The upward closure of a perfect thin class

Abstract

There is a perfect thin ${\Pi }_1^0$ class whose upward closure in the Turing degrees has full measure (and indeed contains every 2-random degree). Thus, in the Muchnik lattice of ${\Pi }_1^0$ classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [S. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic 11 (2005) 1–27].