Elementary equivalence

Remarks

Equivalently, $\mathcal{M} \equiv \mathcal{N}$ when $\mathrm{Th}(\mathcal{M}) = \mathrm{Th}(\mathcal{N})$, where $\mathrm{Th}$ is the set of sentences true in a structure.

Example: $(\Q, <) \equiv (\R, <)$. Both are dense linear orders without endpoints, and the theory of dense linear orders without endpoints is complete.

Non-example: $(\Z, <) \not\equiv (\Q, <)$. The sentence $\forall x \, \forall y \, (x < y \to \exists z \, (x < z \land z < y))$ is true in $(\Q, <)$ and false in $(\Z, <)$, with $x = 0$ and $y = 1$.

Isomorphic structures are elementarily equivalent. The example shows that the converse fails, since $\Q$ and $\R$ have different cardinalities.