86 found
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  1.  53
    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.
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  2.  3
    On the Consistency Strength of Level by Level Inequivalence.Arthur W. Apter - 2017 - Archive for Mathematical Logic 56 (7-8):715-723.
    We show that the theories “ZFC \ There is a supercompact cardinal” and “ZFC \ There is a supercompact cardinal \ Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent.
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  3.  11
    Identity Crises and Strong Compactness : II. Strong Cardinals.Arthur W. Apter & James Cummings - 2001 - Archive for Mathematical Logic 40 (1):25-38.
    . From a proper class of supercompact cardinals, we force and obtain a model in which the proper classes of strongly compact and strong cardinals precisely coincide. In this model, it is the case that no strongly compact cardinal \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} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^\kappa = \kappa^+$\end{document} supercompact.
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  4.  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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  5.  54
    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 (...)
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  6.  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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  7.  64
    Inner Models with Large Cardinal Features Usually Obtained by Forcing.Arthur W. Apter, Victoria Gitman & Joel David Hamkins - 2012 - Archive for Mathematical Logic 51 (3-4):257-283.
    We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ +, another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. (...)
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  8.  16
    Patterns of Compact Cardinals.Arthur W. Apter - 1997 - Annals of Pure and Applied Logic 89 (2-3):101-115.
    We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If (...)
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  9.  11
    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 (...)
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  10.  28
    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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  11.  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 is supercompact up to a (...)
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  12.  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 of measurable cardinals. These results improve and (...)
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  13.  38
    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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  14.  38
    Making All Cardinals Almost Ramsey.Arthur W. Apter & Peter Koepke - 2008 - Archive for Mathematical Logic 47 (7-8):769-783.
    We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...)
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  15.  22
    Coding Into HOD Via Normal Measures with Some Applications.Arthur W. Apter & Shoshana Friedman - 2011 - Mathematical Logic Quarterly 57 (4):366-372.
    We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  16.  17
    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.
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  17.  22
    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 (...)
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  18.  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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  19.  19
    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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  20.  37
    Indestructible Strong Compactness but Not Supercompactness.Arthur W. Apter, Moti Gitik & Grigor Sargsyan - 2012 - Annals of Pure and Applied Logic 163 (9):1237-1242.
  21.  11
    Some Results on Consecutive Large Cardinals.Arthur W. Apter - 1983 - Annals of Pure and Applied Logic 25 (1):1-17.
    We obtain 2 models in which AC is false and in which there are long sequences of consecutive large cardinals.
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  22.  12
    Indestructible Strong Compactness and Level by Level Inequivalence.Arthur W. Apter - 2013 - Mathematical Logic Quarterly 59 (4-5):371-377.
  23.  22
    A Global Version of a Theorem of Ben-David and Magidor.Arthur W. Apter & James Cummings - 2000 - Annals of Pure and Applied Logic 102 (3):199-222.
    We prove a consistency result about square principles and stationary reflection which generalises the result of Ben-David and Magidor [4].
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  24.  38
    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 (...)
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  25.  45
    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 (...)
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  26.  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 the strongly compact cardinals and the cardinals which (...)
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  27.  36
    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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  28.  14
    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 κ-strategically closed forcing.
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  29.  23
    James Cummings. A Model in Which GCH Holds at Successors but Fails at Limits. Transactions of the American Mathematical Society, Vol. 329 , Pp. 1–39. - James Cummings. Strong Ultrapowers and Long Core Models. The Journal of Symbolic Logic, Vol. 58 , Pp. 240–248. - James Cummings. Coherent Sequences Versus Radin Sequences. Annals of Pure and Applied Logic, Vol. 70 , Pp. 223–241. - James Cummings, Matthew Foreman, and Menachem Magidor. Squares, Scales and Stationary Reflection. Journal of Mathematical Logic, Vol. 1 , Pp. 35–98. [REVIEW]Arthur W. Apter - 2002 - Bulletin of Symbolic Logic 8 (4):550-552.
  30.  17
    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 large as desired. (...)
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  31.  20
    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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  32.  25
    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 universe (...)
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  33.  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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  34.  44
    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 strongly compact cardinal κ. We get a negative (...)
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  35.  37
    Ad and Patterns of Singular Cardinals Below Θ.Arthur W. Apter - 1996 - Journal of Symbolic Logic 61 (1):225-235.
