Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914412557762560 |
|---|---|
| author | Džamonja, Mirna |
| author_facet | Džamonja, Mirna |
| contents | 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 $\leκ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$ Džamonja, Mirna Logic 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 $\leκ$. |
| title | Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$ |
| topic | Logic |
| url | https://arxiv.org/abs/2509.19519 |