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The theory $REI_{\alpha}$ has as its language $L=\{E_{\beta}|\beta\leq\alpha\}\cup{\{E_{-1}\}}$, and each $E_{\beta}$ and $E_{-1}$ are binary relation symbols. Let $T$ (=$REI_{\alpha}$) be the theory that states that all $x,y$ are $E_{-1}$ related and that each equivalence class $E_{\beta + 1}$ refines $E_{\beta}$ in to infinitely many classes and $E_{\alpha}(x,y)$ is equivalent to $x=y$ and that for all $\beta<\gamma$, $E_{\gamma}$ refines $E_{\beta}$.

The way I think of (at least certain) models of $T$ is as follows: Assuming that there are $\kappa$-many $\beta$ equivalence classes for each $\beta$ (and $\kappa>|\alpha|$ and $\kappa$ regular for good measure though I'm pretty certain that its irrelevant). Now $\kappa^\alpha$ models $T$, where I think of $f(\beta)$ as giving the $E_{\beta}$ class to which $f$ belongs (this makes sense as there are $\kappa$ many $E_{\beta}$ classes, I can think of them as being enumerated by $\kappa$).

Now I have several questions:

Q1) What do models of $T$ look like in general?

Q2) It this theory complete? And if so why?

Q3) Can I build models $M$ of $T$ s.t. for each limit $\gamma<\alpha$ and for each $a\in{M}$: $E_{\gamma}(x,a)$ iff $E_{\beta}(x,a)$ for all $\beta<\gamma$? And is possible to view such a model as an elementary substructure of the sort of model I have described (or in a similar manner)?

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  • $\begingroup$ I'm a little confused: what exactly are you asking? $\endgroup$ – Noah Schweber Nov 26 '15 at 9:59
  • $\begingroup$ @NoahSchweber: I edited the question a fair bit. I hope the questions are now clearer. $\endgroup$ – user185596 Nov 26 '15 at 16:08
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First of all, the structure on $\kappa^\alpha$ that you describe is not a model of $\text{REI}_\alpha$. If you define $f E_\beta g$ if and only if $f(\beta) = g(\beta)$, then the equivalence relations will be cross-cutting rather than refining. What you actually want to do is take $\kappa^{\alpha+1}$ and for each $\beta\in \{-1\}\cup (\alpha+1)$, define $f E_\beta g$ if and only if $f(\gamma) = g(\gamma)$ for all $\gamma\leq\beta$.

Q1) In general, a model will look like some set of paths through an infinitely branching tree of height $\alpha+1$. Such a model $M$ of size $\kappa$ can always be embedded in the model $\kappa^{\alpha+1}$ described above.

Q2) Yes. You can see this by proving that the theory has quantifier-elimination and that any two models can be embedded (elementarily, by QE) into a common larger model.

Q3) Sure. For example, to cut down the model on $\kappa^{\alpha+1}$ described above to an elementary substructure satisfying your condition, you could, for each limit $\gamma<\alpha$, enumerate the functions $\gamma\to \kappa$ as $(g_\iota)_{\iota<\lambda}$, and let's assume that $\kappa$ is large enough relative to $\alpha$ so that $\lambda<\kappa$. Then require that if a function $f$ in your model agrees with $g_\iota$ on its restriction to $\gamma$, then $f(\gamma) = \iota$. This describes a submodel, which is elementary by QE.

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