>>12132885Just in case you mean this at face value, different models of the reals don't have to be the same sets, in the sense of equality provable via set extensionality. Say rationals are an equivalence class of pairs of naturals.
If Dedekind cuts are subsets of while Cauchy reals are equivalence classes of rationl valued sequences , how would they be the same.
They are indeed significantly different: From a constructive angle, Cauchy reals without explicit modulus of convergence are not provably Cauchy complete set (all limits being in it). And while Dedekind reals are Cauchy complete, even axioms like Strong Collection, Replacement and Exponentiation don't suffice to show that they even form a set.
Dedekind reals are just the decisions on that respect its order (as in, the Dedekind reals sort of are reified predicates themselves). So they are complicated to comprehend (I mean this formally, not colloquially).
I've never cared for that bijection, but I think given a Cauchy real, you should easily be able to come up with a predicate in terms of it. To go from a Dedekind real to a Cauchy real, you define a sequence that zooms in on the cut point, which requires to make a left-right judgement.
