---
tag: 1JJCGNX240
kind: concept
name: Categorical theory
msc: [03C35]
sources:
  - url: https://en.wikipedia.org/wiki/Categorical_theory
same_as:
  wikidata: Q15830473
  wikipedia: 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|ZR7Y5C235P]].
For countable theories, [[Morley's categoricity theorem|V7N84P60SJ]] 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.

## Explanations

### L1 human

A theory is a list of axioms, and it usually has many different models.
A theory is categorical in a size when all its models of that size are copies of one another.
For example, any two countable dense linear orders without endpoints, such as the rationals and the rationals between $0$ and $1$, are isomorphic as orders.
So the theory of such orders is categorical in the countable size, but it is not categorical in larger sizes.