---
tag: ZR7Y5C235P
kind: statement
role: theorem
name: Łoś–Vaught test
msc: [03C35]
uses: [1JJCGNX240, 655K8Y7VS8, 484M0TZ1S7, 0EBACQFAZW]
about: [1JJCGNX240, 655K8Y7VS8]
sources:
  - url: https://en.wikipedia.org/wiki/Łoś–Vaught_test
same_as:
  wikidata: Q25303635
  wikipedia: Łoś–Vaught test
---

## Statement

Let $T$ be a satisfiable theory in a first-order signature $L$, and let $\kappa$ be an infinite [[cardinal|0EBACQFAZW]] with $\kappa \ge \abs{L}$.
Suppose that $T$ has no finite models and that $T$ is [[$\kappa$-categorical|1JJCGNX240]].
Then $T$ is [[complete|655K8Y7VS8]].

## Remarks

Suppose $T$ is not complete, and pick a sentence $\varphi$ such that neither $\varphi$ nor $\neg \varphi$ follows from $T$.
Then $T \cup \set{\varphi}$ and $T \cup \set{\neg \varphi}$ both have models, and these are infinite.
By the [[Löwenheim–Skolem theorem|484M0TZ1S7]], both have models of size $\kappa$.
These two models are not isomorphic, since one satisfies $\varphi$ and the other does not, which contradicts $\kappa$-categoricity.

Example: the theory of dense linear orders without endpoints has no finite models and is $\aleph_0$-categorical, so it is complete.
Similarly, the theory of algebraically closed fields of a fixed characteristic is complete, since it is categorical in every uncountable cardinal.

The hypothesis on finite models is necessary.
Take the empty theory $T = \emptyset$ in the empty signature, whose models are the nonempty sets.
For each infinite $\kappa$, any two sets of size $\kappa$ are in bijection, so $T$ is $\kappa$-categorical.
But $T$ is not complete: the sentence $\forall x \, \forall y \, (x = y)$ holds in a model of size $1$ and fails in a model of size $2$.

## Explanations

### L1 human

A theory is complete when it decides every statement in its language.
The test says that if all infinite models of one size look the same and there are no finite models, the theory decides everything.
It is a quick way to prove completeness for natural theories such as dense orders or algebraically closed fields.

### L2 human

The test turns a structural fact, uniqueness of models of one cardinality, into a logical fact, completeness.
Completeness of a recursively axiomatized theory implies that it is decidable, so the test also gives decision procedures.
It is often combined with a back-and-forth argument for $\aleph_0$-categoricity, or with a dimension argument, such as transcendence degree, for uncountable categoricity.
Its converse fails: the theory of $(\Z, <)$ is complete but categorical in no infinite cardinal.