MEASURING CLUB-SEQUENCES TOGETHER WITH THE CONTINUUM LARGE

Abstract

AbstractMeasuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta < {\omega _1}}}$ with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$, a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$. We answer a question of Justin Moore by building a forcing extension satisfying measuring together with ${2^{{\aleph _0}}} > {\aleph _2}$. The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.