- J. D. Hamkins, “Canonical seeds and Prikry trees,” Journal of 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 = {Journal of 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://jdh.hamkins.org/seeds}, }
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.