## Journal of Symbolic Logic

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

### Relative randomness and real closed fields

#### 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