7 found
Order:
  1.  7
    The Tree Property at First and Double Successors of Singular Cardinals with an Arbitrary Gap.Alejandro Poveda - 2020 - Annals of Pure and Applied Logic 171 (5):102778.
  2.  5
    More on the Preservation of Large Cardinals Under Class Forcing.Joan Bagaria & Alejandro Poveda - forthcoming - Journal of Symbolic Logic:1-34.
    We prove two general results about the preservation of extendible and $C^{}$ -extendible cardinals under a wide class of forcing iterations. As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{}$ -extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{}$ -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$ -definable behaviour of the power-set function on regular cardinals. We show (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  3.  11
    Identity Crisis Between Supercompactness and Vǒpenka’s Principle.Yair Hayut, Menachem Magidor & Alejandro Poveda - 2022 - Journal of Symbolic Logic 87 (2):626-648.
    In this paper we study the notion of $C^{}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{}$ -supercompact but also that the least supercompact is $C^{}$ -supercompact }$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  4. Sigma-Prikry Forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - forthcoming - Journal of Mathematical Logic.
    Journal of Mathematical Logic, Ahead of Print. In Part I of this series [5], we introduced a class of notions of forcing which we call [math]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [math]-Prikry. We proved that given a [math]-Prikry poset [math] and a [math]-name for a nonreflecting stationary set [math], there exists a corresponding [math]-Prikry poset that projects to [math] and kills the stationarity of [math]. In (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  5.  1
    Contributions to the Theory of Large Cardinals Through the Method of Forcing.Alejandro Poveda - 2021 - Bulletin of Symbolic Logic 27 (2):221-222.
    The dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle. In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics.We commence Part I by investigating the Identity Crisis phenomenon in the region comprised (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  6.  7
    The Tree Property at Double Successors of Singular Cardinals of Uncountable Cofinality with Infinite Gaps.Mohammad Golshani & Alejandro Poveda - 2021 - Annals of Pure and Applied Logic 172 (1):102853.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  7.  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