Full Idea
It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic.
Gist of Idea
Semantic consequence is ineffective in second-order logic
Source
Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.-16
A Reaction
I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'.
Related Idea
Idea 10083 A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]