The upward closure of a perfect thin class

Abstract There is a perfect thin Π 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 Π 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].

[1]  Jeff B. Paris.  Measure and minimal degrees , 1977 .

[2]  Peter A. Cholak,et al.  Automorphisms of the lattice of Π₁⁰ classes; perfect thin classes and anc degrees , 2001 .

[3]  Donald A. Martin,et al.  Axiomatizable Theories with Few Axiomatizable Extensions , 1970, J. Symb. Log..

[4]  S. G. Simpson.  An extension of the recursively enumerable Turing degrees , 2007 .

[5]  A. Nies.  Computability and randomness , 2009 .

[6]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[7]  Zofia Adamowicz.  On Maximal Theories , 1991, J. Symb. Log..

[8]  Rodney G. Downey,et al.  Abstract dependence, recursion theory, and the lattice of recursively enumerable filters , 1983, Bulletin of the Australian Mathematical Society.

[9]  Ming Li,et al.  Kolmogorov Complexity and its Applications , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[10]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[11]  Douglas Cenzer,et al.  Countable Thin Pi01 Classes , 1993, Ann. Pure Appl. Log..

[12]  Douglas Cenzer.  ∏10 Classes in Computability Theory , 1999, Handbook of Computability Theory.

[13]  André Nies,et al.  Calibrating Randomness , 2006, Bull. Symb. Log..

[14]  Stephen G. Simpson.  Mass problems and randomness , 2005, Bull. Symb. Log..

[15]  R. Soare,et al.  Π⁰₁ classes and degrees of theories , 1972 .