A fragment of Asperó–Mota’s finitely proper forcing axiom and entangled sets of reals

Tadatoshi Miyamoto, Teruyuki Yorioka Fundamenta Mathematicae 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.

Authors

  • Tadatoshi MiyamotoMathematics
    Nanzan University
    18 Yamazato-cho, Showa-ku
    Nagoya 466-8673, Japan
    e-mail
  • Teruyuki YoriokaDepartment of Mathematics
    Shizuoka University
    Ohya 836
    Shizuoka 422-8529, Japan
    e-mail

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