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March 2005 Relative randomness and real closed fields
Alexander Raichev
J. Symbolic Logic 70(1): 319-330 (March 2005). DOI: 10.2178/jsl/1107298522

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).

Citation

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Alexander Raichev. "Relative randomness and real closed fields." J. Symbolic Logic 70 (1) 319 - 330, March 2005. https://doi.org/10.2178/jsl/1107298522

Information

Published: March 2005
First available in Project Euclid: 1 February 2005

zbMATH: 1090.03014
MathSciNet: MR2119135
Digital Object Identifier: 10.2178/jsl/1107298522

Subjects:
Primary: 03D80 , 68Q30

Keywords: c.a. real , d.c.e. real , real closed field , relative randomness , rK-reducibility

Rights: Copyright © 2005 Association for Symbolic Logic

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Vol.70 • No. 1 • March 2005
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