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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.20466 |
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| _version_ | 1866916943349415936 |
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| author | Kaplan, Eyal |
| author_facet | Kaplan, Eyal |
| contents | We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $κ$ and $τ\leqκ^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $κ$ has exactly $τ$ distinct lifts in $V[G]$, and every normal measure on $κ$ in $V[G]$ arises as such a lift. This version differs from the original Friedman-Magidor theorem in several notable ways. First, the new technique does not involve forcing over canonical inner models or rely on any fine-structural tools or assumptions, allowing it to be applied in the realm of large cardinals beyond the current reach of the inner model program. Second, in the case where $τ\leq κ^+$, all lifts of a normal measure $U\in V$ on $κ$ to $V[G]$ have the same ultrapower. Finally, the technique generalizes to a version of the Friedman-Magidor theorem for extenders. An additional advantage is that the forcing used is notably simple, relying only on nonstationary support product forcing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of normal measures, revisited Kaplan, Eyal Logic We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $κ$ and $τ\leqκ^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $κ$ has exactly $τ$ distinct lifts in $V[G]$, and every normal measure on $κ$ in $V[G]$ arises as such a lift. This version differs from the original Friedman-Magidor theorem in several notable ways. First, the new technique does not involve forcing over canonical inner models or rely on any fine-structural tools or assumptions, allowing it to be applied in the realm of large cardinals beyond the current reach of the inner model program. Second, in the case where $τ\leq κ^+$, all lifts of a normal measure $U\in V$ on $κ$ to $V[G]$ have the same ultrapower. Finally, the technique generalizes to a version of the Friedman-Magidor theorem for extenders. An additional advantage is that the forcing used is notably simple, relying only on nonstationary support product forcing. |
| title | The number of normal measures, revisited |
| topic | Logic |
| url | https://arxiv.org/abs/2507.20466 |