Relative Randomness and Real Closed Fields

Journal of Symbolic Logic 70 (1):319 - 330 (2005)
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 No keywords specified (fix it)
Categories (categorize this paper)
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 10,316
External links
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.
William C. Calhoun (2006). Degrees of Monotone Complexity. Journal of Symbolic Logic 71 (4):1327 - 1341.
Johanna N. Y. Franklin (2010). Subclasses of the Weakly Random Reals. Notre Dame Journal of Formal Logic 51 (4):417-426.
George Barmpalias (2010). Relative Randomness and Cardinality. Notre Dame Journal of Formal Logic 51 (2):195-205.
Arnold W. Miller (1983). Mapping a Set of Reals Onto the Reals. Journal of Symbolic Logic 48 (3):575-584.
Analytics

Monthly downloads

Sorry, there are not enough data points to plot this chart.

Added to index

2010-08-24

Total downloads

1 ( #398,768 of 1,096,453 )

Recent downloads (6 months)

0

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Start a new thread
Order:
There  are no threads in this forum
Nothing in this forum yet.