David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 70 (1):319 - 330 (2005)
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||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Rodney G. Downey & Evan J. Griffiths (2004). Schnorr Randomness. Journal of Symbolic Logic 69 (2):533 - 554.
George Barmpalias (2003). The Approximation Structure of a Computably Approximable Real. Journal of Symbolic Logic 68 (3):885-922.
William C. Calhoun (2006). Degrees of Monotone Complexity. Journal of Symbolic Logic 71 (4):1327 - 1341.
Tamara Servi (2008). Noetherian Varieties in Definably Complete Structures. Logic and Analysis 1 (3-4):187-204.
Johanna N. Y. Franklin (2010). Subclasses of the Weakly Random Reals. Notre Dame Journal of Formal Logic 51 (4):417-426.
Sebastiaan A. Terwijn & Domenico Zambella (2001). Computational Randomness and Lowness. Journal of Symbolic Logic 66 (3):1199-1205.
Kenshi Miyabe (2010). An Extension of van Lambalgen's Theorem to Infinitely Many Relative 1-Random Reals. Notre Dame Journal of Formal Logic 51 (3):337-349.
George Barmpalias (2010). Relative Randomness and Cardinality. Notre Dame Journal of Formal Logic 51 (2):195-205.
Joseph S. Miller (2004). Every 2-Random Real is Kolmogorov Random. Journal of Symbolic Logic 69 (3):907-913.
Ya'acov Peterzil (1993). Reducts of Some Structures Over the Reals. Journal of Symbolic Logic 58 (3):955-966.
Arnold W. Miller (1983). Mapping a Set of Reals Onto the Reals. Journal of Symbolic Logic 48 (3):575-584.
António M. Fernandes & Fernando Ferreira (2002). Groundwork for Weak Analysis. Journal of Symbolic Logic 67 (2):557-578.
Jan E. Holly (1995). Canonical Forms for Definable Subsets of Algebraically Closed and Real Closed Valued Fields. Journal of Symbolic Logic 60 (3):843-860.
Sorry, there are not enough data points to plot this chart.
Added to index2010-08-24
Total downloads1 ( #398,768 of 1,096,453 )
Recent downloads (6 months)0
How can I increase my downloads?