I'm taking a beginner course in mathematical logic, and I was struggling to understand some applications of the downward Löwenheim–Skolem theorem. The version of Löwenheim–Skolem theorem that was presented in the lectures is:
Suppose $L$ is countable first order language, and $\mathcal{A}$ is an $L$-structure. Let $T$ be the theory of $\mathcal{A}$, i.e. $T = Th(\mathcal{A}) =\{ \theta: \mathcal{A} \models \theta \text{ where } \theta \text{ is an L-sentence}\}$. Then, there is a countable model $\mathcal{B}$ such that $\mathcal{B}\models T$.
$\Delta$ was defined to be the sentences which describe a dense linear order with no end points, i.e. $\Delta = \{ \varphi_1, \dots , \varphi_6\}$ where for example $\varphi_1 = \forall x_1, \exists x_2, \ x_2 < x_1$. The proof of the "Los-Vaught test" was given.
(Los-Vaught Test) For every $L$-sentence $\theta$, $\Delta \vdash \theta$ or $\Delta \vdash \lnot \theta$.
The proof starts: Suppose not. Then, $\Delta \cup \{\theta\}$ and $\Delta \cup \{ \lnot\theta\}$ are both consistent. Godel's completeness theorem tells us (the model existence lemma tells us?) that if it is consistent, then it should have a model. Then Löwenheim–Skolem theorem tells us, that we should have a countable model.
I don't understand the last sentence, namely, why the Löwenheim–Skolem theorem tells us that. How can I know that $\Delta \cup \{\theta\}$ and $\Delta \cup \{ \lnot\theta\}$ are theories of some $L$-structure?
