Results for 'Strongly compact cardinal'

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  1.  6
    More on the Least Strongly Compact Cardinal.Arthur W. Apter - 1997 - Mathematical Logic Quarterly 43 (3):427-430.
    We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2k supercompact.
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  2.  10
    Universal Indestructibility for Degrees of Supercompactness and Strongly Compact Cardinals.Arthur W. Apter & Grigor Sargsyan - 2008 - Archive for Mathematical Logic 47 (2):133-142.
    We establish two theorems concerning strongly compact cardinals and universal indestructibility for degrees of supercompactness. In the first theorem, we show that universal indestructibility for degrees of supercompactness in the presence of a strongly compact cardinal is consistent with the existence of a proper class of measurable cardinals. In the second theorem, we show that universal indestructibility for degrees of supercompactness is consistent in the presence of two non-supercompact strongly compact cardinals, each of (...)
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  3.  22
    How Large is the First Strongly Compact Cardinal? Or a Study on Identity Crises.Menachem Magidor - 1976 - Annals of Mathematical Logic 10 (1):33-57.
  4.  44
    Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori. Strong Axioms of Infinity and Elementary Embeddings. Annals of Mathematical Logic, Vol. 13 , Pp. 73–116. - Menachem Magidor. HOW Large is the First Strongly Compact Cardinal? Or A Study on Identity Crises. Annals of Mathematical Logic, Vol. 10 , Pp. 33–57. [REVIEW]Carlos Augusto Di Prisco - 1986 - Journal of Symbolic Logic 51 (4):1066-1068.
  5.  46
    Arthur W. Apter. On the Least Strongly Compact Cardinal. Israel Journal of Mathematics, Vol. 35 , Pp. 225–233. - Arthur W. Apter. Measurability and Degrees of Strong Compactness. The Journal of Symbolic Logic, Vol. 46 , Pp. 249–254. - Arthur W. Apter. A Note on Strong Compactness and Supercompactness. Bulletin of the London Mathematical Society, Vol. 23 , Pp. 113–115. - Arthur W. Apter. On the First N Strongly Compact Cardinals. Proceedings of the American Mathematical Society, Vol. 123 , Pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the Strong Equality Between Supercompactness and Strong Compactness.. Transactions of the American Mathematical Society, Vol. 349 , Pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' Result is Best Possible. Ibid., Pp. 2007–2034. - Arthur W. Apter. More on the Least Strongly Compact Cardinal. Mathematical Logic Quarterly, Vol. 43 , Pp. 427–430. - Arthur W. Apter. Laver Indestructibility and the Class of Compact Cardinals. The Journal of Sy. [REVIEW]James W. Cummings - 2000 - Bulletin of Symbolic Logic 6 (1):86-89.
  6. How Large is the First Strongly Compact Cardinal? Or: A Study on Identity Crises.M. Magidor - 1976 - Annals of Pure and Applied Logic 10 (1):33.
     
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  7. Department of Computer Science. Eotvos University, Rakoczi Ut 5, H-1088 Budapest VIII, Hungary, Kope@ Cs. Elte. Hu. Ten Papers by Arthur Apter on Large Cardinals Arthur W. After. On the Least Strongly Compact Cardinal. Israeljournal of Mathematics, Vol. 35 (1980). Pp. 225-233. [REVIEW]S. Aharon Shelah - 2000 - Bulletin of Symbolic Logic 6:86.
  8.  33
    The Least Measurable Can Be Strongly Compact and Indestructible.Arthur W. Apter & Moti Gitik - 1998 - Journal of Symbolic Logic 63 (4):1404-1412.
    We show the consistency, relative to a supercompact cardinal, of the least measurable cardinal being both strongly compact and fully Laver indestructible. We also show the consistency, relative to a supercompact cardinal, of the least strongly compact cardinal being somewhat supercompact yet not completely supercompact and having both its strong compactness and degree of supercompactness fully Laver indestructible.
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  9.  44
    Exactly Controlling the Non-Supercompact Strongly Compact Cardinals.Arthur W. Apter & Joel David Hamkins - 2003 - Journal of Symbolic Logic 68 (2):669-688.
    We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set (...)
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  10.  18
    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.
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  11.  48
    Diamond (on the Regulars) Can Fail at Any Strongly Unfoldable Cardinal.Mirna Džamonja & Joel David Hamkins - 2006 - Annals of Pure and Applied Logic 144 (1):83-95.
    If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which κ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
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  12.  21
    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 (...)
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  13.  26
    Tallness and Level by Level Equivalence and Inequivalence.Arthur W. Apter - 2010 - Mathematical Logic Quarterly 56 (1):4-12.
    We construct two models containing exactly one supercompact cardinal in which all non-supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals.
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  14.  37
    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 (...)
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  15.  13
    Some Remarks on Indestructibility and Hamkins? Lottery Preparation.Arthur W. Apter - 2003 - Archive for Mathematical Logic 42 (8):717-735.
