Coherent Adequate Forcing and Preserving CH

TL;DR

This paper introduces a new class of forcing posets called coherent adequate type forcings, which preserve the Continuum Hypothesis (CH) while allowing specific set-theoretic constructions, such as adding club subsets of 902.

Contribution

It develops a general framework for forcing with coherent adequate sets and proves that such forcings preserve CH, including a novel forcing for adding a club subset of 902 with finite conditions.

Findings

Coherent adequate type forcings preserve CH.

Existence of a forcing for adding a club subset of 902 with finite conditions.

Framework applicable to various set-theoretic constructions.

Abstract

We develop a general framework for forcing with coherent adequate sets on $H(\lambda)$ as side conditions, where $\lambda \ge \omega_2$ is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of $\omega_2$ with finite conditions while preserving CH, solving a problem of Friedman.