An Iteration Theorem for ${\omega }_1$-preserving Forcings

Abstract

We prove an iteration theorem which guarantees for a wide class of nice iterations of ${\omega }_1$-preserving forcings that ${\omega }_1$ is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a nice iteration of ${\omega }_1$-preserving forcings which force SRP at successor steps and preserves old stationary sets does not collapse ${\omega }_1$.