---
tag: WN43DRPNBX
kind: concept
name: Morley rank
msc: [03C45]
sources:
  - url: https://en.wikipedia.org/wiki/Morley_rank
same_as:
  wikidata: Q6912570
  wikipedia: 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|T7JW5QY6KR]] and in Morley's proof of his categoricity theorem.

## Explanations

### L1 human

Morley rank measures the size of a definable set by how many times one can split it into infinitely many smaller pieces.
A finite nonempty set has rank $0$, since it cannot be split into infinitely many nonempty pieces.
The complex line has rank $1$: it splits into infinitely many points, but its definable subsets are finite or cofinite, so it does not split into infinitely many infinite pieces.
The complex plane has rank $2$, since it splits into infinitely many disjoint lines.