Categorical theory

Summary

A theory is categorical in an infinite cardinal $\kappa$ when it has, up to isomorphism, exactly one model of size $\kappa$. By the Löwenheim–Skolem theorems, a theory with an infinite model has models of many sizes. So categoricity is always relative to a cardinal. Categoricity gives completeness through the Łoś–Vaught test. For countable theories, Morley's categoricity theorem says that categoricity in one uncountable cardinal implies it in all of them. The study of categorical theories was the starting point of stability theory.