### Single Idea 13646

#### [catalogued under 5. Theory of Logic / K. Features of Logics / 6. Compactness]

Full Idea

Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency.

Gist of Idea

Compactness is derived from soundness and completeness

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.79

A Reaction

[this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness.