In mathematical logic, Morley rank, introduced by Michael D. Morley (1965), is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.
Fix a theory T with a model M. The Morley rank of a formula φ defining a definable (with parameters) subset S of M is an ordinal or −1 or ∞, defined by first recursively defining what it means for a formula to have Morley rank at least α for some ordinal α.
For a definable subset of a model M (defined by a formula φ) the Morley rank is defined to be the Morley rank of φ in any ℵ0-saturated elementary extension of M. In particular for ℵ0-saturated models the Morley rank of a subset is the Morley rank of any formula defining the subset.
If φ defining S has rank α, and S breaks up into no more than n < ω subsets of rank α, then φ is said to have Morley degree n. A formula defining a finite set has Morley rank 0.
Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity theorem and in the larger area of model theoretic stability theory.