MathOverflow is a question and answer site for professional mathematicians. 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

One of the ways to define the Morley rank of a definable set is with respect to a model, say $M$, i.e. a set has rank $\alpha+1$ if there are infinitely many definable subsets with parameters in $M$ of rank $\alpha$ (and a similar clause for limit ordinals). One then shows that once $M$ is $\aleph_0$-saturated then considering definable subsets with parameters in elementary extensions of $M$ doesn't change the rank.

However, in some theories, e.g. algebraically closed fields of fixed characteristic, any model will do, i.e. Cantor-Bendxson rank and Morley rank coincide.

What is known about this phenomenon, i.e. in which theories Cantor-Bendixson equals Morley rank in any (not necessary $\aleph_0$-saturated) model?

share|cite|improve this question
Dima, I will add to your question the following remark: the fact that this is true in an algebraically closed field of any characteristic has nothing to do with the field and everything to do with the strong minimality of the theory. When the theory is strongly minimal, then Morley rank is definable in families, so working in an existentially closed model would suffice. – James Freitag Jul 27 '12 at 2:55
Dear James, how is definability of Morley rank related to the fact that one can compute Morley rank using parameters from existentially closed models? – Dima Sustretov Jul 30 '12 at 12:47
Does this mean that in $DCF_0$, say, there are examples of sets such that their Cantor-Bendixson rank (wrt some model) is not the same as their Morley rank? – Dima Sustretov Dec 11 '12 at 20:45
up vote 5 down vote accepted

The distinction between Morley rank as defined by arbitrary formulas and by definable families of formulas is essential. $\aleph_1$- categoricity in particular implies the rank can defined by definable families. Since my 1973 [?] article in the transactions AMS or Shelah's book or say Pillay's geometric model theory book

share|cite|improve this answer
Welcome to MathOverflow, John! – Joel David Hamkins Feb 11 '13 at 12:26
By the way, you can use simple tex here, just write it with dollar signs and slashes as usual. – Joel David Hamkins Feb 11 '13 at 14:53

Maybe the following remark will help you:

For any strongly minimal forumla $\varphi (x)$ and any formula $\psi(x_1,..,x_n,y)$ which implies $\varphi (x_i)$ for all i, the class $ \lbrace b : MR(\psi(x_1,..,x_n,b))=k \rbrace$ is definable for every k.

This is $26.4$ of Tent and Ziegler's book "A course in model theory"

share|cite|improve this answer
Dear Tim, I do not get it how some kind of definability in families might be helpful. Let our definable set contain infinitely many mutually disjoint definable sets of rank $\alpha$. They might be defined all by different formulas $\psi_i(x, b_i)$. We need to prove that there exist infinitely many such formulas with parameters in a given model, probably by changing the parameters $b_i$. The trouble is not so much to control the rank but to ensure that no intersections occur when we change the parameters. – Dima Sustretov Dec 12 '12 at 11:45

In $ACf_p$ or $DCF_0$, one always assume that they are working in a monster model. So saturation still does the job. Note that they are both $\omega$-stable. So saturated model for any regular cardinal exists.

But maybe that is not what you are asking.

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.