80 found
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  1.  23
    Arthur W. Apter & Joel David Hamkins (2002). Indestructibility and the Level-by-Level Agreement Between Strong Compactness and Supercompactness. 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.  12
    Arthur W. Apter, Moti Gitik & Grigor Sargsyan (2012). Indestructible Strong Compactness but Not Supercompactness. Annals of Pure and Applied Logic 163 (9):1237-1242.
  3.  10
    Arthur W. Apter & Moti Gitik (1998). The Least Measurable Can Be Strongly Compact and Indestructible. 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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  4.  1
    Arthur W. Apter (2007). Indestructibility and Level by Level Equivalence and Inequivalence. 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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  5.  6
    Arthur W. Apter & Shoshana Friedman (2011). Coding Into HOD Via Normal Measures with Some Applications. 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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  6.  6
    Arthur W. Apter (2008). Indestructibility and Measurable Cardinals with Few and Many Measures. 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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  7.  19
    Arthur W. Apter (2012). On Some Questions Concerning Strong Compactness. 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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  8.  14
    Arthur W. Apter (2009). Indestructibility and Stationary Reflection. 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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  9.  7
    Arthur W. Apter (2006). Failures of SCH and Level by Level Equivalence. 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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  10.  3
    Arthur W. Apter (1997). Patterns of Compact Cardinals. 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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  11.  25
    Arthur W. Apter & Joel David Hamkins (2003). Exactly Controlling the Non-Supercompact Strongly Compact Cardinals. 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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  12.  10
    Arthur W. Apter (2007). Supercompactness and Level by Level Equivalence Are Compatible with Indestructibility for Strong Compactness. 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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  13.  17
    Arthur W. Apter (2010). Indestructibility, Instances of Strong Compactness, and Level by Level Inequivalence. 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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  14.  20
    Arthur W. Apter (2011). A Remark on the Tree Property in a Choiceless Context. Archive for Mathematical Logic 50 (5-6):585-590.
    We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ ${{\rm ZF} + \neg{\rm AC}_\omega}$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the (...)
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  15.  14
    Arthur W. Apter & James Cummings (2000). Identity Crises and Strong Compactness. 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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  16.  19
    Arthur W. Apter & Peter Koepke (2008). Making All Cardinals Almost Ramsey. 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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  17.  9
    Arthur W. Apter (2011). Indestructibility, HOD, and the Ground Axiom. 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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  18.  15
    Arthur W. Apter (1996). Ad and Patterns of Singular Cardinals Below Θ. 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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  19.  3
    Arthur W. Apter (forthcoming). A Note on Tall Cardinals and Level by Level Equivalence. Mathematical Logic Quarterly:n/a-n/a.
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  20.  4
    Arthur W. Apter (2006). The Least Strongly Compact Can Be the Least Strong and Indestructible. 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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  21.  4
    Arthur W. Apter (2016). Indestructibility and Destructible Measurable Cardinals. Archive for Mathematical Logic 55 (1-2):3-18.
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  22.  30
    Arthur W. Apter, Victoria Gitman & Joel David Hamkins (2012). Inner Models with Large Cardinal Features Usually Obtained by Forcing. 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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  23.  4
    Arthur W. Apter & James Cummings (2001). Identity Crises and Strong Compactness. Archive for Mathematical Logic 40 (1):25-38.
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  24.  3
    Arthur W. Apter (2000). On a Problem of Woodin. 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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  25.  10
    Arthur W. Apter (2001). Some Structural Results Concerning Supercompact Cardinals. 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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  26.  6
    Arthur W. Apter & James Cummings (2008). An L-Like Model Containing Very Large Cardinals. 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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  27.  16
    Arthur W. Apter (1990). Successors of Singular Cardinals and Measurability Revisited. Journal of Symbolic Logic 55 (2):492-501.
