I apologize if this question sounds too philosophical. I am reading about Turing's paper on computability and it got me thinking:
Why do we bother defining the mysterious "undefined things" in-between on the real number line? How important are they?
I agree that it is ok to define anything and work with any structures. But undoubtedly, some structures are more interesting than others.
Does distinguishing between computable and uncomputable real numbers generate any important results in mathematics?
(Edit: Thank you for the answers so far, I understand that we do not have to distinguish computability in the set of real numbers. I am curious what happens if we do (in any way). Has anyone investigated?)