Automorphisms of the Lattice of Π0 1Classes; Perfect Thin Classes and ANC Degrees

• Published in: Transactions of the American Mathematical Society 2001-12, Vol.353 (12), p.4899-4924
• Main Authors: Cholak, Peter, Coles, Richard, Downey, Rod, Herrmann, Eberhard
• Format: Article
• Language: English
• Subjects: Automorphisms; Boolean algebras; Correspondence theory; Logical theorems; Mathematical lattices; Mathematical theorems; Mathematics; Recursion theory; Recursively enumerable sets; Topological theorems
• ISSN: 0002-9947

Π0 1classes are important to the logical analysis of many parts of mathematics. The Π0 1classes 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 Π0 1classes) forms an orbit in the lattice of Π0 1classes; 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 Π0 1classes. We remark that the automorphism result is proven via a Δ0 3automorphism, and demonstrate that this complexity is necessary.