Least upper bounds for minimal pairs of α-R.E. α-degrees

Abstract

The application of priority arguments to study the structure of the upper semilattice of α-r.e. α-degrees for all admissible ordinals α was first done successfully by Sacks and Simpson [5] who proved that there exist incomparable α-r.e. α-degrees. Lerman and Sacks [3] studied the existence of minimal pairs of α-r.e. α-degrees, and proved their existence for all admissible ordinals α which are not refractory. We continue the study of the α-r.e. α-degrees, and prove that no minimal pair of α-r.e. α-degrees can have as least upper bound the complete α-r.e. α-degree.The above-mentioned theorem was first proven for α = ω by Lachlan [1]. Our proof for α = ω differs from Lachlan's in that we eliminate the use of the recursion theorem. The proofs are similar, however, and a knowledge of Lachlan's proof will be of considerable aid in reading this paper.We assume that the reader is familiar with the basic notions or α-recursion theory, which can be found in [2] or [5].Throughout the paper a will be an arbitrary admissible ordinal. We identify a set A ⊆ α with its characteristic function, A(x) = 1 if x ∈ A, and A(x) = 0 if x ∉ A.If A ⊆ α and B ⊆ α, then A ⊕ B will denote the set defined byA ⊕ B(x) = A(y) if x = λ + 2z, λ is a limit ordinal, z < ω and y = λ + z,= B(y) if x = λ + 2z + 1, λ is a limit ordinal, z < ω, and y = λ + z.