Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Mathematical Logic and Foundations → Predicate Logic ↓


Model Theory


Introduction to model theory

Model theory is a branch of mathematical logic that deals with the relationship between formal languages and their interpretations or models. It is primarily concerned with the study of the relationships that exist between a formal language and its structures. In simple terms, model theory explores how sentences in a mathematical language can be true or false depending on what "world" or structure we interpret them in.

Model theory has its roots in the work of mathematicians such as Alfred Tarski, who significantly developed the initial concepts in the mid-20th century. Tarski was important in formalizing the semantics of languages and providing methods for understanding what it means for a statement to be true within a structure.

Basic concepts of predicate logic

Before delving deeper into model theory, it is essential to understand the basics of predicate logic, as this forms the backbone of our exploration. Predicate logic extends propositional logic by including quantifiers and variables, allowing us to express statements about objects.

Structures in predicate logic

A structure in predicate logic is a mathematical object that provides an interpretation for the symbols of a formal language. A structure typically consists of:

  • A nonempty set U called the universe or domain of discourse, which contains the objects we talk about.
  • The interpretations of the function symbols correspond to real functions from U^n to U
  • The interpretations of the relation symbols correspond to subsets of U^n.
  • The interpretations of the fixed symbols correspond to specific elements of U

Languages and formulas

A language in the context of predicate logic includes:

  • A set of relation symbols, each of which has a specified arity (the number of arguments it takes).
  • A set of function symbols, each of which also has a specified arity.
  • A set of fixed symbols.

A formula is a well-formed sequence of symbols that follows the syntactic rules of the language. For example, the formula ∀x (P(x) → Q(x)) expresses that for every element x in the universe, if P(x) is true, then Q(x) is also true.

Understanding the model

In model theory, a model is simply a structure that satisfies a given set of formulas or statements. A model can be thought of as a "world" in which the statements of a language are evaluated for truth.

Visual example: Simple model

Consider a language with a unary relation P and a constant symbol a. The structure of this language might be something like this:

    U = {1, 2, 3} P^A = {1, 3} (the interpretation of P) a^A = 1 (the interpretation of a)
    U = {1, 2, 3} P^A = {1, 3} (the interpretation of P) a^A = 1 (the interpretation of a)
1 2 3

Here, P^A contains elements 1 and 3, which indicates that they satisfy property P. Element 2 does not have this property.

Satisfaction of sources

An important concept in model theory is that of satisfaction. We say that a structure A satisfies a formula φ if φ is true when interpreted in the structure. If A satisfies φ, we write A ⊨ φ.

For example, if our language has a formula P(a), with the structure given above, A ⊨ P(a) since a^A = 1 and 1 is in P^A.

Elementary equivalences and series

Two structures A and B are inherently equivalent (denoted by A ≡ B) if they satisfy the same sentences in our language. This means that any statement that is true in A is also true in B, and vice versa.

Example of elementary equivalence

Let us consider two structures for the language of arithmetic with a single relation =:

    Structure A: U_A = {0, 1, 2, 3} =^A as normal identity Structure B: U_B = {0, 1, 2, 3, 4} =^B as normal identity
    Structure A: U_A = {0, 1, 2, 3} =^A as normal identity Structure B: U_B = {0, 1, 2, 3, 4} =^B as normal identity
0 1 2 3 0 1 2 3 4

While both structures A and B contain the elements 0, 1, 2, and 3 under the identity relation, because B contains an additional element that does not affect the satisfiability of integer arithmetic sentences in the first-order language, A and B are elementarily equivalent in this context.

Brevity and completeness

Two fundamental theorems in model theory are the compactness theorem and the completeness theorem.

Compactness theorem

The compactness theorem states that if every finite subset of a set of first-order sentences has a model, then the whole set will also have a model. This theorem implies that we can often understand infinite structures by examining their finite parts.

Completeness theorem

The completeness theorem, proved by Kurt Gödel, asserts that if a formula is true in every model of a set of sentences, then there is a proof of the formula from these sentences. Essentially, this equates with proving truth in first-order logic.

Ultraproducts and their use

Ultraproducts are a construction in model theory that allows us to create new models from a family of existing models. The construction uses a concept from set theory called ultrafilters.

Basic view of ultraproduct

Given structures A_1, A_2, A_3, ... and an ultrafilter U, the ultraproduct is a structure that "averages" these structures according to U.

A_1 A_2 A_3 , Ultraproduct

The ultraproduct captures properties that are present "almost everywhere" in the given family of structures.

Applications of model theory

Model theory is used in many topics of mathematics. Some notable applications include:

  • Algebra: Model theory has been used to solve problems in algebra, such as the solution of X for the first-order theory of algebraically closed fields.
  • Intermediate logic: Model theory is central to understanding the transition between propositional logic and set theory.
  • Database theory: In computer science, model theory is used to study data models and query languages.

Conclusion

Model theory is an essential part of mathematical logic, providing a framework for understanding how formal languages relate to mathematical structures. By building on the foundations of predicate logic, it provides insight into the nature of mathematical truth and provability. While we have touched on its various aspects and applications, model theory is a vast and expanding field, constantly revealing new connections between different areas of mathematics.


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