Minimal covers and hyperdegrees

Abstract

Every hyperdegree at or above that of Kleene’s O O is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty Σ 1 1 \Sigma _1^1 class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If V = L V = L or a generic extension of L L , then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If O # {O^\# } exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.