Let $\Sigma_\infty$ be a set of axioms in the language $\{\sim\}$ (where $\sim$ is a binary relation symbol) that states:
(i) $\sim$ is an equivalence relation;
(ii) every equivalence class is infinite;
(iii) there are infinitely many equivalence classes.Show that $\Sigma_{\infty}$ admits QE and is complete. (It is given that it is also possible to use Vaught's test to prove completeness.)
I think I have shown that $\Sigma_\infty$ admits QE, but am not sure how to show completeness. There is a theorem, however, that states that if a set of sentences $\Sigma$ has a model and admits QE, and there exists an $L$-structure that can be embedded in every model of $\Sigma$, then $\Sigma$ is complete.
Thanks.