Measuring club sequences, together with the Continuum Hypothesis

TL;DR

This paper constructs models of set theory where the measuring property coexists with the Continuum Hypothesis or a larger continuum, using a novel forcing iteration with symmetry constraints.

Contribution

It introduces a new forcing method that combines measuring with CH or larger continuum, applicable over any ZFC model.

Findings

Constructed a model satisfying measuring and CH.

Produced a model with measuring and a continuum larger than the second uncountable.

Demonstrated the forcing iteration with symmetry constraints is versatile.

Abstract

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and with symmetry constraints. Also, we show that a small variation of this construction produces a model of measuring together with the continuum being larger than the second uncountable cardinal.