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 $\kappa \ge {\omega }_2$, $\text{ISP}(\kappa )$ implies that $\text{SCH}$ 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.