Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

TL;DR

The paper constructs two models of ZFC with identical cardinals and cofinalities, where GCH holds in one model and fails everywhere in the other, starting from large cardinal assumptions.

Contribution

It introduces a cofinality-preserving forcing method to produce models with GCH holding in one and failing everywhere in the other, expanding understanding of GCH's consistency.

Findings

GCH can be made to fail everywhere while preserving cardinals and cofinalities.

A new forcing technique is developed based on large cardinal assumptions.

Models with contrasting GCH behaviors are constructed from the same initial universe.

Abstract

Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.