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srn347 comments on Second-Order Logic: The Controversy - Less Wrong
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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.