Categoricity

A theory T is categorical in a cardinal κ if any two models of T of cardinality κ are isomorphic; categoricity at a particular infinite κ is a strong uniqueness property for models of that size and underpins deep classification results (e.g., Morley's theorem for uncountable categoricity).

The property of a first-order theory that, for a specified infinite cardinal κ, all models of the theory of cardinality κ are isomorphic; equivalently, the theory has exactly one model up to isomorphism in that cardinality.