Elementary equivalence

Two structures M and N (with the same signature) are said to be elementarily equivalent, denoted M ≡ N, if every sentence true in M is true in N and vice versa: for every sentence $$\phi$$,
 * $$ M \models \phi \iff N \models \phi $$.

Equivalently, the complete theory of M is the same as the complete theory of N.

Elementary equivalence is an equivalence relation, strictly weaker than isomorphism. For example, the field of complex numbers is elementarily equivalent to the field of algebraic numbers
 * $$ \mathbb{C} \equiv \overline{\mathbb{Q}} $$.

This follows from the completeness of ACF0. In general if T is a complete theory, then any two models of T are elementarily equivalent. When T is not complete, the elementarily equivalence classes of models of T correspond exactly to the completions of T.