  28.  9
    Arthur W. Apter (2005). An Easton Theorem for Level by Level Equivalence. 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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  29.  37
    Arthur W. Apter & James Cummings (2002). Blowing Up the Power Set of the Least Measurable. Journal of Symbolic Logic 67 (3):915-923.
    We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of [1] by using the optimal hypothesis.
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  30.  7
    Arthur W. Apter (2001). Supercompactness and Measurable Limits of Strong Cardinals. Journal of Symbolic Logic 66 (2):629-639.
    In this paper, two theorems concerning measurable limits of strong cardinals and supercompactness are proven. This generalizes earlier work, both individual and joint with Shelah.
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  31.  3
    Arthur W. Apter (2008). Reducing the Consistency Strength of an Indestructibility Theorem. 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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  32.  3
    Arthur W. Apter (2003). Some Remarks on Indestructibility and Hamkins' Lottery Preparation. Archive for Mathematical Logic 42 (8):717-735.
  33.  17
    Arthur W. Apter (2011). Level by Level Inequivalence Beyond Measurability. 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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  34.  6
    Arthur W. Apter & Peter Koepke (2006). The Consistency Strength of Aleph{Omega} and Aleph_{{Omega}1} Being Rowbottom Cardinals Without the Axiom of Choice. 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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  35.  7
    Arthur W. Apter (2009). Indestructibility Under Adding Cohen Subsets and Level by Level Equivalence. 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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  36.  7
    Arthur W. Apter (2000). A New Proof of a Theorem of Magidor. 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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  37. Arthur W. Apter (2006). Supercompactness and Measurable Limits of Strong Cardinals II: Applications to Level by Level Equivalence. 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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  38.  11
    Arthur W. Apter (2012). Indestructibility, Measurability, and Degrees of Supercompactness. 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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  39.  7
    Arthur W. Apter (1998). Laver Indestructibility and the Class of Compact Cardinals. Journal of Symbolic Logic 63 (1):149-157.
    Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in (...)
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  40.  10
    Arthur W. Apter (1999). On Measurable Limits of Compact Cardinals. Journal of Symbolic Logic 64 (4):1675-1688.
    We extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and (...)
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  41.  6
    Arthur W. Apter (1981). Measurability and Degrees of Strong Compactness. Journal of Symbolic Logic 46 (2):249-254.
    We prove, relative to suitable hypotheses, that it is consistent for there to be unboundedly many measurable cardinals each of which possesses a large degree of strong compactness, and that it is consistent to assume that the least measurable is partially strongly compact and that the second measurable is strongly compact. These results partially answer questions of Magidor on the relationship of strong compactness to measurability.
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  42.  5
    Arthur W. Apter (2010). How Many Normal Measures Can ℵ Ω 1+1 Carry? Mathematical Logic Quarterly 56 (2):164-170.
    Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for ℵ1 to be regular and for ℵmath image to be measurable and to carry precisely τ normal measures, where τ ≥ ℵmath image is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for ℵω +1 using the same hypotheses.
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  43.  5
    Arthur W. Apter (1992). Some New Upper Bounds in Consistency Strength for Certain Choiceless Large Cardinal Patterns. 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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  44.  26
    Arthur W. Apter & Peter Koepke (2010). The Consistency Strength of Choiceless Failures of SCH. 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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  45.  5
    Arthur W. Apter (1985). An AD-Like Model. Journal of Symbolic Logic 50 (2):531-543.
  46. Arthur W. Apter & James Cummings (2000). A Global Version of a Theorem of Ben-David and Magidor. 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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  47.  5
    Arthur W. Apter (2005). Diamond, Square, and Level by Level Equivalence. 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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  48.  1
    Arthur W. Apter (1996). A Cardinal Pattern Inspired by AD. 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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  49.  7
    Arthur W. Apter (2010). Tallness and Level by Level Equivalence and Inequivalence. 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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  50.  3
    Arthur W. Apter (2015). A Universal Indestructibility Theorem Compatible with Level by Level Equivalence. Archive for Mathematical Logic 54 (3-4):463-470.
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