I am attempting to show that if two sentences have the same enumerable models, then they are logically equivalent. I am told that I need to apply the Löwenheim-Skolem theorem (if a set of sentences has a model, then it has an enumerable model) in some way. Is the following the correct line of thought?
Let $\Gamma$ and $\Omega$ be sets of sentences that have the same enumerable model, $\mathcal{M}$. Thus $\mathcal{M}\models\Gamma$ and $\mathcal{M}\models\Omega$. Then (am I missing a step here?) $\mathcal{M}\models\Gamma\cup\Omega$. By the Löwenheim-Skolem theorem, because we have shown there exists a model that makes true the set of sentences $\Gamma\cup\Omega$, we are guaranteed the existence of an enumerable model that makes true $\Gamma\cup\Omega$.
At this point I'm grasping at straws. I am not totally sure how to prove that they are equivalent under every interpretation (or how exactly the L-S theorem comes into play).