A natural-number invariant attached to a definable set of a fixed Morley rank that counts the number of top-rank definable components (up to definable equivalence) appearing in a canonical finite partition.

Principle

If a definable set X has Morley rank α, its Morley degree d is the maximal finite number n such that X can be partitioned into n disjoint definable subsets each of Morley rank α; equivalently it measures the multiplicity of α-ranked components in a canonical decomposition.

Demonstration

In an algebraically closed field, an algebraic variety decomposed into finitely many irreducible components of maximal Zariski dimension has Morley degree equal to the number of those maximal-dimensional irreducible components (when parameters fix the decomposition).

Misapplication

Using Morley degree without first fixing Morley rank, attempting to apply it to sets with infinite rank, or conflating degree with algebraic multiplicity or cardinality; treating it as invariant under arbitrary parameter changes when it depends on definable parameters.

Consequence

Combined with Morley rank, Morley degree gives a fine-grained canonical decomposition of definable sets into a finite union of top-rank pieces, enabling counting arguments, comparison of definable families, and formulations of uniqueness up to finite equivalence.

Reversal

A set of Morley degree 1 presents a single top-dimensional piece and behaves more 'connectedly'; higher Morley degree indicates several top-level components and a form of definable disconnectedness.

Boundary

Defined only for definable sets with well-defined (usually finite) Morley rank; absent or uninformative for sets with infinite rank or in unstable theories where rank theory fails to control structure.

Semantic Tension

Tension between Morley degree and algebraic/geometric multiplicity: degree counts definable top-rank components, not scheme-theoretic multiplicity; also competes conceptually with notions of connectedness and irreducibility in geometry.

Synthesis

Morley degree is the finite numerical companion to Morley rank that counts how many top-rank definable components a set has, completing the rank-based classification by quantifying multiplicity of maximal pieces.