The Medvedev lattice of computably closed sets

Simpson introduced the lattice P of Π 1 classes under Medvedev reducibility. Questions regarding completeness in P are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π 1 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 P does not allow an implication operator (i.e. that P is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.