Chang's Conjecture and semiproperness of nonreasonable posets

Abstract

Let $\mathbb{Q}$ denote the poset which adds a Cohen real then shoots a club through the complement of $([{\omega }_2{]}^{\omega }{)}^V$ with countable conditions. We prove that the version of Strong Chang's Conjecture from \cite{MR2965421} implies semiproperness of $\mathbb{Q}$, and that semiproperness of $\mathbb{Q}$---in fact semiproperness of any poset which is sufficiently \emph{nonreasonable} in the sense of Foreman-Magidor~\cite{MR1359154}---implies the version of Strong Chang's Conjecture from \cite{MR2723878} and \cite{MR1261218}. In particular, semiproperness of $\mathbb{Q}$ has large cardinal strength, which answers a question of Friedman-Krueger~\cite{MR2276627}. One corollary of our work is that the version of Strong Chang's Conjecture from \cite{MR2965421} does not imply the existence of a precipitous ideal on ${\omega }_1$.

arXiv commentAdded a new section (Section 3) about relationship between MM, dagger principle, and the principle ${\text{SCC}}_{\text{gap}}^{\text{cof}}$