The number of measures on very large measurable cardinals

Arthur W. Apter; Eyal Kaplan; Alejandro Poveda

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arXiv · 2603.11710

The number of measures on very large measurable cardinals

Arthur W. Apter·Eyal Kaplan·Alejandro Poveda

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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 As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda.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.

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BibTeX RIS

Arthur W. Apter, Eyal Kaplan, Alejandro Poveda. 2026-03-12. The number of measures on very large measurable cardinals. https://arxiv.org/abs/2603.11710

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