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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