The Rudin-Frolik order and the Ultrapower Axiom

TL;DR

This paper investigates the structure of the Rudin-Frolik order on countably complete ultrafilters assuming the Ultrapower Axiom, revealing that such ultrafilters have finitely many predecessors under this assumption.

Contribution

It establishes that under the Ultrapower Axiom, countably complete ultrafilters have finitely many predecessors in the Rudin-Frolik order, clarifying the order's structure.

Findings

Ultrafilters have finitely many predecessors under the Ultrapower Axiom.

The structure of the Rudin-Frolik order is more constrained assuming the Ultrapower Axiom.

Any wellfounded ultrapower is derived from finitely many ultrapowers.

Abstract

We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom, holds in all known canonical inner models. It turns out that assuming the Ultrapower Axiom, much more about the Rudin-Frolik order can be determined. Our main theorem is that under the Ultrapower Axiom, a countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolik order. In other words, any wellfounded ultrapower (of the universe) is the ultrapower of at most finitely many ultrapowers.