Elementarily

Hence, mathematical logic suggests that the application of mathematical physics to the universe as a whole can generate two different types of multiverse: classes of non-isomorphic but elementarily equivalent models; and classes of non-isomorphic and elementarily inequivalent models. ❋ Gordon McCabe (2009)

In this case, the models of the theory will be mutually non-isomorphic and elementarily inequivalent. ❋ Gordon McCabe (2009)

Recalling that a complete theory T is one in which any sentence s, or its negation Not (s), belongs to the theory T, it follows that every model of a complete theory must be elementarily equivalent. ❋ Gordon McCabe (2009)

Whilst isomorphic models must be elementarily equivalent, there is no need for elementarily equivalent models to be isomorphic. ❋ Gordon McCabe (2009)

Two models of a theory are defined to be elementarily equivalent if they share the same truth-values for all the sentences of the language. ❋ Gordon McCabe (2009)

The class of models of a complete theory will be mutually non-isomorphic, but they will nevertheless be elementarily equivalent. ❋ Gordon McCabe (2009)

(In the case of a complete theory, the models of different cardinality will be elementarily equivalent, even if they are non-isomorphic). ❋ Gordon McCabe (2009)

This description implied among other things that the structures elementarily equivalent to R are exactly the real-closed fields, a class of fields which was already known to the algebraists in its own right. ❋ Hodges, Wilfrid (2009)

B are elementarily equivalent if and only if they have ultrapowers that are isomorphic to each other. ❋ Hodges, Wilfrid (2009)

On the other hand an embedding between elementarily equivalent structures, or even between isomorphic structures, need not be elementary. ❋ Hodges, Wilfrid (2009)

To continue the example of the field R of real numbers: It is often not at all obvious whether two given structures are or are not elementarily equivalent. ❋ Hodges, Wilfrid (2009)

A and B are elementarily equivalent if and only if Th (A) = Th (B). ❋ Hodges, Wilfrid (2009)

If two structures are isomorphic then they share all model-theoretic properties; in particular they are elementarily equivalent. ❋ Hodges, Wilfrid (2009)