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  1. Specialising Trees with Small Approximations I.Rahman Mohammadpour - forthcoming - Journal of Symbolic Logic:1-24.
    Assuming $\mathrm{PFA}$, we shall use internally club $\omega _1$ -guessing models as side conditions to show that for every tree T of height $\omega _2$ without cofinal branches, there is a proper and $\aleph _2$ -preserving forcing notion with finite conditions which specialises T. Moreover, the forcing has the $\omega _1$ -approximation property.
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  • Forcing the Mapping Reflection Principle by finite approximations.Tadatoshi Miyamoto & Teruyuki Yorioka - 2021 - Archive for Mathematical Logic 60 (6):737-748.
    Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, (...)
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  • The approachability ideal without a maximal set.John Krueger - 2019 - Annals of Pure and Applied Logic 170 (3):297-382.
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  • A forcing notion collapsing $\aleph _3 $ and preserving all other cardinals.David Asperó - 2018 - Journal of Symbolic Logic 83 (4):1579-1594.
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