I have read that Gödel's incompleteness theorem does not apply to real closed field theory. But the incompleteness theorem applies only to formal systems, that is systems whose alphabet of symbols and whose set of axioms are all fully defined, and so is not applicable to an abstract theory that does not have any representation as a formal system. True arithmetic is an example of a theory that does not have any representation as a formal system. So my questions is:
Is there any example of a formal system that is a representation of a real number closed field theory and which is complete? Integers are not defined in real closed field theory, which is given as the reason why the incompleteness theorem does not apply. But if there could be a fully defined formal system that would be a representation of real number closed field theory, among other things, the domain of its variables for real numbers would have to be fully defined. It is difficult to see how that domain could be defined without first defining integers and rationals so that irrationals could be defined in terms of such integers and rationals.
The theory of real closed fields is complete - a result of Tarski. This is a first-order theory with the axioms of an ordered field and an axiom scheme saying that a polynomial of odd degree that changes sign has a root - so it is a formal system. The real line is a model of this system.
This does not mean, however, that the only model of real closed fields is the real line. There are still many other models, including subfields of the reals as well as extensions of the reals. All of these models satisfy exactly the same set of sentences.
The completeness result does not fundamentally depend on having any model - it is just a result about the theory (the set of sentences). A formal system does not come equipped with a model.