Destruction or preservation as you like it

  • J. D. Hamkins, “Destruction or preservation as you like it,” Ann.~Pure Appl.~Logic, vol. 91, iss. 2-3, pp. 191-229, 1998.  
    @article {Hamkins98:AsYouLikeIt,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Destruction or preservation as you like it},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {91},
    YEAR = {1998},
    NUMBER = {2-3},
    PAGES = {191--229},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E55 (03E35)},
    MRNUMBER = {1604770 (99f:03071)},
    MRREVIEWER = {Joan Bagaria},
    DOI = {10.1016/S0168-0072(97)00044-4},
    URL = {http://jdh.hamkins.org/asyoulikeit/},
    eprint = {1607.00683},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of ${<}\kappa$-directed closed posets which preserve a supercompact cardinal $\kappa$ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible.

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