    .In this paper, we first prove several general theorems about strongness, supercompactness, and indestructibility, along the way giving some new applications of Hamkins’ lottery preparation forcing to indestructibility. We then show that it is consistent, relative to the existence of cardinals κ<λ so that κ is λ supercompact and λ is inaccessible, for the least strongly compact cardinal κ to be the least strong cardinal and to have its strongness, but not its strong compactness, indestructible under (...)
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  16.  43
    On Some Questions Concerning Strong Compactness.Arthur W. Apter - 2012 - Archive for Mathematical Logic 51 (7-8):819-829.
    A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a (...) compact cardinal κ. We get a negative answer to the first of these questions and positive answers to the second of these questions for a supercompact cardinal κ in the context of the absence of the full Axiom of Choice. (shrink)
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  17.  16
    Universal Partial Indestructibility and Strong Compactness.Arthur W. Apter - 2005 - Mathematical Logic Quarterly 51 (5):524-531.
    For any ordinal δ, let λδ be the least inaccessible cardinal above δ. We force and construct a model in which the least supercompact cardinal κ is indestructible under κ-directed closed forcing and in which every measurable cardinal δ < κ is < λδ strongly compact and has its < λδ strong compactness indestructible under δ-directed closed forcing of rank less than λδ. In this model, κ is also the least strongly compact (...). We also establish versions of this result in which κ is the least strongly compact cardinal but is not supercompact. (shrink)
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  18.  35
    MRP , Tree Properties and Square Principles.Remi Strullu - 2011 - Journal of Symbolic Logic 76 (4):1441-1452.
    We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one (...)
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  19.  21
    Identity Crises and Strong Compactness.Arthur W. Apter & James Cummings - 2000 - Journal of Symbolic Logic 65 (4):1895-1910.
    Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals κ 1 ,..., κ n are so that κ i for i = 1,..., n is both the i th measurable cardinal and κ + i supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.
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  20.  37
    Level by Level Inequivalence Beyond Measurability.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (7-8):707-712.
    We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide.
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  21.  44
    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 (...)
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  22.  16
    Strong Compactness and a Global Version of a Theorem of Ben-David and Magidor.Arthur W. Apter - 2000 - Mathematical Logic Quarterly 46 (4):453-460.
    Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ+ strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, □γ+ω fails but □γ+ω, ω holds. This generalizes a result of Ben-David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.
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  23.  9
    Level by Level Equivalence and Strong Compactness.A. W. Apter - 2004 - Mathematical Logic Quarterly 50 (1):51.
    We force and construct models in which there are non-supercompact strongly compact cardinals which aren't measurable limits of strongly compact cardinals and in which level by level equivalence between strong compactness and supercompactness holds non-trivially except at strongly compact cardinals. In these models, every measurable cardinal κ which isn't either strongly compact or a witness to a certain phenomenon first discovered by Menas is such that for every regular cardinal λ (...)
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  24.  29
    On the Indestructibility Aspects of Identity Crisis.Grigor Sargsyan - 2009 - Archive for Mathematical Logic 48 (6):493-513.
    We investigate the indestructibility properties of strongly compact cardinals in universes where strong compactness suffers from identity crisis. We construct an iterative poset that can be used to establish Kimchi–Magidor theorem from (in The independence between the concepts of compactness and supercompactness, circulated manuscript), i.e., that the first n strongly compact cardinals can be the first n measurable cardinals. As an application, we show that the first n strongly compact cardinals can be the first (...)
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  25.  11
    Superatomic Boolean Algebras Constructed From Strongly Unbounded Functions.Juan Carlos Martínez & Lajos Soukup - 2011 - Mathematical Logic Quarterly 57 (5):456-469.
    Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation (...)
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  26.  74
    The Necessary Maximality Principle for C. C. C. Forcing is Equiconsistent with a Weakly Compact Cardinal.Joel D. Hamkins & W. Hugh Woodin - 2005 - Mathematical Logic Quarterly 51 (5):493-498.
    The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal.
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  27.  11
    Characterizing Strong Compactness Via Strongness.Arthur W. Apter - 2003 - Mathematical Logic Quarterly 49 (4):375.
    We construct a model in which the strongly compact cardinals can be non-trivially characterized via the statement “κ is strongly compact iff κ is a measurable limit of strong cardinals”. If our ground model contains large enough cardinals, there will be supercompact cardinals in the universe containing this characterization of the strongly compact cardinals.
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  28.  18
    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 (...)
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  29.  52
    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 (...)
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  30.  21
    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 (...)
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  31.  29
    Failures of SCH and Level by Level Equivalence.Arthur W. Apter - 2006 - Archive for Mathematical Logic 45 (7):831-838.
    We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is a stationary set of cardinals on which SCH fails. In this model, the structure of the class of supercompact cardinals can be arbitrary.
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  32.  18
    Failure of GCH and the Level by Level Equivalence Between Strong Compactness and Supercompactness.Arthur W. Apter - 2003 - Mathematical Logic Quarterly 49 (6):587.
    We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals.
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  33.  34
    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.
