous ones:

**Zeno’s Infinity Paradoxes:** Dichotomy (quoted from wikipedia)

“Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.”

So what is the starting distance? Can Homer even begin his journey??

** Gabriel’s Horn:**

Take the equation y = 1/x

From x=1 to x infinite, we rotate y around the x-axis. This creates a horn shape with a wide opening of radius 1 at x=1.

The horn has a finite volume and infinite surface area!

So if you fill the horn with a finite amount of paint, to fill the finite volume, it must cover the sides, but they’re infinite, so how can it? So maybe the finite paint has covered an infinite surface area, and you have paint left over, to make up the volume of the shape. So umm… how does that work??

**Hilbert’s Hotel**

A Grand Hotel has an infinite number of rooms and infinite guests. So the rooms to guests ratio is 1 to 1. What happens if someone new comes along? Well the Hotel is full right? Nope. Everyone moves along one room, so room 1 goes to room 2 etc.

What if infinite new guests turn up?? BUT THE HOTEL IS FULL. Noop. Everyone could just go into the room that is double the room number they are already in, to make space for infinite guests.

So do we conclude that ∞ + 1 = ∞

or 2∞ = ∞ ??

isn’t this a good place to introduce aleph-zero rather than the Mobius band, to distinguish between ‘sizes’ of infinities? PS I got called out (rightly, I think) for using a diagram of infinite sets like the one you have – it shows natural within whole within integers within rational – but these sets are actually all the same size (because we can only draw finite sets to represent infinite ones) By an 11yo (sulk!)

There’s a lot I didn’t put in. People have written entire books on infinity! I see what you’re saying about the diagram. The one i used wasn’t written to be used for describing infinity/s, however I think it serves as a nice way of visualising why some infinities are considered bigger than others. I’m curious as to why / how it came about that you were discussing cardinality with an 11 year old! Please tell me more!!