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Maths › 03. Logic and Foundations ›

6.4 Stability

Stability theory studies the complexity of first order theories by analyzing how types behave over parameter sets, and in particular by measuring how many distinct types exist and how these types interact with definable sets. The guiding idea is that a theory should be considered tame if the space of types is controlled, and wild if the space of types grows too quickly or exhibits arbitrary combinatorial patterns. Throughout this...
logic model theory stability types May 2, 2026
Maths › 03. Logic and Foundations ›

6.1 Definable Sets

Definability is one of the central ideas of model theory. Once a first order language and a structure have been fixed, formulas can be used to describe subsets of the domain, relations on the domain, and functions between definable sets. In this way, syntax gives rise to geometry inside a structure: a formula becomes a description of a set of points satisfying a condition. In this section, we work with...
logic model theory definability first order logic May 2, 2026
Maths › 03. Logic and Foundations ›

6.5 Classification

Classification theory studies first order theories by organizing them according to the complexity of their definable sets and types, with the goal of distinguishing theories that admit a systematic structural description from those whose models exhibit uncontrolled combinatorial behavior. The central principle is that the behavior of formulas, when evaluated across different parameter configurations, determines the global structure of models, and therefore one can classify theories by analyzing the combinatorial...
logic model theory classification theory stability May 2, 2026
Maths › 03. Logic and Foundations ›

6.2 Types and Realizations

Types give a precise way to describe all first order properties that an element, or a tuple of elements, may satisfy in a structure. Instead of looking at one formula at a time, a type collects many formulas together and treats them as a possible description of an element. This is useful because model theory often studies not only the elements already present in a structure, but also the kinds...
logic model theory types realizations May 2, 2026
Maths › 03. Logic and Foundations ›

6.3 Saturated Models

Saturated models provide a way to control which types are realized inside a structure, and they serve as canonical large models in which all consistent descriptions over small parameter sets are already realized, so that no further extension is needed to witness those types. Throughout this section, let $L$ be a first order language, let $T$ be an $L$-theory, and let $\mathcal M \models T$ be an $L$-structure with domain...
logic model theory saturated models types May 2, 2026
Maths › 03. Logic and Foundations ›

6. Definability and Types

Definability studies which sets, relations, and functions can be described inside a structure using formulas of a fixed language, and it is one of the main ways model theory connects syntax with the internal geometry of mathematical structures. The chapter begins with definable sets and functions, where a formula with free variables is used to specify a collection of tuples from a model, and this gives a precise way to...
logic model theory definability types May 2, 2026
Maths › 03. Logic and Foundations ›

5.2 Lowenheim Skolem

The Lowenheim Skolem theorems describe how the size of models of a first order theory can be changed without affecting truth of sentences, and they show that if a theory has any infinite model, then it has models of many different infinite sizes, which reveals an important limitation of first order logic in controlling cardinality. Definition 5.10 (Cardinality of a Structure) Let $\mathcal{M}$ be a structure with underlying set $M$....
logic model theory lowenheim skolem cardinality May 2, 2026
Maths › 03. Logic and Foundations ›

5.5 Limitations

First order logic has strong structural properties such as compactness and the Lowenheim Skolem theorems, and these properties impose intrinsic limitations on what can be expressed or uniquely characterized within the language. Definition 5.27 (Expressibility) A class of structures $\mathcal{K}$ is said to be first order definable if there exists a set of first order sentences $\Gamma$ such that: $$ \mathcal{M} \in \mathcal{K} \quad \text{if and only if} \quad \mathcal{M}...
logic model theory limitations expressiveness May 2, 2026
Maths › 03. Logic and Foundations ›

5.1 Compactness

The compactness theorem is one of the central structural results of first order logic, and it expresses a precise relationship between finite and infinite behavior of theories, showing that global satisfiability is completely controlled by finite fragments. Definition 5.1 (Structures and Satisfaction) Let $\mathcal{L}$ be a first order language, and let $\mathcal{M}$ be an $\mathcal{L}$-structure. For a sentence $A$ in $\mathcal{L}$, we write: $$ \mathcal{M} \models A $$ to mean...
logic model theory compactness first order logic May 2, 2026
Maths › 03. Logic and Foundations ›

5.3 Applications

Model theoretic methods, and in particular the compactness theorem and the Lowenheim Skolem theorems, have strong consequences in algebra, because many algebraic structures can be described by first order sentences, and therefore logical principles can be used to construct models and prove existence theorems that are difficult to obtain by direct algebraic arguments. First Order Theories in Algebra Many algebraic structures can be expressed in a first order language. For...
logic model theory algebra compactness applications May 2, 2026
Maths › 03. Logic and Foundations ›

