A note on adding isomorphisms and the pseudointersection number
Corey Bacal Switzer
TL;DR
The paper investigates how Baumgartner's axiom $\mathsf{BA}$ interacts with forcing and cardinal characteristics. It shows that any reasonable forcing that yields $\mathsf{BA}$ necessarily pushes the pseudointersection number $\mathfrak{p}$ beyond $\aleph_1$, by constructing $\aleph_1$-dense sets $A,B$ and analyzing how a generic isomorphism between them induces a pseudointersection for any $\aleph_1$-sized tower via the Cantor–Lebesgue map. The authors define a notion of 'reasonable' forcing for a pair of sets and prove a main lemma asserting that forcing with such posets creates the desired pseudointersection, implying $\mathfrak{p}>\aleph_1$ in the resulting models and contributing to questions of Todorčević and Steprāns–Watson. They also adapt the argument to forcing BA on $2^\omega$ using Medini's forcing, showing a parallel mechanism that yields pseudointersections. The results connect BA with large values of $\mathfrak{p}$ and illustrate how forcing BA interacts with well-studied cardinal characteristics, suggesting new directions for resolving whether BA implies $\mathfrak{p}>\aleph_1$ in full generality.
Abstract
We prove that for every tower $\mathcal T$ there are $\aleph_1$-dense $A$ and $B$ so that any ``reasonable" forcing notion $\mathbb{P}$ -- an adjective that includes all known ones -- for making $A$ and $B$ isomorphic will add a pseudointersection for the tower. This shows in particular that $\mathsf{MA}_{\aleph_1}(σ{\rm -centered})$ holds in all known models of $\mathsf{BA}$, which provides intrigue to well known questions of Todorčević and Steprāns-Watson.