We study the lattice of local operators in Hylandʼs Effective Topos. We show that this lattice is a free completion under internal sups indexed by the natural numbers object, generated by what we call basic local operators.We produce many new local operators and we employ a new concept, sight, in order to analyze these.We show that a local operator identified by A.M. Pitts in his thesis, gives a subtopos with classical arithmetic.

Simpson introduced the lattice of Π0 1 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 Π0 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 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. (shrink)

Let $\mathcal{P}_w$ be the lattice of Muchnik degrees of nonempty $\Pi^0_1$ subsets of $2^\omega$. The lattice $\mathcal{P}$ has been studied extensively in previous publications. In this note we prove that the lattice $\mathcal{P}$ is not Brouwerian.

We characterize the join-irreducible Medvedev degrees as the degrees of complements of Turing ideals, thereby solving a problem posed by Sorbi. We use this characterization to prove that there are Medvedev degrees above the second-least degree that do not bound any join-irreducible degrees above this second-least degree. This solves a problem posed by Sorbi and Terwijn. Finally, we prove that the filter generated by the degrees of closed sets is not prime. This solves a problem posed by Bianchini and Sorbi.