Archive for Mathematical Logic 48 (6):493-513 (2009)
Abstract |
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 n measurable cardinals while the strong compactness of each strongly compact cardinal is indestructible under Levy collapses (our theorem is actually more general, see Sect. 3). A further application is that the class of strong cardinals can be nonempty yet coincide with the class of strongly compact cardinals while strong compactness of any strongly compact cardinal κ is indestructible under κ-directed closed posets that force GCH at κ
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Keywords | Large cardinals Supercompact cardinal Strongly compact cardinals Identity crisis Indestructibility |
Categories | (categorize this paper) |
DOI | 10.1007/s00153-009-0134-6 |
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References found in this work BETA
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.
The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
The Least Measurable Can Be Strongly Compact and Indestructible.Arthur W. Apter & Moti Gitik - 1998 - Journal of Symbolic Logic 63 (4):1404-1412.
Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.
View all 15 references / Add more references
Citations of this work BETA
Superstrong and Other Large Cardinals Are Never Laver Indestructible.Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis & Toshimichi Usuba - 2016 - Archive for Mathematical Logic 55 (1-2):19-35.
Indestructibility When the First Two Measurable Cardinals Are Strongly Compact.Arthur W. Apter - 2022 - Journal of Symbolic Logic 87 (1):214-227.
The Tree Property at the Successor of a Singular Limit of Measurable Cardinals.Mohammad Golshani - 2018 - Archive for Mathematical Logic 57 (1-2):3-25.
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