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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.

Dog_69
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  • See Ordinal and Ordinal arithmetic. And see Ordinals and cardinals. – Mauro ALLEGRANZA Apr 18 '18 at 15:37
  • I had already read the Wiki's article about initial ordinals. I'm going to edit my question to tell you my experience with ordinals and cardinals. – Dog_69 Apr 18 '18 at 15:42
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    Historically $\aleph$ preceded $\omega,$ before the general theory of ordinals and before the modern axiomatization of set theory. You can find a lot about the $\omega_x$ family in "Set Theory: An Introduction To Independence Proofs" by K.Kunen.... I have the first edition. I believe there is a revised second edition....... $\omega_x$ is the infinite cardinal ordinal $y$ such that the set of infinite cardinal ordinals less than $y$ is order-isomorphic to the ordinal $x.$ – DanielWainfleet Apr 18 '18 at 15:47
  • @DanielWainfleet: thanks for the reference. – Dog_69 Apr 18 '18 at 15:48
  • I might be missing something. Are you asking for a historical reference, or a pedagogical reference? – Asaf Karagila Apr 19 '18 at 07:59
  • @AsafKaragila: Maybe the title leads to confusion (it's a joke with the chicken or egg paradox). My question is concerned about the definition of the $\omega$'s: they can be simply notation to emphasize we are considering the aleph's like ordinals or they can be defined intrinsically, without mention aleph's numbers and cardinality. If the second option is possible, I would like you to give me some bibliography where I can find this approach. For example Kunen's book (which I don't like because he defines them via transfinite recursion, and he doesn't distinguish between aleph's and omega's). – Dog_69 Apr 19 '18 at 18:15
  • Yes, your title and general formulation lead to confusion. It led to a comment on my answer regarding historical context, and a history tag being introduced. Don't be funny, be clear. If you can be both, great. If not, always choose clarity. – Asaf Karagila Apr 19 '18 at 19:12
  • Sorry, I couldn't answer yesrteday. I have wirtten the question again. I think now it is very clear. I have also removed the tag ''math history''. – Dog_69 Apr 20 '18 at 09:59

2 Answers2

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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.

Noah Schweber
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  • Related with my comment to your answer here I will say your answer is very good. I didn't realize you can define $\omega_\alpha$ via transfinite recursion using the Hartog number. As before, thanks for your patience and sorry for the inconvenience caused. – Dog_69 Apr 24 '18 at 18:08
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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.

Asaf Karagila
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  • May be the edit helps you to give me a more detailed answer. – Dog_69 Apr 18 '18 at 15:47
  • I have come to repent my "cardinal sin" (maybe you recall) of writing $\aleph_{\aleph_0}$. I had already learned my lesson when I more recently came across it in "Intro to Cardinal Arithmetic" which you had recommended, in a fn on page 45. –  Apr 18 '18 at 18:29
  • But didnt Cantor conceive of $\omega$ and $\aleph$ as two different things the first an ordinal , the second a cardinal number, two different types of transfinite numbers. Only later did von Neumann define the cardinals as initial ordinal making the two notions represented by the same set. That was my understanding of things. – Rene Schipperus Apr 18 '18 at 19:42
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    @Rene: Yes. Cantor didn't think about things in terms of sets like we do today. For him an ordinal was an equivalence class of well-ordering, not a particular set. – Asaf Karagila Apr 19 '18 at 07:58