Aiming to reduce all mathematics, including the concept of number, to logic
I agree with this point of view! My concern is with people who do not share this relatively benign view of infinity, and regard theorems proven using strong systems like ZF with suspicion.
The reason it matters what counts as “logic” is that such skeptics, to the extent that they can discuss matters of proof and soundness at all, must have a core of formally acceptable reasoning whose derived propositions they grant validity to, and therefore the parts of mathematics that can be interpreted in such a system overcome this skepticism.
What is interesting is not “whether” mathematics can be reduced to logic, but “how much” of mathematics can be reduced to logic. There are several ways to get as far as PA that could be deemed “logical”, but doesn’t going further than that entail ontological commitments to infinite objects or collections of some kind?
Is there a plausible way to present second-order logic, or some other system that is arguably “logical”, where the existence of infinite entities follows naturally?