Canonical seeds and Prikry trees

  • J. D. Hamkins, “Canonical seeds and Prikry trees,” J.~Symbolic Logic, vol. 62, iss. 2, pp. 373-396, 1997.  
    @article {Hamkins97:Seeds,
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Canonical seeds and {P}rikry trees},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {The Journal of Symbolic Logic},
    VOLUME = {62},
    YEAR = {1997},
    NUMBER = {2},
    PAGES = {373--396},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E40 (03E05 03E55)},
    MRNUMBER = {1464105 (98i:03070)},
    MRREVIEWER = {Douglas R.~Burke},
    DOI = {10.2307/2275538},
    URL = {http://dx.doi.org/10.2307/2275538},
    }

Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.