[bibtex key=Hamkins97: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.