Model complete theory

In model theory, a theory is called model complete if every embedding of models is an elementary equivalence. This notion was introduced by Abraham Robinson.

Model companion and model completion

A model companion of a theory T is a model complete theory T^* such that every model of T can be embedded in a model of T^* and vice versa. Robinson proved that a theory has at most one model companion.

A model completion for a theory T is a model companion T^* such that for any model M of T, the theory of T^* together with the diagram of M is complete. Roughly speaking, this means every model of T is embeddable in a model of T^* in a unique way.

If T^* is a model companion of T then the following conditions are equivalent:

• T^* is a model completion of T
• T^* has elimination of quantifiers
• T has the amalgamation property.

Examples

• The theory of dense linear orders with a first and last element is complete but not model complete.
• The theory of dense linear orders with two constant symbols is model complete but not complete.

The theory of algebraically closed fields is the model completion of the theory of fields.

• The theory of real closed fields is the model companion for the theory of formally real fields, but is not a model completion.

References

Chang, Chen Chung; Keisler, H. Jerome (1990) [1973], Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3