Effectively closed mass problems and intuitionism

摘要

Let Ps denote the lattice of all strong degrees (or Medvedev degrees) of nonempty Π10 (or effectively closed) subsets of the Cantor space 2 ~ω, and let Pw denote the lattice of all weak degrees (or Muchnik degrees) of nonempty Π10 subsets of 2 ~ω. We show that Ps is not a Brouwer algebra and Pw is not a Heyting algebra. These give solutions to problems presented by Simpson in 2008 and by Terwijn in 2006.