Iterating Generalised Perfect Set Forcing Along Well-Founded Orders
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915933716480000 |
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| author | Džamonja, Mirna |
| author_facet | Džamonja, Mirna |
| contents | Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving $\aleph_1$. In \cite{Property-B} we considered a generalised perfect set forcing with respect to a filter on a cardinal $κ$ satisfying $κ^{<κ}=κ$, which we denoted ${\mathbb P} (\mathcal F)$, and proved that its iteration with supports of size $\leκ$ along any ordinal preserves cardinals up and including $κ^+$.
We show that there is a version of the geometric iteration technique that applies to ${\mathbb P} (\mathcal F)$, to yield that for $κ$ satisfying $κ^{<κ}=κ$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\leκ$ along any well-founded partial order, while preserving cardinals up and including $κ^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10826 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Iterating Generalised Perfect Set Forcing Along Well-Founded Orders Džamonja, Mirna Logic 03E35 Vladimir Kanovei \cite{zbMATH01335192} developed the technique of geometric iteration and used it to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving $\aleph_1$. In \cite{Property-B} we considered a generalised perfect set forcing with respect to a filter on a cardinal $κ$ satisfying $κ^{<κ}=κ$, which we denoted ${\mathbb P} (\mathcal F)$, and proved that its iteration with supports of size $\leκ$ along any ordinal preserves cardinals up and including $κ^+$. We show that there is a version of the geometric iteration technique that applies to ${\mathbb P} (\mathcal F)$, to yield that for $κ$ satisfying $κ^{<κ}=κ$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\leκ$ along any well-founded partial order, while preserving cardinals up and including $κ^+$. |
| title | Iterating Generalised Perfect Set Forcing Along Well-Founded Orders |
| topic | Logic 03E35 |
| url | https://arxiv.org/abs/2604.10826 |