The Ultrapower Axiom implies GCH above a supercompact cardinal

TL;DR

This paper demonstrates that assuming the Ultrapower Axiom, the Generalized Continuum Hypothesis holds above a supercompact cardinal, linking inner model theory principles to set-theoretic continuum hypotheses.

Contribution

It establishes a new connection between the Ultrapower Axiom and GCH above supercompact cardinals, advancing understanding in inner model theory and large cardinal hypotheses.

Findings

GCH holds above a supercompact cardinal under the Ultrapower Axiom

Ultrapower Axiom serves as a sufficient condition for GCH in this context

Provides a new perspective on the relationship between large cardinals and continuum hypotheses

Abstract

We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theory at the level of supercompact cardinals.