6 found
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Jonathan Verner [3]Jonathan L. Verner [3]
  1.  24
    Towers in filters, cardinal invariants, and luzin type families.Jörg Brendle, Barnabás Farkas & Jonathan Verner - 2018 - Journal of Symbolic Logic 83 (3):1013-1062.
    We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection. We prove the following results:Many classical examples of nice tall filters contain no towers.It is consistent that tall analytic P-filters contain towers of arbitrary regular height.It is consistent that all towers generate nonmeager filters, in particular Borel filters do not contain towers.The statement “Every ultrafilter contains towers.” is independent of ZFC.Furthermore, we study many possible logical implications between the existence of towers in filters, (...)
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  2.  29
    Set-theoretic blockchains.Miha E. Habič, Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner & Kameryn J. Williams - 2019 - Archive for Mathematical Logic 58 (7-8):965-997.
    Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings (...)
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  3.  19
    Completely separable mad families and the modal logic of βω.Tomáš Lávička & Jonathan L. Verner - 2022 - Journal of Symbolic Logic 87 (2):498-507.
    We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding in [4], where (...)
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  4.  27
    Completely separable mad families and the modal logic of.Tomáš Lávička & Jonathan L. Verner - 2020 - Journal of Symbolic Logic:1-10.
    We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $. In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $. This improves the results of G. Bezhanishvili and J. Harding in [4], where (...)
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  5.  26
    A Lifting Argument for the Generalized Grigorieff Forcing.Radek Honzík & Jonathan Verner - 2016 - Notre Dame Journal of Formal Logic 57 (2):221-231.
    In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal $\kappa$ from the optimal hypothesis, while adding new unbounded subsets to $\kappa$. In some ways these forcings are closer to the Cohen-type forcings—we show that they are not minimal—but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.
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  6.  6
    Lower Bounds of Sets of P-points.Borisa Kuzeljevic, Dilip Raghavan & Jonathan L. Verner - 2023 - Notre Dame Journal of Formal Logic 64 (3):317-327.
    We show that MAκ implies that each collection of Pc-points of size at most κ which has a Pc-point as an RK upper bound also has a Pc-point as an RK lower bound.
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