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 Veli\v{ c}kovi\'c to the case of semi-proper forcing and this was drafted in \cite{ velickovic2021iteration}. We here straighten the details and further elaborate on Veli\v{ c}kovi\'c'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}).
Categories: math.LO