Types of Models to Know for Model Theory

In Model Theory, different types of models help us understand the structure and behavior of theories. Key concepts include completeness, saturation, and homogeneity, each revealing unique properties that shape how models relate to one another and to their underlying theories.

1. Complete models

   • A model is complete if every sentence in the language is either true or false in the model.
   • Completeness ensures that the model satisfies all the consequences of its axioms.
   • Complete models are often associated with complete theories, which have a unique model up to isomorphism.

2. Saturated models

   • A saturated model can realize all types over any set of parameters of a certain size.
   • Saturation is a measure of the richness of a model, indicating it can accommodate many different structures.
   • These models are crucial in understanding the behavior of theories in model theory.

3. Homogeneous models

   • A homogeneous model is one where any isomorphism between finite subsets can be extended to an automorphism of the entire model.
   • This property ensures that the model looks "the same" from every finite perspective.
   • Homogeneity is important for studying the symmetry and structure of models.

4. Prime models

   • A prime model over a theory is a model that is elementarily embeddable into any other model of the theory.
   • Prime models serve as a "building block" for other models, providing a foundational structure.
   • They are unique up to isomorphism for complete theories.

5. Atomic models

   • An atomic model is one where every type over a finite set of parameters is realized by some element in the model.
   • These models are particularly useful in studying the structure of theories and their types.
   • Atomicity relates closely to the concept of saturation.

6. Universal models

   • A universal model is a model that can embed any other model of the same language.
   • These models are often used to study the relationships between different models and their properties.
   • Universal models help in understanding the limits of expressibility in a given theory.

7. Existentially closed models

   • An existentially closed model is one in which every existential formula that is satisfied in the model is also realized in the model.
   • This property ensures that the model is "complete" with respect to existential statements.
   • Existentially closed models play a key role in the study of algebraic structures.

8. ω-categorical models

   • An ω-categorical model is one that has exactly one countable model up to isomorphism for its theory.
   • These models exhibit a high degree of regularity and predictability in their structure.
   • ω-categoricity is a strong form of categoricity that has implications for the classification of models.

9. Strongly minimal models

   • A strongly minimal model has the property that every definable set is either finite or co-finite.
   • This property leads to a very rigid structure, limiting the complexity of definable sets.
   • Strongly minimal theories are important in the study of simplicity in model theory.

10. Stable models

    • A stable model is one where the number of types over any set of parameters is bounded.
    • Stability is a key concept in understanding the complexity of theories and their models.
    • Stable models often exhibit nice structural properties and are easier to analyze.