I don't know how to prove some class of models is whether an elementary class or an elementary class in the wider sense. How can I prove what it is?

Solution 1:

I'll just give hints, the rest is up to you :

a) Consider the theory $$T_{inf} := T \cup \{\exists x_1 \exists x_2 \dots \exists x_n \bigwedge_{1\leqslant i < j \leqslant n} x_i \neq x_j \ \big| \ n \in \mathbb{N} \}$$ The models of $T_{inf}$ are precisely the infinite models of $T$.

b) assume for contradiction that Kfin is ECΔ. Now consider an ultraproduct, say $U$, of arbitrarily large finite models of $T$ : $U$ is an infinite model of $T$. Since an ECΔ class is closed under ultraproducts, $U$ is also in Kfin, a contradiction.

c) assume for contradiction that Kinf is EC, ie there is a sentence $\sigma$ such that Kinf $ = \operatorname{Mod}(\sigma)$. Then the models in Kfin are precisely the models of $T \cup \{\neg \sigma\}$ : hence Kfin is ECΔ, a contradiction to point b).