We use cardinal numbers to count how many elements are in a set. Another thing we think of numbers for is listing elements. That is, we put things in order: first, second, third, and so on.
We identified a cardinal number as an isomorphism class of sets. Ordinal numbers work much the same way, but we use sets equipped with well-orders. Now we don’t allow all the functions between two sets. We just consider the order-preserving functions. If and are two well-ordered sets, a function preserves the order if whenever then . We consider two well-ordered sets to be equivalent if there is an order-preserving bijection between them, and define an ordinal number to be an equivalence class of well-ordered sets under this relation.
If two well-ordered sets are equivalent, they must have the same cardinality. Indeed, we can just forget the order structure and we have a bijection between the two sets. This means that two sets representing the same ordinal number also represent the same cardinal number.
Now let’s just look at finite sets for a moment. If two finite well-ordered sets have the same number of elements, then it turns out they are order-equivalent too. It can be a little tricky to do this straight through, so let’s sort of come at it from the side. We’ll use finite ordinal numbers to give a model of the natural numbers. Since the finite cardinals also give such a model there must be an isomorphism (as models of between finite ordinals and finite cardinals. We’ll see that the isomorphism required by the universal property sends each ordinal to its cardinality. If two ordinals had the same cardinality, then this couldn’t be an isomorphism, so distinct finite ordinals have distinct cardinalities. We’ll also blur the distinction between a well-ordered set and the ordinal number it represents.
So here’s the construction. We start with the empty set, which has exactly one order. It can seem a little weird, but if you just follow the definitions it makes sense: any relation from to itself is a subset of , and there’s only one of them. Reading the definitions carefully, it uses a lot of “for every”, but no “there exists”. Each time we say “for every” it’s trivially true, since there’s nothing that can make it false. Since we never require the existence of an element having a certain property, that’s not a problem. Anyhow, we call the empty set with this (trivial) well-ordering the ordinal . Notice that it has (cardinal number) zero elements.
Now given an ordinal number we define . That is, each new number has the set of all the ordinals that came before it as elements. We need to put a well-ordering on this set, which is just the order in which the ordinals showed up. In fact, we can say this a bit more concisely: if . More explicitly, each ordinal number is an element of every one that comes after it. Also notice that each time we make a new ordinal out of the ones that came before it, we add one new element. The successor function here adds one to the cardinality, meaning it corresponds to the successor in the cardinal number model of . This gives a function from the finite ordinals onto the finite cardinals.
What’s left to check is the universal property. Here we can leverage the cardinal number model and this surjection of finite ordinals onto finite cardinals. I’ll leave the details to you, but if you draw out the natural numbers diagram it should be pretty clear how to how that the universal property is satisfied.
The upshot of all of this is that finite ordinals, like finite cardinals, give another model of the natural numbers, which is why natural numbers seem to show up when we list things.