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The 22 pages that link to Compactness theorem, each with the reason it gives.

• First-order logicRelated: Compactness yields existence results that cannot be obtained by checking only one finite fragment.

• Model theoryRelated: It turns finite satisfiability into existence of a model for the whole theory.

• Logical consequenceRelated: Compactness constrains which consequence relations first-order semantics can support.

• Universal quantificationRelated: It reveals limits on what collections of universal constraints can enforce in first-order logic.

• Second-order logicCompared with: Compactness fails for full second-order consequence.

• Gödel's completeness theoremRelated: Completeness and compactness are equivalent over standard first-order logic.

• CompletenessRelated: Compactness is closely related to first-order completeness and follows from it.

• UltrafilterRelated: Ultraproducts provide a standard proof by assembling models along an ultrafilter.

• Forcing (mathematics)Compared with: It is another model-theoretic tool, but does not construct extensions by adjoining generic sets.

• Löwenheim–Skolem theoremRelated: Compactness and Löwenheim–Skolem together yield broad model-existence and size results.

• Peano axiomsRelated: Compactness helps explain why first-order axioms cannot rule out nonstandard arithmetic models.

• Van der Waerden's theoremRelated: Compactness transfers finite interval guarantees into the infinite positive integers.

• Elementary equivalenceRelated: It turns finite satisfiability into models that can share all first-order sentences.

• Existence theoremRelated: Compactness can turn finite consistency into the existence of a model.

• Boolean prime ideal theoremRelated: For propositional logic, compactness is another equivalent formulation of this choice principle.

• Paris–Harrington theoremRelated: Compactness helps connect finite combinatorial assertions with formal theories and their models.

• Kőnig's lemmaRelated: Tree-based proofs use the lemma to assemble compatible finite satisfactions into an infinite model.

• Deduction theoremCompared with: It is a major model-theoretic result, unlike this proof-transformation principle.

• Craig interpolationCompared with: Compactness concerns finite satisfiability, not factoring an entailment through shared vocabulary.

• De Bruijn–Erdős theorem (graph theory)Related: Graph coloring can be encoded as a first-order theory, so logical compactness yields the theorem.

• Lindström's theoremRelated: One of the two properties that jointly restrict the strength of a logic.

• Trakhtenbrot's theoremCompared with: Compactness relies on arbitrary models and does not preserve finite satisfiability.