Suppose $K$ and $K'$ are two classes of structures in the same language which are both elementary classes. Is their union also an elementary class? Note, the classes do not have to be finitely axiomatizable, they just have to be axiomatizable.
Is the union of two elementary classes an elementary class
model-theory
Related Solutions
Some hints.
For #1, say $K$ is axiomatized by $A$ and also by a finite set $F$. Let $\phi$ be the conjunction of the formulas in $F$. Then $A \vdash \phi$; apply compactness.
For #2, say $K$ is axiomatized by $A$ and $K^c$ is axiomatized by $B$. Assuming that $K$ is not finitely axiomatizable, you can show that $A \cup B$ is consistent, which is a contradiction.
For #3, this is a relatively standard compactness exercise. You idea works, you just have to prove it. You can apply #1: if the class was finitely axiomatizable it would be axiomatizable by a subset of your given set of axioms.
I think it's helpful to think about problems like this from a topological perspective.
Ignoring set/class issues, consider the topology on the class $\mathbb{S}_L$ of all $L$-structures generated by the base $$\{\{\mathcal{M}: \mathcal{M}\models\varphi\}:\varphi\in Sent_L\}.$$ Trivially we have "axiomatizable $\iff$ closed." Much more significantly, by the compactness theorem we have "finitely axiomatizable $\iff$ clopen," and in fact this is just a restatement of the compactness theorem. (Note that if we were to do this with a non-compact logic in place of $\mathsf{FOL}$ all we would be guaranteed is "finitely axiomatizable $\color{red}{\implies}$ clopen.")
The fact that we can topologically characterize finite axiomatizability immediately gives a strong negative answer to your second question on general topological grounds: if I partition a clopen set into two open pieces, each of those pieces is clopen. However, the first question is left untouched by this reasoning and in fact ultimately has a positive answer:
Consider the language $L=\{A,B\}$ consisting of two unary relations. Let $K$ be the class of $L$-structures in which there are at least as many $A$s as $B$s, and let $K'$ be the class of $L$-structures in which there are at least as many $B$s as $A$s. We have $$K\cup K'=\mathbb{S}_L,$$ but it's easy to show that both $K$ and $K'$ are infinitely axiomatizable.
Best Answer
If $K$ and $K'$ are axiomatized by the sets $\Sigma$ and $\Sigma'$ of sentences, then $K\cup K'$ is axiomatized by $\{\alpha\lor\beta:\alpha\in\Sigma\text{ and }\beta\in\Sigma'\}$.