Boolean ultrapowers, the Bukovsky-Dehornoy phenomenon, and iterated ultrapowers

TL;DR

This paper explores the relationship between Boolean ultrapowers, iterated ultrapowers, and the Bukovsky-Dehornoy phenomenon, revealing conditions under which certain ultrapower constructions align with Boolean ultrapowers and their implications.

Contribution

It demonstrates that the length ω iterated ultrapower by a normal ultrafilter is a Boolean ultrapower and identifies conditions where longer iterations differ, extending understanding of ultrapower models.

Findings

The length ω iterated ultrapower by a normal ultrafilter is a Boolean ultrapower.

Longer ultrapower iterations with different ultrafilters can differ from Boolean ultrapowers.

A criterion involving a simple skeleton determines when the Bukovsky-Dehornoy phenomenon holds.

Abstract

We show that while the length $\omega$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $\omega$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct limit model by the critical sequence as the Bukovsky-Dehornoy phenomenon, and we develop a sufficient criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers. Assuming that the canonical generic filter over the Boolean ultrapower model has what we call a continuous representation, we show that the Boolean model consists precisely of those members of the intersection model that have continuously and eventually uniformly represented codes.