Infinity and beyond

The set of natural numbers is infinite: 1, 2, 3…. The set of real numbers is also infinite: 0.1, 0.11, 0.12, 0.2… but it’s larger than the infinity of natural numbers. Georg Cantor devised an elegant argument to prove these different infinities.

Cantor's diagonal argument
Cronholm144, CC BY-SA 3.0, via Wikimedia Commons

Okay, I’m a little nervous whenever I try to explain dense mathematical concepts on this website, because I’m not a math teacher and it’s difficult to simplify these ideas without impoverishing their meaning. Nevertheless, today I’m going to try to explain Cantor’s diagonal method. When Cantor proposed this theory in 1891 it was widely ridiculed; today there’s a whole Wikipedia page dedicated to criticisms (see the links below) but outside of some pseudo-mathematical cranks it’s pretty widely accepted.

Take natural numbers. These are the counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. Obviously, there’s no end to these natural numbers – you can keep counting up forever and never reach a natural end. So, taken together, the natural numbers are infinite.

Next, let’s consider a series of sets. A set here is a collection of numbers, a long (potentially infinite) list. In my example, the first set is the set of natural numbers:

Set 1: 1, 2, 3, 4, 5, 6, 7, 8...

The second set is just the odd numbers:

Set 2: 1, 3, 5, 7...

The third set is just the even numbers:

Set 3: 2, 4, 6, 8...

The fourth set is a random collection of numbers that I just chose for the hell of it:

Set 4: 18, 67, 22, 1, 72, 5...

I can just keep making new lists like this forever. In other words, the number of possible sets is infinite. But, and this is the really crucial part: it’s even more infinite than the infinity of natural numbers.

To demonstrate this, let’s create a new set. From my sets above, we’ll take the first number from the first set, the second number from the second set, the third number from the third set, and so on. This is the new set:

Set A: 1, 3, 6, 1...

Now, I’m going to change every single one of those numbers to something different – here, by adding 1 to each number.

Set B: 2, 4, 7, 2...

This final set is different from every set that came before it. Because of the way we constructed it – taking bits from all of the other sets and changing them – it is literally impossible for any existing set in our infinite series of sets to be the same as this one. And that will always be true, no matter how many more sets we create.

What does all this mean? The set of natural numbers is infinite, but we can keep counting upwards as long as we like. The set of infinite sets is infinite, but it’s an infinity we cannot even approach by counting upwards in the same way. In mathematical terms, the set of natural numbers is “countably infinite.” Our set of infinite sets is bigger than that: it’s “uncountably infinite.”

Phew. Still with me? The final kick in the pants for a single type of infinity: it turns out that the set of real numbers – that is, natural numbers, negative numbers, fractions, and irrational numbers like pi – is uncountably infinite. Numberphile has a nice illustration of this:

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