Automorphisms of the lattice of $\Pi _1^0$ classes; perfect thin classes and anc degrees

Abstract

$\Pi _1^0$ classes are important to the logical analysis of many parts of mathematics. The $\Pi _1^0$ 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 relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of $\Pi _1^0$ classes) forms an orbit in the lattice of $\Pi _1^0$ classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of $\Pi _1^0$ classes. We remark that the automorphism result is proven via a $\Delta _3^0$ automorphism, and demonstrate that this complexity is necessary.