User:IssaRice/Construction of the real numbers

something that i have seen clarified exactly once is why constructing the real numbers (for instance via dedekind cuts or equivalence classes of cauchy sequences of rationals) constitutes a proof that $\mathbf{R}$ (or a complete ordered fields or whatever you want to call it) exists. the reason is this (?): to show that something exists, one way to do this is to show that a model (in the model theory sense) of it exists.