A property holding between models with the same signature when the models have identical theories, i.e., for a deductive system $\text{L}$,

• • ${\mathfrak{M}}_0$ and ${\mathfrak{M}}_1$ are elementarily equivalent if $T{h}_{\text{L}}\left({\mathfrak{M}}_0\right)=T{h}_{\text{L}}\left({\mathfrak{M}}_1\right)$

Elementary equivalence is a weaker property than isomorphism. Important cases of elementary equivalent but not isomorphic models are the natural numbers with addition and multiplication $\left(〈\mathbb{N},+,\times 〉\right)$ and a non-standard model of true arithmetic that extends $\mathbb{N}$ by adding infinitely many non-standard elements greater than any standard natural number.