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The importance of Π1 0 classes in effective randomness

Published online by Cambridge University Press:  12 March 2014


George Barmpalias
Affiliation:
School of Mathematics, Statistics, and Computer Science, Victoria University, Po Box 600 Wellington, New Zealand, E-mail: George.Barmpalias@mcs.vuw.ac.nz, URL: http.//www.mcs.vuw.ac.nz/~georgeb/
Andrew E.M. Lewis
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK, E-mail: andy@aemlewis.co.uk, URL: http://www.aemlewis.co.uk
Keng Meng Ng
Affiliation:
School of Mathematics, Statistics, and Computer Science, Victoria University, Po Box 600 Wellington, New Zealand, E-mail: keng.meng.ng@mcs.vuw.ac.nz, URL: http://www.mcs.vuw.ac.nz/~selwyn/

Abstract

We prove a number of results in effective randomness, using methods in which Π1 0 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.


Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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