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  1. Arthur W. Apter (2013). Indestructible Strong Compactness and Level by Level Inequivalence. Mathematical Logic Quarterly 59 (4-5):371-377.
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  2. Arthur W. Apter & Brent Cody (2013). Consecutive Singular Cardinals and the Continuum Function. Notre Dame Journal of Formal Logic 54 (2):125-136.
    We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ holds, $\kappa$ and $\kappa^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\mathrm (...)
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  3. Arthur W. Apter (2012). Indestructibility, Measurability, and Degrees of Supercompactness. Mathematical Logic Quarterly 58 (1):75-82.
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  4. 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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  5. Arthur W. Apter, Moti Gitik & Grigor Sargsyan (2012). Indestructible Strong Compactness but Not Supercompactness. Annals of Pure and Applied Logic 163 (9):1237-1242.
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  6. 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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  7. 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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  8. Arthur W. Apter (2011). Indestructibility, HOD, and the Ground Axiom. Mathematical Logic Quarterly 57 (3):261-265.
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  9. 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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  10. Arthur W. Apter & Shoshana Friedman (2011). Coding Into HOD Via Normal Measures with Some Applications. Mathematical Logic Quarterly 57 (4):366-372.
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  11. Arthur W. Apter (2010). How Many Normal Measures Can W1+1 Carry? Mathematical Logic Quarterly 56 (2):164-170.
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  12. Arthur W. Apter (2010). How Many Normal Measures Can ℵ Ω 1+1 Carry? Mathematical Logic Quarterly 56 (2):164-170.
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  13. 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. Arthur W. Apter (2010). Tallness and Level by Level Equivalence and Inequivalence. Mathematical Logic Quarterly 56 (1):4-12.
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  15. 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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  16. Arthur W. Apter & Grigor Sargsyan (2010). An Equiconsistency for Universal Indestructibility. 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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  17. Arthur W. Apter (2009). Indestructibility and Stationary Reflection. Mathematical Logic Quarterly 55 (3):228-236.
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  18. Arthur W. Apter (2009). Indestructibility Under Adding Cohen Subsets and Level by Level Equivalence. Mathematical Logic Quarterly 55 (3):271-279.
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  19. 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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  20. Arthur W. Apter (2008). Reducing the Consistency Strength of an Indestructibility Theorem. Mathematical Logic Quarterly 54 (3):288-293.
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  21. 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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  22. 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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  23. Arthur W. Apter & Grigor Sargsyan (2008). Universal Indestructibility for Degrees of Supercompactness and Strongly Compact Cardinals. 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 which exhibits a significant amount of indestructibility (...)
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  24. Arthur W. Apter (2007). Indestructibility and Level by Level Equivalence and Inequivalence. Mlq 53 (1):78-85.
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  25. 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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  26. 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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  27. 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.
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  28. 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.
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  29. 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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  30. Arthur W. Apter & Peter Koepke (2006). The Consistency Strength of InlineEquation ID=" IEq1"> EquationSource Format=" TEX"> ImageObject Color=" BlackWhite" FileRef=" 15320065ArticleIEq1. Gif" Format=" GIF" Rendition=" HTM" Type=" Linedraw"/> And. [REVIEW] Archive for Mathematical Logic 45 (6):721-738.
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  31. Arthur W. Apter & Grigor Sargsyan (2006). Identity Crises and Strong Compactness III: Woodin Cardinals. [REVIEW] 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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  32. Arthur W. Apter (2005). An Easton Theorem for Level by Level Equivalence. Mathematical Logic Quarterly 51 (3):247-253.
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  33. 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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  34. Arthur W. Apter (2005). On a Problem of Foreman and Magidor. Archive for Mathematical Logic 44 (4):493-498.
    A question of Foreman and Magidor asks if it is consistent for every sequence of stationary subsets of the ℵ n ’s for 1≤n<ω to be mutually stationary. We get a positive answer to this question in the context of the negation of the Axiom of Choice. We also indicate how a positive answer to a generalized version of this question in a choiceless context may be obtained.
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  35. Arthur W. Apter (2005). Universal Partial Indestructibility and Strong Compactness. Mathematical Logic Quarterly 51 (5):524-531.
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  36. Arthur W. Apter & Grigor Sargsyan (2004). Jonsson-Like Partition Relations and J: V → V. Journal of Symbolic Logic 69 (4):1267 - 1281.
    Working in the theory "ZF + There is a nontrivial elementary embedding j : V $\rightarrow$ V", we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal $\mu (...)
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  37. Arthur W. Apter (2003). Characterizing Strong Compactness Via Strongness. Mathematical Logic Quarterly 49 (4):375.
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  38. Arthur W. Apter (2003). Some Remarks on Indestructibility and Hamkins' Lottery Preparation. 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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  39. 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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  40. Arthur W. Apter (2002). Aspects of Strong Compactness, Measurability, and Indestructibility. Archive for Mathematical Logic 41 (8):705-719.
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  41. Arthur W. Apter (2002). [Omnibus Review]. Bulletin of Symbolic Logic 8 (4):550-552.
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  42. Arthur W. Apter (2002). Review of J. Cummings, A Model in Which GCH Holds at Successors but Fails at Limits; Strong Ultrapowers and Long Core Models; Coherent Sequences Versus Radin Sequences; and J. Cummings, M. Foreman, and M. Magidor, Squares, Scales and Stationary Reflection. [REVIEW] Bulletin of Symbolic Logic 8 (4):550-552.
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  43. 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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  44. 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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  45. J. Cummings & Arthur W. Apter (2002). REVIEWS-Four Papers. Bulletin of Symbolic Logic 8 (4):550-551.
     
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  46. Arthur W. Apter (2001). REVIEWS-Three Papers on the Tree Property. Bulletin of Symbolic Logic 7 (2):28-168.
     
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  47. Arthur W. Apter (2001). Review: Uri Abraham, Aronszajn Trees on $Mathscr{N}2$ and $Mathscr{N}3$; James Cummings, Matthew Foreman, The Tree Property; Menachem Magidor, Saharon Shelah, The Tree Property at Successors of Singular Cardinals. [REVIEW] Bulletin of Symbolic Logic 7 (2):283-285.
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  48. 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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  49. 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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  50. Arthur W. Apter & Mirna Džamonja (2001). Some Remarks on a Question of D. H. Fremlin Regarding Ε-Density. Archive for Mathematical Logic 40 (7):531-540.
    We show the relative consistency of ℵ1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω1 satisfying a certain density condition necessarily contain all (...)
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