---
tag: 655K8Y7VS8
kind: concept
name: Complete theory
msc: [03C35, 03C07]
specializes: [92SWXST1ZM]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_theory
same_as:
  wikidata: Q3508260
  wikipedia: Complete theory
---

## Summary

A complete theory is a satisfiable [[theory|92SWXST1ZM]] that decides every sentence of its language: it implies the sentence or its negation.
All models of a complete theory are [[elementarily equivalent|P3B1V182N4]].
The theory of a single structure is always complete, and proving that a list of axioms is complete is a central task of model theory.

## Explanations

### L1 human

A complete theory leaves no first-order question open.
The axioms of groups are not complete: they do not decide whether $x \cdot y = y \cdot x$ holds for all $x$ and $y$.
The axioms of dense linear orders without endpoints are complete: every first-order question about such an order has the same answer in $(\Q, <)$ and in $(\R, <)$.