The Galvin property under the Ultrapower Axiom
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914188116361216 |
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| author | Benhamou, Tom Goldberg, Gabriel |
| author_facet | Benhamou, Tom Goldberg, Gabriel |
| contents | We continue the study of the Galvin property from \cite{bgs} and \cite{Benhamou2}. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non p-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from \cite{Benhamou_Dobrinen}. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a $κ$-complete ultrafilter has the Galvin property if and only if it is an iterated sum of $p$-points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15078 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Galvin property under the Ultrapower Axiom Benhamou, Tom Goldberg, Gabriel Logic We continue the study of the Galvin property from \cite{bgs} and \cite{Benhamou2}. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound non p-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from \cite{Benhamou_Dobrinen}. Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a $κ$-complete ultrafilter has the Galvin property if and only if it is an iterated sum of $p$-points. |
| title | The Galvin property under the Ultrapower Axiom |
| topic | Logic |
| url | https://arxiv.org/abs/2306.15078 |