Is the axiomatic approach to defining $\mathbb{R}$ rigorous? Some authors define $\mathbb{R}$ axiomatically. That is, they assume there exists a set $\mathbb{R}$ with binary operations $\cdot$ and $+$ such that the field axioms are satisfied, an order $\leq$ satisfying various axioms and the least upper bound property.
My question is, simply, is such an approach rigorous? The existence of this set is taken for granted!  
I have read through some of the answers to this question and the answers to this suggest to me that the only rigorous (from the set theoretic point of view) way to develop $\mathbb{R}$ is constructively (that is, defining $\mathbb{N}$, then $\mathbb{Z}$, then $\mathbb{Q}$ and then using dedekind cuts/cauchy sequences to define $\mathbb{R}$). 
 A: Well, the axiomatic approach "works" but only a posteriori, not a priori.
The point is that one can prove the following two claims:
1) There exists an ordered field which satisfies the Least Upper Bound Property (such a field is called Dedekind-complete)
2) Any two such fields are isomorphic (as ordered fields)
Note that proving claim 1 amounts to come up with some construction based on "earlier constructed" objects based on set theory (the natural numbers, then the rationals, etc..)
These two assertions justify the use of the axiomatic approach; In particular, since any two models will be isomorphic, there is "nothing more" beyond the axioms, from the prespective of the theory of ordered fields.
Putting it differently, any claim which can be phrased using $\cdot,+,-0,1,<$ which holds in one model will be true in every model. 
Of course, two different models do not need to be identical as sets (i.e the details of the construction can be different), but the differences won't have any meaning from the perspective that interest us.
This  last point is quite important, and in fact occurs many times in mathematics; For example, in his book "Naive Set Theory" , Paul Halmos mentions that the exact constructionof the ordered pair $(x,y)$ doe snot matter at all, what matters are the properties we want it to satisy.
In some sense, we do not care how exactly it is constructed, since the differences won't affect any aspects which interest us.
Other examples of this phenomenon arise in algebra, for example in the concept of tensor products. One usually states an "axiomatic definition" in the form a universal property, and then proves that:
1) Any two objects satisfying this property are isomorphic
2) There exists an object with the required property
After the two claims were proven, one can safely forget the details of the actual construction (i.e the proof of the second claim)
