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 Reference
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]