Categoricity
Categoricity is a concept from model theory describing when a theory has a unique model, up to isomorphism, in a given cardinality. More precisely, a theory T is categorical in a cardinal κ if any two models of T with cardinality κ are isomorphic. When a theory is complete, categoricity in κ means there is exactly one model of size κ up to isomorphism.
A cornerstone result is Morley’s categoricity theorem. It states that if a complete first-order theory in a
The categoricity spectrum of a theory refers to the set of cardinals κ for which the theory is
Key examples include the theory of algebraically closed fields of fixed characteristic (ACF_p), which is categorical
Notes: Categoricity is a central theme in stability theory and has influenced the development of classification