Algorithmically approximable numbers - theory?

Question

You could consider a real number to be "algorithmically approximable" if there is an algorithm that produces a sequence of rational numbers that converges against this number. This would obviously include algebraic numbers, and numbers like $\pi$ and $e$. The set of these numbers would be countable.

Is there a developed theory of this kind of numbers (or similarly defined numbers)? Some "real" analysis, maybe?

Answer 1

The term for these number is the computable numbers. In general the resulting fields have the usual names except with the word "computable" in front of them. Of particular note is the development of computable analysis. Computable group theory and computable combinatorics (which is mostly interested in graph theory and computable cardinals) also an ongoing field of research.