Revised support iterations and CH

Abstract

Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham's problem on the existence of uncountable non-constuctible sequences all of whose proper initial segments are constructible, Friedman's problem on whether every 2-coloring of $S^2_0=\{α<ω_2\colon\cf(α)=ω\}$ has an uncountable sequentially closed homogeneous subset, and existence of a precipitous normal filter on ${\omega }_2$ with $S_0^2\in \mathcal{F}$. The posets which Shelah uses in these constructions are Prikry forcing, Namba forcing, and the forcing consisting of closed countable subsets of $S^{\ast }$ under reverse end-extension, where $S^{\ast }$ is a fixed stationary co-stationary subset of $S_0^2$. Shelah establishes different preservation theorems for each of these three posets (the theorem for Namba forcing is particularly intricate). We establish a general preservation theorem for a variant of RCS iterations which includes all three posets in a straightforward way.