Baumgartner's Axiom and Small Posets
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918263352459264 |
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| author | Marun, Pedro Shelah, Saharon Switzer, Corey Bacal |
| author_facet | Marun, Pedro Shelah, Saharon Switzer, Corey Bacal |
| contents | We contribute to the study of $\aleph_1$-dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is consistent with $\textsf{MA}$ that there exists an $\aleph_1$-dense set of reals $A$ so that, in any cardinal-preserving generic extension by a forcing of size $\aleph_1$, $A$ and $A^*$ do not contain uncountable subsets which are order isomorphic. This strengthens a result of Avraham and the second author and yields a different proof of a theorem of Moore and Todorcevic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Baumgartner's Axiom and Small Posets Marun, Pedro Shelah, Saharon Switzer, Corey Bacal Logic 03E05, 03E35, 03E50 We contribute to the study of $\aleph_1$-dense sets of reals, a mainstay in set theoretic research since Baumgartner's seminal work in the 70s. In particular, we show that it is consistent with $\textsf{MA}$ that there exists an $\aleph_1$-dense set of reals $A$ so that, in any cardinal-preserving generic extension by a forcing of size $\aleph_1$, $A$ and $A^*$ do not contain uncountable subsets which are order isomorphic. This strengthens a result of Avraham and the second author and yields a different proof of a theorem of Moore and Todorcevic. |
| title | Baumgartner's Axiom and Small Posets |
| topic | Logic 03E05, 03E35, 03E50 |
| url | https://arxiv.org/abs/2512.21247 |