Ketonen Order in Ultrafilter Hierarchies

• Ketonen order is defined as a strict, well-founded partial ordering on σ-complete ultrafilters that extends the Mitchell order via sequence and factor embedding formulations.
• It integrates algebraic and game-theoretic approaches by relating to the Lipschitz order, thus broadening analysis of ultrafilter reducibility and measure hierarchies.
• Under the Ultrapower Axiom, the Ketonen order becomes linear and well-ordered, while in ZFC its separation phenomena highlight rich independence and structural diversity.

The Ketonen order is a strict well-founded partial ordering on the class of all σ-complete ultrafilters. Introduced in a modern form by Gabriel Goldberg, it extends the classical Mitchell order—originally defined for normal measures on measurable cardinals—to arbitrary σ-complete ultrafilters, and captures a fine structural hierarchy among large cardinal measures. The Ketonen order plays a central role in the study of ultrafilter reducibility, the interplay between algebraic and game-theoretic orderings, and the impact of large cardinal axioms such as the Ultrapower Axiom (UA) and its variants (Kaplan, 14 Dec 2025).

1. Preliminaries: σ-complete Ultrafilters and the Mitchell Order

A σ-complete ultrafilter $U$ on a measurable cardinal $\kappa$ is an ultrafilter closed under intersections of fewer than $\kappa$ sets. Equivalently, $U$ arises from an elementary embedding $j_U : V \to M_U$ with critical point $\kappa$. For normal measures $U, W$ on the same $\kappa$, the classical Mitchell order $U \triangleleft W$ is defined by $U \in M_W$. The factorization of ultrapower embeddings

$$j_W : V \rightarrow M_W, \qquad j_U^{M_W} : M_W \rightarrow M_U^{M_W}$$

is such that $j_U^{M_W} \circ j_W = j_U$.

2. Definition and Formulations of the Ketonen Order

The Ketonen order, denoted $<_k$, provides a strict hierarchy among σ-complete ultrafilters $U$ and $W$ on cardinals $\kappa_U$ and $\kappa_W$. The conditions $U <_k W$ are given equivalently by:

1. Sequence Formulation: There is a $W$-measure one set $I \in W$ and a sequence $\{ U_i : i \in I \}$ of σ-complete ultrafilters each "concentrating on" $i$ (i.e., for all $X, Y \subseteq \kappa_U$ with $X \cap i = Y \cap i$, $X \in U_i \Leftrightarrow Y \in U_i$), such that for every $X \subseteq \kappa_U$:

$$X \in U \iff \{ i \in I : X \cap i \in U_i \} \in W.$$

2. Factor Embedding Formulation: There exists a σ-complete ultrafilter $U^* \in M_W$ and an elementary embedding $k : M_U \to M_{U^*}^{M_W}$ satisfying:

$$k \circ j_U = j_{U^*}^{M_W} \circ j_W, \qquad k([\mathrm{Id}]_U) < j_{U^*}([\mathrm{Id}]_W),$$

where $[\mathrm{Id}]_U$ denotes the equivalence class of the identity in $M_U$.

For normal measures, the Ketonen order $<_k$ coincides with the Mitchell order: $U <_k W \iff U \in M_W \iff U \triangleleft W$.

3. Structural Properties

The Ketonen order enjoys several fundamental properties:

• Strictness: $U <_k W$ implies $U \ne W$.
• Transitivity: If $U <_k V$ and $V <_k W$, then $U <_k W$.
• Well-foundedness: There are no infinite descending $<_k$-chains; that is, sequences $\ldots <_k U_2 <_k U_1 <_k U_0$ do not exist.

These properties are established by induction on the rank of ultrafilters, using the non-existence of non-trivial elementary embeddings of the universe into itself to rule out infinite $<_k$-descending sequences. The result is that $<_k$ constitutes a strict well-founded partial order on all σ-complete ultrafilters (Kaplan, 14 Dec 2025).

