A note on the Ketonen order and Lipschitz reducibility between ultrafilters

TL;DR

This paper investigates the relationship between the Ketonen and Lipschitz orders on ultrafilters, demonstrating that they can differ under certain set-theoretic axioms, contrary to previous assumptions.

Contribution

It proves that the Ketonen and Lipschitz orders on ultrafilters do not necessarily coincide under the Weak Ultrapower Axiom, challenging prior beliefs.

Findings

The Lipschitz order extends the Ketonen order.

Under UA, the two orders coincide.

They can differ under weaker axioms.

Abstract

In his study of the Ultrapower Axiom (UA), Goldberg revealed a connection between UA and the determinacy of certain games that witness Lipschitz reducibility between ultrafilters. In particular, he analyzed the relationship between the Ketonen and Lipschitz orders - two natural extensions of the Mitchell order from normal measures to arbitrary $\sigma$-complete ultrafilters - and proved that the Lipschitz order extends the Ketonen order. He further observed that under UA the two orders coincide. Goldberg asked if it's consistent that the orders differ from each other. We show that the answer is positive. In fact, even the Weak Ultrapower Axiom does not imply that the Ketonen and Lipschitz orders coincide.