Generics for Mathias forcing over general Turing ideals

TL;DR

This paper extends the study of Mathias forcing generics from the case of computable sets to general Turing ideals, revealing how the properties of these ideals influence the computational complexity and information encoding in the generics.

Contribution

It provides a classification of generics over Turing ideals based on their computability-theoretic properties and introduces new coding techniques and results about introreducibility.

Findings

Generics over non-trivial Turing ideals can encode complex information.

A classification of generics based on ideal properties is established.

New results on introreducibility and the existence of certain degrees are presented.

Abstract

In Mathias forcing, conditions are pairs $(D,S)$ of sets of natural numbers, in which $D$ is finite, $S$ is infinite, and $\max D < \min S$. The Turing degrees and computational characteristics of generics for this forcing in the special (but important) case where the infinite sets $S$ are computable were thoroughly explored by Cholak, Dzhafarov, Hirst, and Slaman~\cite{CDHS-2014}. In this paper, we undertake a similar investigation for the case where the sets $S$ are members of general countable Turing ideals, and give conditions under which generics for Mathias forcing over one ideal compute generics for Mathias forcing over another. It turns out that if $\mathcal{I}$ does not contain only the computable sets, then non-trivial information can be encoded into the generics for Mathias forcing over $\mathcal{I}$. We give a classification of this information in terms of computability-theoretic properties of the ideal, using coding techniques that also yield new results about introreducibility. In particular, we extend a result of Slaman and Groszek and show that there is an infinite $\Delta^0_3$ set with no introreducible subset of the same degree.