Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal κ, then a cardinal with that property exists below κ.

What does "that is witnessed by a structure of limited rank" mean? What exactly is limited rank?

share|cite|improve this question
up vote 5 down vote accepted

What is meant here is a property $P$ such that a cardinal $\kappa$ satisfies $P$ iff there is an $\alpha>\kappa$ such that $V_\alpha\models\Psi(\kappa)$ for some appropriate $\Psi$ (that, of course, depends on $P$).

Supercompactness and strongness are not like that, since both require the existence of arbitrarily large measures with certain properties. On the other hand, $3$-hugeness, although in consistency strength much higher than supercompactness, is verified "locally" in the sense indicated in the previous paragraph.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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