##  [Categoricity](/categoricity) 

  ##  [Categoricity](https://puremath.quantumdictionary.io/categoricity-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

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).

 

 

 

 

 





 

 



 ##  [Categoricity](https://algebra.quantumdictionary.io/categoricity-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The property of a first-order theory that, for a specified infinite cardinal κ, all models of the theory of cardinality κ are isomorphic; equivalently, the theory has exactly one model up to isomorphism in that cardinality.