Baumgartner's Axiom and Small Posets

Pedro Marun; Saharon Shelah; Corey Bacal Switzer

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arXiv · 2512.21247

Baumgartner's Axiom and Small Posets

Pedro Marun·Saharon Shelah·Corey Bacal Switzer

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Abstract

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.

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BibTeX RIS

Pedro Marun, Saharon Shelah, Corey Bacal Switzer. 2025-12-24. Baumgartner's Axiom and Small Posets. https://arxiv.org/abs/2512.21247

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