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In Appendix B to their Model Theory, Chang and Keisler list some problems and conjectures that, at the time of publication, were unsolved. A few of them take imperative form, for instance:

"Develop a theory of models which stresses the order type of the model $\mathfrak{A} = \langle A, <, \dots\rangle$ rather than the cardinality of the set $A$."

Has anyone worked on this?

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I don't have my copy with me right now, but this does seem related to this post:… – James Freitag Aug 18 '10 at 17:47
up vote 6 down vote accepted

I don't think many model theorists have worked on this. Granted, I'm a little unclear what Chang and Keisler were asking here, but here's one possible precisification:

Question: Suppose we are given a (complete?) theory T in a language with a binary relation < such that T proves "< is a strict linear ordering." Try to develop a theory of the "order-type spectrum" $I(\alpha, T)$, which is defined as the number of nonisomorphic models $M$ of $T$ such that $(M, <^M)$ has order type $\alpha$.

You could start by trying to think about what it means for $T$ to be "$\alpha$-categorical" for some order type $\alpha$, meaning, what are necessary and sufficient conditions for $I(\alpha, T)$ to equal $1$? (I have no idea whether anybody has investigated this before.) For example, if $T$ proves that the ordering is dense without endpoints and $\eta$ is the order type of the rationals, then $I(\eta, T) = 1$ if and only if $T$ is $\omega$-categorical, since the complete theory of $(\mathbb{Q}, <)$ itself is $\omega$-categorical.

An immediate complication I see to this project is that I don't know if there is any good analogue of the upward Lowenheim-Skolem theorem. It seems like it would be difficult to answer the question: "Given a theory $T$, for which infinite order types $\alpha$ is $I(\alpha, T) \neq 0$?" (The corresponding question for cardinalities of the universe for a $T$ with infinite models is trivial, by Lowenheim-Skolem.) For example: your theory $T$ could force the order type of any model to not be Dedekind complete (e.g. take the complete theory of an densely-ordered ring with a unary predicate for a proper convex subring).

Are there Morley-like categoricity theorems? If $T$ is countable, say, and $I(\alpha, T) = 1$ for some uncountable order type $\alpha$, can we conclude that $I(\beta, T) \leq 1$ for every uncountable $\beta$? Possibly this could be an interesting question; offhand I have no idea what the answer is.

Illustrating the difficulties of this, here's a paper just on the possible order types of a particular theory, PA:

"Order-types of models of Peano arithmetic," by Andrey Bovykin and Richard Kaye, pp. 275-285 of Logic and Algebra, edited by Yi Zhang with a preface by Oleg Belegradek, Contemporary Mathematics 302, AMS.

A different interpretation of the original question would be: given a particular order type $\alpha$, investigate structures with order type $\alpha$. Along these lines, many model theorists (such as Pillay, van den Dries, Wilkie, and others) have been studying expansions of the ordered field of the real numbers under the rubric of "o-minimal theories," though generally the interest has been in definable sets rather than models per se. Chris Miller is an example of an o-minimalist who has done a significant amount of research just on structures expanding the field of reals; check out his webpage for some state-of-the-art papers in this area.

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It might be hard to figure out what a plausible Morley-like theorem would be. If T is an o-minimal theory in a countable language and the underlying order of an uncountable model is saturated, then the model is saturated. So for these orders we would have $I(\alpha,T)=1$, while this seems unlikely for most orders--and I expect there are some results of Shelah's proving special cases of this. – Dave Marker Aug 19 '10 at 14:08
Here's a question that I don't know the answer to off hand. Can you characterize the order types that arise as reducts of real closed fields? – Dave Marker Aug 19 '10 at 14:16
@Dave: Do you have examples of order types which are dense, without endpoints, and 2-homogeneous (any two increasing pairs (a,b) and (c,d) are in the same orbit of the order-automorphism group), but which could not be the reduct of a real-closed field? – John Goodrick Aug 19 '10 at 16:31
@John: I suppose that an $\aleph_1$-like dense linear order is 2-homogeneous, but certainly can't be the order type of a real closed field. – Dave Marker Aug 20 '10 at 0:17

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