Iterating Semi-proper Forcing using Virtual Models

Abstract

By a virtual model, we mean a model of set theory which is elementary in its transitive closure. Virtual models are first used by Neeman \cite{neeman2014forcing} to iterate forcing. That paper is concerned with proper forcing. The method was then adjusted by Veličković to the case of semi-proper forcing and this was drafted in \cite{velickovic2021iteration}. We here straighten the details and further elaborate on Veličković's method. The first section collects facts about virtual model, the second section describes the iteration, and the third one illustrates the method in the case of getting saturation of ${\mathsf{NS}}_{{\omega }_1}$ (loosely relying on \cite{schindler2016nsomega1}).