I'm taking a beginner course in mathematical logic. In the proof of some properties of non-standard natural numbers, the lecturer has used the downward Löwenheim–Skolem theorem, which I didn't understand. The version of Löwenheim–Skolem theorem that I know 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 $\mathcal{B}\models T$.
In the lecture, the Peano Arithmetic Axioms are expressed in logical formulas. These axioms are denoted $P_1, \dots, P_n$. We define a model $\mathcal{N} =\langle \mathbb{N}, <, \operatorname{succ}, 0\rangle$. We define the set $\Sigma:= \{c>t_m (0): m \in \mathbb{N}\}$, where $c,0$ are $L$-constants, and $t_m:= \operatorname{succ}(\operatorname{succ}(\dots \operatorname{succ}(0)\dots)$, where $\operatorname{succ}$ has been applied $m $ times.
In the lecture, we prove that $\Sigma \cup Th(\mathcal{N})$ has a model, which is not the model $\mathcal{N}$. Then, we use the "downward Löwenheim–Skolem theorem to conclude that there is a countable model $\mathcal{M} \models \Sigma \cup Th(\mathcal{N})$". This is the part that I don't understand. No additional explanation was given, so this is probably something simple that I'm missing here. Can anyone help me understand this bit? (How can I know that $\Sigma \cup Th(\mathcal{N})$ is some theory of some $L$-structure, which is a condition of being able to use the Löwenheim–Skolem theorem?)
Let $\mathcal{A}$ be a model of $\Sigma \cup Th(\mathcal{N})$. Then $Th(\mathcal{A})$ contains $\Sigma \cup Th(\mathcal{N})$. So, if $\mathcal{M}$ is a countable model of $Th(\mathcal{A})$, it is also a model of $\Sigma \cup Th(\mathcal{N})$.