🧠Model Theory Unit 10 – Omitting Types Theorem and Prime Models

Key Concepts and Definitions

• Type: A set of formulas in a fixed number of free variables that is consistent with a theory
• Isolated type: A type that is realized by a unique element in every model of the theory
• Omitted type: A type that is not realized in some model of the theory
• Prime model: A model that can be elementarily embedded into any other model of the same theory
• Atomic model: A model where every type realized in the model is isolated
• Countable model: A model whose underlying set is countable
• Elementary embedding: A function between two models that preserves all first-order formulas
  • Injective function that preserves truth values of formulas
  • Preserves and reflects satisfiability of types

Omitting Types Theorem: Statement and Significance

• States that under certain conditions, there exists a countable model of a theory that omits a given set of non-isolated types
  • Conditions: Theory is countable, complete, and has a countable language
  • Types to be omitted must be non-isolated and countable
• Significance: Allows construction of models with specific properties by omitting certain types
• Provides a method for proving the existence of models with desired characteristics
• Fundamental result in model theory with applications in various areas of mathematics
• Connects the concepts of types, countable models, and non-isolated types
• Demonstrates the power of first-order logic in controlling the structure of models
• Used in the study of prime models and atomic models

Understanding Prime Models

• Definition: A model that can be elementarily embedded into any other model of the same theory
• Existence: Not all theories have prime models
  • Theories with prime models are called model complete
• Uniqueness: When a prime model exists, it is unique up to isomorphism
• Atomic models: Prime models are always atomic models
  • Every type realized in a prime model is isolated
• Constructing prime models: Prime models can be constructed by omitting all non-isolated types
  • Application of the Omitting Types Theorem
• Role in model theory: Prime models serve as a canonical model for a theory
  • Minimal model with respect to elementary embeddings
• Relationship to saturated models: Prime models and saturated models are dual notions
  • Saturated models: Models that realize all types over small subsets

Connections Between Omitting Types and Prime Models

• Omitting types is a key technique for constructing prime models
  • Omit all non-isolated types to ensure the model is atomic
• Prime models can be characterized by the property of omitting non-isolated types
  • A model is prime if and only if it omits all non-isolated types
• The existence of prime models is closely related to the ability to omit types
  • Theories with prime models must have the property that all non-isolated types can be omitted simultaneously
• Omitting types and prime models are central to the study of model completeness
  • A theory is model complete if and only if it has a prime model
• The uniqueness of prime models is a consequence of the omitting types property
  • Any two models that omit the same set of non-isolated types are isomorphic

Proof Techniques and Strategies

• Constructing models by omitting types
  • Start with a countable, complete theory with a countable language
  • Identify the non-isolated types to be omitted
  • Use the Omitting Types Theorem to construct a countable model omitting these types
• Proving the existence of prime models
  • Show that the theory is model complete
  • Demonstrate that all non-isolated types can be omitted simultaneously
  • Use the Omitting Types Theorem to construct a model omitting all non-isolated types
• Proving uniqueness of prime models
  • Suppose there are two prime models of the same theory
  • Show that both models omit the same set of non-isolated types
  • Conclude that the models are isomorphic by the uniqueness property of omitting types
• Proving a theory is not model complete
  • Find two models of the theory that are not elementarily equivalent
  • Show that one model cannot be elementarily embedded into the other
  • Conclude that the theory does not have a prime model

Applications in Model Theory

• Studying the structure of models
  • Prime models provide a canonical representation of a theory
  • Omitting types allows for the construction of models with specific properties
• Classification of theories
  • Model completeness: Theories with prime models
  • $\omega$-categorical theories: Theories with a unique countable model up to isomorphism
• Connections to other areas of mathematics
  • Algebra: Constructing prime models of algebraic theories
  • Geometry: Studying prime models of geometric theories
  • Analysis: Investigating prime models of theories in functional analysis
• Foundations of mathematics
  • Independence results: Constructing models of set theory that omit certain types
  • Non-standard models: Using prime models to study non-standard models of arithmetic

Common Pitfalls and Misconceptions

• Confusing isolated and non-isolated types
  • Isolated types are realized by a unique element in every model
  • Non-isolated types may not be realized in some models
• Assuming all theories have prime models
  • Not all theories are model complete
  • There are theories without prime models (dense linear orders)
• Misunderstanding the role of countability
  • The Omitting Types Theorem requires a countable language and countable types
  • Prime models may not exist for uncountable theories
• Overlooking the importance of elementary embeddings
  • Prime models are defined in terms of elementary embeddings
  • Isomorphisms between prime models must be elementary embeddings
• Confusing prime models and saturated models
  • Prime models are minimal with respect to elementary embeddings
  • Saturated models are maximal with respect to realizing types

Practice Problems and Examples

• Determine whether the theory of dense linear orders has a prime model
  • Show that the theory is not model complete
  • Conclude that there is no prime model
• Construct a prime model of the theory of algebraically closed fields of characteristic 0
  • Show that the theory is model complete
  • Use the Omitting Types Theorem to construct a model omitting all non-isolated types
• Prove that the theory of infinite sets is $\omega$-categorical
  • Show that the theory has a unique countable model up to isomorphism
  • Use the Omitting Types Theorem to construct the countable model
• Find an example of a theory that is not model complete but has a prime model
  • Consider the theory of infinite sets with a unary predicate
  • Construct a prime model by omitting non-isolated types
• Prove that the theory of dense linear orders without endpoints is not model complete
  • Find two models that are not elementarily equivalent (rationals and reals)
  • Show that one cannot be elementarily embedded into the other