A property of a first-order theory (in a countable language) that it has exactly one model of countable cardinality up to isomorphism.

Principle

Complete structural determination at the countable level: the theory fixes the isomorphism type of any countable model, so countable models cannot differ nontrivially.

Demonstration

The theory of dense linear orders without endpoints (the order type of the rationals) is ω-categorical because every countable dense linear order without endpoints is isomorphic to the rationals, hence there is a unique countable model up to isomorphism.

Misapplication

Confusing ω-categoricity with categoricity in uncountable cardinals or assuming ω-categoricity implies uniqueness of models in all infinite sizes; ω-categoricity only controls the countable case.

Consequence

Strong classification of definable sets and types in countable models: ω-categorical theories have well-behaved automorphism groups and often admit finite combinatorial descriptions of n-types, aiding model-theoretic analysis.

Reversal

The opposite is having many nonisomorphic countable models; a theory might be complete yet permit continuum-many nonisomorphic countable models.

Boundary

Typically formulated for countable languages and concerns models of cardinality ℵ0; it does not assert anything about models of uncountable cardinalities or about computable presentations except insofar as additional hypotheses are given.

Semantic Tension

Tension with categoricity in larger cardinals: ω-categoricity is about uniqueness in countable size, while other notions (e.g., κ-categoricity) address different cardinalities and may behave very differently.

Synthesis

Ω-Categoricity means the theory completely determines the isomorphism class of its countable models; it is a strong rigidity property at the ℵ0 level that yields tight control over definable structure while leaving larger-cardinal behavior independent.