18 found
Order:
  1.  12
    Chain Conditions of Products, and Weakly Compact Cardinals.Assaf Rinot - 2014 - Bulletin of Symbolic Logic 20 (3):293-314,.
    The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis of (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   11 citations  
  2.  6
    Knaster and Friends II: The C-Sequence Number.Chris Lambie-Hanson & Assaf Rinot - 2020 - Journal of Mathematical Logic 21 (1):2150002.
    Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  3.  10
    A Microscopic Approach to Souslin-Tree Constructions, Part I.Ari Meir Brodsky & Assaf Rinot - 2017 - Annals of Pure and Applied Logic 168 (11):1949-2007.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  4.  51
    A Relative of the Approachability Ideal, Diamond and Non-Saturation.Assaf Rinot - 2010 - Journal of Symbolic Logic 75 (3):1035-1065.
    Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   9 citations  
  5.  7
    More Notions of Forcing Add a Souslin Tree.Ari Meir Brodsky & Assaf Rinot - 2019 - Notre Dame Journal of Formal Logic 60 (3):437-455.
    An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree.In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  6.  5
    A Microscopic Approach to Souslin-Tree Construction, Part II.Ari Meir Brodsky & Assaf Rinot - 2021 - Annals of Pure and Applied Logic 172 (5):102904.
    In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  7.  8
    A Cofinality-Preserving Small Forcing May Introduce a Special Aronszajn Tree.Assaf Rinot - 2009 - Archive for Mathematical Logic 48 (8):817-823.
    It is relatively consistent with the existence of two supercompact cardinals that a special Aronszajn tree of height ${\aleph_{\omega_1+1}}$ is introduced by a cofinality-preserving forcing of size ${\aleph_3}$.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  8.  4
    Souslin Trees at Successors of Regular Cardinals.Assaf Rinot - 2019 - Mathematical Logic Quarterly 65 (2):200-204.
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark   1 citation  
  9.  11
    The Eightfold Way.James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot & Dima Sinapova - 2018 - Journal of Symbolic Logic 83 (1):349-371.
    Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  10.  5
    On Guessing Generalized Clubs at the Successors of Regulars.Assaf Rinot - 2011 - Annals of Pure and Applied Logic 162 (7):566-577.
    König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of a (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  11.  9
    On the Consistency Strength of the Milner–Sauer Conjecture.Assaf Rinot - 2006 - Annals of Pure and Applied Logic 140 (1):110-119.
    In their paper from 1981, Milner and Sauer conjectured that for any poset P,≤, if , then P must contain an antichain of cardinality κ. The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin’s Maximum and holds for all singular λ (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  12.  4
    Same Graph, Different Universe.Assaf Rinot - 2017 - Archive for Mathematical Logic 56 (7-8):783-796.
    May the same graph admit two different chromatic numbers in two different universes? How about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Gödel’s constructible universe, for every uncountable cardinal \ below the first fixed-point of the \-function, there exists a graph \ satisfying the following:\ has size and chromatic number \;for every infinite cardinal \, there exists a cofinality-preserving \-preserving forcing extension in which \=\kappa \).
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  13. Knaster and Friends III: Subadditive Colorings.Chris Lambie-Hanson & Assaf Rinot - forthcoming - Journal of Symbolic Logic:1-48.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  14. Sigma-Prikry Forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - forthcoming - Journal of Mathematical Logic.
    In Part I of this series [5], we introduced a class of notions of forcing which we call Σ-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardi...
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  15.  33
    Antichains in Partially Ordered Sets of Singular Cofinality.Assaf Rinot - 2007 - Archive for Mathematical Logic 46 (5-6):457-464.
    In their paper from 1981, Milner and Sauer conjectured that for any poset $\langle P,\le\rangle$ , if $cf(P,\le)=\lambda>cf(\lambda)=\kappa$ , then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark  
  16.  8
    Saharon Shelah. Middle Diamond. Archive for Mathematical Logic, Vol. 44 , Pp. 527–560. - Saharon Shelah. Diamonds. Proceedings of the American Mathematical Society, Vol. 138 , No. 6, Pp. 2151–2161. - Martin Zeman. Diamond, GCH and Weak Square. Proceedings of the American Mathematical Society, Vol. 138 , No. 5, Pp. 1853–1859. [REVIEW]Assaf Rinot - 2010 - Bulletin of Symbolic Logic 16 (3):420-423.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  17.  2
    On the Ideal J[Κ].Assaf Rinot - 2022 - Annals of Pure and Applied Logic 173 (2):103055.
  18.  1
    Sigma-Prikry Forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - forthcoming - Journal of Mathematical Logic:2150019.
    In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark