---
tag: V7N84P60SJ
kind: statement
role: theorem
name: Morley's categoricity theorem
msc: [03C35, 03C45]
uses: [1JJCGNX240, 655K8Y7VS8, 0W7FBYS1F1, 0EBACQFAZW, MT8TDVPBXB]
about: [1JJCGNX240]
sources:
  - url: https://en.wikipedia.org/wiki/Categorical_theory
---

## Statement

Let $T$ be a [[complete theory|655K8Y7VS8]] in a countable first-order signature, with infinite models.
If $T$ is [[$\kappa$-categorical|1JJCGNX240]] for some [[uncountable|0W7FBYS1F1]] [[cardinal|0EBACQFAZW]] $\kappa$, then $T$ is $\kappa$-categorical for every uncountable cardinal $\kappa$.

## Remarks

Michael Morley proved the theorem in 1965, answering a question of Jerzy Łoś.
A key step shows that a theory categorical in an uncountable cardinal is $\omega$-stable: over each countable set of parameters there are only countably many complete types.
Such a theory has a strongly minimal formula, possibly with parameters.
Each model is determined up to isomorphism by the dimension of the set that this formula defines, as a vector space is determined by its dimension.

Example: the theory of [[algebraically closed fields|MT8TDVPBXB]] of a fixed characteristic is categorical in every uncountable cardinal, with transcendence degree as the dimension.
The theory of infinite vector spaces over $\Q$ is another example.

The countability of the signature matters in this form.
Saharon Shelah extended the theorem: a complete theory $T$ categorical in some cardinal $\kappa > \abs{T}$ is categorical in every cardinal $\kappa > \abs{T}$.
The theorem says nothing about $\aleph_0$.
The theory of algebraically closed fields of characteristic $0$ is uncountably categorical but not $\aleph_0$-categorical.
The theory of dense linear orders without endpoints is $\aleph_0$-categorical but not uncountably categorical.

## Explanations

### L1 human

Some theories have only one model of each uncountable size, up to isomorphism.
Morley's theorem says that, for a countable language, having only one model of a single uncountable size already forces this for every uncountable size.
Algebraically closed fields are the typical case: such a field is fixed by its characteristic and how many independent transcendental elements it has.

### L2 human

The proof introduced tools that became the core of stability theory: Morley rank, $\omega$-stability, and prime models over sets.
A countable complete theory with infinite models has one of four patterns of categoricity.
It is categorical in all infinite cardinals, only in $\aleph_0$, only in the uncountable ones, or in none.
Baldwin and Lachlan later showed that an uncountably categorical theory has either one countable model or countably infinitely many.