🧠Model Theory Unit 12 – Interpretations and Definability

Key Concepts and Definitions

• Model theory studies mathematical structures and their properties using formal languages and logical tools
• A structure $\mathcal{A}$ consists of a domain (universe) $A$ and a set of relations, functions, and constants defined on $A$
• A language $\mathcal{L}$ is a set of symbols used to describe structures, including logical connectives, quantifiers, variables, and non-logical symbols (relations, functions, and constants)
• An $\mathcal{L}$-structure is a structure that interprets the symbols of the language $\mathcal{L}$
• A theory $T$ is a set of sentences (formulas with no free variables) in a language $\mathcal{L}$
• A model of a theory $T$ is an $\mathcal{L}$-structure that satisfies all the sentences in $T$
• An interpretation is a way of defining one structure inside another using formulas of the language
• Definability refers to the ability to express a relation, function, or set using a formula in the language of the structure

Interpretations in Model Theory

• Interpretations allow us to study the relationships between different structures and theories
• An interpretation of a structure $\mathcal{A}$ in a structure $\mathcal{B}$ is a way of defining the domain, relations, functions, and constants of $\mathcal{A}$ using formulas in the language of $\mathcal{B}$
• The interpretation preserves the truth of formulas, meaning that if a formula holds in $\mathcal{A}$, its interpreted version holds in $\mathcal{B}$
• Interpretations can be used to prove results about one structure by working in another structure that may be easier to understand or has more desirable properties
• The composition of interpretations is also an interpretation, allowing for the study of chains of related structures
• Interpretations play a crucial role in model-theoretic proofs and constructions, such as the proof of the compactness theorem and the construction of saturated models
• The existence of an interpretation between structures can provide insights into their similarities and differences, as well as the expressive power of their respective languages

Types of Interpretations

• A definable interpretation is one in which the domain and all the relations, functions, and constants of the interpreted structure are defined by formulas in the language of the interpreting structure
• A parameter-definable interpretation allows the use of parameters (elements of the interpreting structure) in the defining formulas
• A bi-interpretation is a pair of interpretations between two structures, each of which is definable in the other
  • Bi-interpretations establish a strong connection between the two structures, showing that they are essentially the same up to definability
• A mutual interpretation is a generalization of bi-interpretation, where two structures interpret each other, but the compositions of the interpretations may not be definable
• An interpretation is called effective if there is an algorithm that, given a formula in the language of the interpreted structure, produces a formula in the language of the interpreting structure that defines the interpretation of the original formula
• Interpretations with parameters are a generalization of definable interpretations, allowing the use of elements from the interpreting structure as parameters in the defining formulas
• Interpretations modulo a definable equivalence relation are used when the desired interpretation is not definable on the entire domain of the interpreting structure but can be defined on a quotient structure obtained by a definable equivalence relation

Definability and Its Importance

• Definability is a central concept in model theory that captures the idea of expressing a relation, function, or set using a formula in the language of the structure
• A relation $R$ on a structure $\mathcal{A}$ is definable if there exists a formula $\varphi(x_1, \ldots, x_n)$ such that $R = \{(a_1, \ldots, a_n) \in A^n : \mathcal{A} \models \varphi(a_1, \ldots, a_n)\}$
• Definable sets and functions are those whose graphs are definable relations
• The definable sets of a structure form a Boolean algebra, closed under union, intersection, and complement
• Definability is closely related to the expressive power of the language and the complexity of the structure
• The study of definability helps to understand the limitations and capabilities of formal languages in describing mathematical objects
• Definable relations and functions are preserved under isomorphisms, making them intrinsic properties of the structure
• The notion of definability can be relativized to a subset of the structure, leading to the concept of definability with parameters

