> Hence the equivalence I claimed is a precise statement
They're obviously equivalent.
My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.
I do appreciate the reference to ring localization -- will have to look at that further.
They're obviously equivalent.
My point is that what's easily noticeable in one incarnation of the theory is different than what's easily noticeable in the other incarnation (or if you prefer, expressible), and switching our language for the same abstract structure can highlight different interesting features of it. And further, there's still utility to using topological perspectives and language to discuss the integers or naturals, even if it's equivalent to set theory.
I do appreciate the reference to ring localization -- will have to look at that further.