How to show that the intersection of two elementary classes is a elementary class

Question

I am wondering how I can show that the intersection between two elementary classes is a elementary class.

I have two elementary classes A and B, $A,B = (M|M\models\Gamma)$ and I want to give a proof that the intersection $A\cap B$ is a new elementary class C which is the class of the models found in both A and B.

I see in this situation two approaches, either assume that this is true and prove it or assume that C is not elementary and derive a contradiction, am I on the right track with this? Or what is the best approach?

Thanks in advance!

Answer 1

I suspect you are overthinking this. Suppose $A=\{\mathcal{M}:\mathcal{M}\models\Gamma\}$ and $B=\{\mathcal{N}:\mathcal{N}\models\Theta\}$ for some sets of sentences $\Gamma,\Theta$. Elements of $A\cap B$ are exactly structures which satisfy both $\Gamma$ and $\Theta$ at once; do you see a set of sentences $\Delta$ whose models are exactly the models of both $\Gamma$ and $\Theta$?

HINT: don't work too hard ...

Just consider the union of $\Gamma$ and $\Theta$: $$\Delta=\Gamma\cup\Theta.$$

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A more interesting example concerns unions of elementary classes. For a toy example, suppose $A=\{\mathcal{M}:\mathcal{M}\models\gamma\}$ and $B=\{\mathcal{N}:\mathcal{N}\models\theta\}$ for some single sentences $\gamma$ and $\theta$. Then $A\cup B=\{\mathcal{A}: \mathcal{A}\models\gamma\vee\theta\}$. Do you see how to extend this to sets of sentences in place of infinite sentences?