Switch to: References

Add citations

You must login to add citations.
  1. Global Singularization and the Failure of SCH.Radek Honzik - 2010 - Annals of Pure and Applied Logic 161 (7):895-915.
    We say that κ is μ-hypermeasurable for a cardinal μ≥κ+ if there is an embedding j:V→M with critical point κ such that HV is included in M and j>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V* where F is realised on all V-regular (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • An Easton Like Theorem in the Presence of Shelah Cardinals.Mohammad Golshani - 2017 - Archive for Mathematical Logic 56 (3-4):273-287.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  • An Equiconsistency for Universal Indestructibility.Arthur W. Apter & Grigor Sargsyan - 2010 - Journal of Symbolic Logic 75 (1):314-322.
    We obtain an equiconsistency for a weak form of universal indestructibility for strongness. The equiconsistency is relative to a cardinal weaker in consistency strength than a Woodin cardinal. Stewart Baldwin's notion of hyperstrong cardinal. We also briefly indicate how our methods are applicable to universal indestructibility for supercompactness and strong compactness.
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Indestructibility of Vopěnka’s Principle.Andrew D. Brooke-Taylor - 2011 - Archive for Mathematical Logic 50 (5-6):515-529.
    Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  • Fragile Measurability.Joel Hamkins - 1994 - Journal of Symbolic Logic 59 (1):262-282.
    Laver [L] and others [G-S] have shown how to make the supercompactness or strongness of κ indestructible by a wide class of forcing notions. We show, alternatively, how to make these properties fragile. Specifically, we prove that it is relatively consistent that any forcing which preserves $\kappa^{<\kappa}$ and κ+, but not P(κ), destroys the measurability of κ, even if κ is initially supercompact, strong, or if I1(κ) holds. Obtained as an application of some general lifting theorems, this result is an (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   13 citations  
  • On Closed Unbounded Sets Consisting of Former Regulars.Moti Gitik - 1999 - Journal of Symbolic Logic 64 (1):1-12.
    A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Laver Sequences for Extendible and Super-Almost-Huge Cardinals.Paul Corazza - 1999 - Journal of Symbolic Logic 64 (3):963-983.
    Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  • Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness.Arthur W. Apter & Joel David Hamkins - 2002 - Journal of Symbolic Logic 67 (2):820-840.
    Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.
    Direct download (13 more)  
     
