I have three first-order theories $A,B,C$ at hand such that
every model of $A$ either satisfies $B$ or satisfies $C$ (or both).
Presumably, none of these theories is finitely axiomatizable. I am a little uncomfortable with this because I don't know how to write the displayed implication entirely syntactically (which in particular means models are not mentioned).
Can one rewrite the displayed statement above in terms of first-order syntactical notions only?
If not, are there simple (well-known) examples of such $A,B$ and $C$ that can help comfort my mind?
Let theory $B$ have axioms (or theorems) $\theta_i$, and let $C$ have axioms $\varphi_j$, where $i$ and $j$ range over suitable index sets.
Form all possible sentences of the shape $\beta \lor \gamma$, where $\beta$ ranges over all finite conjunctions of the $\theta_i$, and $\gamma$ ranges over all finite conjunctions of the $\varphi_i$.