Full Idea
In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory.
Gist of Idea
Tarski and Vaught established the equivalence relations between first-order structures
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Reference
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.282
A Reaction
This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic.