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 ACF_{0}. 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*.