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08 February 2025
 
  » arxiv » 1605.0296

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Chang's Conjecture and semiproperness of nonreasonable posets
Sean D. Cox ;
Date 1 May 2016
AbstractLet $mathbb{Q}$ denote the poset which adds a Cohen real then shoots a club through the complement of $ig( [omega_2]^omega ig)^V$ 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 $omega_1$.
Source arXiv, 1605.0296
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