13 found
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  1.  12
    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.  8
    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.
  3.  7
    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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  4.  25
    Notes on the Partition Property of {\ Mathcal {P} _\ Kappa\ Lambda}.Yoshihiro Abe & Toshimichi Usuba - 2012 - Archive for Mathematical Logic 51 (5-6):575-589.
    We investigate the partition property of ${\mathcal{P}_{\kappa}\lambda}$ . Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that ${\mathcal{P}_{\kappa}\lambda}$ carries a (λ κ , 2)-distributive normal ideal without the partition property, then λ is ${\Pi^1_n}$ -indescribable for all n < ω but not ${\Pi^2_1}$ -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of ${\mathcal{P}_{\kappa}\lambda}$ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over (...)
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  5.  21
    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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  6.  7
    Two-Cardinal Versions of Weak Compactness: Partitions of Pairs.Pierre Matet & Toshimichi Usuba - 2012 - Annals of Pure and Applied Logic 163 (1):1-22.
  7.  5
    New Combinatorial Principle on Singular Cardinals and Normal Ideals.Toshimichi Usuba - 2018 - Mathematical Logic Quarterly 64 (4-5):395-408.
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  8.  4
    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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  9.  24
    Hierarchies of Ineffabilities.Toshimichi Usuba - 2013 - Mathematical Logic Quarterly 59 (3):230-237.
  10.  10
    Subtlety and Partition Relations.Toshimichi Usuba - 2016 - Mathematical Logic Quarterly 62 (1-2):59-71.
  11.  4
    Local Saturation of the Non-Stationary Ideal Over Pκλ.Toshimichi Usuba - 2007 - Annals of Pure and Applied Logic 149 (1):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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  12. Bounded Dagger Principles.Toshimichi Usuba - 2014 - Mathematical Logic Quarterly 60 (4-5):266-272.
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  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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