Applications of the Magidor Iteration to Ultrafilter Theory

TL;DR

This paper explores the application of Magidor iteration to ultrafilter theory, characterizing sums of ultrafilters and analyzing implications for ultrapower axioms and ultrafilter properties.

Contribution

It provides new characterizations of ultrafilter sums after Magidor iteration and distinguishes between weak and strong ultrapower axioms.

Findings

Weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom.

Constructs a non-rigid ultrapower.

Creates two ultrafilters with identical ultrapowers on different cardinals.

Abstract

We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower Axiom. We also construct a non-rigid ultrapower and two uniform ultrafilters on different cardinals that have the same ultrapower.