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

Abstract

Abstract. Π 0 1 classes are important to the logical analysis of many parts of mathematics. The Π 0 1 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 Π 0 1 classes) forms an orbit in the lattice of Π 0 1 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 Π 0 1 classes. We remark that the automorphism result is proven via a ∆ 0 3 automorphism, and demonstrate that this complexity is necessary.

Degree invariance in the Π10classes

“…For all n > 2, the high n and non-low" c.e. degrees are invariant overall-This greatly expands the known invariant classes of S'n, as the only other examples are {0} and the collection of array noncomputable (anc) degrees, the latter shown by Cholak, Coles, Downey, and Herrmann [5] via perfect thin ITj classes. Note that though the present result is for fn, the techniques are squarely in the realm ofg\…”

“…ideals. As shown in Cholak, Coles, Downey, and Herrmann [5] and our [17], these three structures are computably isomorphic in a natural way, though when one moves from I"^ classes to ideals or vice-versa, order is reversed.…”

Calibrating Randomness

“…degrees and are an invariant orbit for the high c.e. degrees (see Cholak, Coles, Downey, and Herrmann [25]).…”

“…25 (Downey, Griffiths, and Reid [33]). A set A is Kurtz random iff K M (A n) n − O(1) for each computably layered machine M .…”

Mass Problems and Randomness

“…There is a large literature on thin perfect Π 0 1 subsets of 2 going back to Martin/Pour-El [38]. See Downey/Jockusch/Stob [15,16] and Cholak et al [11]. Typically, thin perfect Π 0 1 subsets of 2 are constructed by means of priority arguments.…”