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  1. Martin’s maximum revisited.Matteo Viale - 2016 - Archive for Mathematical Logic 55 (1-2):295-317.
    We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM++ makes the Π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_2}$$\end{document}-fragment of the theory of Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to (...)
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  • 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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  • Guessing models and the approachability ideal.Rahman Mohammadpour & Boban Veličković - 2020 - Journal of Mathematical Logic 21 (2):2150003.
    Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call GM+ holds. This principle implies ISP and ISP, and hence th...
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  • Indestructibility of some compactness principles over models of PFA.Radek Honzik, Chris Lambie-Hanson & Šárka Stejskalová - 2024 - Annals of Pure and Applied Logic 175 (1):103359.
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  • Small embedding characterizations for large cardinals.Peter Holy, Philipp Lücke & Ana Njegomir - 2019 - Annals of Pure and Applied Logic 170 (2):251-271.
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  • Strong tree properties for small cardinals.Laura Fontanella - 2013 - Journal of Symbolic Logic 78 (1):317-333.
    An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.
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  • On the universality of the nonstationary ideal.Sean D. Cox - 2018 - Mathematical Logic Quarterly 64 (1-2):103-117.
    Burke proved that the generalized nonstationary ideal, denoted by NS, is universal in the following sense: every normal ideal, and every tower of normal ideals of inaccessible height, is a canonical Rudin‐Keisler projection of the restriction of NS to some stationary set. We investigate how far Burke's theorem can be pushed, by analyzing the universality properties of NS with respect to the wider class of ‐systems of filters introduced by Audrito and Steila. First we answer a question of Audrito and (...)
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  • PFA and Ideals on $\omega_{2}$ Whose Associated Forcings Are Proper.Sean Cox - 2012 - Notre Dame Journal of Formal Logic 53 (3):397-412.
    Given an ideal $I$ , let $\mathbb{P}_{I}$ denote the forcing with $I$ -positive sets. We consider models of forcing axioms $MA(\Gamma)$ which also have a normal ideal $I$ with completeness $\omega_{2}$ such that $\mathbb{P}_{I}\in \Gamma$ . Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on $\omega_{2}$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ obtained from a supercompact cardinal. (...)
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  • Namba forcing, weak approximation, and guessing.Sean Cox & John Krueger - 2018 - Journal of Symbolic Logic 83 (4):1539-1565.
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  • Forcing axioms, approachability, and stationary set reflection.Sean D. Cox - 2021 - Journal of Symbolic Logic 86 (2):499-530.
    We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.
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