First some definitions: For a set $\Sigma$ of $\mathcal{L}$-sentences, $Mod(\Sigma)$ denotes the class of all models that satisfy $\Sigma$. For a class $\mathcal{M}$ of models, we say it is $EC$ if $\mathcal{M} = Mod(\sigma)$ for some sentence $\sigma$ and $EC_\Delta$ if $\mathcal{M} = Mod(\Sigma)$.
Problem. Let $T$ be a theory having arbitrary large finite models. (For example, $T$ can be axioms for groups, or fields, or linear orderings.) Let $\mathcal{K}_{inf} = \{ M : M \models T, \mbox{Card}(|M|)\geq \infty \}$ and $\mathcal{K}_{fin} = \{ M : M \models T, \mbox{Card}(|M|)< \infty \}$.
Question.
(a) What does it mean by 'finite models'? Model's universe finite? Or a finite number of models?
(b) $\mathcal{K}_{inf}$ is $EC_\Delta$
(c) $\mathcal{K}_{fin}$ is $EC_\Delta$
(d) $\mathcal{K}_{inf}$ is $EC$
Obviously I don't want some full detailed solution. It's just I can't even start on the problems.
Give some suggestions and hints. Should I use Compactness Theorem?
Ok, here are some hints:
b) Can you express "the domain of M is infinite" as a first order theory? Call this theory $\text{Th}(\infty)$. Then the class of models of $T\cup \text{Th}(\infty)$ is $\mathcal{K}_{\text{inf}}$.
c) Here you can use compactness. Suppose $\mathcal{K}_{\text{fin}}$ is the class of models of $T^*$. Then $T^*\cup \text{Th}(\infty)$ is inconsistent. Do you see a problem here?
d) Use part c): if you had a single sentence picking out $\mathcal{K}_{\text{inf}}$, you could find a theory picking out $\mathcal{K}_{\text{fin}}$.