Morley rank
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 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.
The Morley rank is then defined to be α if it is at least α but not at least α + 1, and is defined to be ∞ if it is at least α for all ordinals α, and is defined to be −1 if S is empty.
For a 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. A formula with Morley rank 1 and Morley degree 1 is called strongly minimal. A strongly minimal structure is one where the trivial formula x = x is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of Morley's categoricity theorem and in the larger area of stability theory (model theory).
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