Mitchell rank for supercompactness

TL;DR

This paper introduces a Mitchell rank for supercompact cardinals, explores how to manipulate this rank through forcing, and discusses implications for measurable and strongly compact cardinals.

Contribution

It defines a Mitchell rank for supercompactness and develops forcing techniques to adjust this rank and related properties.

Findings

Can force to reduce the Mitchell rank of supercompact cardinals

Provides methods to control the degree of measurability via forcing

Extends results to strongly compact cardinals

Abstract

This paper defines a Mitchell rank for supercompact cardinals. If $\kappa$ is a $\theta$-supercompact cardinal then $o_{\theta-sc}(\kappa) = \sup \{ o_{\theta-sc}(\mu) + 1 \ | \ \mu \in m(\kappa)\}$, where $m(\kappa)$ is the collection of normal fine measures on $P_{\kappa}\theta$. We show how to force to kill the degree of a measurable cardinal $\kappa$ to any specified degree which is less than or equal to the degree of $\kappa$ in the ground model. We will also show how to softly kill the Mitchell rank for supercompactness of any supercompact cardinal so that in the forcing extension it is any desired degree less than or equal to its degree in the ground model, along with some results concerning strongly compact cardinals.