In this paper the methods of
recursive function theory are applied to certain classes of real numbers as determined
by their Dedekind cuts or by their binary expansions. Instead of considering recursive
real numbers as in constructive analysis, we examine real numbers whose
lower Dedekind cut is a recursively enumerable (r.e.) set of rationals, since
the r.e. sets constitute the most elementary nontrivial class which includes
nonrecursive sets. The principal result is that the sets A of natural numbers which
“determine” such real numbers α (in the sense that the characteristic function of A
corresponds to the binary expansion of α) may be very far from being r.e., and
may even be cohesive. This contrasts to the case of recursive real numbers,
where A is recursive if and only if the corresponding lower Dedekind cut is
recursive.
