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Relative randomness and real closed fields

Published online by Cambridge University Press:  12 March 2014

Alexander Raichev
Alexander Raichev*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln DR, Madison, Wisconsin 53706, USA, E-mail: raichev@math.wisc.edu
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Abstract

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).

Keywords

relative randomnessreal closed fieldrK-reducibilityd.c.e. realc.a. real
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 319 - 330
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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