Abstract

I would like to survey a series of beautiful and almost mysterious properties of Prikry sequences which have analogues for other Prikry type forcings. The first of these is Mathias’ characterization of Prikry sequences as those that are almost contained in every set of measure 1 with respect to the normal ultrafilter being used for the forcing. This is the key to the second property, which is that the sequence of critical points when forming iterated ultrapowers by that ultrafilter is a Prikry sequence over the limit model. Using this, it is not hard to conclude that Prikry sequences are maximal, in the sense that they almost contain every other Prikry sequence present in their forcing extension. Another phenomenon is that the forcing extension of the limit model by the critical sequence is the same as the intersection of the finite iterates. I will show another canonical representation of that model. Yet another property is that the limit model can be realized as a single Boolean ultrapower. Most of these results were known for Prikry forcing, and I will show that some of them carry over to certain variants of Prikry forcing and Magidor forcing.