# Contepts

## Idea

In model theory, given a cardinal $\kappa$, a theory is $\kappa$-categorical (or categorical in cardinality $\kappa$), if it has precisely one isomorphism class of models of cardinality $\kappa$.

## Properties

The Morley categoricity theorem says that a first-order theory $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.

Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.

## References

Revised on January 4, 2015 08:37:43 by Urs Schreiber (127.0.0.1)