Isomorphism Classes of Generating Sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915904655196160 |
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| author | Benhamou, Tom Cummings, James Goldberg, Gabriel Hayut, Yair Poveda, Alejandro |
| author_facet | Benhamou, Tom Cummings, James Goldberg, Gabriel Hayut, Yair Poveda, Alejandro |
| contents | We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $σ$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $ω$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$.
On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $κ$ is supercompact, and for a $κ^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}κ$-directed closed $κ^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $κ$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_κ<2^κ$, answering questions from \cite{Benhamou_Goldberg2025}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isomorphism Classes of Generating Sets Benhamou, Tom Cummings, James Goldberg, Gabriel Hayut, Yair Poveda, Alejandro Logic We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $σ$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $ω$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $κ$ is supercompact, and for a $κ^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}κ$-directed closed $κ^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $κ$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_κ<2^κ$, answering questions from \cite{Benhamou_Goldberg2025}. |
| title | Isomorphism Classes of Generating Sets |
| topic | Logic |
| url | https://arxiv.org/abs/2504.18381 |