17 found
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  1. Some Remarks on Openly Generated Boolean Algebras.Sakaé Fuchino - 1994 - Journal of Symbolic Logic 59 (1):302-310.
    A Boolean algebra B is said to be openly generated if {A: A ≤rc B, |A| = ℵ0} includes a club subset of [ B]ℵ0 . We show: (V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1). (MA+(σ-closed)). Every L∞ℵa -free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove (...)
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  2.  16
    Sticks and Clubs.Sakaé Fuchino, Saharon Shelah & Lajos Soukup - 1997 - Annals of Pure and Applied Logic 90 (1):57-77.
    We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club (...)
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  3.  23
    On L∞Κ-Free Boolean Algebras.Sakaé Fuchino, Sabine Koppelberg & Makoto Takahashi - 1992 - Annals of Pure and Applied Logic 55 (3):265-284.
    We study L∞κ-freeness in the variety of Boolean algebras. It is shown that some of the theorems on L∞κ-free algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean algebras. But we also investigate properties such as the ccc of L∞κ-free Boolean algebras which have no counterpart in the varieties above.
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  4.  4
    Strong Downward Löwenheim–Skolem Theorems for Stationary Logics, I.Sakaé Fuchino, André Ottenbreit Maschio Rodrigues & Hiroshi Sakai - forthcoming - Archive for Mathematical Logic:1-31.
    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 \ 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 Diagonal Reflection Principle for internally clubness of S. Cox. We (...)
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  5.  20
    On Potential Embedding and Versions of Martin's Axiom.Sakaé Fuchino - 1992 - Notre Dame Journal of Formal Logic 33 (4):481-492.
  6.  5
    The Set-Theoretic Multiverse as a Mathematical Plenitudinous Platonism Viewpoint.Sakaé Fuchino - 2012 - Annals of the Japan Association for Philosophy of Science 20:49-54.
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  7.  39
    On the Weak Freese–Nation Property of ?(Ω).Sakaé Fuchino, Stefan Geschke & Lajos Soukupe - 2001 - Archive for Mathematical Logic 40 (6):425-435.
    Continuing [6], [8] and [16], we study the consequences of the weak Freese-Nation property of (?(ω),⊆). Under this assumption, we prove that most of the known cardinal invariants including all of those appearing in Cichoń's diagram take the same value as in the corresponding Cohen model. Using this principle we could also strengthen two results of W. Just about cardinal sequences of superatomic Boolean algebras in a Cohen model. These results show that the weak Freese-Nation property of (?(ω),⊆) captures many (...)
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  8.  16
    Partial Orderings with the Weak Freese-Nation Property.Sakaé Fuchino, Sabine Koppelberg & Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (1):35-54.
    A partial ordering P is said to have the weak Freese-Nation property if there is a mapping tf : P → [P]0 such that, for any a, b ε P, if a b then there exists c ε tf∩tf such that a c b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, (...)
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  9.  11
    Models of Real-Valued Measurability.Sakae Fuchino, Noam Greenberg & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 142 (1):380-397.
    Solovay’s random-real forcing [R.M. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory , Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428] is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay’s model that do not follow from the existence of real-valued measurability.
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  10.  41
    Destructibility of Stationary Subsets of Pκλ.Sakaé Fuchino & Greg Piper - 2005 - Mathematical Logic Quarterly 51 (6):560-569.
    For a regular cardinal κ with κ<κ = κ and κ ≤ λ , we construct generically a subset S of {x ∈ Pκλ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense but the stationarity of S can be destroyed by a κ+-c. c. forcing ℙ* which does not add any new element of Pκλ . Actually ℙ* can be chosen so that ℙ* is κ-strategically closed. However we show that such (...)
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  11.  17
    On the Weak Freese-Nation Property of Complete Boolean Algebras.Sakaé Fuchino, Stefan Geschke, Saharon Shelah & Lajos Soukup - 2001 - Annals of Pure and Applied Logic 110 (1-3):89-105.
    The following results are proved: In a model obtained by adding ℵ 2 Cohen reals , there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. Modulo the consistency strength of a supercompact cardinal , the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. If a weak form of □ μ and cof =μ + hold for each μ >cf= ω , then the weak Freese-Nation property of 〈 P (...)
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  12.  6
    On< I> L_< Sub>∞ Κ-Free Boolean Algebras.Sakaé Fuchino, Sabine Koppelberg & Makoto Takahashi - 1992 - Annals of Pure and Applied Logic 55 (3):265-284.
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  13.  31
    On the Simplicity of the Automorphism Group ofP(Ω)/Fin.Sakaé Fuchino - 1992 - Archive for Mathematical Logic 31 (5):319-330.
    We prove that the automorphism group ofP(ω)/fin remains simple if ℵ2 Cohen reals are added to a model of ZFC+CH.
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  14.  7
    A Reflection Principle As a Reverse-Mathematical Fixed Point Over the Base Theory ZFC.Sakaé Fuchino - 2017 - Annals of the Japan Association for Philosophy of Science 25:67-77.
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  15.  4
    Mathematics and Set Theory:数学と集合論.Sakaé Fuchino - 2018 - Journal of the Japan Association for Philosophy of Science 46 (1):33-47.
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  16.  1
    [Challenged by [Challenged by [Challenged by the Incompleteness Theorems]]] - a Review of K. Godel, "Uber Fromal Unentscheidbare Satze der Principia Mathematica Und Verwandter Systeme I", Translated and Commented by Susumu Hayashi and Mariko Yasugi , / Kazuyuki Tanaka, "Challenged by Godel".Sakaé Fuchino - 2013 - Journal of the Japan Association for Philosophy of Science 41 (1):63-80.
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  17. 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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