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  1. Incompatible Bounded Category Forcing Axioms.David Asperó & Matteo Viale - 2022 - Journal of Mathematical Logic 22 (2).
    We introduce bounded category forcing axioms for well-behaved classes Γ. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe...
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  • Superstrong and Other Large Cardinals Are Never Laver Indestructible.Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba - 2016 - Archive for Mathematical Logic 55 (1-2):19-35.
    Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing Q∈Vθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...)
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  • Strongly Uplifting Cardinals and the Boldface Resurrection Axioms.Joel David Hamkins & Thomas A. Johnstone - 2017 - Archive for Mathematical Logic 56 (7-8):1115-1133.
    We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost-hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
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  • First‐Order Undefinability of the Notion of Transfinitely Uplifting Cardinals.Kentaro Fujimoto - forthcoming - Mathematical Logic Quarterly.
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  • Combining Resurrection and Maximality.Kaethe Minden - 2021 - Journal of Symbolic Logic 86 (1):397-414.
    It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes. The extent to which resurrection and maximality overlap is explored via the local maximality principle.
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  • Infinite Forcing and the Generic Multiverse.Giorgio Venturi - 2020 - Studia Logica 108 (2):277-290.
    In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.
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  • Diagonal Reflections on Squares.Gunter Fuchs - 2019 - Archive for Mathematical Logic 58 (1-2):1-26.
    The effects of the forcing axioms \, \ and \ on the failure of weak threaded square principles of the form \\) are analyzed. To this end, a diagonal reflection principle, \, and it implies the failure of \\) if \. It is also shown that this result is sharp. It is noted that \/\ imply the failure of \\), for every regular \, and that this result is sharp as well.
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  • On a Class of Maximality Principles.Daisuke Ikegami & Nam Trang - 2018 - Archive for Mathematical Logic 57 (5-6):713-725.
    We study various classes of maximality principles, \\), introduced by Hamkins :527–550, 2003), where \ defines a class of forcing posets and \ is an infinite cardinal. We explore the consistency strength and the relationship of \\) with various forcing axioms when \. In particular, we give a characterization of bounded forcing axioms for a class of forcings \ in terms of maximality principles MP\\) for \ formulas. A significant part of the paper is devoted to studying the principle MP\\) (...)
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  • 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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  • Hierarchies of Resurrection Axioms.Gunter Fuchs - 2018 - Journal of Symbolic Logic 83 (1):283-325.
    I analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings. I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies (...)
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  • Perfect Subsets of Generalized Baire Spaces and Long Games.Philipp Schlicht - 2017 - Journal of Symbolic Logic 82 (4):1317-1355.
    We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from (...)
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  • On Resurrection Axioms.Konstantinos Tsaprounis - 2015 - Journal of Symbolic Logic 80 (2):587-608.
    The resurrection axioms are forms of forcing axioms that were introduced recently by Hamkins and Johnstone, who developed on earlier ideas of Chalons and Veličković. In this note, we introduce a stronger form of resurrection and show that it gives rise to families of axioms which are consistent relative to extendible cardinals, and which imply the strongest known instances of forcing axioms, such as Martin’s Maximum++. In addition, we study the unbounded resurrection postulates in terms of consistency lower bounds, obtaining, (...)
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  • Absoluteness Via Resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.
    The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms (...)
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  • Virtual Large Cardinals.Victoria Gitman & Ralf Schindler - 2018 - Annals of Pure and Applied Logic 169 (12):1317-1334.
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