Second order logic can also arithmatise sentences, and also has fixed points. So the usual proofs carry over about the 1st incompleteness theorem. But there’s an easier way to see this. There can’t be any computable procedure to check if a second order sentence is valid or not, because if there was we could check if PA->Theorem and therefore decide Peano Arithmetic and therefore the Halting problem.
Second order logic can also arithmatise sentences, and also has fixed points. So the usual proofs carry over about the 1st incompleteness theorem. But there’s an easier way to see this. There can’t be any computable procedure to check if a second order sentence is valid or not, because if there was we could check if PA->Theorem and therefore decide Peano Arithmetic and therefore the Halting problem.