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APAL
2007

Randomness and the linear degrees of computability

9 years 1 months ago
Randomness and the linear degrees of computability
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 such a characterization. Such hopes, however, are in vain since no real is of maximal -degree.
Andrew E. M. Lewis, George Barmpalias
Added 08 Dec 2010
Updated 08 Dec 2010
Type Journal
Year 2007
Where APAL
Authors Andrew E. M. Lewis, George Barmpalias
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