You don’t know enough set theory and model theory. Of course there are different models of the real numbers. There are non isomorphic models of the first 4 axioms of Euclidean geometry even. A given set of axioms can have lots of different models.
The widely accepted first order axioms of the natural numbers have nonstandard models. Of course each such model has an isomorphic initial segment. There are two things going on. There is the set of axioms and there are models for those axioms. I think in your usage: “define real number” corresponds to “give the axioms for the real numbers”. I’m not sure though.
Non standard models of arithmetic really takes me back. Anyway, axiom/models is the common terminology though I recall a rather long rant that postulate is better once you start talking about multiple incompatible systems.
Please consider that the poster telling you that you do not know enough to comment intelligently about this topic may be correct.
> Anyway, axiom/models is the common terminology though I recall a rather long rant that postulate is better once you start talking about multiple incompatible systems.
Model has a very specific definition in this area of mathematics. This sentence is nonsense.
I’ve seen some truly bizarre definitions of the reals which are internally consistent and include things like the lowest possible number.