Editing Elementary equivalence (section)

{{Short description|Concept in model theory}}
{{Inline citations|date=February 2023}}
In [[model theory]], a branch of [[mathematical logic]], two [[Structure (mathematical logic)|structure]]s ''M'' and ''N'' of the same [[Signature (mathematical logic)|signature]] ''σ'' are called '''elementarily equivalent''' if they satisfy the same [[First-order logic|first-order]] [[Sentence (mathematical logic)|''σ''-sentence]]s.

If ''N'' is a [[Substructure (mathematics)|substructure]] of ''M'', one often needs a stronger condition. In this case ''N'' is called an '''elementary substructure''' of ''M'' if every first-order ''σ''-formula ''φ''(''a''<sub>1</sub>,&nbsp;…,&nbsp;''a''<sub>''n''</sub>) with parameters ''a''<sub>1</sub>,&nbsp;…,&nbsp;''a''<sub>''n''</sub> from ''N'' is true in ''N'' if and only if it is true in&nbsp;''M''.
If ''N'' is an elementary substructure of ''M'', then ''M'' is called an '''elementary extension''' of&nbsp;''N''. An [[embedding#Universal algebra and model theory|embedding]] ''h'':&nbsp;''N''&nbsp;→&nbsp;''M'' is called an '''elementary embedding''' of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of&nbsp;''M''.

A substructure ''N'' of ''M'' is elementary if and only if it passes the '''Tarski–Vaught test''': every first-order formula ''φ''(''x'',&nbsp;''b''<sub>1</sub>,&nbsp;…,&nbsp;''b''<sub>''n''</sub>) with parameters in ''N'' that has a solution in ''M'' also has a solution in&nbsp;''N'' when evaluated in&nbsp;''M''. One can prove that two structures are elementarily equivalent with the [[Ehrenfeucht–Fraïssé games]].

Elementary embeddings are used in the study of [[large cardinals]], including [[rank-into-rank]].