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I'm studying Model Theory: an introduction by David Marker and more specifically doing the exercises of chapter 2 ("Basic techniques").

In several of these exercises, one is asked to build new models from previous ones, and I'm finding myself trying to use over and over the same argument: show that the diagrams/elementary diagrams of the previous models form a consistent theory with compactness arguments and then take a model of the whole theory. Are there any other constructions which should be labelled "basic techniques" in order to do the job?

To be more specific, here is some exercise that I tried to solve.

Let $T$ be a consistent theory, $\Gamma = \{\phi : \phi \text{ is a $\forall\exists$-sentence and $T\vdash\phi$}\}$, and $\mathcal M\models \Gamma$.

a) Show that there is $\mathcal N\models T$ such that if $\psi$ is a $\exists\forall$-sentence and $\mathcal M\models \psi$, then $\mathcal N\models \psi$.

b) Show that there is $\mathcal N'\supseteq\mathcal M$ with $\mathcal N' \equiv \mathcal N$.

c) Show that there is $\mathcal M'\supseteq \mathcal N'$ such that $\mathcal M\prec \mathcal M'$.

For the first one, I take $T' = T \cup \{\phi : \text{$\phi$ is a $\exists\forall$-sentence and $\mathcal M\models \phi$}\}$. If this was incoherent, then there would be $\exists \bar v\forall \bar w\phi_1,\ldots,\exists \bar v\forall \bar w\phi_n$ in the rhs set such that $T\vdash \bigvee_{i=1}^n \neg \exists \bar v\forall \bar w \phi_i(\bar v,\bar w)$. I guess there is a contradiction here, but I'm unable to find it.

For the second one, taking $T' = \mathrm{Diag}(\mathcal N)\cup \{\phi(\bar m) : \phi \text{ is a qf-formula and $\bar m\in\mathcal M$}\}$ should work, I guess.

Is it the way this exercise should be solved? For the third question, I'd take the set of qf-sentences true in $\mathcal N'$ and the elementary diagram of $\mathcal M$, but I haven't verified it yet. Thanks for your time!

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