Tag Archives: trees

Degrees of rigidity for Souslin trees

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[bibtex key=FuchsHamkins2009:DegreesOfRigidity]

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

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[bibtex key=Hamkins97:Seeds]

Applying the seed concept to Prikry tree forcing β„™πœ‡, I investigate how well β„™πœ‡ preserves the maximality property of ordinary Prikry forcing and prove that β„™πœ‡ Prikry sequences are maximal exactly when πœ‡ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if πœ‡ is a strongly normal supercompactness measure, then β„™πœ‡ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.