February 20
Ben-zion Weltsch,
Rutgers University
Maximality of Prikry-Type Sequences
Prikry forcing was devised by Karel Prikry to show that, given large cardinals, one can make a regular cardinal singular while having it remain a cardinal. The cofinal sequence added by Prikry forcing is called the Prikry sequence. The Prikry sequence is maximal in the sense that any other generic sequence is, modulo a finite initial segment, a subsequence of the Prikry sequence. We call this property the maximality property. A key to showing this property is using the normal ultrafilter associated with the singularized large cardinal. In this talk, we discuss the maximality property for Prikry forcings of various ultrafilters. We present partial results on a conjecture of Woodin on maximality for supercompact Prikry forcing, Prikry forcings without the maximality property, and intermediate models of Prikry forcings.