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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2509.19519 |
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- We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set forcing. The work is modelled after Baumgartner's Axiom A and his proof that iterations with countable support of the same preserve $\aleph_1$. In honour of James E. Baumgartner, the property introduced here is called Property B$(κ)$. The known additional difficulties when forcing at cardinals higher than $\aleph_1$ make for a less general theorem and a more complex theorem on the iteration, which is not an iteration theorem in the classical sense. The results extend to other cardinals $κ$ such that $κ^{<κ}=κ$, in place of $\aleph_1$. We give examples of individual forcings that have Property B$(κ)$ and their products. In particular, we introduce a correct version of the generalised perfect set forcing, which we call Perfect Set Forcing with Respect to a Filter. We give its basic properties and show that for the right kind of filter $\mathcal F$ this kind of forcing is iterable with supports of size $\leκ$.