5.4 Nonstandard Models

Nonstandard models arise when a first order theory admits models that satisfy all its axioms but differ from the intended or standard interpretation, and their existence follows naturally from the compactness theorem and the Lowenheim Skolem theorems. Definition 5.22 (Standard Model) A standard model is a structure that is intended to represent the usual interpretation of a theory. For example, the standard model of arithmetic is: $$ \mathbb{N} = {0,1,2,\dots}...
logic model theory nonstandard models compactness May 2, 2026
Maths › 03. Logic and Foundations ›

5. Compactness and Completeness

Compactness and completeness are two central results that explain why first order logic is both powerful and limited, since they connect formal proof, semantic truth, and the existence of models in a precise way. The chapter begins with the compactness theorem, which says that a set of first order sentences has a model whenever every finite subset has a model, and this result shows that global satisfiability can often be...
logic model theory compactness completeness May 2, 2026
Maths › 03. Logic and Foundations ›

4.5 Examples

Model theory treats many familiar mathematical objects as structures for suitable first order languages, and this viewpoint allows groups, rings, fields, ordered sets, graphs, and geometric systems to be studied using a common formal framework. Algebraic Structures Algebraic structures are among the most natural examples of first order structures, because they are usually described by a domain together with operations and distinguished constants. A group may be represented in the...
logic model theory structures algebra geometry May 2, 2026
Maths › 03. Logic and Foundations ›

4.4 Isomorphism

Structures are often presented using particular sets, symbols, and operations, but many mathematical questions concern the structure itself rather than the specific names of its elements. Isomorphism formalizes this idea by identifying structures that have the same internal organization, even when their underlying domains are different. Definition 4.31 (Isomorphism) Let $\mathcal A$ and $\mathcal B$ be structures for the same signature. An isomorphism from $\mathcal A$ to $\mathcal B$ is...
logic model theory isomorphism invariants May 2, 2026
Maths › 03. Logic and Foundations ›

4.2 Substructures

A structure may contain smaller structures inside it, provided that the smaller domain is closed under the operations of the language and carries the relations inherited from the larger structure. Definition 4.13 (Substructure) Let $\mathcal A$ and $\mathcal B$ be structures for the same signature, and let $A$ and $B$ be their underlying domains. We say that $\mathcal A$ is a substructure of $\mathcal B$, written: $$ \mathcal A \subseteq...
logic model theory substructures embeddings May 2, 2026
Maths › 03. Logic and Foundations ›

4.1 Languages

Model theory studies mathematical structures by describing them using formal languages, and the first step is to specify precisely what symbols are available and how they are organized, since the expressive power of a theory depends on the language in which it is written. Languages A first order language is determined by a collection of symbols that are used to build terms and formulas, and these symbols are grouped according...
logic model theory languages signatures May 2, 2026
Maths › 03. Logic and Foundations ›

4. Structures and Models

Model theory studies mathematical structures through the formal languages used to describe them, and it asks how much information about a structure can be expressed by first order formulas. A structure consists of a domain together with interpretations of the symbols in a language, and this framework allows groups, rings, fields, ordered sets, graphs, and many other mathematical objects to be treated within one common logical setting. The chapter begins...
logic model theory structures models May 2, 2026
Maths › 03. Logic and Foundations ›

4.3 Elementary Equivalence

Two structures may differ as sets and operations, yet satisfy exactly the same first order sentences, and this situation is captured by the notion of elementary equivalence, which compares structures at the level of logical expressibility rather than at the level of concrete representation. Definition 4.23 (Theory of a Structure) Let $\mathcal A$ be a structure for a fixed signature. The theory of $\mathcal A$, denoted: $$ \mathrm{Th}(\mathcal A) $$...
logic model theory elementary equivalence theories May 2, 2026
Maths › 03. Logic and Foundations ›

2. First-Order Logic

First-order logic extends propositional logic by allowing formulas to speak about objects, properties of objects, and relations between objects, rather than treating every statement as an indivisible unit. The language contains variables, constants, function symbols, predicate symbols, logical connectives, and quantifiers, and these symbols allow mathematical statements to be expressed with much greater internal structure than in propositional logic. The first part of the chapter defines terms, predicates, and formulas,...
logic first-order logic model theory foundations May 2, 2026