    Export citation  
     
    Bookmark   17 citations  
  • Projective Well-Orderings and Bounded Forcing Axioms.Andrés Eduardo Caicedo - 2005 - Journal of Symbolic Logic 70 (2):557-572.
    In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Projective Well-Orderings and Bounded Forcing Axioms.Andrés Eduardo Caicedo - 2005 - Journal of Symbolic Logic 70 (2):557 - 572.
    In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark  
  • The Axiom of Infinity and Transformations J: V → V.Paul Corazza - 2010 - Bulletin of Symbolic Logic 16 (1):37-84.
    We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  • The Wholeness Axiom and Laver Sequences.Paul Corazza - 2000 - Annals of Pure and Applied Logic 105 (1-3):157-260.
    In this paper we introduce the Wholeness Axiom , which asserts that there is a nontrivial elementary embedding from V to itself. We formalize the axiom in the language {∈, j } , adding to the usual axioms of ZFC all instances of Separation, but no instance of Replacement, for j -formulas, as well as axioms that ensure that j is a nontrivial elementary embedding from the universe to itself. We show that WA has consistency strength strictly between I 3 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   13 citations  
  • Killing the $GCH$ Everywhere with a Single Real.Sy-David Friedman & Mohammad Golshani - 2013 - Journal of Symbolic Logic 78 (3):803-823.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  • Kurepa Trees and Namba Forcing.Bernhard König & Yasuo Yoshinobu - 2012 - Journal of Symbolic Logic 77 (4):1281-1290.
    We show that strongly compact cardinals and MM are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding ‘regressive' $\lambda$-Kurepa trees in either case. We argue that the destruction of regressive Kurepa trees requires a non-standard application of MM. As a corollary, we find a consistent example of an $\omega_2$-closed poset that is not forcing equivalent to any $\omega_2$-directed-closed poset.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • On Foreman’s Maximality Principle.Mohammad Golshani & Yair Hayut - 2016 - Journal of Symbolic Logic 81 (4):1344-1356.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Identity Crises and Strong Compactness III: Woodin Cardinals. [REVIEW]Arthur W. Apter & Grigor Sargsyan - 2005 - Archive for Mathematical Logic 45 (3):307-322.
    We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the n th strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Indestructibility, Instances of Strong Compactness, and Level by Level Inequivalence.Arthur W. Apter - 2010 - Archive for Mathematical Logic 49 (7-8):725-741.
    Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ + strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which ${A = \emptyset}$ . The first of these contains a supercompact cardinal κ and (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • The Lottery Preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
    The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   49 citations  
  • More on Simple Forcing Notions and Forcings with Ideals.M. Gitik & S. Shelah - 1993 - Annals of Pure and Applied Logic 59 (3):219-238.
    It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • A Laver-Like Indestructibility for Hypermeasurable Cardinals.Radek Honzik - 2019 - Archive for Mathematical Logic 58 (3-4):275-287.
    We show that if \ is \\)-hypermeasurable for some cardinal \ with \ \le \mu \) and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model \ in which the \\)-hypermeasurability of \ is indestructible by the Cohen forcing at \ of any length up to \ is \\)-hypermeasurable in \). The preservation of hypermeasurability is useful for subsequent arguments. The construction of \ is based on the ideas of Woodin and Cummings :1–39, (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Normal Measures on a Tall Cardinal.Arthur W. Apter & James Cummings - 2019 - Journal of Symbolic Logic 84 (1):178-204.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • The Tree Property at Double Successors of Singular Cardinals of Uncountable Cofinality.Mohammad Golshani & Rahman Mohammadpour - 2018 - Annals of Pure and Applied Logic 169 (2):164-175.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  • Indestructibility and Measurable Cardinals with Few and Many Measures.Arthur W. Apter - 2008 - Archive for Mathematical Logic 47 (2):101-110.
    If κ < λ are such that κ is indestructibly supercompact and λ is measurable, then we show that both A = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries the maximal number of normal measures} and B = {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ carries fewer than the maximal number of normal measures} are unbounded (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • Laver and Set Theory.Akihiro Kanamori - 2016 - Archive for Mathematical Logic 55 (1-2):133-164.
  • Reflecting Stationary Sets and Successors of Singular Cardinals.Saharon Shelah - 1991 - Archive for Mathematical Logic 31 (1):25-53.
    REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   28 citations  
  • A Note on Tall Cardinals and Level by Level Equivalence.Arthur W. Apter - 2016 - Mathematical Logic Quarterly 62 (1-2):128-132.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Indestructibility Properties of Remarkable Cardinals.Yong Cheng & Victoria Gitman - 2015 - Archive for Mathematical Logic 54 (7-8):961-984.
  • The Least Strongly Compact Can Be the Least Strong and Indestructible.Arthur W. Apter - 2006 - Annals of Pure and Applied Logic 144 (1):33-42.
    We construct two models in which the least strongly compact cardinal κ is also the least strong cardinal. In each of these models, κ satisfies indestructibility properties for both its strong compactness and strongness.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Superstrong and Other Large Cardinals Are Never Laver Indestructible.Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba - 2016 - Archive for Mathematical Logic 55 (1-2):19-35.
  • Indestructibility and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2007 - Mathematical Logic Quarterly 53 (1):78-85.
    If κ < λ are such that κ is indestructibly supercompact and λ is 2λ supercompact, it is known from [4] that {δ < κ | δ is a measurable cardinal which is not a limit of measurable cardinals and δ violates level by level equivalence between strong compactness and supercompactness}must be unbounded in κ. On the other hand, using a variant of the argument used to establish this fact, it is possible to prove that if κ < λ are (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  • Tall Cardinals.Joel D. Hamkins - 2009 - Mathematical Logic Quarterly 55 (1):68-86.
    A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of a strong cardinal. Any tall cardinal (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  • Weak Square Bracket Relations for P Κ (Λ).Pierre Matet - 2008 - Journal of Symbolic Logic 73 (3):729-751.
    We study the partition relation $X@>{\rm w}>>[Y]_{p}^{2}$ that is a weakening of the usual partition relation $X\rightarrow [Y]_{p}^{2}$ . Our main result asserts that if κ is an uncountable strongly compact cardinal and $\germ{d}_{\kappa}\leq \lambda ^{<\kappa}$ , then $I_{\kappa,\lambda}^{+}@>{\rm w}>>[I_{\kappa,\lambda}^{+}]_{\lambda <\kappa}^{2}$ does not hold.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark  
  • Supercompactness and Level by Level Equivalence Are Compatible with Indestructibility for Strong Compactness.Arthur W. Apter - 2007 - Archive for Mathematical Logic 46 (3-4):155-163.
    It is known that if $\kappa < \lambda$ are such that κ is indestructibly supercompact and λ is 2λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  • Large Cardinals Need Not Be Large in HOD.Yong Cheng, Sy-David Friedman & Joel David Hamkins - 2015 - Annals of Pure and Applied Logic 166 (11):1186-1198.
  • Singular Cardinals and the Pcf Theory.Thomas Jech - 1995 - Bulletin of Symbolic Logic 1 (4):408-424.
  • Indestructibility, HOD, and the Ground Axiom.Arthur W. Apter - 2011 - Mathematical Logic Quarterly 57 (3):261-265.
    Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ κ is inaccessible. We show it is also (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Some New Upper Bounds in Consistency Strength for Certain Choiceless Large Cardinal Patterns.Arthur W. Apter - 1992 - Archive for Mathematical Logic 31 (3):201-205.
    In this paper, we show that certain choiceless models of ZF originally constructed using an almost huge cardinal can be constructed using cardinals strictly weaker in consistency strength.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  • Strongly Unfoldable Cardinals Made Indestructible.Thomas A. Johnstone - 2008 - Journal of Symbolic Logic 73 (4):1215-1248.
    I provide indestructibility results for large cardinals consistent with V = L, such as weakly compact, indescribable and strongly unfoldable cardinals. The Main Theorem shows that any strongly unfoldable cardinal κ can be made indestructible by <κ-closed. κ-proper forcing. This class of posets includes for instance all <κ-closed posets that are either κ -c.c, or ≤κ-strategically closed as well as finite iterations of such posets. Since strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, the Main Theorem therefore makes (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   9 citations  
  • Indestructibility and Stationary Reflection.Arthur W. Apter - 2009 - Mathematical Logic Quarterly 55 (3):228-236.
    If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ -strategically closed forcing and λ is weakly compact, then we show thatA = {δ < κ | δ is a non-weakly compact Mahlo cardinal which reflects stationary sets}must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  • Indestructibility Under Adding Cohen Subsets and Level by Level Equivalence.Arthur W. Apter - 2009 - Mathematical Logic Quarterly 55 (3):271-279.
    We construct a model for the level by level equivalence between strong compactness and supercompactness in which the least supercompact cardinal κ has its strong compactness indestructible under adding arbitrarily many Cohen subsets. There are no restrictions on the large cardinal structure of our model.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation