Categorical theory

A first-order theory is categorical in a cardinality if all its models of that size are isomorphic. This property measures how completely the theory determines structures at that size.

First-order theory: A set of sentences in first-order logic that specifies conditions on mathematical structures. Categoricity is a property of a theory's models, not of a single structure.
Löwenheim–Skolem theorem: A pair of results showing how first-order theories relate to models of different infinite cardinalities. These results constrain categoricity across sizes.
Categoricity of dense linear orders: The theory of dense linear orders without endpoints has one countable model up to isomorphism. The rational order exemplifies countable categoricity without categoricity in every size.
Categorical theory in cardinality κ: A theory with exactly one model up to isomorphism of cardinality κ. This is the precise size-indexed form of categoricity.
Model (mathematical logic): A mathematical structure in which every sentence of a given theory is true. The theory's models are the structures whose isomorphism types categoricity compares.
Downward Löwenheim–Skolem theorem: A theorem giving suitably small elementary substructures of infinite first-order structures. It helps rule out categoricity in uncountable sizes when a theory has an infinite model.
Categoricity of algebraically closed fields: Algebraically closed fields of a fixed characteristic are classified up to isomorphism by transcendence degree. At each uncountable cardinality, the theory has a unique model for each fixed characteristic.
Countably categorical theory: A theory with exactly one countable model up to isomorphism. Countable categoricity is the most familiar instance of size-specific categoricity.
Isomorphism: A bijective structure-preserving map whose inverse also preserves structure. Categoricity identifies models up to this notion of structural sameness.
Upward Löwenheim–Skolem theorem: A theorem guaranteeing larger models for first-order theories with infinite models. Larger models need not be isomorphic to models at smaller cardinalities.

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