Note: If you need a refresher on well-orderings, ordinals, and cardinals, check out the Ordinals and Cardinals page.

When I was in elementary school, a tiny part of the mathematics curriculum, a few minutes in fourth grade, was the distinction between ‘ordinal numbers’ and ‘cardinal numbers.’ We were vaguely told that ordinal numbers dealt with order and cardinal numbers dealt with amount, and things were pretty much left at that. I remember feeling vaguely dissatisfied with this lesson and, looking back at it, I recognize it as representative of a somewhat common occurrence in the post-New Math American classroom. A very interesting topic (in this case, ordinals vs. cardinals) was brought up, and basically none of what makes it interesting was discussed.

In this case, the basic issue is that we talked about ordinals and cardinals (and everything else in mathematics) solely in the context of finite numbers (this is entirely understandable; we were, after all, in fourth grade). And, in the realm of the finite, ordinals and cardinals are pretty much the same! Given, say, four indistinguishable objects, there is only one essentially distinct way to well-order them. There’s a first element, a second, a third, and a fourth. That’s it! Each finite cardinal corresponds to a finite ordinal, and vice versa, and no conceptual clarity is gained from making a distinction between the two.

(The reader may rightly accuse me of being a little harsh here and of purposely conflating the common and technical usages of the term ‘ordinal number.’ The ‘ordinal numbers’ we saw in elementary school were of the form ‘first,’ ‘second,’ ‘third,’ etc. I still maintain, though, that the difference we were taught between finite ordinals and finite cardinals is uninteresting and misleading. The mere act of counting a finite set imposes an order on it, and the type of this order is unique. It is of course important to learn the words ‘first,’ ‘second,’ ‘third,’ etc., but that is not what this lesson was doing. (I’d also be happy to be proven wrong about this – if you had a very different experience, either as a student or as a teacher, let me know!))

The situation changes as soon as we reach the infinite. Given a countable infinity of indistinguishable objects (one for each natural number), there are many essentially distinct ways of well-ordering them. One can, for example, order them like the natural numbers (this corresponds to the ordinal $\omega$). One could set one object aside, order the others like the natural numbers, and then put the set-aside object at the end (this corresponds to the ordinal $\omega + 1$ and is easily seen to be distinct from the first ordering, as it has a largest element while the first does not). One could divide the objects into two infinite sets, order them both like the natural numbers, and put one set after the other (this corresponds to the ordinal $\omega + \omega$. There are in fact uncountably many different ways to well-order this set of objects, each corresponding to a different ordinal $\alpha$, where $\alpha < \omega_1$ (recall that $\omega_1$ is the least uncountable ordinal).

The world of countable ordinals is a fascinating and confusing place, and some people spend much of their careers studying just these objects. Let me give you one example. We will first need some notation. Suppose that $\alpha, \beta$, and $\gamma$ are ordinals and $n$ is a natural number. Recall that, as a set, an ordinal is considered to be the set of all ordinals smaller than it. Also, recall that $[\gamma]^n$ stands for the set of all subsets of $\gamma$ of size $n$. Then

$\gamma \rightarrow (\alpha, \beta)^n$

stands for the following assertion: whenever $f$ is a function, $f:[\gamma]^n \rightarrow \{0,1\}$, there is either a set $X \subseteq \gamma$ of order-type $\alpha$ such that, for all $x \in [X]^n$, $f(x) = 0$, or there is a set $Y \subseteq \gamma$ of order-type $\beta$ such that, for all $y \in [Y]^n$, $f(y) = 1$. This is thus a statement in Ramsey Theory, introduced in a previous post. In that post, we proved the Infinite Ramsey Theorem, a special case of which can be rephrased in our new terminology as:

for all $n < \omega$, $\omega \rightarrow (\omega, \omega)^n$.

This statement becomes false when we replace $\omega$ by $\omega_1$. In fact, it fails already for $n=2$, which we denote by

$\omega_1 \not\rightarrow (\omega_1, \omega_1)^2$.

There is a lot to say about this that we won’t bring up here. For example, in our next post, we will consider the question of whether there are any cardinals $\kappa > \omega$ for which

$\kappa \rightarrow (\kappa, \kappa)^2$

is true. There is also a great deal to be said about slightly weakening the above false statement about $\omega_1$ and asking:

for which ordinals $\alpha < \omega_1$ is $\omega_1 \rightarrow (\omega_1, \alpha)^2$ true?

This question gets quite complicated but is fairly well understood. However, as soon as we replace the ‘2’ with a ‘3’, our current understanding becomes quite poor. For example, the following seemingly basic question is wide open:

Is it the case that, for all $\alpha < \omega_1$, $\omega_1 \rightarrow (\alpha, 4)^3$ is true?

The answer is ‘yes’ for all $\alpha \leq \omega^2 + 1$ and ‘no’ for $\alpha = \omega_1$. In between, we have no idea.

For an extensive tour of the countable ordinals, I highly recommend you take a look at the recent series at John Baez’s excellent blog, Azimuth (the first installment is here). Also, make sure to visit David Madore’s delightful interactive visualization of the countable ordinals.