Gap forcing: generalizing the Lévy-Solovay theorem

  • J. D. Hamkins, “Gap forcing: generalizing the Lévy-Solovay theorem,” Bull.~Symbolic Logic, vol. 5, iss. 2, pp. 264-272, 1999.  
    @article{Hamkins99:GapForcingGen,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Gap forcing: generalizing the {L}\'evy-{S}olovay theorem},
    JOURNAL = {Bull.~Symbolic Logic},
    FJOURNAL = {The Bulletin of Symbolic Logic},
    VOLUME = {5},
    YEAR = {1999},
    NUMBER = {2},
    PAGES = {264--272},
    ISSN = {1079-8986},
    MRCLASS = {03E40 (03E55)},
    MRNUMBER = {1792281 (2002g:03106)},
    MRREVIEWER = {Carlos A.~Di Prisco},
    DOI = {10.2307/421092},
    URL = {http://dx.doi.org/10.2307/421092},
    month = {June},
    eprint = {math/9901108},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

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