A fragment of Asperó–Mota’s finitely proper forcing axiom and entangled sets of reals
Volume 251 / 2020
Fundamenta Mathematicae 251 (2020), 35-68
MSC: 03E35, 03E50.
DOI: 10.4064/fm814-11-2019
Published online: 27 March 2020
Abstract
We introduce a fragment ${\sf PFA}^{\text {s-fin}}({\omega _1})$ of Asperó–Mota’s finitely proper forcing axiom ${\sf PFA}^{\rm fin}({\omega _1})$. ${\sf PFA}^{\text {s-fin}}({\omega _1})$ implies some consequences of ${\sf PFA}^{\rm fin}({\omega _1})$, for example ${\sf MA} _{\aleph _1}$ and the assertion that any two Aronszajn trees are club-isomorphic. For each integer $k\geq 2$, it is consistent that ${\sf PFA}^{\text {s-fin}}({\omega _1})$ holds, there exists a $k$-entangled set of reals, and $2^{\aleph _0}=\aleph _2$. This extends Abraham–Shelah’s theorem that Martin’s Axiom does not imply that any two $\aleph _1$-dense sets of reals are isomorphic.
