> I'm pointing out that the nature of R is independent of its definition, and that any mathematical formal system you invent to construct R (up to isomorphism) is just a description. R exists independently on its own.

Isn't that a bit metaphysical and open to interpretation? I would say R only meaningfully exists as defined. A mathematical "truth" is either an axiom or it has a proof?

(Not that I think that makes a difference for the original argument about 1)