model complete theory

{{Short description|Concept in model 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 T is an \aleph_0-categorical theory, then it always has a model companion.{{sfn|Saracino|1973}}{{sfn|Simmons|1976}}

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:{{sfn|Chang|Keisler|2012}}

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

Examples

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 that embeds into any model of T, then T is complete.{{sfn| Marker|2002}}

Notes

{{reflist}}

References

  • {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=1990|orig-year=1973|publisher=Elsevier|edition=3rd|series=Studies in Logic and the Foundations of Mathematics|isbn=978-0-444-88054-3}}
  • {{Cite book|last1=Chang|first1=Chen Chung|author1-link=Chen Chung Chang|last2=Keisler|first2=H. Jerome|author2-link=Howard Jerome Keisler|title=Model Theory|year=2012|orig-year=1990|publisher=Dover Publications|edition=3rd|series=Dover Books on Mathematics|pages=672|isbn=978-0-486-48821-9}}
  • {{cite book|last=Hirschfeld|first=Joram|last2=Wheeler|first2=William H.|chapter=Model-completions and model-companions|title=Forcing, Arithmetic, Division Rings|series=Lecture Notes in Mathematics|publisher=Springer|volume=454|pages=44–54|year=1975|isbn=978-3-540-07157-0|mr=0389581|doi=10.1007/BFb0064085}}
  • {{cite book | last=Marker | first=David | title= Model Theory: An Introduction | publisher=Springer-Verlag|location= New York| year=2002 | isbn=0-387-98760-6| series=Graduate Texts in Mathematics 217}}
  • {{cite journal

|last=Saracino

|first=D.

|title=Model Companions for ℵ0-Categorical Theories

|journal=Proceedings of the American Mathematical Society

|volume=39

|issue=3

|date=August 1973

|pages=591–598

}}

  • {{cite journal

|last=Simmons

|first=H.

|title=Large and Small Existentially Closed Structures

|journal=Journal of Symbolic Logic

|volume=41

|issue=2

|year=1976

|pages=379–390

}}

{{Mathematical logic}}

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Category:Mathematical logic

Category:Model theory