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  34.  16
    Supercompactness and Measurable Limits of Strong Cardinals II: Applications to Level by Level Equivalence.Arthur W. Apter - 2006 - Mathematical Logic Quarterly 52 (5):457-463.
    We construct models for the level by level equivalence between strong compactness and supercompactness in which for κ the least supercompact cardinal and δ ≤ κ any cardinal which is either a strong cardinal or a measurable limit of strong cardinals, 2δ > δ+ and δ is < 2δ supercompact. In these models, the structure of the class of supercompact cardinals can be arbitrary, and the size of the power set of κ can essentially be made as (...)
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  35.  24
    An Easton Theorem for Level by Level Equivalence.Arthur W. Apter - 2005 - Mathematical Logic Quarterly 51 (3):247-253.
    We establish an Easton theorem for the least supercompact cardinal that is consistent with the level by level equivalence between strong compactness and supercompactness. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals. We also briefly indicate how our methods of proof yield an Easton theorem that is consistent with the level by level equivalence between strong compactness and supercompactness in a (...)
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  36.  19
    Diamond, Square, and Level by Level Equivalence.Arthur W. Apter - 2004 - Archive for Mathematical Logic 44 (3):387-395.
    We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional combinatorial properties. In particular, in this model, ♦ δ holds for every regular uncountable cardinal δ, and below the least supercompact cardinal κ, □ δ holds on a stationary subset of κ. There are no restrictions in our model on the structure of the class of supercompact cardinals.
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  37.  22
    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.
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  38.  16
    Inaccessible Cardinals, Failures of GCH, and Level-by-Level Equivalence.Arthur W. Apter - 2014 - Notre Dame Journal of Formal Logic 55 (4):431-444.
    We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta $, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend (...)
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  39.  7
    Large Cardinals and Ramifiability for Directed Sets.R. Hinnion & O. Esser - 2000 - Mathematical Logic Quarterly 46 (1):25-34.
    The notion of “ramifiability” , usually applied to cardinals, can be extended to directed sets and is put in relation here with familiar “large cardinal” properties.
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  40.  11
    Indestructible Strong Compactness and Level by Level Inequivalence.Arthur W. Apter - 2013 - Mathematical Logic Quarterly 59 (4-5):371-377.
  41.  14
    An L-Like Model Containing Very Large Cardinals.Arthur W. Apter & James Cummings - 2008 - Archive for Mathematical Logic 47 (1):65-78.
    We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals.
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  42.  26
    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 (...)
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  43.  58
    Elementary Chains and C (N)-Cardinals.Konstantinos Tsaprounis - 2014 - Archive for Mathematical Logic 53 (1-2):89-118.
    The C (n)-cardinals were introduced recently by Bagaria and are strong forms of the usual large cardinals. For a wide range of large cardinal notions, Bagaria has shown that the consistency of the corresponding C (n)-versions follows from the existence of rank-into-rank elementary embeddings. In this article, we further study the C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardinals, giving some improved consistency bounds while, at the same time, addressing questions which had been left open. In addition, (...)
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  44.  31
    Localizing the Axioms.Athanassios Tzouvaras - 2010 - Archive for Mathematical Logic 49 (5):571-601.
    We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus ${\in}$ -induction scheme. ZFC+ “there is (...)
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  45.  46
    Generic Embeddings Associated to an Indestructibly Weakly Compact Cardinal.Gunter Fuchs - 2010 - Annals of Pure and Applied Logic 162 (1):89-105.
    I use generic embeddings induced by generic normal measures on that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower works much like it does when κ is a Woodin limit of (...)
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  46.  4
    Square Below a Non-Weakly Compact Cardinal.Hazel Brickhill - forthcoming - Archive for Mathematical Logic:1-18.
    In his seminal paper introducing the fine structure of L, Jensen proved that under \ any regular cardinal that reflects stationary sets is weakly compact. In this paper we give a new proof of Jensen’s result that is straight-forward and accessible to those without a knowledge of Jensen’s fine structure theory. The proof here instead uses hyperfine structure, a very natural and simpler alternative to fine structure theory introduced by Friedman and Koepke.
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  47.  18
    Proper Forcing Extensions and Solovay Models.Joan Bagaria & Roger Bosch - 2003 - Archive for Mathematical Logic 43 (6):739-750.
    We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength (...)
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  48.  5
    Partition Complete Boolean Algebras and Almost Compact Cardinals.Peter Jipsen & Henry Rose - 1999 - Mathematical Logic Quarterly 45 (2):241-255.
    For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated . It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ ≥ K. For (...)
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  49.  28
    Strongly Minimal Groups in the Theory of Compact Complex Spaces.Matthias Aschenbrenner, Rahim Moosa & Thomas Scanlon - 2006 - Journal of Symbolic Logic 71 (2):529 - 552.
    We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.
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  50.  11
    Tameness From Large Cardinal Axioms.Will Boney - 2014 - Journal of Symbolic Logic 79 (4):1092-1119.
    We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. (...)
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