Blog Archives

Topic Archive: Boolean ultrapowers

CUNY Logic WorkshopFriday, February 5, 20162:00 pmGC 6417

Gunter Fuchs

Boolean ultrapowers in set theory

The City University of New York

Boolean ultrapowers can serve to explain phenomena that arise in the context of iterated ultrapowers, such as the genericity of the critical sequence over the direct limit model. The examples we give are ultrapowers formed using the complete Boolean algebras of Prikry forcing, Magidor forcing, or a generalization of Prikry forcing. Boolean ultrapowers can also be viewed as a direct limit of ultrapowers, and we present some criteria for when the intersection of these ultrapowers is equal to the generic extension of the Boolean ultrapower, thus arriving at a generalization of a phenomenon first observed by Bukovsky and Dehornoy in the context of Prikry forcing.

Set theory seminarFriday, September 18, 201510:00 amGC 3212

Gunter Fuchs

Boolean ultrapowers and the Bukovsky-Dehornoy phenomenon

The City University of New York

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.

Set theory seminarFriday, April 24, 201510:00 amGC 6417

Joseph Van Name

When is a Boolean ultrapower an ultrapower?

CUNY Borough of Manhattan Community College

The Boolean ultrapower construction is a natural generalization of the classical ultrapower construction, but the Boolean ultrapower uses an ultrafilter on a complete Boolean algebra instead of a set. It was initially unknown as to whether in ZFC there exists a Boolean ultrapower which is not always isomorphic to a classical ultrapower. This problem was resolved in 1976 by Bernd and Sabine Koppelberg who constructed a Boolean ultrapower which is not an ultrapower. On the other hand, there does not seem to be any reference in the mathematical literature to atomless Boolean ultrapowers which are isomorphic to classical ultrapowers.

We shall first generalize the notion of a Boolean ultrapower to the notion of a BPA-ultrapower which is in a sense the most general ultrapower construction. Then by applying a result of Joel David Hamkins which characterizes the Boolean ultrapowers which are classical ultrapowers, we shall investigate examples of Boolean ultrapowers which are not classical ultrapowers as well as Boolean ultrapowers which are classical ultrapowers. For instance, I claim that under GCH every complete atomless Boolean algebra has an ultrafilter which gives rise to a Boolean ultrapower which is not a classical ultrapower. On the other hand, using the Keisler-Shelah isomorphism theorem, we may construct Boolean ultrapowers in ZFC on a fairly general class of Boolean algebras which are classical ultrapowers.