After you’d been playing for a while she gave you another bag containing some blocks with the a number painted on one side of each of them; 0, 1, 4, and 9. Feeling playful you put the horse on block 0, the sheep on block 1, the cow on block 4 and the pig on block 9. Your mother walked back into the room and asked you how many blocks you have and you answered ‘4′ immediately. You didn’t have to count them this time because you already knew how many animals there were and you’d just paired the animals and blocks together in a one-to-one fashion.

Flash forward to now and matching items in different sets is still useful because you can see whether one set is bigger, smaller or the same size as another set. If you’re counting items in a finite set you will typically match them with a section of the “natural” numbers; 0, 1, 2, 3, 4, and so on, but what happens if you have a set containing an infinite number of items? Well, you can continue using the natural numbers; because you can always add 1 to them they never end. Often you might think of the last number in the list of natural numbers as being infinity (although you should realise that infinity itself doesn’t exist as a number in any real sense). So you see that ∞ + 1 = ∞, and this can be extrapolated to give ∞ + n = ∞ and ∞ - n = ∞.

Now think of the square numbers; 1*1 = 1, 2*2 = 4, 3*3 = 9 and so on. At a first glance you might think that there are more natural numbers than square numbers because they consist of all the square numbers and all the non-square numbers (such as 2, 3, 5, 7, 8, or 10). However, the number of square numbers can’t be finite because they are formed by taking a natural number and multiplying it by itself, and you know that there are an infinite number of natural numbers. Consider the implications of how you’ve formed the square numbers: they have been paired one-to-one with the natural numbers. This means that there are just as many squares as there are naturals.

Now consider that if you take a large enough sequential set of numbers from the naturals that contains both square and non-square numbers, say { 2, 3, 4, 5, 6, 7 }, you would expect to find more non-square numbers than square numbers, so there are at least as many non-square numbers as square numbers and at most the as many as the natural numbers. As the number of squares is the same as the number of naturals (using the matching method), this means that there are also as many non-square numbers as there are squares or naturals.

Infinity doesn’t behave like a regular number, and addition and subtraction don’t work in the usual way; you’ve just seen that an infinite set of numbers (the naturals) less another infinite set (the squares) is still infinite (the non-squares). So now you know that ∞ + ∞ = ∞ and ∞ - ∞ = ∞.

Now consider the set of integers which is comprised of the natural numbers and their negative counterparts; {…, -3, -2, -1, 0, 1, 2, 3, …}. This time you might expect twice as many integers as natural numbers because we’re repeating every number twice, but they can still be paired off with the naturals: 0 → 0, 1 →1, -1 →2, 2 →3, -2 →4, and so on. This can be extrapolated to give m * ∞ = ∞; multiplication with infinity doesn’t work in the normal way either.

So, what about ∞ * ∞ ? Think of the rational numbers, or the fractions as they’re better known, such as ½, ⅓, ¼, or ¾. If you were to set up a graph with the natural numbers on x-axis and also on the y-axis and at each whole number pair you let the value equal y/x then you can generate all the rationals:

All the fractions you can name will be in this grid somewhere, for example 1234/5678 will be at y = 1234, x = 5678.

Notice that by listing the fractions this we’re still including some numbers twice; 1/1 and 2/2 are the same number. To avoid this we simply ignore any fractions that are not fully cancelled down. Then we start at 1 and zig-zag back and forth across the grid we’ve just created as shown in the example. This creates a one-to-one mapping as we’ve done before, which means that there are as many rationals as there are natural numbers. Further, you can now see that ∞ * ∞ = ∞.

What can you try after multiplication? What about exponents, where you raise one number to another power; 2³ = 8 or 5² = 25 for example. After all the examples above you might think that 2^∞ = ∞, but you’d be wrong. That number is bigger than the infinity we’ve been working with.

In the 19th Century Georg Cantor developed most of the work I’ve described so far. He’s considered the father of set theory, which are collections of objects like the bag of toys, or the naturals, integers and rationals you’ve seen so far. Having developed sets he was also the first person to come up with the notion of being able to match things in order to count them.  Then he used this to develop the (pretty amazing) examples you’ve seen above.

First you need to know that the set of real numbers contains all the rational numbers you’ve already seen, and also the irrational numbers, like π or √2 which you can’t express as a fraction. Cantor demonstrated that the real numbers can’t be matched one-to-one with the natural numbers using an argument similar to this:

Consider any decimal expansion like 0.10549815007… In order to generate all the numbers between 0 and 1 you would need to list all the numbers of the form 0.x₁x₂x₃x₄…x_∞ and replace each of the x_i with a number between 0 and 9. You should also exclude the possibility that all of the x_i=0 or 9 since these just equal 0 or 1 respectively.

Now in a similar way to when you listed the rationals we set up a grid so that you can reference each of the x_i in the decimal expansion:

0.x₁₁x₁₂x₁₃x₁₄…
0.x₂₁x₂₂x₂₃x₂₄…
0.x₃₁x₃₂x₃₃x₃₄…
0.x₄₁x₄₂x₄₃x₄₄…

Now it’s set up you should be able to count through the grid in the same way that you counted through the rationals. The trouble is that unlike the rationals where you had listed every number, you have not listed every number in the reals. To see this consider the following number:

Let Y = y₁y₂y₃y₄… and set y₁ to be different from x₁₁, y₂ to be different from x₂₂, y₃ to be different from x₃₃, and so on. You now have a number that is different from the first number in the list, different from the second number in the list, and the third, and the fourth, and so on. In fact, you’ve just created a new number that wasn’t in the list before and you could generate new numbers like this forever.

So we can’t even count the numbers between zero and one, or any other interval of the real numbers, no matter how many we choose. That means that there are more real numbers than natural numbers and you’ve now found out about a new type of infinity.

To avoid confusion between the types of infinity Cantor labelled the size of set of natural numbers pronounced Aleph 0, and the size of the set of real numbers pronounced Aleph 1. You can generate even bigger infinities by raising 2 to the power of Aleph_n, an infinity of bigger infinities in fact!

Finally Cantor wondered whether there was a number between and or whether = . Cantor believed the latter and spent years trying to prove it. Sadly it was a bugger of a question, and only thanks to Kurt Gödel (the guy who developed the incompleteness theorems) and Paul Cohen do we now know that the question is undecidable, that is it is neither provable nor disprovable from the standard axioms used in set theory. There could well be types of infinity we’ll never know about.