If you care about doing maths in the logic, or at least about arithmetics, it seems completeness of first-order logic doesn’t really give any guarantees anyway. Logicians know this better than me, and most still prefer first-order logic — with few arguing at the side for second-order logic. Why?

In particular, it seems in fact that all proofs in first-order logic are also valid in second-order logic, and more facts become true (for instance for arithmetics) and might be provable.

First-order logic is better because first-order arithmetic does not describe naturals correctly? In either logic, there are properties of naturals one can’t prove — depending on the logic, either because the logic is incomplete (second-order), or because the property of interest only holds for true naturals and fails for other non-standard models (first-order). IIRC, according to Peter Smith’s book, that’s the case for Gödel’s sentence.

It’d be bad if first-order proofs became invalid second-order proofs, but that seems impossible

(and this abstract suggests it is impossible: http://consequently.org/writing/ptm-second-order/). Warning: I refer to proofs *in* the logic, not to proofs about the logic, so the post is not a counterexample.

Finally, second-order logic is boring for model theorists, because Löwenheim-Skolem theorems fail. But that could be provocatively phrased as follows: second-order logic doesn’t have certain bugs that you can build an entire branch of mathematics on. (Almost: there might be model theory not depending on Löwenheim-Skolem’s theorems, not sure).

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