PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Logic and Foundations → Model Theory ↓


Structures and Theories in Model Theory


Model theory is a field of mathematical logic that deals with the relations between formal languages and their interpretations or models. The main components in model theory are structures and theories. Structures are special mathematical constructions, while theories are collections of statements in a formal language.

Understanding the structures

A structure is a specific way of interpreting the symbols of a formal language. It consists of a non-empty set called the universe and the interpretations of the function symbols, relation symbols, and constant symbols that fit within this universe.

Example of a simple structure

Consider a structure for the language of arithmetic. The language consists of symbols for numbers (0, 1, 2, ...), operations (+, -) and relations (=, <, >).

universe: { Z } // the set of all integers
Functions: + (addition), - (subtraction)
Relations: = (equality), <, > (order)

Here, the language of arithmetic is interpreted on integers. Addition and subtraction are interpreted as usual, and relations are standard integer comparisons.

Representation with diagrams

0 1 2

This diagram represents a structure with elements 0, 1, and 2. The arrows represent the process of adding one to a number within the universe of integers.

Theory: A collection of statements

A theory is a set of sentences in a formal language. These sentences express truths about a particular structure or class of structures.

Example of the principle in arithmetic

Consider the following sentences:

(1) ∀x (x + 0 = x)
(2) ∀x ∀y (x + y = y + x)
(3) ∀x ∃y (x + y = 0)

Each of the statements given here is part of a theorem of arithmetic:

  • (1) states that adding 0 to any number x gives x.
  • (2) shows that the addition is commutative.
  • (3) states that for every number x there exists a number y whose sum is 0. This shows the existence of additive inverses.

Linking theories to structures

An important aspect of model theory is to understand how theories are related to structures. A structure is a model of a theory if every sentence in the theory is true within that structure.

For example, the standard structure of the integers forms a model of the theory of arithmetic presented above since all sentences are valid for integers.

Types of structures and models

We can classify structures and models based on their properties and their relation to theories:

Primary structures

Elementary structures truly satisfy each and all of the axioms of a given theory. Elementary substructures occur when a structure is a subset and satisfies all the axioms in the same way.

An exemplary elementary structure is the set of rational numbers within the structure of real numbers. Both satisfy the axioms of a complete ordered field without anomalies.

Visual representation: elementary substructure

Reals Rational

This shows that the set of rational numbers is a subset (substructure) of the real numbers. The blue circle shows an elementary substructure, which maintains all arithmetic operations and properties within the larger structure of the real numbers.

Consistency and completeness

A consistent theory is one in which no contradictions arise, i.e. both the statement and its negation cannot be proven. Completeness refers to the aspect where all possible statements in the language are in principle either provable or unprovable.

Stability

A classical example of this is Peano arithmetic, which is known to be consistent but not necessarily complete due to Gödel's incompleteness theorems.

Consistency principle: ∀x(x + 0 = x)
Inconsistent if: ∃x (x + 0 ≠ x)

Case studies and applications

Model theory not only provides deep insights into mathematical structures, but also extends to other areas and applications such as:

Arithmetic logic

Applications in mathematical logic often use structures and principles to establish rigor, such as classification into particular theories exhibiting isomorphisms between models.

Consider a finite model theory applied in computational areas to assess the solvability of certain logical problems.

Set theory

Model theory helps analyze various set theories, such as Zermelo–Fraenkel, in order to establish the basics of cardinality and ordering.

The study of various infinite cardinals often employs principles of model theory to define and distinguish the size of structures.

Conclusion: Building a bridge between language and interpretation

The interaction between structures and theories allows mathematicians and logicians to explore the consistency, interpretability, and scope of mathematical logic. Understanding these concepts well not only adds clarity to mathematical theory, but also expands its utility and application across a variety of disciplines.

This subtle balance between abstract formalism and concrete interpretation highlights the versatility and centrality of model theory in modern mathematics.


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