14 found
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  1.  57
    The downward directed grounds hypothesis and very large cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
    A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, the mantle, the intersection of all grounds, must be a model of ZFC. V has only set many grounds if and only if the mantle is a ground. We also show that if the universe (...)
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  2.  46
    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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  3.  19
    Strongly compact cardinals and the continuum function.Arthur W. Apter, Stamatis Dimopoulos & Toshimichi Usuba - 2021 - Annals of Pure and Applied Logic 172 (9):103013.
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  4.  26
    On the existence of skinny stationary subsets.Yo Matsubara, Hiroshi Sakai & Toshimichi Usuba - 2019 - Annals of Pure and Applied Logic 170 (5):539-557.
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  5.  19
    Extendible cardinals and the mantle.Toshimichi Usuba - 2019 - Archive for Mathematical Logic 58 (1-2):71-75.
    The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.
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  6.  44
    Hierarchies of ineffabilities.Toshimichi Usuba - 2013 - Mathematical Logic Quarterly 59 (3):230-237.
  7.  28
    Two-cardinal versions of weak compactness: Partitions of pairs.Pierre Matet & Toshimichi Usuba - 2012 - Annals of Pure and Applied Logic 163 (1):1-22.
  8.  17
    New combinatorial principle on singular cardinals and normal ideals.Toshimichi Usuba - 2018 - Mathematical Logic Quarterly 64 (4-5):395-408.
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  9.  23
    Subtlety and partition relations.Toshimichi Usuba - 2016 - Mathematical Logic Quarterly 62 (1-2):59-71.
    We study the subtlety of a cardinal κ and of. We show that, under a certain large cardinal assumption, it is consistent that is subtle for some but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of can be characterized by a certain partition relation on. We also study the property of faintness of κ, and the subtlety of with the strong inclusion.
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  10.  14
    Bounded dagger principles.Toshimichi Usuba - 2014 - Mathematical Logic Quarterly 60 (4-5):266-272.
    For an uncountable cardinal κ, let be the assertion that every ω1‐stationary preserving poset of size is semiproper. We prove that is a strong principle which implies a strong form of Chang's conjecture. We also show that implies that is presaturated.
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  11.  55
    Splitting stationary sets in.Toshimichi Usuba - 2012 - Journal of Symbolic Logic 77 (1):49-62.
    Let A be a non-empty set. A set $S\subseteq \mathcal{P}(A)$ is said to be stationary in $\mathcal{P}(A)$ if for every f: [A] <ω → A there exists x ∈ S such that x ≠ A and f"[x] <ω ⊆ x. In this paper we prove the following: For an uncountable cardinal λ and a stationary set S in \mathcal{P}(\lambda) , if there is a regular uncountable cardinal κ ≤ λ such that {x ∈ S: x ⋂ κ ∈ κ} is (...)
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  12.  17
    Local saturation of the non-stationary ideal over Pκλ.Toshimichi Usuba - 2007 - Annals of Pure and Applied Logic 149 (1-3):100-123.
    Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of such that , the ideal generated by the non-stationary ideal over together with , is λ+-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis . We also show that in our model we can make -saturated, where S is the (...)
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  13.  13
    Roles of Large Cardinals in Set Theory.Toshimichi Usuba & Hiroshi Fujita - 2012 - Journal of the Japan Association for Philosophy of Science 39 (2):83-92.
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  14.  58
    Notes on the partition property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_\kappa\lambda}$$\end{document}. [REVIEW]Yoshihiro Abe & Toshimichi Usuba - 2012 - Archive for Mathematical Logic 51 (5-6):575-589.
    We investigate the partition property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document}. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\kappa}\lambda}$$\end{document} carries a (λκ, 2)-distributive normal ideal without the partition property, then λ is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^1_n}$$\end{document}-indescribable for all n < ω but not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...)
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