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5. Theory of Logic / K. Features of Logics / 6. Compactness

[satisfaction by satisfying the finite subsets]

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