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On the indestructibility aspects of identity crisis

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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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Authors and Affiliations

  1. Group in Logic and the Methodology of Science, University of California, Berkeley, CA, 94720, USA

    Grigor Sargsyan

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  1. Grigor Sargsyan
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Correspondence to Grigor Sargsyan.

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The author wishes to thank Arthur Apter for introducing him to the subject of this paper and to set theory in general. Some of the main ideas of this paper have their roots in the author’s undergraduate years when the author was taking a reading course with Apter. Those days were among the most enjoyable days of the author’s undergraduate years. The author would also like to thank the anonymous referee for helpful suggestions.

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Sargsyan, G. On the indestructibility aspects of identity crisis. Arch. Math. Logic 48, 493–513 (2009). https://doi.org/10.1007/s00153-009-0134-6

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  • DOI: https://doi.org/10.1007/s00153-009-0134-6

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