Guessing models imply the singular cardinal hypothesis

Abstract

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ 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 and Hachtman-Sinapova.