Countable support iterations and large continuum

Abstract

We prove that any countable support iteration formed with posets with ${\omega }_2$-p.i.c.\ has ${\omega }_2$-c.c., assuming CH in the ground model and assuming also that ${\omega }_1$ is not collapsed. This improves earlier results of Shelah by removing the restriction on the length of the iteration. Thus, we solve the problem of obtaining a large continuum via such forcing iterations.