[dooglius]

Agreed; e.g. if you prove something about the real numbers, the matter of how R is constructed out of your axiomatic system doesn't matter

[jfarmer]

The picture isn't quite so clean in the constructive context, which is what many of these proof systems are rooted in, e.g., https://mathoverflow.net/questions/236483/difference-between...

[pas]

there are questions where the abstraction of real numbers becomes leaky, and some axioms (or their lack) poke through.

https://en.wikipedia.org/wiki/Axiom_of_choice#Real_numbers