It matters because it means second order arithmetic isn't a finite object in the same way first order effective theories are where you might have "infinitely" m

It matters because it means second order arithmetic isn't a finite object in the same way first order effective theories are where you might have "infinitely" many axioms but they are packaged up in a finite object, an algorithm. The whole point of a formal system, its formality, is that it doesn't require any semantic notions to describe it. Second order arithmetic with its full model is no such thing. It is inherently infinitary and therefore not formal at all. And it's a treacherous object. C.f. Richard's paradox which demonstrates that "subset of the integers" is by no means a naive idea.