An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha = \emptyset$. Countable incompleteness is equivalent to $(\omega,\omega)$-regularity, and for every $\kappa$, there is always an $(\omega,\kappa)$-regular ultrafilter. The failure of $(\omega,\omega_1)$-regularity for an ultrafilter on $\omega_1$ is consistent relative to large cardinals, and carries some large cardinal strength.

In the 1970s, Kanamori proved that if $\lambda$ is singular, then every ultrafilter on $\lambda^+$ is $(\lambda,\lambda^+)$-regular. Question: Is it known whether Kanamori's result can be improved to, "If $\lambda$ is singular, then every ultrafilter on $\lambda^+$ is $(\kappa,\lambda^+)$-regular for some $\kappa < \lambda$"? What about in the case $\lambda=\aleph_\omega$?

  • 1
    $\begingroup$ Hi Monroe. You should take a look at Lipparini's work, particularly the nice More on regular and decomposable ultrafilters in ZFC. $\endgroup$ Apr 3, 2014 at 21:24

1 Answer 1


Assuming large cardinals, the answer is no even for $\aleph_\omega$.

Given a supercompact cardinal, Ben-David and Magidor constructed a model in which $\aleph_{\omega+1}$ carries a uniform indecomposable ultrafilter $U$, which means that $U$ is $\theta$-indecomposable for $\aleph_0<\theta<\aleph_\omega$.

For regular $\theta$, $\theta$-indecomposability means that the ultrafilter is closed closed under intersections of decreasing sequences of length $\theta$, and for ultrafilters this is equivalent to the failure of $(\theta,\theta)$-regularity. [due to Kanamori or Ketonen --- I'll need to check.]

Said another way, this means that any point-$<\theta$ family of sets from the ultrafilter has size $<\theta$, and hence certainly $(\theta,\aleph_{\omega+1})$-regularity fails.

The ultrafilter in the Ben-David/Magidor paper is $\aleph_n$-indecomposable whenever $0<n<\omega$, hence it cannot be $(\kappa,\aleph_{\omega+1})$ regular for any $\kappa<\aleph_\omega$.

[Note: In the other direction, the answer is surely yes if $V=L$ and the proof ought to go by showing that a counterexample implies some variety of stationary reflection, but I haven't had time to investigate.]

  • 2
    $\begingroup$ I think Prikry proved that if $V=L$ then every uniform ultrafilter on any infinite $\kappa$ is $(\omega,\kappa)$-regular. $\endgroup$ Apr 4, 2014 at 16:02
  • $\begingroup$ Thanks, Andreas! I'll fill in the references as I find them! $\endgroup$ Apr 4, 2014 at 16:07
  • $\begingroup$ Oops. I found the Prikry reference, "On a problem of Gillman and Keisler" [Ann. Math. Logic 2 (1970) 179-187], but according to the review on MathSciNet, it only proves the case $\kappa=\aleph_1$. $\endgroup$ Apr 4, 2014 at 17:13
  • $\begingroup$ In the Kanamori-Magidor "Evolution of Large Cardinal Axioms" paper, the question is mentioned as still open. $\endgroup$ Apr 4, 2014 at 17:16
  • 2
    $\begingroup$ According to the MathSciNet review of Dieter Donder's paper "Regularity of ultrafilters and the core model" [Israel J. Math. 63 (1988) 289-322] all uniform ultrafilters on $\kappa$ are $(\omega,\kappa)$-regular provided there is no inner model with a measurable cardinal and the core model computes $\kappa^+$ correctly. (Presumably this refers to the original Jensen-Dodd core model.) So it seems we're OK with room to spare if $V=L$. $\endgroup$ Apr 4, 2014 at 17:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.