Is there a general theorem that would state, with proper sanitization, that most known results regarding the use of real numbers can actually be used when considering only computable reals? Or is there a proper characterisation of results that remain valid when considering only the computable reals? A side question is whether results concerning computable reals can be proved without having to consider all real, or anything that is not computable. I am thinking specifically of calculus and mathematical analysis, but my question is in no way limited to that.
Actually, I suppose there is a hierarchy of computable reals corresponding to the Turing hierarchy (Is that correct?). Then, more abstractedly, is there an abstract theory of real (I am not sure what the terminology should be), for which a number of results could be proved, that would apply to the traditional real numbers, but also to computable reals, and to any level of the Turing hierarchy of computable reals, if it exists.
Then my question could possibly be stated as: Is there a characterization of results that will apply in the abstract theory of reals when they have been proved for traditionnal reals. And, could these results be proved directly in the abstract theory, without considering traditional reals.
I am also interested in understanding how and when these theories of reals diverge.
P.S. I do not know where to fit this in my question. I realised that a good deal of the mathematics on the reals have been generalized with topology. So it may be that the answer to my question, or part of it, can be found there. But there may also be more to it.