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).

Principle

Uniqueness of a model up to isomorphism at a specified size constrains the combinatorial and algebraic structure of models and often forces stability or other tameness conditions that permit structural classification.

Demonstration

The theory of dense linear orders without endpoints is ℵ0-categorical: up to isomorphism there is exactly one countable dense order without endpoints (the rationals with their order), so the theory has a unique countable model.

Misapplication

Assuming categoricity in one cardinal implies categoricity in others; for example, countable categoricity does not automatically yield uncountable categoricity, and categorical behavior depends on the language and signature.

Consequence

Categoricity yields strong model-theoretic conclusions such as completeness, control of automorphism groups, and often stability; in many cases it allows complete structural descriptions of models of the given cardinality.

Reversal

A theory that is not categorical at κ admits multiple non-isomorphic models of cardinality κ, indicating diversity of possible structures and failure of rigid classification at that size.

Boundary

Categoricity is defined relative to a cardinal and a fixed language; it does not assert anything about models of other cardinalities and can be sensitive to expansions or reducts of the language.

Semantic Tension

Tension between categoricity and completeness or model-completeness: categoricity concerns uniqueness of models at a size, whereas completeness concerns deciding sentences; a theory can be complete without being categorical and vice versa.

Synthesis

Categoricity means uniqueness of models up to isomorphism at a given cardinality and, when present, acts as a powerful hypothesis that forces structural regularity and enables deep classification theorems.