THE UPWARD CLOSURE OF A PERFECT THIN CLASS

Abstract. There is a perfect thin Π0 1 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2-random degree.) Thus, in the Muchnik lattice of Π0 1 classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [15]. 1