Few new reals

TL;DR

This paper presents a novel forcing method that adds only  reals to build models of CH with strong negations of Club Guessing, answering a question of Moore.

Contribution

It introduces a new forcing construction that adds few reals while preserving CH and achieving strong negation of Club Guessing, which was previously unresolved.

Findings

Constructs models of CH with  new reals.

Achieves Measuring together with CH.

Provides a finite-support weak forcing iteration with side conditions.

Abstract

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only $\aleph_1$-many of them. Using this approach, we build a model in which a very strong form of the negation of Club Guessing at $\omega_1$ known as Measuring holds together with CH, thereby answering a well-known question of Moore. This construction can be described as a finite-support weak forcing iteration with side conditions consisting of suitable graphs of sets of models with markers. The CH-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by the edges in the graph.