Single Idea 13626

[catalogued under 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=]

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]