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Single Idea 13630

[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness ]

Full Idea

It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures.

Gist of Idea

Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures

Source

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

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.-12

A Reaction

The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade?

Related Idea

Idea 13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]

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]