Journal of Symbolic Logic

Relative randomness and real closed fields

Alexander Raichev

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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).

Article information

Source
J. Symbolic Logic, Volume 70, Issue 1 (2005), 319-330.

Dates
First available in Project Euclid: 1 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1107298522

Digital Object Identifier
doi:10.2178/jsl/1107298522

Mathematical Reviews number (MathSciNet)
MR2119135

Zentralblatt MATH identifier
1090.03014

Subjects
Primary: 03D80: Applications of computability and recursion theory 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]

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

Citation

Raichev, Alexander. Relative randomness and real closed fields. J. Symbolic Logic 70 (2005), no. 1, 319--330. doi:10.2178/jsl/1107298522. https://projecteuclid.org/euclid.jsl/1107298522


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