Les corps algébriquement clos (resp. réel clos) sont les modèles existentiellement clos de la théorie des corps (resp. des corps ordonnés) dans le langage des anneaux L ann, resp. des anneaux ordonnés L ann ? { }) ,
, Une extension élémentaire de N est une extension M de N qui satisfait les mêmes formules (à paramètres dans N )
Supposons que la classe ? des modèles existentiellement clos de T soit élémentaire. Alors on peut montrer que la théorie de ? est modèle-complète ,
, Une théorie T * est le modèle-compagnon d'une théorie T si les modèles de T * sont exactement les modèles existentiellement clos de T . Si de plus T a la propriété d'amalgation
La théorie des corps algébriquement clos (resp. réel clos) est la modèle-complétion ,
, Une théorie T élimine les quantificateurs si toute formule du langage de T est logiquement équivalente modulo T à une formule sans quanteur
, Un théorie T universelle admet une modèle-complétion T * si et seulement si T * est un modèle-compagnon de T et si T * élimine les quantificateurs
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