Is it possible to have an $L_{\omega_1,\omega}$-sentence $\phi$ in a vocabulary that includes $<$ that satisfies the following?

  1. $<$ is a linear order on a definable subset;
  2. $\phi$ is $\aleph_1$-categorical;
  3. in the unique model of $\phi$ of size $\aleph_1$, $<$ is also of size $\aleph_1$.
  • $\begingroup$ Perhaps we can get to "no" by omitting types. The right tools for this might be in John Baldwin's Categoricity (from the AMS or online), or @DaveMarker's primer on infinitary logic at homepages.math.uic.edu/~marker/inf.pdf. $\endgroup$ – Matt F. Jun 22 '16 at 12:41
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    $\begingroup$ Did you have a look at jstor.org/stable/2274053 ? $\endgroup$ – TimZ Jul 15 '16 at 9:25
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    $\begingroup$ I would imagine that quite a bit is known about this (but I haven't been able to make much of the Grossberg-Shelah paper). Do you know if requiring the domain of < to be the entire model makes a difference? Two minor observations : (1) The long line would appear to be an example, if the question were about $L_{\omega_1,\omega}(Q)$ (2) If there is a model such that for some countable subset of the domain of $<$ uncountably many gaps were filled, and $2^{\aleph_{0}} < 2^{\aleph_{1}}$, then there would be $2^{\aleph_{1}}$ many such linear orders. This is similar to Theorem 5.8 of ....... $\endgroup$ – Paul Larson Jul 19 '16 at 21:41
  • $\begingroup$ ...... my paper with Baldwin (users.miamioh.edu/larsonpb/galois_types_august17_15.pdf). So unless absoluteness of $\aleph_{1}$-categority for sentences of $L_{\omega_{1}, \omega}$ fails, you'd need an example where there is no such countable set. This may not help, of course. $\endgroup$ – Paul Larson Jul 19 '16 at 21:45
  • $\begingroup$ @TimZ: Thank you for the paper. Indeed, it has interesting results, but I can not see how they help with the question at hand. Their conclusion is that certain $T$'s have the maximum number of models, but their assumption is that for every $\mu<\mu^*(\lambda,\kappa)$, there is a model $M_\mu$ and some $\phi$ defines a well-order of type $\mu$ on a subset of $M_\mu$. We have nothing of this sort here. Our $<$ is just a linear order. I can not see how to pull the well-order required. (cont.) $\endgroup$ – Ioannis Souldatos Jul 21 '16 at 22:51

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