Logic Colloquium 2004, Tutorial on Geometric Stability Theory, Morley Rank

Morley rank

Let φ(x) be a formula in the (tuple of) variable(s) possibly with parameters from the model M. The Morley rank of φ, RM(φ), is defined by the following conditions:

RM(φ) ≥ 0 ⇔ M satisfies (∃ x) φ
RM(φ) ≥ &lambda (for λ a limit ordinal) ⇔ (∀ α < λ) RM(φ) ≥ α
RM(φ) ≥ α + 1 ⇔ over some elementary extension N of M there is an infinite sequence ψ₀, ψ₁, … of formulae which are pairwise inconsistent for which &psi_i → φ and RM(ψ_i) ≥ α for every index i

If RM(φ) ≥ α for every ordinal α, then we say that RM(φ) = ∞.
If RM(φ) = α, then the Morley degree of φ, dM(φ), is the maximal m for which one can find a sequence ψ₁, …, ψ_m of pairwise contradictory formulae each of which implies φ and has Morley rank α.