I’d like to begin a series addressing the age-old school-boy question, namely,

I’ve found that most people, even mathematically-inclined folk, are contented with the answer “$\infty$” – and so was I, until I learned about ordinal and surreal numbers – these are classes of numbers which contain all of the regular counting numbers $\mathbb{N} = \{ 0, 1, 2, 3, 4,… \}$, as well as more exotic elements, some of which we will meet today. When I learned about them in high school, I thought these number systems might help us to extend sums like these, after all, I had seen how infinite sums such as:

Had been conquered by clever proofs like the one below:

So why couldn’t the notion of an infinite sum be extended even further to include any arbitrary sequence? In this series I would like to catalog all of the approaches which I’ve seen for answering this specific question – yes, surprisingly, there are different answers depending on the field of mathematics you choose to study. I’ll begin with the most natural place, which is the study of Set Theory and in particular the theory of Ordinal numbers. Upon my presentation, it may well seem like the theory we explore will be able to answer this question, but by the end I’ll introduce some problems with this perspective.

Set Theory

Set theory refers to the current foundations of mathematics. At one point, the foundations undergirding mathematics were Geometry, in particular Euclidean Geometry, but a series of revelations and crises, as well as some remarkable results by one Georg Cantor, set the stage for the modern foundations we have today, these are a collection of axioms known as ZFC Set Theory. ZFC stands for Zermelo, Frankel, and Choice, the first two were leading pioneers in the foundations of set theory, and Choice refers to an axiom of Choice. There are advocates for competing foundations, namely Category Theory and Homotopy Type Theory (HoTT). Before we can answer the question above, we ought to first answer even more basic questions, such as, what is meant by “$1$,” what is meant by “$+$?” Before we can do that, we need to discuss Axioms.

If you have small children, or were once an inquisitive child yourself, you know that, inevitably, they discover that you can ask “why?” ad infinitum in response to any answer you give. This sequence of questions, though potentially infinite, is always cut off at some finite stage, usually by an adult that’s lost his patience, exclaiming, “BECAUSE I SAID SO!”

And so it is much the same in mathematics, for every result one can ask, “but why is this so?” - until we meet an axiom. An axiom is an assumption, something which is asserted to be true without proof. It is should be “obvious” or “intuitive” to some degree. ZFC Set theory is a collection of axioms, it is the collection of axioms in some sense, because all of mathematics is done with these assumptions. We won’t get into what these axioms are, that would be a whole series of its own, but I feel compelled to mention that this is the system mathematicians have adopted. We’re going to adopt a “naive” version of set theory, for our purposes it’ll suffice, at least for now.

What is a number?

We’ll now construct the natural numbers. To a set theorist, everything is a set, and that includes numbers. First, \(0 = \varnothing = \{ \}\), that is, $0$ is the empty set. It would be an insult to call any other the empty set, so he can have it. From there, we proceed in stages.

And on and on it goes. For any natural number $n$, we have \(n = \{ n-1, n-2, \ldots, 3, 2, 1, 0 \}\). In general, if you are at stage $n$, say, you can get to $n^{+} = n+1$ by taking

This operation is called the “successor” function, and with it, we can now define addition.

What is addition?

We’ll write $+(a, b)$ in place of $a+b$, but they should signify the same thing to you. You’ll see why this choice is made in a moment. Here is the definition, we’ll go through an example to make it clear, because when you first see it it looks quite baroque.

Example: computing 3 + 2

We compute $+(3,2)$ using the recursive definition of addition:

Since $2 \neq 0$, we apply the second case:

Again, since $1 \neq 0$:

Now we reach the base case:

Substituting back step by step, we obtain:

Therefore, we get the happy result, $3 + 2 = 5$. Aren’t you glad you’ve read this blog?

1+1+1+...?

There is no good read to stop our procedure for generating numbers at any finite stage, in fact, we can form the familiar collection:

And then take its successor:

These numbers, and indeed the infinitude of numbers which come afterwards, ($\omega+2, 2 * \omega, \omega^{2}, \omega^{3}, \omega^{\omega}, $ etc…) form a collection known as the ordinal numbers, which are an extension of the natural numbers $\mathbb{N}$ with similar properties, such as that you can add, multiple, exponentiate, and so on with these numbers and still stay within the class. One can check that $+(\omega, 1) = \omega+1$ as one would expect, but, consider the interesting sum: $+(1, \omega)$. How can we go about computing this sum? $\omega$ is not the successor of any number, (there is no $\omega-1$ in the ordinals, just as there is no $0-1$ in $\mathbb{N}$) and it is not $0$, so it doesn’t fit within either of our two cases. What we must do is extend our definition as follows:

Here, a limit ordinal is an ordinal like $\omega$ in that it is not the successor of any ordinal. With this definition in hand, we see that

That isn’t a typo. $1+\omega = \omega \neq \omega+1$. That’s quite a surprising result! In light of all this, it might feel like the following should be true:

There are some problems with this. For starters, we haven’t quite defined what it means to take an “infinite sum” with the ordinals. We can take a finite number of sums, as many as we please, but we have yet to define how we would go about taking an infinte sum. Supposing we did have a working definition of infinite sums, and given what we have just shown above, we can derive some nice properties, for instance the equality:

But,

So in some sense, infinite addition should not commute with finite addition. This might look promising, and gives us some intuition for the result above. In the next post, I’ll talk about some attempts to define this infinite addition operation, and we’ll learn about multiplication in the ordinal numbers.