Chang’s conjecture and semiproperness of nonreasonable posets

Article

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 Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) implies semiproperness of \(\mathbb {Q}\), and that semiproperness of \(\mathbb {Q}\)—in fact semiproperness of any poset which is sufficiently nonreasonable in the sense of Foreman and Magidor (Ann Pure Appl Log 76(1):47–97, 1995)—implies the version of strong Chang’s conjecture from Woodin (The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, 2nd ed., vol 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010) and Todorčević (Conjectures of Rado and Chang and cardinal arithmetic, Kluwer Acadamic, Dordrecht, 1993). In particular, semiproperness of \(\mathbb {Q}\) has large cardinal strength, which answers a question of Friedman and Krueger (Trans. Am. Math. Soc. 359(5):2407–2420, 2007). One corollary of our work is that the version of Strong Chang’s Conjecture from Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) does not imply the existence of a precipitous ideal on \(\omega _1\).

Keywords

Semiproper forcing Chang’s conjecture Reasonable forcing Forcing axioms Martin’s maximum Stationary reflection 

Mathematics Subject Classification

03E05 03E35 03E55 03E57 03E65 

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA

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