🧠Model Theory Unit 4 – Theories and Models

Key Concepts and Terminology

• 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 $\phi$ implies a sentence $\psi$, then there exists a sentence $\theta$ in the common language of $\phi$ and $\psi$ such that $\phi$ implies $\theta$ and $\theta$ implies $\psi$
• 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.