# Blog Archives

# Topic Archive: Prikry-type forcing

# Why is Prikry forcing subcomplete?

Subcomplete forcing was introduced by Jensen as a class of forcings which do not add reals, but may change cofinalities to $\omega$, unlike proper forcing. In this talk I will show that Prikry forcing is subcomplete.

# Boolean ultrapowers and the Bukovsky-Dehornoy phenomenon

I will present a criterion for when an ultrafilter on a Boolean algebra gives rise to the Bukovsky-Dehornoy phenomenon, namely that the intersection of all intermediate ultrapowers is equal to the the Boolean model. Time permitting, I will show that the Boolean algebras of Prikry and Magidor forcing satisfy the strong Prikry property, and that these forcings come with a canonical imitation iteration whose limit model is the Boolean ultrapower by a very canonical ultrafilter on their respective Boolean algebras.

# Mutual Stationarity and Prikry-type forcings

Mutual stationarity is a property first introduced by Foreman and Magidor to study saturation properties of nonstationary ideals. Given a sequence $\langle\kappa_i : i < \lambda\rangle$ of regular cardinals, a sequence $\langle S_i: i < \lambda\rangle$ with $S_i \subseteq \kappa_i$ stationary for every $i$, is mutually stationary iff there are stationarily many subsets $A \subseteq \sup_{i < \lambda} \kappa_i$ s.t. $\sup(A \cap \kappa_i) \in S_i$ for all $i$ with $\kappa_i \in A$. Consider this second property of a sequence $\langle\kappa_i : i < \lambda\rangle$: there is a forcing $P$ that changes $\text{cof}(\kappa_i)$ to $\eta_i$ without changing cofinalities or cardinalites of ordinals below $\inf{\kappa_i : i < \lambda}$. We want to discuss how, and why, these properties are related.