Compactness theorem

A set of first-order sentences has a model if and only if every finite subset has a model. It lets finite satisfiability establish satisfiability for an entire theory.

First-order logic: A formal system with variables, quantifiers, predicates, and rules for reasoning about objects and relations. The theorem applies to sets of sentences in this logic.
Nonstandard model of arithmetic: A model of arithmetic containing elements larger than every standard natural number. Adding a constant larger than each numeral is finitely satisfiable, so compactness yields such a model.
Löwenheim–Skolem paradox: The apparent tension between set theories describing uncountable sets and having countable models. Compactness combines with Löwenheim–Skolem results to expose this model-theoretic phenomenon.
Completeness theorem: A theorem connecting semantic validity with derivability in a formal proof system. For first-order logic, completeness supplies a proof route to compactness; the two properties are distinct.
Satisfiability: The property of a formula or sentence set having an interpretation that makes every member true. Compactness equates satisfiability of a theory with satisfiability of all its finite subsets.
Downward Löwenheim–Skolem theorem: A theorem stating that a first-order theory with an infinite model has models of smaller infinite cardinalities, subject to its language size. Compactness and Löwenheim–Skolem together produce countable models of theories with infinite models.
Categorical theory: A theory whose models are all isomorphic at a specified cardinality. Compactness constrains categoricity across infinite cardinalities in countable first-order languages.
Propositional compactness theorem: The result that a set of propositional formulas is satisfiable whenever all its finite subsets are satisfiable. It is the propositional counterpart, with a simpler proof using truth assignments.
Ultraproduct: A structure formed from a family of structures by identifying sequences according to an ultrafilter. Ultraproducts provide a direct proof by combining models of finite fragments.
Nonstandard analysis: A framework for analysis using extensions of the real numbers that include infinitesimal and infinite elements. Compactness helps construct elementary extensions containing infinitesimals.

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