A theorem in model theory stating that a complete first-order theory in a countable language that is categorical in some uncountable cardinal κ is categorical in all uncountable cardinals; in other words, categoricity in one uncountable cardinal implies categoricity across the entire uncountable spectrum.

Principle

Unicity at one uncountable size propagates: the structural rigidity implied by categoricity in a single uncountable cardinal, together with completeness and countability of the language, forces a unique isomorphism type for models in every uncountable cardinality.

Demonstration

For example, the theory of algebraically closed fields of fixed characteristic is categorical in every uncountable cardinal: being categorical in one uncountable cardinal (because models are determined uniquely up to transcendence degree) gives, by Morley's theorem, categoricity in all uncountable cardinals, illustrating the transfer phenomenon.

Misapplication

Applying Morley's theorem to countable cardinals (expecting categoricity in ℵ0 from an uncountable instance) or omitting the hypothesis that the language is countable and the theory complete; the theorem specifically concerns first-order completeness and uncountable cardinalities in a countable language.

Consequence

Morley's theorem underlies deep classification results: categoricity in an uncountable cardinal implies strong structural properties such as ω-stability and ultimately enables a fine-grained analysis of models (e.g. existence of orthogonality, analysis by minimal types) across uncountable sizes.

Reversal

The inverse scenario is a theory that is categorical in some cardinals but not in others (especially failing to be categorical in any uncountable cardinal); such behavior can occur when hypotheses of Morley's theorem are not satisfied or in non-first-order contexts.

Boundary

Requires a complete first-order theory in a countable language and the hypothesis of categoricity in at least one uncountable cardinal; it does not address categoricity in the countable case and does not directly extend to uncountable languages or to logics beyond first order without further hypotheses.

Semantic Tension

Morley's global transfer of categoricity interacts with local notions like ω-categoricity and strong minimality: categoricity in uncountable cardinals implies tameness properties (ω-stability, decomposition into minimal types) but those local notions do not automatically yield uncountable categoricity without the global hypothesis.

Synthesis

Morley's theorem links a single uncountable instance of uniqueness to uniform uniqueness across all uncountable cardinalities for complete first-order theories in countable languages: it converts an instance of rigidity into a sweeping classification principle that drives much of modern stability theory.