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.

Principle

Uniqueness of structure at a given size: if a theory is categorical in κ then any two κ-sized models cannot be distinguished by structure-preserving bijections and therefore share the same invariant-theoretic description.

Demonstration

Example: the theory of dense linear orders without endpoints is categorical in the countable cardinality (its countable model is order-isomorphic to the rationals); by contrast, the theory of algebraically closed fields of fixed characteristic is categorical in every uncountable cardinality, so models of the same uncountable size are determined up to isomorphism by their transcendence degree.

Misapplication

Asserting that categoricity in one cardinal implies categoricity in all cardinals without qualification; or conflating categoricity with completeness (a complete theory may still have many non-isomorphic models of a given cardinality).

Consequence

When a theory is categorical in a cardinal, one gains powerful structural control: invariants classify models, types and automorphism groups are constrained, and classification-theoretic methods (e.g., stability and ranks) often become applicable.

Reversal

A non-categorical theory at κ admits two or more non-isomorphic models of size κ; the inversion highlights the existence of genuinely different structures with identical axioms at that size.

Boundary

Categoricity is a statement about models of a particular cardinality (usually infinite); it does not by itself assert properties in other cardinalities, nor does it assert the existence of a model of that cardinality unless existence is separately given.

Semantic Tension

Categoricity vs completeness: completeness fixes truth-values of sentences across all models but says nothing about the number of non-isomorphic models at a fixed size; conversely, categoricity fixes isomorphism class at a size but not necessarily sentence-by-sentence determinacy across languages or other sizes.

Synthesis

Categoricity captures when a theory yields a unique structural universe at a specified infinite size: it is the principle that, under the theory's axioms and for that cardinal, there is one canonical model up to isomorphism, producing rigid classification and strong model-theoretic consequences.