2

I know that in the early stages of set theory, e.g., by Cantor and I think largely until von Neumann's formalization of the notion of "ordinal" within his set theory, ordinals were often treated as distinct from sets -- they were the types of well-ordered sets, but not identified with them. I've occasionally seen modern mentions of ordinals that suggest a similar view.

For example, in the answer to Difference Between Cantor's Ordinals and Von Neumann Ordinals?, Asaf Karagila mentions as an aside that a Cantorian conception of ordinals could correspond to taking ordinals to be an abstract category (I think I've seen the claim elsewhere, but this was just the first mention I could find). Similarly, when discussing forcing there's frequent mention of models of $\mathsf{ZFC}$ with the same ordinals. Much like talk of "the real $\in$ relation", this is at least suggestive of something like a set theory transcendent conception of ordinals.

My question is are there modern implementations of a "theory of ordinals" that are independent of any set theory (allowing, perhaps, for "artifactual" uses of set theory to provide the model theory)? I suspect that something in the realm of order theory or category theory would fit the bill, but I don't know either area well enough to navigate my way to such a theory.

Dennis
  • 2,627
  • 2
    Ordinals are a special case of order types, namely, ordinals are the order types of well-ordered sets. For order types in general, the modern understanding is pretty much the same as Cantor's; Von Neumann's realization of ordinals does not generalize to general order types. An order type is just an "abstract" ordered set, the same sort of thing as an "abstract group". – bof Jan 12 '18 at 01:21
  • @bof Thanks for the lead! A question, if you'd be so kind as to indulge: does the failure of realization result from the fact that order isomorphism isn't wholly set-theoretic, i.e., that such isomorphisms don't typically reduce to bijective functions on sets due to "additional structure"? Or does the failure reflect a more radical departure from set-theoretic thinking? – Dennis Jan 12 '18 at 01:29
  • 2
    I don't think the fact that Von Neumann ordinals don't generalize to "Von Neumann order types" is in any particular need of explanation. It's the usual state of affairs in mathematics that abstract objects don't have a canonical realization in terms of sets: there is no canonical realization of the Klein four-group, or the dodecahedron, or the real number system in terms of sets, as far as I know; well-ordered sets are a very special case. But what do I know, I'm just a dilletante. Let's see what the experts have to say. – bof Jan 12 '18 at 01:41
  • 2
    I'm about 99% sure Asaf didn't mean "category" in the sense of category theory in that answer. It's also not clear to me what you consider to be "set theory" as opposed to "independent of any set theory". Can you give a concrete example of something you would consider "independent of any set theory"? – Eric Wofsey Jan 12 '18 at 04:24
  • @EricWofsey I suppose I’m thinking mainly of not within the explicit context of a background of extensional set-like objects (with the paradigm obviously being ZFC). Maybe something with a more “intensional” flavor. Or perhaps divorced from foundational concerns altogether and perhaps thinking of order as more part of something like the “logical framework”. – Dennis Jan 12 '18 at 04:26
  • @Eric: I actually did. In the sense that when you think about a category, you often think of its structure after quotient by the "isomorphic objects" equivalence relation. In that sense, all trivial groups are the same, and all groups with exactly five members are the same, etc. Cantor—at least according to the quote I mention—treated well-ordered sets the same way, not as specific sets, but as abstract orders with some relations between them (embeddings, to be accurate) modulo the isomorphism relation. – Asaf Karagila Jan 25 '18 at 07:46
  • (But I do agree that this question is somewhat vague, especially in asking what are set theories. Do you need set theory to develop category theory in the first place? Generally speaking, a basic course in set theory has a lot of basic theorems about well-orders which are pretty divorced from a logical and foundational frameworks, e.g. transfinite induction and recursion and consequences thereof like the comparability of well-orders and so on.) – Asaf Karagila Jan 25 '18 at 07:48
  • @AsafKaragila do you have suggestions for how the question might be clarified? The basic idea I’m trying to get at is that, e.g., Cantor (and I believe either Zermelo or Bernays) developed free standing theories of ordinals with their own axiomatizations. Of course set-theoretic model theory wasn’t as ubiquitous then. But they seemed to conceive of the theory of ordinals as wholly separate from the set theory they were developing — “conceptually independent” or something. Maybe part of a theory of relations? Any thoughts as to how this could be made more concrete? – Dennis Jan 25 '18 at 08:13
  • 1
    Generally I would put these into the order theory box, not theory of relations. As far as a theory which is free standing, I honestly don't know how much further you can reach beyond the basics. There is probably more to say, yes. Cantor normal form, Hessenberg sums, etc. And these are probably of some interest, but I can't quite find much more to say. If I am not mistaken, Sierpinski had a book called Ordinal Numbers, or maybe some other guy from the Polish school. Or maybe it was Cardinal and Ordinal Numbers... I don't remember now. But I would check that out. – Asaf Karagila Jan 25 '18 at 08:46

0 Answers0