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  1. The lottery preparation.Joel David Hamkins - 2000 - Annals of Pure and Applied Logic 101 (2-3):103-146.
    The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...)
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  • 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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  • 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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  • 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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  • On measurable limits of compact cardinals.Arthur W. Apter - 1999 - 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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  • A note on tall cardinals and level by level equivalence.Arthur W. Apter - 2016 - Mathematical Logic Quarterly 62 (1-2):128-132.
    Starting from a model “κ is supercompact” + “No cardinal is supercompact up to a measurable cardinal”, we force and construct a model such that “κ is supercompact” + “No cardinal is supercompact up to a measurable cardinal” + “δ is measurable iff δ is tall” in which level by level equivalence between strong compactness and supercompactness holds. This extends and generalizes both [, Theorem 1] and the results of.
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  • 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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  • Level by level equivalence and strong compactness.Arthur 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 > κ, κ is λ strongly compact iff κ is (...)
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  • 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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  • Forcing the Least Measurable to Violate GCH.Arthur W. Apter - 1999 - Mathematical Logic Quarterly 45 (4):551-560.
    Starting with a model for “GCH + k is k+ supercompact”, we force and construct a model for “k is the least measurable cardinal + 2k = K+”. This model has the property that forcing over it with Add preserves the fact k is the least measurable cardinal.
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  • 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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  • 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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  • 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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  • 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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