The Magidor Iteration and Restrictions of Ultrapowers to the Ground Model

Abstract

We study the Magidor iteration of Prikry forcings below a measurable limit of measurables $\kappa$. We first characterize all the normal measures $\kappa$ carries in the generic extension, building on and extending the main result of \cite{ben2014forcing}. Then, for every such normal measure, we prove that the restriction of its ultrapower, from the generic extension to the ground model, is an iterated ultrapower of $V$ by normal measures. This is done without core model theoretic assumptions; $ \mbox{GCH}_{\leq κ} $ in the ground model suffices.