The structure of the Mitchell order - I

TL;DR

This paper introduces tame orders, a broad class of well-founded orders, and demonstrates their realization as Mitchell orders on measurable cardinals under weaker consistency assumptions than previously required.

Contribution

It establishes that tame orders of size at most  can be realized as Mitchell orders on measurable cardinals with weaker assumptions than the known bounds.

Findings

Tame orders are a wide class of well-founded orders.

Any tame order of size  can be realized as a Mitchell order.

Realization requires weaker consistency assumptions than previous results.

Abstract

We isolate here a wide class of well founded orders called tame orders and show that each such order of cardinality at most $\kappa$ can be realized as the Mitchell order on a measurable cardinal $\kappa$, from a consistency assumption weaker than $o(\kappa) = \kappa^+$.