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Single Idea 13646

[from 'Foundations without Foundationalism' by Stewart Shapiro, in 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.