The directedness of the Rudin-Keisler order at measurable cardinals
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908768959201280 |
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| author | Hayut, Yair Poveda, Alejandro |
| author_facet | Hayut, Yair Poveda, Alejandro |
| contents | The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $λ\leq κ$, the following theories are equiconsistent modulo ZFC: (1) $κ$ is a measurable cardinal with $o(κ)=λ^+$ (resp. $o(κ)=κ$). (2) The Rudin-Keisler order restricted to the set of $κ$-complete (non-principal) ultrafilters on $κ$ is $λ^+$-directed (resp. $κ^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $λ$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The directedness of the Rudin-Keisler order at measurable cardinals Hayut, Yair Poveda, Alejandro Logic The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $λ\leq κ$, the following theories are equiconsistent modulo ZFC: (1) $κ$ is a measurable cardinal with $o(κ)=λ^+$ (resp. $o(κ)=κ$). (2) The Rudin-Keisler order restricted to the set of $κ$-complete (non-principal) ultrafilters on $κ$ is $λ^+$-directed (resp. $κ^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $λ$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed. |
| title | The directedness of the Rudin-Keisler order at measurable cardinals |
| topic | Logic |
| url | https://arxiv.org/abs/2601.10614 |