Morley rank

Summary

Morley rank is an ordinal-valued dimension for definable sets in a first-order structure. A definable set has rank at least $\alpha + 1$ when it contains infinitely many pairwise disjoint definable subsets of rank at least $\alpha$. In an algebraically closed field, the Morley rank of a constructible set equals its dimension in the sense of algebraic geometry. A complete theory in a countable language is $\omega$-stable exactly when every formula has ordinal Morley rank. The notion is central in stable theories and in Morley's proof of his categoricity theorem.