On Ultrapowers and Cohesive Ultrafilters
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910696945483776 |
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| author | Benhamou, Tom |
| author_facet | Benhamou, Tom |
| contents | We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of ultrafilters. Secondly, we improve two theorems of Kanamori \cite{Kanamori1978} from the 70's. We then study the point spectrum and the depth spectrum of an ultrafilter, and give a simple positive answer to Kanamori's question \cite[Question 2]{Kanamori1978} starting from a supercompact cardinal. We also prove that a positive answer requires more than $o(κ)=κ^{++}$. Finally, we prove several consistency results regarding the point and depth spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Ultrapowers and Cohesive Ultrafilters Benhamou, Tom Logic We characterize the Tukey order, the Galvin property/ Cohesive ultrafilters from \cite{Kanamori1978} in terms of ultrapowers. We use this characterization to measure the distance between the Tukey order and other well-known orders of ultrafilters. Secondly, we improve two theorems of Kanamori \cite{Kanamori1978} from the 70's. We then study the point spectrum and the depth spectrum of an ultrafilter, and give a simple positive answer to Kanamori's question \cite[Question 2]{Kanamori1978} starting from a supercompact cardinal. We also prove that a positive answer requires more than $o(κ)=κ^{++}$. Finally, we prove several consistency results regarding the point and depth spectrum. |
| title | On Ultrapowers and Cohesive Ultrafilters |
| topic | Logic |
| url | https://arxiv.org/abs/2410.06275 |