Methods of Proving Definability

• To prove that a relation or function is definable, one needs to find a formula in the language of the structure that defines it
• The method of quantifier elimination can be used to prove definability by showing that every formula in the language is equivalent to a quantifier-free formula
  • Structures with quantifier elimination (such as the field of real numbers) have a particularly simple definable sets and functions
• The method of back-and-forth arguments can be used to prove definability by establishing an isomorphism between the structure and another structure where the relation or function is known to be definable
• The method of interpretation can be used to prove definability by interpreting the structure in another structure where the relation or function is definable
• The method of automorphism arguments can be used to prove definability by showing that the relation or function is invariant under all automorphisms of the structure
• The method of compactness can be used to prove definability by constructing a sequence of definable approximations to the relation or function and using the compactness theorem to obtain a definable limit
• The method of ultrapowers can be used to prove definability by embedding the structure into an ultrapower where the relation or function becomes definable

Undefinability and Its Implications

• Undefinability occurs when a relation, function, or set cannot be expressed using a formula in the language of the structure
• The existence of undefinable relations or functions in a structure indicates that the language is not expressive enough to capture all the relevant properties of the structure
• Undefinability results often rely on the use of automorphisms, showing that the relation or function is not invariant under some automorphism of the structure
• The undefinability of certain relations or functions can be used to prove the existence of non-standard models of a theory (models that are not isomorphic to the intended model)
• Undefinability can also be used to establish independence results, showing that a particular statement cannot be proved or disproved within a given theory
• The study of undefinability helps to delineate the boundaries of what can be expressed in a formal language and what requires stronger axioms or more expressive languages
• Undefinability results can have implications for the decidability and complexity of theories, as they may indicate the presence of inherently complex or uncomputable relations or functions

Applications in Mathematics and Logic

• Model theory has numerous applications in various branches of mathematics, including algebra, geometry, number theory, and analysis
• In algebra, model theory is used to study the properties of algebraic structures (groups, rings, fields) and their connections to first-order theories
  • For example, the theory of algebraically closed fields is complete and has quantifier elimination, which has important consequences for the study of polynomial equations
• In geometry, model theory is used to investigate the logical foundations of geometric theories and the relationships between different geometries
  • For instance, the theory of real closed fields provides a model-theoretic characterization of Euclidean geometry
• In number theory, model theory is used to study the logical properties of arithmetic and its extensions
  • Gödel's incompleteness theorems, which have profound implications for the foundations of mathematics, are based on model-theoretic arguments
• In analysis, model theory is used to study the logical properties of real and complex numbers, as well as the structures arising in functional analysis
  • The theory of o-minimal structures, which includes the real field with exponentiation, has important applications in real analytic geometry
• In logic, model theory is a fundamental tool for investigating the properties of formal systems and their semantics
  • Model theory plays a central role in the study of decidability, completeness, and the classification of theories
• Model-theoretic methods have also found applications in computer science, particularly in the areas of database theory, verification, and constraint satisfaction
  • The study of finite model theory, which focuses on structures with finite domains, has important connections to computational complexity theory

Common Challenges and Misconceptions

• One common challenge in model theory is the construction of models with specific properties, such as saturated models or models realizing certain types
  • These constructions often involve intricate arguments using tools like compactness, ultraproducts, and back-and-forth methods
• Another challenge is the study of theories with limited expressive power, such as those without the equality symbol or with restricted quantifier usage
  • These theories may exhibit unusual behavior and require specialized techniques for their analysis
• A common misconception is that model theory is only concerned with first-order logic, when in fact it encompasses a wide range of logics, including infinitary, second-order, and higher-order logics
• There is also a misconception that model theory is purely abstract and disconnected from other areas of mathematics
  • In reality, model theory has deep connections to various branches of mathematics and has led to significant advances in fields like algebra, geometry, and number theory
• The study of definability and interpretations can be challenging due to the intricate nature of the arguments involved and the need to work with complex formulas and structures
• The concept of interpretation is sometimes confused with the notion of embedding, which is a stronger condition requiring the preservation of all formulas, not just those in the language of the interpreted structure
• The distinction between definable and undefinable relations or functions can be subtle and may depend on the specific language and axioms being considered
  • It is important to be precise about the context in which definability is being discussed