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. It is model complete but not complete.
• 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
• Hirschfeld, Joram; Wheeler, William H. (1975), "Model-completions and model-companions", Forcing, Arithmetic, Division Rings, Lecture Notes in Mathematics, vol. 454, Berlin / Heidelberg: Springer, pp. 44–54, doi:10.1007/BFb0064085, ISBN 978-3-540-07157-0, ISSN 1617-9692, MR0389581 {{citation}}: Text "DOI 10.1007/BFb0064082 (book)" ignored (help)