Model complete theory

Summary

A theory is model complete when every embedding between two of its models is elementary. Then a smaller model agrees with a larger one on all first-order statements about its own elements. Abraham Robinson introduced the notion as an algebraic counterpart of quantifier elimination, which implies it. Model completeness is equivalent to the property that every formula is equivalent to an existential formula. The theory of real closed fields in the language of rings is a standard model complete theory without quantifier elimination.