Iterative forcing and hyperimmunity in reverse mathematics

TL;DR

This paper introduces a modularized forcing technique in reverse mathematics that preserves hyperimmunity, enabling the separation of theorems over omega-models similar to previous methods.

Contribution

It presents a new modular framework for forcing in reverse mathematics that preserves hyperimmunity, matching prior separation results.

Findings

The framework effectively separates theorems over omega-models.

It generalizes previous preservation of definitions techniques.

The method is adaptable for various separation problems.

Abstract

The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions.