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Single Idea 13646

[filed under theme 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 Ref

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.

The 17 ideas with the same theme [satisfaction by satisfying the finite subsets]:

9995 Proof in finite subsets is sufficient for proof in an infinite set [Enderton]
10771 Compactness is important for major theories which have infinitely many axioms [Tharp]
10772 Compactness blocks infinite expansion, and admits non-standard models [Tharp]
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
13841 Why should compactness be definitive of logic? [Boolos, by Hacking]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD]
17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
10977 Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) [Read]
10976 Compactness makes consequence manageable, but restricts expressive power [Read]
10974 Compactness is when any consequence of infinite propositions is the consequence of a finite subset [Read]
10975 Compactness does not deny that an inference can have infinitely many premisses [Read]
17867 If a concept is not compact, it will not be presentable to finite minds [Almog]