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Eliezer_Yudkowsky comments on Second-Order Logic: The Controversy - Less Wrong
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Second order logic may be able to express all expressible statements in 3rd order logic, 4th order, up to any finite Nth order, but perhaps there exist ∞th order statements that 2nd order logic cannot express. Albeit, such statements could never be fully decidable and would thus be, at best, semidecidable or co-semidecidable. This may not be complete, but science works the same way.
I don't think that infinitary logic is the same as ω-order logic.
Wouldn't ω-order logic be a subset of infinitary logic? Or do I have it backwards?
I don't think they have anything to do with each other. Infinitary logic is first-order logic with infinite proof lengths. Second-order logic is finite proof lengths with quantification over predicates. I don't know if there's any particular known relation between what these two theories can express.