Randomness and the linear degrees of computability

@article{Lewis2007RandomnessAT,
title={Randomness and the linear degrees of computability},
author={Andrew E. M. Lewis and George Barmpalias},
journal={Ann. Pure Appl. Logic},
year={2007},
volume={145},
pages={252-257}
}

We show that there exists a real α such that, for all reals β, if α is linear reducible to β (α ≤` β, previously denoted α ≤sw β) then β ≤T α. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no `-complete ∆2 real. Upon realizing that quasi-maximality does not characterize the random reals—there exist reals which are not random but which are of quasi-maximal `-degree—it is then natural to ask whether maximality could provide… CONTINUE READING