It is a common misconception that infinity is the biggest thing there is. Whenever you think of infinity, there is something even more unimaginably vast than the unimaginable vastness of infinity. There is an even bigger infinity.

Aleph Null

Aside from sounding really dramatic, aleph null (written as ℵ₀) is the fancy maths name for the smallest type of infinity, it is sometimes called the countable or listable infinity. This is the amount of whole numbers, and the size of infinity that I talked about in this post. In that post I talked briefly about equivalent infinities in the second ball game. A more mathematical way of showing infinities are equal is being able to pair up each item in one infinite set with each item in another.

As this is called the listable infinity, it is unsurprising that it is possible to list all the items in a set this size. For example, we can see that the whole numbers are this size, because we can list them:

• 1
• 2
• 3
• 4
• 5
• ...

There are several other sets of numbers that are easy to list, the square numbers, prime numbers, even numbers and several others. But there are some other groups of numbers that are slightly harder to list, how about the rationals - that is every number that can be written as a fraction?

Well, we can arrange every fraction in a table, where the column is the number on top and the row the number on the bottom - we can then draw a line through the table that will pass through every cell and simply move along the line, adding each fraction to our list as we reach it.

Aleph One

It may be tempting to ask if there is anything we can't list, if there is only aleph null. However, I did say at the start of this post that there is more than one different size of infinity, which means there must be something we can't put in a list. Spoilers - it's the real numbers.

The real numbers consist of all the numbers on the number line - any infinite or non-infinite string of repeat or non-repeating digits is a real number. As you can imagine, there are a lot of these numbers. But are there really more than an infinite amount of real numbers?

Let us assume that the amount of real numbers is aleph null, therefore we can put them in a list. I have made an attempt below. However, I can show that however good at making lists you are, you’ll have missed an infinite number of real numbers!

If we can find a number that isn’t on our list, we know our list is incomplete. One way to show a number isn’t on the list is to make sure that is to different to every other number in at least one digit. We can construct a number with exactly this property. To make our new number take the first digit of the first number in the list, add one to it unless it’s nine, then replace it with a zero. This will be the first digit of our new number. As it is different to the first number in the first digit, it is not the first number on our list. Repeat this step with the second digit of the second number, we now know our new number is not the second number on our list. As real numbers are infinitely long, we can repeat this process for every single number in our list, which gives us a new number not on the list.

• 3.465 ...
• 1.254 ...
• 5.145 ...
• 8.014 ...
• 4.255 ...

Other Infinities

This begs one important question, how many different sizes of infinity are there? Is aleph one the biggest infinity? Is there an infinity between aleph null and aleph one? We don't know and we will never know. It has been shown that it is impossible to prove any of these questions true or false within set theory. That doesn't mean they don't have answers, it just means we can only ever guess.

However, all is not lost; we can make modifications to set theory to allow this problem to be solved. There is an issue with this method - all mathematical proofs are built on axioms, by adding some we can get a solution to the above questions. The only trouble is, axioms are supposed to be self-evident, there is no proof for them. The axioms we have to add to get a solution to these problems are not self-evident.

What's worse, is that there are two candidates - one can be used to show that the above two infinites are all there is; the other can show there are an endless number of infinites where the number of infinities is greater than any other infinity. These are rather different results and we have no way of knowing which is true. This is a rather big problem for mathematicians.