The Medvedev lattice of computably closed sets

Simpson introduced the lattice of Π01 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π01 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.

[1]  Douglas Cenzer,et al.  Density of the Medvedev lattice of Π01 classes , 2003, Arch. Math. Log..

[2]  Stephen Binns.  A splitting theorem for the Medvedev and Muchnik lattices , 2003, Math. Log. Q..

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

[4]  Andrea Sorbi,et al.  Some Quotient Lattices of the Medvedev Lattice , 1991, Math. Log. Q..

[5]  Andrea Sorbi.  Some Remarks on the Algebraic Structure of the Medvedev Lattice , 1990, J. Symb. Log..

[6]  P. Odifreddi.  The theory of functions and sets of natural numbers , 1989 .

[7]  P. Odifreddi.  Classical recursion theory , 1989 .

[8]  Stephen G. Simpson,et al.  Embeddings into the Medvedev and Muchnik lattices of Π01 classes , 2004, Arch. Math. Log..

[9]  Andrea Sorbi,et al.  Embedding Brouwer Algebras in the Medvedev Lattice , 1991, Notre Dame J. Formal Log..

[10]  H. Priestley,et al.  Distributive Lattices , 2004 .

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

[12]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[13]  Carl G. Jockusch,et al.  Countable retracing functions and $\Pi_2{}^0$ predicates. , 1969 .

[14]  A. Kucera.  Measure, Π10-classes and complete extensions of PA , 1985 .

[15]  Birgit Truust Poulsen.  The Medvedev lattice of degrees of difficulty , 1970 .

[16]  Sebastiaan Terwijn.  Constructive Logic and the Medvedev Lattice , 2006, Notre Dame J. Formal Log..

[17]  V. Jankov.  The Calculus of the Weak "law of Excluded Middle" , 1968 .

[18]  Sebastiaan A. Terwijn,et al.  Complexity and Randomness , 2003 .

[19]  E. Z. Skvortsova.  A faithful interpretation of the intuitionistic propositional calculus by means of an initial segment of the Medvedev lattice , 1988 .

[20]  Robert I. Soare,et al.  Degrees of members of Π10 classes , 1972 .