D.R.E. Degrees and the Nondiamond Theorem

If we study the boolean algebra generated by the recursively enumerable sets we are naturally led to the difference hierarchy of Ershov [6,7,8]. If a set A can be represented as (A1 — A2) U ... U (An_x — An) for r.e. sets An c An_x £ ... £ Ax we say that A is «-r.e. In particular A is r.e. if A is 1-r.e. A set A is called d.r.e. if it is 2-r.e. (so that A = B— C with B and C r.e.). The reader should note that a set A is n-r.e. if and only if there is a recursive function / such that for all x, \ims f[x,s) — A(x), J{x,0) = 0 and \{s:J{x,s + 1) =Ax,s)}\ ^ n. In the obvious way, we call a degree a A?-r.e. (d.r.e.) if it contains an n-r.e. (respectively, d.r.e.) set. It is natural to ask the extent to which the w-r.e.—and in particular the d.r.e.—degrees (Dn) resemble the r.e. degrees (R). Early results (such as the fact that the «-r.e. degrees have no minimal elements, and—as Jockusch observed—the d.r.e. degrees are not complemented) stressed the differences between the d.r.e. degrees and the A" degrees and the similarities between the d.r.e. degrees and the r.e. degrees. In particular, it was open whether the d.r.e. degrees and the r.e. degrees were elementarily equivalent (see, for example, [5]). Answering a question of Sacks (see [13]), both Harrington (unpublished) and Lachlan and Soare (unpublished) show that D2 and R are not elementarily equivalent by showing that 0' is a minimal cover in D2. Cooper, Lempp and Watson [3] verified a conjecture of Arslanov [1] to produce another elementary difference at the two quantifier level (in the language L( U , n ,0,1)). Specifically, they showed that the following sentence holds in D2: