$\Delta^{0}_{2}$ Degrees and transfer theorems
Rodney G. Downey · Illinois Journal of Mathematics · 1987
The main goal of this paper is to demonstrate how weak troth table/ Turing degree "transfer" techniques may be used to obtain information about the A2 (Turing) degrees.Such techniques have previously been applied by Ladner-Sasso [13], Stob [18] and others to obtain information about R, the r.e.T-degrees.The best known example of this phenomenon is Ladner and Sasso's [13] use of contiguous degrees to show that every nonzero r.e.degree has a predecessor with the anticupping property.Let D denote the degrees, W the r.e.weak truth table (W-)degrees and D w the weak truth table degrees.Modifying the Ladner-Sasso analysis to A2 degrees, we shall give a new and relatively easy proof of a result independently proved by Cooper [5] and Slaman and Steel[16] about structural interactions of R and D: THEOREM A. :la, b R(0 :lb W(O < b < a and Vc Dw(a < c tj b a < c))).Theorem B also implies that the elementary theory of (for example) the weak truth table degrees below 0, and the A2 degrees are different (since Posner and Robinson [15] have shown that the nonzero T-degrees below 0' all cup to 0').