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 $\omega_1$.