Effectively closed mass problems and intuitionism

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Abstract

Let Ps denote the lattice of all strong degrees (or Medvedev degrees) of nonempty Π10 (or effectively closed) subsets of the Cantor space 2ω, and let Pw denote the lattice of all weak degrees (or Muchnik degrees) of nonempty Π10 subsets of 2ω. 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.

MSC

03D30
03G10

Keywords

Mass problems
Effective closed sets
Π10 classes
Intuitionism
Brouwer algebras
Heyting algebras

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