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 ≤ ℓ β , previously denoted as 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 ℓ -complete Δ 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 ℓ -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 ℓ -degree.