For the definition of a semicomputable real, see An Introduction to Kolmogorov Complexity and its Applications by Li and Vitanyi (1997). In fact, it is not true that every Cauchy sequence for completion of rationals is semicomputable, we can not complete rationals by semicomputable Cauchy sequences.
My question:
how could we, in a Constructivist way based on Cauchy sequences, complete the rational number system? Specifically, how to overcome the non-computable problem of some (in fact, uncountably many) Cauchy sequences by which to complete the rational system?
Since we complete the rational number system in several ways like Cauchy seqences, Dedekind cuts or even by continued fractions, what is the common ideas (of approach, etc) among them?
