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Chang's Conjecture and semiproperness of nonreasonable posets | Sean D. Cox
; | Date: |
1 May 2016 | Abstract: | Let $mathbb{Q}$ 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 | Services: | Forum | Review | PDF | Favorites |
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