Infinite decreasing chains in the Mitchell order

TL;DR

This paper investigates the structure of the Mitchell order in the context of rank-to-rank extenders, revealing the existence of infinite decreasing sequences and limitations on their length.

Contribution

It demonstrates the existence of transitive decreasing sequences of extenders of any countable length and shows no such sequence exists of length ω₁ in the ill-founded case.

Findings

Existence of transitive Mitchell order decreasing sequences of any countable length.

No such decreasing sequence of length ω₁ exists.

Provides initial understanding of the Mitchell order's structure in the ill-founded case.

Abstract

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length $\omega_1$.