categoricity in cardinality

Categoricity in logic refers to a theory having only one unique model, up to isomorphism. This means that if two models of the theory are structurally identical, they are considered the same. The Löwenheim-Skolem theorem has significant implications for categoricity. It states that if a theory has an infinite model, it has models of all infinite cardinalities. This implies that no theory with an infinite model can be categorical, as models of different cardinalities cannot be isomorphic. However, a theory can be categorical in specific infinite cardinalities, meaning all models of that particular cardinality are isomorphic. A key discovery by Michael Morley states that if a theory is categorical in one uncountable cardinality, it is categorical in all uncountable cardinalities.12