Noninitial segments of the α-degrees

Abstract

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.