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  1. Closed Maximality Principles and Generalized Baire Spaces.Philipp Lücke - 2019 - Notre Dame Journal of Formal Logic 60 (2):253-282.
    Given an uncountable regular cardinal κ, we study the structural properties of the class of all sets of functions from κ to κ that are definable over the structure 〈H,∈〉 by a Σ1-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of ZFC together with large cardinal axioms. In this paper, we present other canonical extensions of ZFC that provide a strong structure theory for these classes. These axioms are (...)
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  • Special subsets of the generalized Cantor space and generalized Baire space.Michał Korch & Tomasz Weiss - 2020 - Mathematical Logic Quarterly 66 (4):418-437.
    In this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver‐null sets. (...)
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  • κ‐Madness and definability.Haim Horowitz & Saharon Shelah - 2022 - Mathematical Logic Quarterly 68 (3):346-351.
    Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal κ, there are no κ‐mad families.
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  • A null ideal for inaccessibles.Sy-David Friedman & Giorgio Laguzzi - 2017 - Archive for Mathematical Logic 56 (5-6):691-697.
    In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^\kappa $$\end{document}, κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} inaccessible, and study its associated ideal of null sets and notion of measurability. This issue was addressed by Shelah ), arXiv:0904.0817, Problem 0.5) and concerns the definition of a forcing which is κκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
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  • More zfc inequalities between cardinal invariants.Vera Fischer & Dániel T. Soukup - 2021 - Journal of Symbolic Logic 86 (3):897-912.
    Motivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa =\lambda ^+$ for some $\lambda \geq \omega $ and $\mathfrak {b}=\kappa ^+$ then $\mathfrak {a}_e=\mathfrak {a}_p=\kappa ^+$. If, additionally, $2^{<\lambda }=\lambda $ then $\mathfrak {a}_g=\kappa ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak {d}$ in terms of (...)
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  • Strong Measure Zero Sets on for Inaccessible.Nick Steven Chapman & Johannes Philipp Schürz - forthcoming - Journal of Symbolic Logic:1-31.
    We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence (...)
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  • Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions.David Asperó, Tapani Hyttinen, Vadim Kulikov & Miguel Moreno - 2019 - Notre Dame Journal of Formal Logic 60 (4):665-682.
    Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal κ. We show the consistency of Eλ-clubλ++,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλλ++ in the space λ++, being continuously reducible to Eλ+-club2,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλ+λ++ in the space 2λ++. Then we show that for κ ineffable Ereg2,κ, the relation of equivalence modulo (...)
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  • Higher dimensional cardinal characteristics for sets of functions.Corey Bacal Switzer - 2022 - Annals of Pure and Applied Logic 173 (1):103031.
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  • The cofinality of the strong measure zero ideal for κ inaccessible.Johannes Philipp Schürz - 2023 - Mathematical Logic Quarterly 69 (1):31-39.
    We investigate the cofinality of the strong measure zero ideal for κ inaccessible and show that it is independent of the size of 2κ.
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