    Using Steel's recent result that assuming AD, in L[R] below Θ, κ is regular $\operatorname{iff} \kappa$ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ 1.
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  36.  13
    The First Measurable Cardinal Can Be the First Uncountable Regular Cardinal at Any Successor Height.Arthur W. Apter, Ioanna M. Dimitriou & Peter Koepke - 2014 - Mathematical Logic Quarterly 60 (6):471-486.
  37.  11
    Reducing the Consistency Strength of an Indestructibility Theorem.Arthur W. Apter - 2008 - Mathematical Logic Quarterly 54 (3):288-293.
    Using an idea of Sargsyan, we show how to reduce the consistency strength of the assumptions employed to establish a theorem concerning a uniform level of indestructibility for both strong and supercompact cardinals.
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  38.  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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  39.  9
    A Cardinal Pattern Inspired by AD.Arthur W. Apter - 1996 - Mathematical Logic Quarterly 42 (1):211-218.
    Assuming Con, a model in which there are unboundedly many regular cardinals below Θ and in which the only regular cardinals below Θ are limit cardinals was previously constructed. Using a large cardinal hypothesis far beyond Con, we construct in this paper a model in which there is a proper class of regular cardinals and in which the only regular cardinals in the universe are limit cardinals.
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  40.  17
    Some Structural Results Concerning Supercompact Cardinals.Arthur W. Apter - 2001 - Journal of Symbolic Logic 66 (4):1919-1927.
    We show how the forcing of [5] can be iterated so as to get a model containing supercompact cardinals in which every measurable cardinal δ is δ + supercompact. We then apply this iteration to prove three additional theorems concerning the structure of the class of supercompact cardinals.
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  41.  7
    Strong Cardinals Can Be Fully Laver Indestructible.Arthur W. Apter - 2002 - Mathematical Logic Quarterly 48 (4):499-507.
    We prove three theorems which show that it is relatively consistent for any strong cardinal κ to be fully Laver indestructible under κ-directed closed forcing.
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  42.  15
    The Consistency Strength of $$\Aleph{\Omega}$$ and $$\Aleph_{{\Omega}1}$$ Being Rowbottom Cardinals Without the Axiom of Choice.Arthur W. Apter & Peter Koepke - 2006 - Archive for Mathematical Logic 45 (6):721-737.
    We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω 2-Rowbottom cardinal carrying an ω 2-Rowbottom filter and ω 1 is regular” has the same consistency strength as the theory “ZFC + There exist ω 1 measurable cardinals”. We (...)
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  43.  5
    [Omnibus Review].Arthur W. Apter - 2002 - Bulletin of Symbolic Logic 8 (4):550-552.
  44.  10
    On a Problem of Woodin.Arthur W. Apter - 2000 - Archive for Mathematical Logic 39 (4):253-259.
    A question of Woodin asks if $\kappa$ is strongly compact and GCH holds for all cardinals $\delta < \kappa$ , then must GCH hold everywhere. We get a negative answer to Woodin's question in the context of the negation of the Axiom of Choice.
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  45.  21
    Indestructibility, Measurability, and Degrees of Supercompactness.Arthur W. Apter - 2012 - Mathematical Logic Quarterly 58 (1):75-82.
    Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that A1 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ+ supercompact} is unbounded in κ. If in addition λ is 2λ supercompact, then A2 = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is δ+ supercompact} is unbounded in κ as well. The large cardinal (...)
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  46.  11
    Removing Laver Functions From Supercompactness Arguments.Arthur W. Apter - 2005 - Mathematical Logic Quarterly 51 (2):154.
    We show how the use of a Laver function in the proof of the consistency, relative to the existence of a supercompact cardinal, of both the Proper Forcing Axiom and the Semiproper Forcing Axiom can be eliminated via the use of lottery sums of the appropriate partial orderings.
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  47.  22
    A New Proof of a Theorem of Magidor.Arthur W. Apter - 2000 - Archive for Mathematical Logic 39 (3):209-211.
    We give a new proof using iterated Prikry forcing of Magidor's theorem that it is consistent to assume that the least strongly compact cardinal is the least supercompact cardinal.
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  48.  40
    The Consistency Strength of Choiceless Failures of SCH.Arthur W. Apter & Peter Koepke - 2010 - Journal of Symbolic Logic 75 (3):1066-1080.
    We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...)
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  49.  23
    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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  50.  12
    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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