The Magidor Iteration and Restrictions of Ultrapowers to the Ground Model

TL;DR

This paper investigates the effects of Magidor iteration of Prikry forcings on measurable cardinals, characterizing measures in the extension and analyzing ultrapower restrictions without core model assumptions.

Contribution

It characterizes all normal measures in the generic extension and shows that ultrapower restrictions are iterated ultrapowers of the ground model, extending previous results.

Findings

All normal measures in the extension are characterized.

Ultrapower restrictions are iterated ultrapowers of the ground model.

Results hold under GCH below the measurable limit.

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 \kappa} $ in the ground model suffices.