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13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1) |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
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. | |
From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref) | |
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'. |