12.2 Definable sets and functions
5 min read•july 30, 2024
Definable sets and functions are the backbone of model theory, bridging syntax and semantics. They allow us to describe subsets and mappings within structures using first-order formulas, revealing the expressive power of our logical language.
These concepts are crucial for comparing structures and theories. By studying definable sets and functions, we can explore interpretations between different mathematical contexts and analyze properties like stability and quantifier elimination.
Definable Sets and Functions
Fundamental Concepts of Definability
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- Definable sets comprise subsets of a structure's universe described using first-order formulas in the structure's language
- Definable functions represent mappings between elements of a structure through first-order formulas defining their graph
- Definability depends on the specific structure and its language
- Different structures may have varying definable sets and functions
- Definable sets and functions connect syntactic descriptions (formulas) with semantic objects (subsets and mappings)
- Parameters in formulas enable definition of sets and functions dependent on specific structure elements
- Studying definable sets and functions reveals the expressive power of first-order logic within a given structure
Mathematical Representation of Definability
- Set A is definable in structure M if a formula φ(x) exists where A = {a ∈ M : M ⊨ φ(a)}
- Function f is definable in structure M if its graph {(x, y) : f(x) = y} forms a in M × M
- Formulas with free variables define sets (single free variable) or relations/functions (multiple free variables)
- Defining formula complexity measures the definable set or function complexity
- Complexity factors include quantifier rank and number of variables
- Boolean combinations of definable sets correspond to logical combinations of defining formulas
- Parametric definability creates families of sets or functions using additional free variables as parameters
- Logical connectives and quantifiers in formulas generate various definable set classes
- Examples include open sets, closed sets, and Borel sets in topological structures
Characterizing Definable Sets
Formula-Based Characterization
- Definable sets utilize first-order formulas in the structure's language for description
- Formulas with free variables define sets by specifying conditions elements must satisfy
- Complexity of defining formulas reflects the intricacy of the definable set
- Higher quantifier rank or more variables indicate increased complexity
- Boolean operations on definable sets correspond to logical operations on formulas
- Union of sets relates to disjunction of formulas
- Intersection of sets relates to conjunction of formulas
- Parametric formulas allow for defining families of sets with additional variables
- Example: defines the set of pairs (a, b) where a < b in an ordered structure
Topological and Algebraic Characterization
- Definable sets in certain structures possess specific topological properties
- In o-minimal structures, definable sets decompose into finitely many cells
- Algebraic properties of definable sets relate to the underlying structure
- In algebraically closed fields, definable sets correspond to algebraic varieties
- Definable sets form a Boolean algebra, a sublattice of the structure's universe power set
- Preservation of definability under between structures
- If two structures are elementarily equivalent, they have the same definable sets
- Tarski-Vaught test determines if a subset forms an elementary substructure based on definability
- Elimination of quantifiers and model connect to the study of definable sets
- Theories with quantifier elimination have particularly simple definable sets
Properties of Definable Sets
Closure Properties
- Definable sets exhibit closure under finite Boolean operations
- Union of definable sets remains definable
- Intersection of definable sets stays definable
- Complement of a definable set is definable
- Image and preimage of definable sets under definable functions maintain definability
- If f is a and A is a definable set, f(A) and f^(-1)(A) are definable
- Definable sets form a Boolean algebra
- This algebra is a sublattice of the power set of the structure's universe
- Elementary equivalence between structures preserves the property of being definable
- If M ≡ N and A is definable in M, a corresponding definable set exists in N
Special Properties in Specific Structures
- O-minimal structures yield definable sets with nice topological and geometric properties
- Finite decomposition into cells (cell decomposition theorem)
- Definable sets have finitely many connected components
- Definable sets in algebraically closed fields correspond to constructible sets
- Finite Boolean combinations of algebraic varieties
- Tarski-Vaught test provides conditions for elementary substructures based on definability
- A subset A ⊆ M is an elementary substructure if for every formula φ(x,y) and a ∈ A, if M ⊨ ∃xφ(x,a), then there exists b ∈ A such that M ⊨ φ(b,a)
- Elimination of quantifiers in theories leads to simpler descriptions of definable sets
- Example: In the theory of real closed fields, every definable set is a finite union of intervals and points
Definability for Structure Relationships
Interpretations and Translations
- Definable sets and functions enable comparison of expressive power between structures and theories
- Interpretations between structures formalize using definable sets and functions
- Allow translation of properties between different mathematical contexts
- Example: Interpreting the complex numbers in the real plane using definable sets and functions
- Definable extend definability to the space of complete types
- Provide tools for analyzing model-theoretic properties (stability, NIP)
- Definable choice functions explore the relationship between definability and the axiom of choice
- Important in studying the interplay between definability and set theory
Preservation and Invariance
- Definability preservation under model-theoretic constructions is key in stability theory
- Ultraproducts and elementary extensions often maintain definability properties
- Definable sets and functions are crucial in studying algebraic and topological properties
- Dimension theory in o-minimal structures relies heavily on definable sets
- Definable sets in algebraically closed fields correspond to constructible sets
- Automorphism groups of structures often analyzed through invariant definable sets and functions
- Galois correspondence between definable sets and automorphism groups
- Example: In differentially closed fields, the definable sets invariant under the derivation correspond to algebraic differential equations
Key Terms to Review (18)
Abraham Robinson: Abraham Robinson was a mathematician best known for his work in model theory, particularly for developing non-standard analysis, which introduced rigorous treatment of infinitesimals. His contributions helped shape the understanding of structures in mathematical logic and advanced the foundational aspects of model theory.
