You seem to keep forgetting that the whole notion of “second-order logic” does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It’s just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say “all possible collections that include some objects and exclude others, but still include and exclude only individual objects” as “all objects”.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense.
Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a “model” for the natural numbers in toto, whatever that could mean.
You seem to keep forgetting that the whole notion of “second-order logic” does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It’s just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say “all possible collections that include some objects and exclude others, but still include and exclude only individual objects” as “all objects”.
Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a “model” for the natural numbers in toto, whatever that could mean.