Archive for Mathematical Logic 48 (6):493-513 (2009)

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
Keywords Large cardinals  Supercompact cardinal  Strongly compact cardinals  Identity crisis  Indestructibility
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DOI 10.1007/s00153-009-0134-6
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Set Theory.T. Jech - 2005 - Bulletin of Symbolic Logic 11 (2):243-245.
The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.

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