Model complete theory

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

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. Not every theory is model-companionable, e.g. theory of groups. However if T is an ${\displaystyle \aleph _{0}}$-categorical theory, then it always has a model companion.^[1][2]

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:^[3]

• 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

• Any theory with elimination of quantifiers is model 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, each containing an infinite number of elements.
• 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.

• 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.

If T is a model complete theory and there is a model of T that embeds into any model of T, then T is complete.^[4]

• 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.

• Chang, Chen Chung; Keisler, H. Jerome (2012) [1990]. Model Theory. Dover Books on Mathematics (3rd ed.). Dover Publications. p. 672. ISBN 978-0-486-48821-9.

• Hirschfeld, Joram; Wheeler, William H. (1975). "Model-completions and model-companions". Forcing, Arithmetic, Division Rings. Lecture Notes in Mathematics. Vol. 454. Springer. pp. 44–54. doi:10.1007/BFb0064085. ISBN 978-3-540-07157-0. MR 0389581.

• Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. New York: Springer-Verlag. ISBN 0-387-98760-6.

• Saracino, D. (August 1973). "Model Companions for ℵ₀-Categorical Theories". Proceedings of the American Mathematical Society. 39 (3): 591–598.

• Simmons, H. (1976). "Large and Small Existentially Closed Structures". Journal of Symbolic Logic. 41 (2): 379–390.