The number of measures on very large measurable cardinals
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917334904471552 |
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| author | Apter, Arthur W. Kaplan, Eyal Poveda, Alejandro |
| author_facet | Apter, Arthur W. Kaplan, Eyal Poveda, Alejandro |
| contents | 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 As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda.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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11710 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The number of measures on very large measurable cardinals Apter, Arthur W. Kaplan, Eyal Poveda, Alejandro Logic 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. |
| title | The number of measures on very large measurable cardinals |
| topic | Logic |
| url | https://arxiv.org/abs/2603.11710 |