A Gitik iteration with nearly Easton factoring

Abstract

We reprove Gitik’s theorem that if the GCH holds and $o(\gk)=\gk+1$ then there is a generic extension in which $\gk$ is still measurable and there is a closed unbounded subset C of $\gk$ such that every $ν\in C$ is inaccessible in the ground model. Unlike the forcing used by Gitik, the iterated forcing $\radin_\gl+1$ used in this paper has the property that if $\gl$ is a cardinal less then $\gk$ then $\radin_\gl+1$ can be factored in V as $\radin_\gk+1=\radin_\gl+1\times\radin_\gl+1,\gk$ where $\card{\radin_\gl+1}\le\gl⁺$ and $\radin_\gl+1,\gk$ does not add any new subsets of $\gl$.