Three types of infinity

Numbers have multiple purposes, some of which are

• Locating things in a list, such as the first person to walk on the moon, the second star to the right, the seventh film in the Fast and Furious franchise. These are all answers to the question, “Which one?”
• Counting things, such as 1 green sheep, 7 dwarfs, 101 dalmations. These are all answers to the question, “How many?”
• Locating things in time or space, such as 6.5 km away, 500 miles away, 5 years ago. These are all answers to the questions, “Where?” or “When?”

There are other purposes, such as measuring or comparing or having something to think about, but they aren’t what I feel like highlighting right now.

The purposes of number give you various models to help think about what numbers are doing when you operate on them, and even if a number began life as a location, you can think about it as a size if that helps. So, even though you know that starting at the 10th person in the list then moving forward three people and starting at the 3rd person in the list and moving forward 10 people are very different, if you think of them as combining collections of people passed so far, you can tell they’re the same.

But infinity is different. The three purposes of locating in a list, counting, and locating in space don’t line up so neatly with each other when you’ve stopped using ordinary numbers you can reach in time. That causes the infinities that go with different purposes to be different to each other.

[Disclaimer: I am not going to go into all the mathematical or philosophical detail possible here, and I’m not going to use all the formal terminology. That’s not the point of this post. So if you’re a mathematician or philosopher who knows about this stuff, think carefully about what I’m trying to achieve here before getting upset at me. And if you want to find out more, I’m sure a quick internet search will provide all the details you need.]

Infinity of location

First, the infinity represented by the symbol ∞. That infinity is a location. When you draw a line and mark it out with numbers, you’re using numbers as locations. You can go forwards using positive numbers and backwards using negative numbers. And what if you go forwards forever, following all the way to the end of the line? That location is called “infinity” and we label it with the symbol ∞. If you go backwards forever you get to the location “minus infinity” or -∞. So with a number line going both ways, there are actually two infinite locations of ∞ and -∞. These infinities aren’t numbers in quite the same way as other numbers, because really they are only locations and can’t also represent counts of how many things there are

So, thinking of numbers as locations in space produces ∞ and -∞. What if you think of numbers as serving the purposes of counting or listing? What infinites do you get then?

Infinity of counting

When you count literal things like sheep, dwarfs, and dogs, there is always a perfectly good number to count them with. Those numbers are called the natural numbers. But some abstract things don’t have a nice natural number for how many there are, most notably, the very numbers you use to count with. How many natural numbers are there? This is another type of infinity. And its name is… however many natural numbers there are. There isn’t a name, really, though when someone says “infinitely many” the size of the collection natural numbers usually what they mean. Many mathematicians use the Hebrew letter aleph with a zero as subscript – ℵ₀ – to represent this quantity.

This isn’t the only quantity that is infinite. Think about all the sets of natural numbers such as the set with just 1 in it, the set of just the digits 1 to 9, the set of even numbers, or the set of prime numbers. How many possible sets of natural numbers are there? That’s another infinity that’s even bigger than ℵ₀.

These infinities are called the transfinite cardinals. This is in reference to the word cardinality, which is the size of a set.

Infinity of listing

What about the first purpose of numbers I mentioned earlier – listing things and knowing where in the list you are? Well there’s infinities for that too.

Imagine listing things in order: first, second, third, fourth, … and your list was somehow longer than all the natural numbers you had available, what number would you say after you’d gotten through all the natural numbers and still needed another one? Well some mathematicians call that ω, a lower case omega. What about the next thing in your list? Well that would be ω+1. It doesn’t have it’s own name, it’s just the number after ω.

That’s the fundamental thing about numbers used for listing: every number has a number right after it, and that’s exactly what “+1” means. In that sense, the two numbers in an addition mean different things. When I say “10+3”, that means, start at the 10th object in the list, and then go 3 positions after that, which arrives at the 13th object in the list. The first number is where to start, and the second number is how many positions forward in the list to go, and the final answer is where you end up. With this interpretation, it’s actually surprising that 10+3 is the same as 3+10, though if your numbers are sizes of collections of things, it’s not as surprising.

With infinite listing numbers, the connecton to sizes is broken, and so it’s not actually true that you can do addition in any order you want. For example, ω+1 is the number after ω. But 1+ω is the number ω steps beyond 1, which is in fact just ω. It doesn’t matter where you start in the natural numbers, you still have infinitely many of them to go to get past them all, so going that far just gets you to the end, not any further. (Yes I know how confusing that sentence is. Sorry. Infinities are confusing.) So 1+ω=ω ≠ ω+1.

These numbers with their weird order-matters addition are called the transfinite ordinals. That goes with the fact that they started life as ordinal numbers: first, second, third etc.

It’s worth noting that this weirdness with addition having different meanings in different orders doesn’t happen with the transfinite cardinals, because combining sets definitely doesn’t have an order. (Though there is other weirdness: ℵ₀+1 and 1+ℵ₀ are the same, yes, but they’re both the same as ℵ₀. This is why you can still fit an extra person in Hotel Infinity even though it’s full.) And with ∞, well you can’t add anything to that at all. It’s a place that isn’t really a number.

Conclusion

So there you go. There’s at least three different infinities to describe the number just beyond the ones we’re familiar with: ω, ℵ₀, and ∞. They are the next number in the list, the size of the counting numbers, and the location at the end of the number line. We need them because each purpose of number has a different way to imagine what the numbers are and so a different way to imagine what is beyond beyond the numbers. And that different purpose dictates the sorts of operations that are possible on these infinities too.

This deep connection to purpose and metaphor in maths is something that really appeals to me.