The Medvedev lattice of computably closed sets

Simpson introduced the lattice P of 0 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 0 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.

• Type: Journal
• Year: 2006
• Where: AML
• Authors: Sebastiaan Terwijn
• Keywords: AML 2006, Medvedev Degrees, Medvedev Reducibility, Questions