User:IssaRice/Construction of the real numbers

something that i have seen clarified exactly once to me (and never in a textbook!) 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.

"The main way to prove that something is consistent is to produce a model of it." https://math.stackexchange.com/a/120024/35525

"It takes a good deal of mathematical sophistication to even appreciate why someone would want to prove that the theory of the real numbers is consistent, and even more sophistication to appreciate why we can do so by making a “model” of the theory." [1]

https://math.stackexchange.com/questions/2044775/prove-that-a-theory-gamma-is-consistent-if-and-only-if-there-is-a-structure This is actually just a restatement of the soundness and completeness theorems: a set of sentences is satisfiable (has a model) iff it is consistent.

also why doesn't constructing a term model prove existence of R?