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 countable language is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals. In particular, such theories are uncountably categorical and exhibit stability and a controlled model-theoretic structure. Morley’s theorem explains why uncountable categoricity is a strong global property rather than a property that can hold only in one large cardinal.

The categoricity spectrum of a theory refers to the set of cardinals κ for which the theory is κ-categorical. In the countable-language setting, Morley’s theorem implies a dichotomy: a theory is either non-categorical in all uncountable cardinals or categorical in every uncountable cardinal.

Key examples include the theory of algebraically closed fields of fixed characteristic (ACF_p), which is categorical in every uncountable cardinal, and the theory of a pure infinite set with equality, which is categorical in every infinite cardinal. These illustrate how uncountable categoricity can sharply constrain the structure of models.

Notes: Categoricity is a central theme in stability theory and has influenced the development of classification theory, with extensions to broader logics and frameworks beyond first-order logic.