Non-Normal Magidor-Radin Types of Forcings
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arXiv
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| Format: | Preprint |
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2024
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| author | Benhamou, Tom Poveda, Alejandro |
| author_facet | Benhamou, Tom Poveda, Alejandro |
| contents | We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}.
Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $ω$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $κ$ to $ω_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $κ$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $κ$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $κ$ to $ω_1$ while introducing a Cohen subset of $κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16704 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Normal Magidor-Radin Types of Forcings Benhamou, Tom Poveda, Alejandro Logic We develop the non-normal variations of two classical Prikry-type forcings; namely, Magidor and Radin forcings. We generalize the fact that the non-normal Prikry forcing is a projection of the extender-based to a coordinate of the extender to our forcing and the Radin/Magidor-Radin-extender-based forcing from \cite{CarmiMagidorRadin,CarmiRadin}. Then, we show that both the non-normal variation of Magidor and Radin forcings can add a Cohen generic function to every limit point of cofinality $ω$ of the generic club. Second, we show that this phenomenon is limited to the cases where the forcings are not designed to change the cofinality of a measurable $κ$ to $ω_1$. Specifically, in the above-mentioned circumstances these forcings do not project onto any $κ$-distributive forcing. We use that to conclude that the extender-based Radin/Magidor-Radin forcing does not add fresh subsets to $κ$ as well. In the second part of the paper we focus on the natural non-normal variation of Gitik's forcing from \cite[\S3]{GitikNonStationary}. Our main result shows that this poset can be employed to change the cofinality of a measurable cardinal $κ$ to $ω_1$ while introducing a Cohen subset of $κ$. |
| title | Non-Normal Magidor-Radin Types of Forcings |
| topic | Logic |
| url | https://arxiv.org/abs/2405.16704 |