We show that for any real number, the class of real numbers less random
than it, in the sense of rK-reducibility,
forms a countable real closed subfield of the real ordered field.
This generalizes the well-known fact that the computable reals form a real
closed field.
¶
With the same technique we show that
the class of differences of computably enumerable reals (d.c.e. reals) and
the class of computably approximable reals (c.a. reals)
form real closed fields.
The d.c.e. result was also proved nearly simultaneously and
independently by Ng (Keng Meng Ng, Master's Thesis,
National University of Singapore, in preparation).
¶
Lastly, we show that the class of d.c.e. reals is properly contained
in the class or reals less random than ω (the halting probability),
which in turn is properly contained in the class of c.a. reals, and that
neither the first nor last class is a randomness class
(as captured by rK-reducibility).