The number of measures on very large measurable cardinals

TL;DR

This paper investigates the possible quantities of normal measures on large measurable cardinals using the Ultrapower Axiom, extending classical results and providing new flexibility in the number of measures.

Contribution

It introduces new methods to determine the number of normal measures on large cardinals without inner model techniques, combining the Ultrapower Axiom with classical results.

Findings

The first $n$ measurable cardinals can have any prescribed pattern of normal measures.

The first measurable above a supercompact can have any given number of normal measures.

Strengthens existing theorems by Goldberg--Woodin and Goldberg, Osinski, and Poveda.

Abstract

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first $n$ measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.