Automorphisms 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 relating automorphisms, invariance and thin classes. Our main results are an analog of the 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 the collection of perfect thin classes (a notion which is definable in the lattice of Π1 classes) form an orbit in the lattice of Π1 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 Π1 classes. We remark that the automorphism result is proven via a ∆3 automorphism, and demonstrate that this complexity is necessary.