Algebraic Closure: Algebraic closure is a field extension in which every non-constant polynomial has a root. This concept is crucial as it allows us to understand the completeness of fields in terms of polynomial equations and their solutions. It plays an essential role in various areas, linking the properties of fields, defining structures, and providing a foundation for applications in algebraic geometry.
Compactness Theorem: The Compactness Theorem states that if every finite subset of a set of first-order sentences is satisfiable, then the entire set is satisfiable. This theorem highlights a fundamental relationship between syntax and semantics in first-order logic, allowing us to derive important results in model theory and its applications across mathematics.
Completeness: Completeness is a property of a logical system that indicates every statement that is true in all models of the system can be proven from its axioms. This means there are no true statements about the structures that can't be derived using the rules of the theory, linking it closely to consistency and the nature of models.
Definable Function: A definable function is a mapping between elements of a structure that can be uniquely specified by a formula in the language of that structure. This concept is crucial in understanding how certain sets and functions can be described within a model, connecting it to the nature of realizations and omissions of types, the study of fields, and the characterization of definable sets.
Definable Group: A definable group is a group that can be described or characterized by a specific logical formula within a given structure, allowing its elements and operations to be precisely identified using the language of model theory. This concept plays a crucial role in connecting algebraic structures with logical descriptions, leading to applications in various fields like algebraic geometry and the study of definable sets and functions.
Definable Set: A definable set is a subset of a model that can be precisely described using a formula from the language of the theory, allowing us to distinguish the elements of that set based on specific properties. The concept of definability connects to various aspects such as the relationships between models, how types can be realized, and the behavior of algebraic structures.
Elementary Equivalence: Elementary equivalence refers to the property where two structures satisfy the same first-order sentences or formulas. This means that if one structure satisfies a certain first-order statement, the other structure must also satisfy that statement, leading to deep implications in model theory and its applications in various fields.
Elementary Extension: An elementary extension is a model that extends another model in such a way that every first-order statement true in the original model remains true in the extended model. This concept is significant because it allows for the preservation of certain logical properties and structures while expanding the universe of discourse, making it a crucial idea in understanding various relationships between models.
Embedding: An embedding is a type of structure-preserving map between two mathematical structures that allows one to understand how one structure can be viewed as a substructure of another. This concept is vital for comparing structures in model theory, where embeddings can reveal relationships and similarities between different models.
First-order formula: A first-order formula is a logical expression that includes quantifiers, variables, and predicates, allowing it to express properties of elements in a structure. These formulas form the foundation of first-order logic, enabling the description of sets and functions within mathematical structures through logical relationships and operations. They can be used to define properties that are either true or false depending on the specific interpretation of the underlying structure.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or models, that respects the operations defined in those structures. It connects different mathematical structures by allowing for a translation of elements and operations from one structure to another while maintaining their properties.
Isomorphism: An isomorphism is a structure-preserving map between two mathematical structures that demonstrates a one-to-one correspondence between their elements, meaning that the structures are essentially the same in terms of their properties and relationships. This concept not only highlights similarities between different structures but also helps in understanding how different theories relate to each other.
Kurt Gödel: Kurt Gödel was an Austrian-American logician, mathematician, and philosopher, best known for his incompleteness theorems which fundamentally changed our understanding of mathematical logic and formal systems. His work has profound implications in areas such as the consistency and completeness of theories, as well as the limitations of axiomatic systems in model theory.
Löwenheim-Skolem Theorem: The Löwenheim-Skolem Theorem states that if a first-order theory has an infinite model, then it has models of all infinite cardinalities. This theorem highlights important properties of first-order logic and models, demonstrating that certain structures can always be found, regardless of the size of the domain.
Saturation: Saturation refers to a property of models in model theory where a model is considered saturated if it realizes all types that are consistent with its theory. This concept connects various features of model theory, including how models can be extended and the behavior of definable sets and functions within those models. Saturation plays a significant role in understanding the complexity and richness of models and their relationships to theories and types.
Second-order formula: A second-order formula is a type of logical statement in mathematical logic that allows quantification over not just individual variables but also over predicates or sets. This enables the expression of more complex properties and relationships than first-order formulas, making second-order logic more expressive for defining sets and functions within a structure.
Types: In model theory, a type is a collection of formulas that describe the possible behaviors of a particular element or set of elements within a given structure. Types help in understanding how certain elements can relate to others and form the basis for defining concepts like definability, omitting types, and imaginary elements.