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C (n)-cardinals

Archive for Mathematical Logic 51 (3-4):213-240 (2012)

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  1. Weakly Remarkable Cardinals, Erdős Cardinals, and the Generic Vopěnka Principle.Trevor M. Wilson - 2019 - Journal of Symbolic Logic 84 (4):1711-1721.
    We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is not ${{\rm{\Delta (...)
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  • Elementary Chains and C (N)-Cardinals.Konstantinos Tsaprounis - 2014 - Archive for Mathematical Logic 53 (1-2):89-118.
    The C (n)-cardinals were introduced recently by Bagaria and are strong forms of the usual large cardinals. For a wide range of large cardinal notions, Bagaria has shown that the consistency of the corresponding C (n)-versions follows from the existence of rank-into-rank elementary embeddings. In this article, we further study the C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardinals, giving some improved consistency bounds while, at the same time, addressing questions which had been left open. In addition, we (...)
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  • The Large Cardinals Between Supercompact and Almost-Huge.Norman Lewis Perlmutter - 2015 - Archive for Mathematical Logic 54 (3-4):257-289.
    I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j:V→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j: V \to M}$$\end{document} such that M is closed under sequences of length sup{j|f:κ→κ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sup\{{j\,|\,f: \kappa \to \kappa}\}}$$\end{document}. Some of the other (...)
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  • A Model of the Generic Vopěnka Principle in Which the Ordinals Are Not Mahlo.Victoria Gitman & Joel David Hamkins - 2019 - Archive for Mathematical Logic 58 (1-2):245-265.
    The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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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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  • On Colimits and Elementary Embeddings.Joan Bagaria & Andrew Brooke-Taylor - 2013 - Journal of Symbolic Logic 78 (2):562-578.
    We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as $\alpha$-strongly compact and $C^{(n)}$-extendible cardinals.
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  • On the Symbiosis Between Model-Theoretic and Set-Theoretic Properties of Large Cardinals.Joan Bagaria & Jouko Väänänen - 2016 - Journal of Symbolic Logic 81 (2):584-604.
  • On Extensions of Supercompactness.Robert S. Lubarsky & Norman Lewis Perlmutter - 2015 - Mathematical Logic Quarterly 61 (3):217-223.
  • Subcomplete Forcing Principles and Definable Well-Orders.Gunter Fuchs - 2018 - Mathematical Logic Quarterly 64 (6):487-504.
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  • On C-Extendible Cardinals.Konstantinos Tsaprounis - 2018 - Journal of Symbolic Logic 83 (3):1112-1131.
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  • Ultrahuge cardinals.Konstantinos Tsaprounis - 2016 - Mathematical Logic Quarterly 62 (1-2):77-87.
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