# 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.