Chang's Conjecture and semiproperness of nonreasonable posets

TL;DR

This paper explores the relationship between Chang's Conjecture and the semiproperness of certain nonreasonable posets, establishing equivalences and implications that connect large cardinal assumptions with forcing properties.

Contribution

It proves that a strong form of Chang's Conjecture implies the semiproperness of a specific poset and shows that semiproperness of nonreasonable posets implies the conjecture, linking large cardinals to forcing.

Findings

Strong Chang's Conjecture implies semiproperness of a Cohen real adding poset.

Semiproperness of nonreasonable posets implies the Strong Chang's Conjecture.

Semiproperness of certain posets has large cardinal strength, answering a question by Friedman-Krueger.

Abstract

Let $\mathbb{Q}$ denote the poset which adds a Cohen real then shoots a club through the complement of $\big( [\omega_2]^\omega \big)^V$ 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 $\omega_1$.