Why do we define ordinals the way we do?

Question

I am studying set theory and have recently encountered the concept of ordinals. The way I understand them, we are trying to classify all well ordered sets (which have desirable inductive properties) into equivalence classes w.r.t. a relation that preserves the order structure, and have a canonical representative for each equivalence classes, but since these turn out to be proper classes and not sets, we call them order types.

And it turns out that if we take the von Neumann ordinals $\left(\left\{ \emptyset,\left\{ \emptyset\right\} ,\left\{ \emptyset,\left\{ \emptyset\right\} \right\} ,\ldots,\omega,\omega\cup\left\{ \omega\right\} ,\ldots\right\} \right)$ then we can prove that any well ordered set is isomorphic to exactly one of these, called its order type.

So my question is what's the motivation for the further abstraction of defining an ordinal as a transitive set which is well ordered w.r.t. $\in $ ? Can't we simply use the fact that the von Neumann ordinals possess these properties? Is there any context where we use ordinals that aren't the von Neumann ordinals? Or is my understanding as presented above somehow flawed?

Answer 1

Your misunderstanding is that you think "ordinals" and "von Neumann ordinals" means different things, such that only some of the ordinals in set theory would be von Neumann ordinals.

Since there is really no such distinction in a formal development of set theory, you become confused.

In other words, the sets you're thinking of when you write $$\emptyset,\{\emptyset\},\left\{ \emptyset,\left\{ \emptyset\right\} \right\} ,\ldots,\omega, \omega \cup \left\{ \omega\right\} ,\ldots$$ are exactly the same sets as the transitive sets that are well-ordered by $\in$.

---

If we want to be a bit more abstract, we could say that an "ordinal number" means "whatever it is that two well-ordered sets have in common when they are order isomorphic". That would be similar to saying that a "natural number" is whatever two finite sets have in common when they have equally many elements.

This is conceptually simple, but not technically convenient. If we want ordinal numbers to be set-theoretic objects (such that we can make them elements of sets, etc), then we need to specify a particular set to represent each of them within set theory.

Von Neumann proposed to represent each isomorphism class of well-orders by the transitive set that is ordered in this particular way by $\in$. This representation is what is known as the "von Neumann ordinals". It's not a particular selection of ordinals; it's a representation that works for all of them.

The von Neumann representation works so well that it has become pretty much universal in set theory.

The reason why we don't just define them as "$\emptyset,\{\emptyset\},\left\{ \emptyset,\left\{ \emptyset\right\} \right\} ,\ldots,\omega, \omega \cup \left\{ \omega\right\} ,\ldots$" is that for rigorous proofs about them we need a definition that doesn't depend on the reader figuring out for himself exactly what is implied by the suggestive notation "$\ldots$" that appears in it.

Answer 2

I want to suggest an answer by posing a more general question.

One way to view what you know is that the von Neumann construction is a set theoretic model that captures the structure of all ordinals (defined abstractly), and that model is universal (perhaps not the right word): any other model is isomorphic.

But the abstract definition is still important. By analogy, why bother defining a complete ordered field when you can just use the field of Dedekind cuts of rationals, or the field of equivalence classes of Cauchy sequences of rationals?

Answer 3

How are you defining the von Neumann ordinals if not as a transitive set well-ordered by $\in$? We need some precise definition of the notion, even if it's intuitively clear what we mean.

Now there are other definitions of von Neumann ordinal out there, my personal favorite being "hereditarily transitive set." However, it's useful to have multiple equivalent definitions - if you prefer, you can think of this as one definition and then a multi-part characterization theorem - since some are more convenient in different contexts (e.g. when developing forcing, "hereditarily transitive set" often makes arguments much snappier than "transitive set well-ordered by $\in$").