Cover Image
QR Code

RELATIVE RANDOMNESS AND REAL CLOSED FIELDS.

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...

Full description

Published in:Journal of Symbolic Logic Vol. 70; no. 1; pp. 319 - 331
Main Author: Raichev, Alexander
Format: Article
Published: Cambridge University Press Mar2005
Subjects:
Arithmetic
Probability theory
Mathematical logic
Logic
Online Access:View this record in EBSCOhost
fields @attributes:
  recordID: 1
pdfLink:
plink: https://search.ebscohost.com/login.aspx?direct=true&db=hlh&AN=16301994&site=ehost-live
header:
  @attributes:
    shortDbName: hlh
    uiTerm: 16301994
    longDbName: Humanities International Complete
    uiTag: AN
  controlInfo:
    bkinfo:
    jinfo:
      jid:
        00224812
        3TY
      jtl: Journal of Symbolic Logic
      issn: 00224812
      maglogo: N
    pubinfo:
      dt: Mar2005
      vid: 70
      iid: 1
      pid: 15979
      pub: Cambridge University Press
    artinfo:
      ui:
        16301994
        10.2178/jsl/1107298522
      ppf: 319
      ppct: 12
      formats:
      tig:
        atl: RELATIVE RANDOMNESS AND REAL CLOSED FIELDS.
      aug:
        au: Raichev, Alexander
        affil: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, Wisconsin 53706, USA
      su:
        Arithmetic
        Probability theory
        Mathematical logic
        Logic
      sug:
        subj:
          Arithmetic
          Probability theory
          Mathematical logic
          Logic
      ab: 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).
      pubtype: Academic Journal
      doctype: Article
      src: R
    language: English
    refInfo:
    copyright:
      @attributes:
        flag: Y
      dt:
        @attributes:
          year: 2005
    holdings:
      @attributes:
        islocal: N