Abstract
Simpson introduced the lattice Open image in new window of Π01 classes under Medvedev reducibility. Questions regarding completeness in Open image in new window 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 Open image in new window does not allow an implication operator (i.e. that Open image in new window is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.
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