Model complete theory

In model theory, a first-order theory is called model complete if every embedding of models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.

Model companion and model completion

A companion of a theory T is a theory T* such that every model of T can be embedded in a model of T* and vice versa.

A model companion of a theory T is a companion of T that is model complete. 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 the amalgamation property.

If T also has universal axiomatization, both of the above are also equivalent to:

• T* has elimination of quantifiers

Examples

• Any theory with elimination of quantifiers is 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 model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes.
• The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains).
• The theory of real closed fields, in the language of rings, is the model companion for the theory of formally real fields, but is not a model completion.

Non-examples

• The theory of dense linear orders with a first and last element is complete but not model complete.
• The theory of groups (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

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 454, Springer, pp. 44–54, doi:10.1007/BFb0064085, ISBN 978-3-540-07157-0, MR 0389581