Thin classes of separating sets
نویسنده
چکیده
There are various definitions for a Martin–Pour-El theory in the literature. We isolate two such definitions: weak Martin–Pour-El theories (which correspond to perfect thin Π1 classes) and strong Martin–Pour-El theories (which correspond to thin classes of separating sets). By concentrating on constructions of appropriate Π1 classes, rather than on direct constructions of the theories, we show how the extend versions of well known results about weak Martin–Pour-El theories to strong Martin–Pour-El theories. In addition, we consider how the degrees of complete consistent extensions of a strong Martin–Pour-El theory relate to the degree of the theory. Finally, we give a new restriction on the degrees of sets occurring in a thin Π1 class and prove that this restriction is strictly stronger than previously known results.
منابع مشابه
Immunity and Non-Cupping for Closed Sets
We extend the notion of immunity to closed sets and to Π1 classes in particular in two ways: immunity meaning the corresponding tree has no infinite computable subset, and tree-immunity meaning it has no infinite computable subtree. We separate these notions from each other and that of being special, and show separating classes for computably inseparable c.e. sets are immune and perfect thin cl...
متن کاملSeparating partial normality classes with weighted composition operators
In this article, we discuss measure theoretic characterizations for weighted composition operators in some operator classes on $L^{2}(Sigma)$ such as, $n$-power normal, $n$-power quasi-normal, $k$-quasi-paranormal and quasi-class$A$. Then we show that weighted composition operators can separate these classes.
متن کاملAutomorphisms of the Lattice of Π1 Classes; Perfect Thin Classes and Anc Degrees
Π1 classes are important to the logical analysis of many parts of mathematics. The Π1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relatin...
متن کامل