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  1. Destructibility of the Tree Property at ${\Aleph _{\Omega + 1}}$.Yair Hayut & Menachem Magidor - 2019 - Journal of Symbolic Logic 84 (2):621-631.
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  • The Tree Property at א Ω+2.Sy-David Friedman & Ajdin Halilović - 2011 - Journal of Symbolic Logic 76 (2):477 - 490.
    Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).
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  • The Tree Property at ℵ Ω+1.Dima Sinapova - 2012 - Journal of Symbolic Logic 77 (1):279-290.
    We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].
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  • The Tree Property and the Failure of the Singular Cardinal Hypothesis at ℵω2.Dima Sinapova - 2012 - Journal of Symbolic Logic 77 (3):934-946.
    We show that given ù many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω²+1 and the SCH fails at ℵ ω².
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  • The Tree Property Up to אω+1.Itay Neeman - 2014 - Journal of Symbolic Logic 79 (2):429-459.
  • Squares and Narrow Systems.Chris Lambie-Hanson - 2017 - Journal of Symbolic Logic 82 (3):834-859.
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  • A Remark on the Tree Property in a Choiceless Context.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (5-6):585-590.
    We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ ${{\rm ZF} + \neg{\rm AC}_\omega}$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the (...)
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  • Aronszajn Trees and the Successors of a Singular Cardinal.Spencer Unger - 2013 - Archive for Mathematical Logic 52 (5-6):483-496.
    From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis fails, there is a bad scale at κ and κ ++ has the tree property. In particular this model has no special κ +-trees.
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  • Regular Ultrafilters and Finite Square Principles.Juliette Kennedy, Saharon Shelah & Jouko Väänänen - 2008 - Journal of Symbolic Logic 73 (3):817-823.
    We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle $\square _{\lambda ,D}^{\mathit{fin}}$ introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ / D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which Mλ / D and Nλ / D are non-isomorphic. The question of the existence of such ultrafilters and models was (...)
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  • Fragility and Indestructibility II.Spencer Unger - 2015 - Annals of Pure and Applied Logic 166 (11):1110-1122.
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  • 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.
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  • $$I_0$$ I 0 and Combinatorics at $$\Lambda ^+$$ Λ +.Shi Xianghui & Trang Nam - 2017 - Archive for Mathematical Logic 56 (1-2):131-154.
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  • Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
  • The Tree Property and the Failure of SCH at Uncountable Cofinality.Dima Sinapova - 2012 - Archive for Mathematical Logic 51 (5-6):553-562.
    Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ +.
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  • On the Indestructibility Aspects of Identity Crisis.Grigor Sargsyan - 2009 - Archive for Mathematical Logic 48 (6):493-513.
    We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi–Magidor theorem from (in The independence between the concepts of compactness and supercompactness, circulated manuscript), i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact cardinals can be the first n measurable cardinals while the strong (...)
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  • The Tree Property Belowℵω⋅2.Spencer Unger - 2016 - Annals of Pure and Applied Logic 167 (3):247-261.
  • The Eightfold Way.James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot & Dima Sinapova - 2018 - Journal of Symbolic Logic 83 (1):349-371.
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  • Reflection of Stationary Sets and the Tree Property at the Successor of a Singular Cardinal.Laura Fontanella & Menachem Magidor - 2017 - Journal of Symbolic Logic 82 (1):272-291.
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  • The Tree Property at the Successor of a Singular Limit of Measurable Cardinals.Mohammad Golshani - 2018 - Archive for Mathematical Logic 57 (1-2):3-25.
    Assume \ is a singular limit of \ supercompact cardinals, where \ is a limit ordinal. We present two methods for arranging the tree property to hold at \ while making \ the successor of the limit of the first \ measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \ with the failure of SCH at \. This extends results of Neeman and Sinapova. The second method is also used to (...)
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  • Power-Like Models of Set Theory.Ali Enayat - 2001 - Journal of Symbolic Logic 66 (4):1766-1782.
    A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| for each a ∈ M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ 1 -like model. Coupled with Chang's two cardinal theorem this implies (...)
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  • The Strong Tree Property and Weak Square.Yair Hayut & Spencer Unger - 2017 - Mathematical Logic Quarterly 63 (1-2):150-154.
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  • Fragility and Indestructibility of the Tree Property.Spencer Unger - 2012 - Archive for Mathematical Logic 51 (5-6):635-645.
    We prove various theorems about the preservation and destruction of the tree property at ω 2. Working in a model of Mitchell [9] where the tree property holds at ω 2, we prove that ω 2 still has the tree property after ccc forcing of size ${\aleph_1}$ or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we prove from a supercompact cardinal that (...)
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  • Cellularity and the Structure of Pseudo-Trees.Jennifer Brown - 2007 - Journal of Symbolic Logic 72 (4):1093 - 1107.
    Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals cT, lT, ϕT, and μT: roughly, cT is the "tallness" of T; lT is the "width" of T; ϕ is the number of "points of finite branching" in T; and μ is the number of "sections of no branching" in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by (...)
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