Non-stationary support iterations of Prikry Forcings and Restrictions of Ultrapower Embeddings to the Ground Model

TL;DR

This paper investigates the structure of nonstationary-support Prikry forcings below a measurable cardinal, characterizing the normal measures in the extension and analyzing ultrapower embeddings restricted to the ground model.

Contribution

It provides a detailed analysis of normal measures and ultrapower embeddings in the context of nonstationary-support Prikry forcings, avoiding core-model theoretic methods.

Findings

Characterization of normal measures in the generic extension.

Every restriction of ultrapower embeddings is an iterated ultrapower.

A sufficient condition for the definability of these restrictions in the ground model.

Abstract

We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal \kappa, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to \kappa suffices.