Undecidability and initial segments of the wtt-degrees ≤0′

Abstract

We have now shown that every non-zero degree below deg _wtt (G) coincides with deg(G _ρ) for some ρ ε Π_n. Since all of the positive requirements have been satisfied, we see that the lattice Π_n has been embedded as an initial segment of the Δ ⁰₂ wtt-degrees above a minimal degree. In fact the embedding is in the wtt-degrees ≤_wtt 0′. To see that this is true, we need only check that there is a recursive function b(x) such that |{s:G _s+1(x)≠G _s (x)}|≤b(x). But this is the case because of the nesting of the trees. Only action taken for the sake of satisfying P ₀ can change G _s (0). This can happen at most twice. The only possible actions which can change G(x) are taken for the sake of satisfying some P _e where e ≤ x, or improving the e-state of some T _e,s (σ), where e+|σ|≤x. A simple induction shows that this is bounded by (2+n+|E _n |+2)^{2 x+1 −1}.

The authors' research was supported by NSF grants DMS-8705818 and DMS-8601048, respectively and MSRI. In addition, Shore was supported by a grant from the U.S.-Israel Bi-national Science Foundation.