Model theory explores mathematical structures using formal languages and logical systems. It delves into models, which are structures satisfying axioms, and theories, which are sets of sentences describing classes of models. Key concepts include signatures, interpretations, satisfiability, and completeness.
The field emerged in the early 20th century, with contributions from Skolem, Tarski, and Robinson. Fundamental principles like the Löwenheim-Skolem theorems and compactness theorem form its foundation. Model theory has applications in algebra, geometry, and number theory, with ongoing research in stability theory and o-minimality.
Model theory studies mathematical structures and their properties using formal languages and logical systems
A model is a mathematical structure that satisfies a set of axioms or sentences in a formal language
Signature (or vocabulary) of a model consists of the non-logical symbols used to define the model, such as constant symbols, function symbols, and relation symbols
An interpretation assigns meaning to the symbols in a signature by mapping them to elements, functions, and relations in a model
A theory is a set of sentences (or axioms) in a formal language that describes a class of models
Satisfiability refers to the property of a model making a sentence or set of sentences true
Completeness of a theory means that every sentence in the language is either provable from the theory or its negation is provable
Compactness theorem states that if every finite subset of a set of sentences has a model, then the entire set has a model
Historical Context and Development
Model theory emerged as a distinct branch of mathematical logic in the early 20th century
The work of mathematicians such as Thoralf Skolem, Alfred Tarski, and Abraham Robinson laid the foundations for model theory
Gödel's completeness theorem (1929) established the connection between semantic and syntactic notions in first-order logic, a crucial result for model theory
Tarski's work on the concept of truth in formal languages and the definition of logical consequence played a significant role in the development of model theory
The 1950s and 1960s saw the introduction of important concepts such as ultraproducts, saturated models, and the Löwenheim-Skolem theorems
The development of stability theory in the 1970s by Saharon Shelah led to a classification of theories based on their model-theoretic properties
Model theory has since found applications in various areas of mathematics, including algebra, geometry, and number theory
Fundamental Principles of Model Theory
Model theory studies the relationship between formal languages and their interpretations in mathematical structures
The Löwenheim-Skolem theorems establish the existence of models of different cardinalities for a given theory
The downward Löwenheim-Skolem theorem states that if a theory has an infinite model, then it has a model of any smaller infinite cardinality
The upward Löwenheim-Skolem theorem states that if a theory has an infinite model, then it has models of arbitrarily large cardinality
The compactness theorem is a fundamental result in model theory, ensuring the existence of models for consistent theories
The completeness theorem connects the semantic notion of logical consequence with the syntactic notion of provability in first-order logic
Quantifier elimination is a property of some theories where every formula is equivalent to a quantifier-free formula, simplifying the study of models
The interpolation theorem states that if a sentence ϕ implies a sentence ψ, then there exists a sentence θ in the common language of ϕ and ψ such that ϕ implies θ and θ implies ψ
The definability of sets and functions within a model is a central concern in model theory, as it relates to the expressive power of the formal language
Types of Models and Their Applications
Finite models are structures with a finite domain, often used in computer science and combinatorics
Example: Graphs, finite fields, and Boolean algebras
Infinite models are structures with an infinite domain, such as the natural numbers, real numbers, or any other infinite set
Countable models are infinite models with a countable domain, such as the rational numbers or the algebraic numbers
Uncountable models are infinite models with an uncountable domain, such as the real numbers or the complex numbers
Saturated models are models that realize all types over small subsets, providing a rich structure for studying the properties of a theory
Prime models are the "smallest" models of a theory, embedding elementarily into any other model of the theory
Atomic models are models where all types realized in the model are principal, meaning they can be defined by a single formula
Homogeneous models are models where any isomorphism between finitely generated substructures extends to an automorphism of the entire model
Formal Languages and Structures
A formal language consists of an alphabet of symbols and a set of formation rules for constructing well-formed formulas
First-order logic is the most commonly used formal language in model theory, allowing for the use of variables, quantifiers, and logical connectives
The signature of a language specifies the non-logical symbols used, such as constant symbols, function symbols, and relation symbols
Structures (or models) provide interpretations for the symbols in a language, assigning meaning to the constants, functions, and relations
Theories are sets of sentences in a formal language that describe the properties of a class of structures
Example: The theory of groups, the theory of fields, and the theory of dense linear orders
Axiomatization is the process of selecting a set of sentences (axioms) that characterize a class of structures up to isomorphism
Completeness and consistency are important properties of theories, ensuring that the theory has a model and does not prove contradictory statements
Axioms and Proof Systems
Axioms are sentences in a formal language that are assumed to be true and serve as the starting point for deriving other sentences
Logical axioms are the axioms of the underlying logic, such as first-order logic, which include tautologies and rules for quantifiers
Non-logical axioms are specific to a particular theory and describe the properties of the structures being studied
Example: The group axioms, the field axioms, and the axioms of Peano arithmetic
Inference rules specify how new sentences can be derived from existing ones, such as modus ponens and universal instantiation
A proof is a sequence of sentences, each of which is either an axiom or follows from previous sentences by an inference rule
Soundness of a proof system means that every sentence provable in the system is true in all models of the theory
Completeness of a proof system means that every sentence true in all models of the theory is provable in the system
The compactness theorem and the completeness theorem are fundamental results connecting proof systems and models
Model Construction and Preservation
Model construction techniques are used to build models with specific properties or to prove the existence of models for a given theory
The Henkin construction is a method for building a model of a consistent theory by extending the language with new constant symbols and ensuring the existence of witnesses for each formula
Ultraproducts are a way of combining a family of models into a single model using an ultrafilter, preserving certain properties of the original models
Elementary substructures are substructures that satisfy the same first-order sentences as the original structure, preserving the truth of all formulas
Elementary extensions are superstructures that satisfy the same first-order sentences as the original structure, allowing for the addition of new elements while preserving the properties of the original model
Homomorphisms and embeddings are functions between structures that preserve the interpretations of the symbols in the language
Isomorphisms are bijective homomorphisms that preserve the structure in both directions, establishing a strong notion of equivalence between models
Automorphisms are isomorphisms from a structure to itself, forming a group that captures the symmetries of the model
Advanced Topics and Current Research
Stability theory studies the classification of theories based on the behavior of their models, particularly the number of types realized in a model
Stable theories have a well-behaved notion of independence and admit a dimension theory for their models
Unstable theories, such as the theory of random graphs, exhibit more complex behavior and require different tools for their analysis
O-minimality is a property of ordered structures where every definable subset of the domain is a finite union of intervals, leading to a tame geometry
The Keisler-Shelah isomorphism theorem states that two elementarily equivalent structures have isomorphic ultrapowers, providing a powerful tool for studying the relationship between models
Model-theoretic forcing is a technique for constructing models with specific properties by iteratively adding new elements to the structure
The Hrushovski construction is a method for building new structures with prescribed properties, such as a strongly minimal set with a non-trivial pregeometry
Applications of model theory to other areas of mathematics include:
Number theory: The study of definable sets in arithmetic structures and the model theory of valued fields
Algebraic geometry: The model theory of algebraically closed fields and the study of definable sets in geometric structures
Combinatorics: The use of model-theoretic techniques in graph theory and the study of finite structures
Current research in model theory includes the development of new tools for studying specific classes of structures, the application of model-theoretic techniques to other areas of mathematics, and the investigation of the connections between model theory and other fields such as computer science and physics.