13 found
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  1.  13
    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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  2.  39
    Chang’s Conjecture and Weak Square.Hiroshi Sakai - 2013 - Archive for Mathematical Logic 52 (1-2):29-45.
    We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of ${\square_{\omega_1, 2}}$ , i.e. Chang’s Conjecture is consistent with ${\square_{\omega_1, 2}}$.
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  3.  50
    On Proofs of the Incompleteness Theorems Based on Berry's Paradox by Vopěnka, Chaitin, and Boolos.Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai - 2012 - Mathematical Logic Quarterly 58 (4-5):307-316.
    By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...)
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  4.  8
    The weakly compact reflection principle need not imply a high order of weak compactness.Brent Cody & Hiroshi Sakai - 2020 - Archive for Mathematical Logic 59 (1-2):179-196.
    The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \ then there is a forcing extension preserving (...)
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  5.  7
    On Katětov and Katětov–Blass Orders on Analytic P-Ideals and Borel Ideals.Hiroshi Sakai - 2018 - Archive for Mathematical Logic 57 (3-4):317-327.
    Minami–Sakai :883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders. In the course of the proof of (...)
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  6.  4
    Strong Downward Löwenheim–Skolem Theorems for Stationary Logics, I.Sakaé Fuchino, André Ottenbreit Maschio Rodrigues & Hiroshi Sakai - 2021 - Archive for Mathematical Logic 60 (1-2):17-47.
    This note concerns the model theoretic properties of logics extending the first-order logic with monadic second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem down to <ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<\aleph _2$$\end{document} for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into four principles. The strongest of these four is shown to be equivalent to the conjunction of CH and the (...)
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  7.  33
    Semistationary and Stationary Reflection.Hiroshi Sakai - 2008 - Journal of Symbolic Logic 73 (1):181-192.
    We study the relationship between the semistationary reflection principle and stationary reflection principles. We show that for all regular cardinals Λ ≥ ω₂ the semistationary reflection principle in the space [Λ](1) implies that every stationary subset of $E_{\omega}^{\lambda}\coloneq \{\alpha \in \lambda \,|\,{\rm cf}(\alpha)=\omega \}$ reflects. We also show that for all cardinals Λ ≥ ω₃ the semistationary reflection principle in [Λ](1) does not imply the stationary reflection principle in [Λ](1).
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  8.  8
    Simple Proofs of $${\Mathsf{SCH}}$$ SCH From Reflection Principles Without Using Better Scales.Hiroshi Sakai - 2015 - Archive for Mathematical Logic 54 (5-6):639-647.
    We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.
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  9.  1
    A Variant of Shelah's Characterization of Strong Chang's Conjecture.Sean Cox & Hiroshi Sakai - 2019 - Mathematical Logic Quarterly 65 (2):251-257.
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  10.  2
    On the Set-Generic Multiverse.Sy-David Friedman, Sakaé Fuchino & Hiroshi Sakai - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 109-124.
    The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.In Sects. 2 and 3 of this note, we give a proof of Bukovsky’s theorem in a modern setting ). In Sect. 4 we check that the multiverse of set-generic extensions (...)
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  11.  8
    Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum.Sakaé Fuchino, André Ottenbreit Maschio Rodrigues & Hiroshi Sakai - 2021 - Archive for Mathematical Logic 60 (3):495-523.
    Continuing, we study the Strong Downward Löwenheim–Skolem Theorems of the stationary logic and their variations. In Fuchino et al. it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters \. This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size \. We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation (...)
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  12.  14
    Generalized Prikry Forcing and Iteration of Generic Ultrapowers.Hiroshi Sakai - 2005 - Mathematical Logic Quarterly 51 (5):507-523.
    It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters.
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  13.  12
    Semistationary Reection and Weak Square.Hiroshi Sakai - 2008 - Journal of Symbolic Logic 73 (1):181-192.