The internal relation

TL;DR

This paper investigates the internal relation, a generalization of the Mitchell order, analyzing its properties and limitations, and proving a conjecture related to rank-to-rank cardinals.

Contribution

It introduces the internal relation, examines its potential illfoundedness, and proves Steel's conjecture on rank-to-rank cardinals.

Findings

The internal relation can be illfounded under certain conditions.

Constraints on illfoundedness are established.

Steel's conjecture on rank-to-rank cardinals is proven.

Abstract

This paper explores various generalizations of the Mitchell order focusing mostly on a generalization called the internal relation. The internal relation lacks the implicit strength requirement in the definition of the Mitchell order, and therefore can fail to be wellfounded. We establish some constraints on the illfoundedness of the internal relation, which leads to a proof of a conjecture of Steel regarding rank-to-rank cardinals.