Guessing models imply the singular cardinal hypothesis

Krueger, John

doi:10.1090/proc/14739

Guessing models imply the singular cardinal hypothesis
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Proc. Amer. Math. Soc. 147 (2019), 5427-5434 Request permission

Abstract:

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega _2$, $\mathsf {ISP}(\kappa )$ implies that $\mathsf {SCH}$ holds above $\kappa$, and (3) forcing posets which have the $\omega _1$-approximation property also have the countable covering property. These results solve open problems of Viale [Ann. Pure Appl. Logic 163 (2012), no. 11, 1660–1678] and Hachtman and Sinapova [J. Symb. Log. 84 (2019), no. 2, 713–725].

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