Degrees of rigidity for Souslin trees

  • G. Fuchs and J. D. Hamkins, “Degrees of rigidity for Souslin trees,” J.~Symbolic Logic, vol. 74, iss. 2, pp. 423-454, 2009.  
    AUTHOR = {Fuchs, Gunter and Hamkins, Joel David},
    TITLE = {Degrees of rigidity for {S}ouslin trees},
    JOURNAL = {J.~Symbolic Logic},
    FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {74},
    YEAR = {2009},
    NUMBER = {2},
    PAGES = {423--454},
    ISSN = {0022-4812},
    CODEN = {JSYLA6},
    MRCLASS = {03E05},
    MRNUMBER = {2518565 (2010i:03049)},
    MRREVIEWER = {Stefan Geschke},
    URL = {},
    doi = {10.2178/jsl/1243948321},
    eprint = {math/0602482},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    file = F,

We investigate various strong notions of rigidity for Souslin trees, separating them under Diamond into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under Diamond that there is a group whose automorphism tower is highly malleable by forcing.

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 = {},

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.