4. Relationship with the Lipschitz Order

The Lipschitz order $<_L$ is a game-theoretic extension, defined using a $\kappa$-length game $G_\kappa(W, U)$ between two players I and II, constructed as follows: At each round $\alpha < \kappa$, I plays $a(\alpha) \in \{0,1\}$, II follows with $b(\alpha) \in \{0,1\}$, producing subsets $a, b \subseteq \kappa$ at the end. Player II wins if $[a \in W \Leftrightarrow b \in U]$.

$U <_L W$ holds if Player I has a winning strategy in both $G_\kappa(W, U)$ and $G_\kappa(P(\kappa)\setminus W, U)$. Equivalently, $U$ is super-Lipschitz reducible to $W$: there exists $f : P(\kappa) \to P(\kappa)$ with

$f(A) \cap (\xi+1) = f(A \cap \xi) \cap (\xi+1)$

for every $\xi < \kappa$, and $f^{-1}[W] = U$.

Goldberg proved that $U <_k W$ always implies $U <_L W$ via explicit construction of the corresponding super-Lipschitz function using the sequence $\{ U_i \}$. However, the converse may fail without stronger axiomatic assumptions (Kaplan, 14 Dec 2025).

5. Linearity and Equivalence under the Ultrapower Axiom

The Ultrapower Axiom (UA) asserts that for any two σ-complete ultrafilters $U, W$ there exist embeddings into ultrapower models that can be aligned, ensuring that $M_{W^*}^{M_U} = M_{U^*}^{M_W}$ and $j_{W^*}^{M_U} \circ j_U = j_{U^*}^{M_W} \circ j_W$ for suitable $U^*$, $W^*$ in the respective models. Under UA, Goldberg established:

• The Ketonen order $<_k$ is linear: every pair of σ-complete ultrafilters is $<_k$-comparable.
• $<_k$ and $<_L$ coincide and become well-orderings (i.e., every pair is comparable, and all chains are well-founded).

Absent UA, examples show that $<_k$ and $<_L$ may differ, and $<_L$ may fail to be linear (Kaplan, 14 Dec 2025).

6. Separation Results and Independence Phenomena

Kaplan established the consistency—in ZFC plus the existence of a measurable cardinal—that $<_k$ and $<_L$ do not coincide. Using an Easton-support product forcing (with $Q_\alpha = \{0,1\}$ for inaccessibles), he constructed pairs of normal measures $V, W$ on $\kappa$ such that:

• $V <_L W$,
• Neither $V <_k W$ nor $W <_k V$.

A similar separation arises under Weak UA (the weakening of UA compatible with $\neg$UA). Using non-stationary-support product forcing, one obtains $\sigma$-complete ultrafilters $V, W$ with $V <_L W$ but $V$, $W$ $<_k$-incomparable. These constructions rely on manipulation of the generics added by the forcing and the failure of certain embeddings to align (Kaplan, 14 Dec 2025).

Axiom System	Separation Possible?	Example Construction
ZFC + Measurable Cardinal	Yes	Easton-support product forcing
Weak UA + ¬UA	Yes	Non-stationary-support product
UA	No; orders coincide	Via embedding alignment

7. Open Questions and Research Directions

Several open questions remain:

• Does linearity of the Lipschitz order $<_L$ (in the absence of UA) force UA itself? If every pair of σ-complete ultrafilters is $<_L$-comparable, must UA hold?
• Is $<_L$ necessarily well-founded in ZFC? Current examples show non-linearity but do not produce infinite descending $<_L$-chains.
• What large cardinal strength is required for determinacy of all games $G_\kappa(W, U)$? Studying fragments of UA via fragments of determinacy for these games is ongoing.

This suggests a deep interconnection between the algebraic structure of ultrafilters, game-theoretic principles, forcing constructions, and large cardinal axioms. The ongoing investigation clarifies the hierarchy of ultrafilter reducibility and the role of determinacy in clarifying the fine structure of the higher infinite (Kaplan, 14 Dec 2025).