Indestructibility and the level-by-level agreement between strong compactness and supercompactness

  • J. D. Apter Arthur W.~and Hamkins, “Indestructibility and the level-by-level agreement between strong compactness and supercompactness,” J.~Symbolic Logic, vol. 67, iss. 2, pp. 820-840, 2002.  
    @ARTICLE{ApterHamkins2002:LevelByLevel,
    AUTHOR = {Apter, Arthur W.~and Hamkins, Joel David},
    TITLE = {Indestructibility and the level-by-level agreement between strong compactness and supercompactness},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {67},
    YEAR = {2002},
    NUMBER = {2},
    PAGES = {820--840},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1905168 (2003e:03095)},
    MRREVIEWER = {Carlos A.~Di Prisco},
    DOI = {10.2178/jsl/1190150111},
    URL = {http://dx.doi.org/10.2178/jsl/1190150111},
    eprint = {math/0102086},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

Can a supercompact cardinal $\kappa$ 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 $\kappa$, 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.