The lottery preparation

  • J. D. Hamkins, “The lottery preparation,” Ann.~Pure Appl.~Logic, vol. 101, iss. 2-3, pp. 103-146, 2000.  
    @article {Hamkins2000:LotteryPreparation,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {The lottery preparation},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {101},
    YEAR = {2000},
    NUMBER = {2-3},
    PAGES = {103--146},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E55 (03E40)},
    MRNUMBER = {1736060 (2001i:03108)},
    MRREVIEWER = {Klaas Pieter Hart},
    DOI = {10.1016/S0168-0072(99)00010-X},
    URL = {http://dx.doi.org/10.1016/S0168-0072(99)00010-X},
    eprint = {math/9808012},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal $\kappa$, for example, becomes fully indestructible by $\kappa$-directed closed forcing; a strong cardinal $\kappa$ becomes indestructible by less-than-or-equal-$\kappa$-strategically closed forcing; and a strongly compact cardinal $\kappa$ becomes indestructible by, among others, the forcing to add a Cohen subset to $\kappa$, the forcing to shoot a club $C$ in $\kappa$ which avoids the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context.

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