A note on adding isomorphisms and the pseudointersection number

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 ${\mathsf{MA}}_{{\aleph }_1}(\ddot{I}ƒ-\mathrm{centered})$ holds in all known models of $\mathsf{BA}$, which provides intrigue to well known questions of Todorčević and Steprāns-Watson.

11 pages, submitted