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.

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. Not every theory is model-companionable, e.g. theory of groups. However if is an -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]:

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 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.

Sufficient condition for completeness of model-complete theories

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

Notes
  1. D. Saracino. Model Companions for ℵ0-Categorical Theories. Proceedings of the American Mathematical Society Vol. 39, No. 3 (Aug., 1973), pp. 591–598
  2. H. Simmons. Large and Small Existentially Closed Structures. J. Symb. Log. 41 (2): 379–390 (1976)
  3. Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover Publications. pp. 672 pages.
  4. David Marker (2002). Model Theory: An Introduction. Springer-Verlag New York.

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
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