I've been reading David Marker's Introduction to Model Theory, and found Vaught's two cardinal theorem (4.3.34): if a theory $T$ has a $(\kappa,\lambda)$-model, where $\kappa > \lambda \geq \aleph_0$, then $T$ has an $(\aleph_1,\aleph_0)$-model.
The proof is made using Vaught pairs, and uses the following: existence of $(\kappa,\lambda)$-model $\Rightarrow$ existence of Vaught pair $\Rightarrow$ existence of countable Vaught pair $\Rightarrow$ existence of $(\aleph_1,\aleph_0)$-model.
However, when constructing the $(\aleph_1,\aleph_0)$-model from a countable Vaught pair it uses the formula that defines the Vaught pair (which may have parameters), and asserts that the same formula can be used to define the $(\aleph_1,\aleph_0)$-model (which should not have parameters).
So according to the theorem, a Vaught pair of Dense linear order without endpoints should assure the existence of a $(\aleph_1,\aleph_0)$-model. I've been able to expose a Vaught pair of DLO w/o endpoints.
1. Is there an $(\aleph_1,\aleph_0)$-model of the theory?
My guess is not, since seems to me that all elements have the same type over $\emptyset$.
2. If the proof of the theorem is wrong, is there an easy workaround? (Not using saturated models, but Vaught pairs)
3. If the proof of the theorem is correct, why can you use a formula with parameters to construct the $(\aleph_1,\aleph_0)$-model?


You're right that Marker is sloppy on this point. However, if you work through the proofs, you'll see that the formulas witnessing the $(\kappa,\lambda)$ model and the Vaughtian pair and the $(\aleph_1,\aleph_0)$-model are all the same. So if you start with a formula without parameters witnessing the $(\kappa,\lambda)$ model, you'll get a Vaughtian pair and an $(\aleph_1,\aleph_0)$-model without parameters. On the other hand, if the formula in your Vaughtian pair has parameters, you'll get a formula with parameters witnessing the $(\aleph_1,\aleph_0)$-model, or, if you prefer, you can add the parameters as constants to the language and get a true $(\aleph_1,\aleph_0)$-model.

  • $\begingroup$ You're right. That makes Marker's proof of Morley's categoricity theorem sloppy as well, because it relies on the Baldwin Lachlan characterization of $\kappa$-categorical theories, and states that the existence of a Vaughtian pair implies the existence of a $(\aleph_1,\aleph_0)$-model and hence a $(\kappa,\aleph_0)$-model (with $\omega$-stability). So to make it work, it's necessary to show that $(\kappa,\lambda)$-models with parameters also contradict the $\kappa$-categoricity of the theory (without constants). $\endgroup$ – Dietrichr022 Jun 5 '15 at 3:37

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