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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 ›

1.5 Soundness and Completeness

Propositional logic can be studied in two complementary ways. The semantic approach studies truth under valuations, while the proof theoretic approach studies formal derivations from axioms and inference rules. Soundness and completeness explain how these two approaches are related. Semantic Consequence Let $\Gamma$ be a set of formulas, and let $A$ be a formula. We say that $A$ is a semantic consequence of $\Gamma$ if every valuation that makes all...
logic propositional logic proof theory soundness completeness May 2, 2026
Maths › 00. General Mathematics ›

02.3 Consistency and Completeness

2.3 Consistency and Completeness A formal system is judged partly by what it can prove and partly by what it avoids proving. Two central properties are consistency and completeness. Consistency means the system does not prove contradictions. A system is consistent if there is no statement P such that both P and ¬P are provable. not both: ⊢ P and ⊢ ¬P An inconsistent system is unusable for ordinary mathematics...
mathematics logic foundations consistency completeness May 2, 2026