The failure of GCH at a degree of supercompactness

TL;DR

This paper explores the consistency strength of supercompact cardinals where the Generalized Continuum Hypothesis fails, establishing equivalences with tallness properties in large cardinals.

Contribution

It determines the large cardinal consistency strength of supercompact cardinals with GCH failure at a certain cardinal, linking it to tallness properties.

Findings

Existence of a supercompact cardinal with GCH failure is equiconsistent with a supercompact and tall cardinal.

Provides foundational facts about tallness with closure in large cardinals.

Abstract

We determine the large cardinal consistency strength of the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$. Indeed, we show that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that $2^\lambda \geq \theta$ is equiconsistent with the existence of a $\lambda$-supercompact cardinal that is also $\theta$-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.