Can $\omega_\alpha$ be defined indepentdly of $\aleph$ numbers?

Question

The question is simply. Can you define $\omega_\alpha$ in an ''intrinsical'' way? Let me explain a little bit. In Kenen's book $\omega_\alpha$ is the same as the aleph's and they are defined via transfinite recursion. In Wikipedia they are defined as the inital ordinals of $\aleph$.

I would like to define them in an intrinsecal way, without mention $\aleph$ numbers (I would like to avoid cardinality but I guess it will be impossible). For example $\omega_0$ is the set of natural numbers. No cardinality, no $\aleph$'s. If I'm not wrong, $\omega_1$ can be defined as the supremum of the class of all ordinals which have cardinality $\aleph_0$. I would have to prove that it is in fact a set, but I'm sure this site is pleny of this kind of questions.

If I repeat the same definition for $\omega_\alpha$ I would need the concept of $\aleph_{\alpha-1}$. So at the end of the they $\omega_\alpha$ would need the aleph's and the couldn't be defined in a ''pure (primordial)'' way.

So, can they be defined in more ways? Where (in what reference) can I find this definition?

Comment 1. I'm using the cardinal assignment of von Neumann. Thus for me a cardinal is also an ordinal.

Comment 2. The reason of my question is I haven't find a ''pure'' definition of such numbers. In most books they are defined together with the $\aleph$'s and it is just a notation to emphasize we are considering ordinals.

Comment 3. I'm not interested in historical approach to these concepts. You can include a little of history in your answers but please focus in the definition.

Answer 1

Yes, there is indeed such an approach. If $\alpha$ is an ordinal, then $\omega_\alpha$ is the unique ordinal $\beta$ with the following two properties:

• $\beta$ is an infinite initial ordinal: there is no bijection between $\beta$ and any smaller ordinal. (That is, $\beta$ is a cardinal - but we don't want to use that word.)

• The set of infinite initial ordinals $<\beta$ is in order-preserving bijection with $\alpha$.

For example, $\omega_0$ is the unique infinite initial ordinal with no infinite initial ordinals less than it (that is, the set of initial ordinals less than it has ordertype $0$) - this is just the first infinite initial ordinal.

---

Note, however, that in giving this description we've already hit upon the idea of cardinals. So the two notions are still entangled in some way. Note, however, that the same entanglement is present when in your question you define $\omega_1$ as the supremum of the countable ordinals: what does "countable" mean? We have to talk about bijections at some level in this process.

Answer 2

Generally $\omega_\alpha$ comes in first. Because ordinals precede the cardinals in the order of things. It is easier to talk about ordinals and well-orders before talking about the $\aleph$ numbers themselves. Especially since the indices of the $\aleph$ numbers are in fact ordinals.

But from a pedagogical point of view, cardinals often come in first, so we can define them under an axiomatic view that they are well-ordered, so $\aleph_{\alpha+1}$ is the successor of $\aleph_\alpha$, and then define $\omega_\alpha$ as the corresponding ordinals.

In any case, as long as you don't write something like $\aleph_{\aleph_0}$, and as long as you remember that while being the same set $\omega_\alpha$ would denote the ordinal and $\aleph_\alpha$ the cardinal (and this is important when you have different arithmetic), you're fine with either way of doing things.