# The Linearity of the Mitchell Order

## TL;DR

This paper demonstrates that under the Ultrapower Axiom, the Mitchell order is linear for specific ultrafilter classes, extending known results and generalizing Solovay's lemma to singular cardinals.

## Contribution

It proves the linearity of the Mitchell order for certain ultrafilters using a weak comparison principle and generalizes Solovay's lemma to singular cardinals.

## Key findings

- Mitchell order is linear on normal ultrafilters
- Linearity extends to Dodd solid ultrafilters
- Assuming GCH, generalized normal ultrafilters also exhibit linear Mitchell order

## Abstract

We show from a weak comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on certain kinds of ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. In the process we prove a generalization of Solovay's lemma to singular cardinals.