Editing Elementary equivalence (section)

==Elementarily equivalent structures==

Two structures ''M'' and ''N'' of the same signature&nbsp;''σ'' are '''elementarily equivalent''' if every first-order sentence (formula without free variables) over&nbsp;''σ'' is true in ''M'' if and only if it is true in ''N'', i.e. if ''M'' and ''N'' have the same [[complete theory|complete]] first-order theory.
If ''M'' and ''N'' are elementarily equivalent, one writes ''M''&nbsp;≡&nbsp;''N''.

A first-order [[theory (mathematical logic)|theory]] is complete if and only if any two of its models are elementarily equivalent.

For example, consider the language with one binary relation symbol '<'.  The model '''R''' of [[real numbers]] with its usual order and the model '''Q''' of [[rational numbers]] with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense [[linear ordering]]. This is sufficient to ensure elementary equivalence, because the theory of [[Dense order|unbounded dense linear orderings]] is complete, as can be shown by the [[Łoś–Vaught test]].

More generally, any first-order theory with an infinite model has non-isomorphic, elementarily equivalent models, which can be obtained via the [[Löwenheim–Skolem theorem]]. Thus, for example, there are [[Non-standard model of arithmetic|non-standard models]] of [[Peano arithmetic]], which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model.