Small forcing creates neither strong nor Woodin cardinals

  • J. D. Hamkins and W. Woodin, “Small forcing creates neither strong nor Woodin cardinals,” Proc.~Amer.~Math.~Soc., vol. 128, iss. 10, pp. 3025-3029, 2000.  
    @article {HamkinsWoodin2000:SmallForcing,
    AUTHOR = {Hamkins, Joel David and Woodin, W.~Hugh},
    TITLE = {Small forcing creates neither strong nor {W}oodin cardinals},
    JOURNAL = {Proc.~Amer.~Math.~Soc.},
    FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {128},
    YEAR = {2000},
    NUMBER = {10},
    PAGES = {3025--3029},
    ISSN = {0002-9939},
    CODEN = {PAMYAR},
    MRCLASS = {03E35 (03E55)},
    MRNUMBER = {1664390 (2000m:03121)},
    MRREVIEWER = {Carlos A.~Di Prisco},
    DOI = {10.1090/S0002-9939-00-05347-8},
    URL = {http://dx.doi.org/10.1090/S0002-9939-00-05347-8},
    eprint = {math/9808124},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.

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