15 found
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  1.  24
    A refinement of the Ramsey hierarchy via indescribability.Brent Cody - 2020 - Journal of Symbolic Logic 85 (2):773-808.
    We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $. By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide (...)
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  2.  22
    Characterizations of the weakly compact ideal on Pλ.Brent Cody - 2020 - Annals of Pure and Applied Logic 171 (6):102791.
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  3.  25
    Higher indescribability and derived topologies.Brent Cody - 2023 - Journal of Mathematical Logic 24 (1).
    We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of [Formula: see text]-indescribability and [Formula: see text]-indescribability of a cardinal [Formula: see text] for all [Formula: see text]. In this context, universal [Formula: see text] formulas exist, there is a normal ideal associated to [Formula: see text]-indescribability and the notions of [Formula: (...)
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  4.  15
    Ideal Operators and Higher Indescribability.Brent Cody & Peter Holy - forthcoming - Journal of Symbolic Logic:1-39.
    We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al., and by the first author.
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  5.  27
    The weakly compact reflection principle need not imply a high order of weak compactness.Brent Cody & Hiroshi Sakai - 2020 - Archive for Mathematical Logic 59 (1-2):179-196.
    The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \ then there is a forcing extension preserving (...)
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  6.  11
    Two-Cardinal Derived Topologies, Indescribability and Ramseyness.Brent Cody, Chris Lambie-Hanson & Jing Zhang - forthcoming - Journal of Symbolic Logic:1-29.
    We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived (...)
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  7.  15
    Forcing a □(κ)-like principle to hold at a weakly compact cardinal.Brent Cody, Victoria Gitman & Chris Lambie-Hanson - 2021 - Annals of Pure and Applied Logic 172 (7):102960.
  8.  11
    Adding a Nonreflecting Weakly Compact Set.Brent Cody - 2019 - Notre Dame Journal of Formal Logic 60 (3):503-521.
    For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...)
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  9.  54
    Easton’s theorem in the presence of Woodin cardinals.Brent Cody - 2013 - Archive for Mathematical Logic 52 (5-6):569-591.
    Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous (...)
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  10.  43
    The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact.Brent Cody, Moti Gitik, Joel David Hamkins & Jason A. Schanker - 2015 - Archive for Mathematical Logic 54 (5-6):491-510.
    We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}-supercompact, for any desired θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\theta}$$\end{document}. In addition, we prove several global results showing how the entire class of weakly compactcardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or (...)
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  11.  29
    Easton's theorem for Ramsey and strongly Ramsey cardinals.Brent Cody & Victoria Gitman - 2015 - Annals of Pure and Applied Logic 166 (9):934-952.
  12.  5
    Two-cardinal ideal operators and indescribability.Brent Cody & Philip White - 2024 - Annals of Pure and Applied Logic 175 (8):103463.
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  13.  41
    On supercompactness and the continuum function.Brent Cody & Menachem Magidor - 2014 - Annals of Pure and Applied Logic 165 (2):620-630.
    Given a cardinal κ that is λ-supercompact for some regular cardinal λ⩾κ and assuming GCH, we show that one can force the continuum function to agree with any function F:[κ,λ]∩REG→CARD satisfying ∀α,β∈domα F. Our argument extends Woodinʼs technique of surgically modifying a generic filter to a new case: Woodinʼs key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the (...)
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  14.  26
    Consecutive Singular Cardinals and the Continuum Function.Arthur W. Apter & Brent Cody - 2013 - Notre Dame Journal of Formal Logic 54 (2):125-136.
    We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ holds, $\kappa$ and $\kappa^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\mathrm (...)
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  15.  42
    The failure of GCH at a degree of supercompactness.Brent Cody - 2012 - Mathematical Logic Quarterly 58 (1):83-94.
    We determine the large cardinal consistency strength of the existence of a λ-supercompact cardinal κ such that equation image fails at λ. Indeed, we show that the existence of a λ-supercompact cardinal κ such that 2λ ≥ θ is equiconsistent with the existence of a λ-supercompact cardinal that is also θ-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.
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