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. • The Morley rank is at least 0 if
S is non-empty. • For
α a
successor ordinal, the Morley rank is at least
α if in some
elementary extension N of
M, the set
S has countably infinitely many disjoint definable subsets
Si, each of rank at least
α − 1. • For
α a non-zero
limit ordinal, the Morley rank is at least
α if it is at least
β for all
β less than
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 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. 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 model theoretic
stability theory. ==Examples==